A world-class program incorporating the highly effective Readiness-Engagement-Mastery model of instructional design
Enhanced support for effective implementation of Readiness-Engagement-Mastery pedagogy
Digital PR1ME Mathematics Teaching Hub for additional teaching resources and online professional development
TM Mathematics is a world-class program that works for every student and teacher. It incorporates:
• the teaching and learning best practices from the global top performers in international studies such as Trends in International Mathematics and Science Study (TIMSS) and Programme for International Student Assessment (PISA): Singapore, Hong Kong and Republic of South Korea, and
• Singapore’s Mathematics Curriculum Framework in which mathematical problem solving is the central focus.
Turn to the pages listed below to understand how TM Mathematics:
• supports lear ning to mastery of all students with a pedagogical framework and instructional design based on proven teaching and learning practices,
• integrates assessment for learning so that every child can succeed, and
• offers a comprehensive, accessible suite of teaching and learning resources for flexibility in planning and instruction, and lear ning.
Supports learning to mastery of all students because it incorporates a pedagogical framework and instructional design based on proven teaching and learning practices of global top-performing education systems.
The central focus of the TM Mathematics Framework is problem solving. Learning progressions ensure focus and coherence in content using an instructional design that incorporates the Readiness-Engagement-Mastery model.
Learning experiences based on the Readiness-Engagement-Mastery instructional model
Learning mathematics via problem solving
Development and communication of mathematical thinking and reasoning
Learning mathematics by doing mathematics
Focused and coherent curriculum based on learning progression principles
Integrates assessment for learning to enable every child to succeed.
Offers a comprehensive, accessible suite of teaching and learning resources for flexibility in planning, instruction and lear ning.
The instructional design of each chapter comprises learning experiences that consistently involve three phases of learning: Readiness, Engagement, and Mastery so that teaching and learning mathematics is effective, measurable and diagnostic.
Because mathematical knowledge is cumulative in nature, a student’s readiness to learn new concepts or skills is vital to learning success.
Let’s Remember systematically assesses students’ grasp of the required prior knowledge and provides an accurate evaluation of their readiness to learn new concepts or skills.
The objective and chapter reference for each task are listed so that teachers can easily reteach the relevant concepts from previous chapters or grades.
Recall:
1. Converting time from the 12-hour clock notation to the 24-hour clock notation (CB4 Chapter 15)
2. Converting time from the 24-hour clock notation to the 12-hour clock notation (CB4 Chapter 15)
Explore encourages mathematical curiosity and a positive learning attitude by getting students to recall the requisite prior knowledge, set learning goals and track their learning as they progress through the unit.
3. Finding the duration of a time interval given time in 24-hour clock notation (CB4 Chapter 15)
EXPLORE
Have students read the word problem on CB p. 278. Discuss with students the following questions:
•Do you have friends or family living in other parts of the world?
•Do they live in a different time zone ? What is the time difference from where you are living?
•Have you tried to communicate with them before? What was the mode of communication? How was your experience like?
Have students form groups to complete the tasks in columns 1 and 2 of the table. Let students know that they do not have to solve the word problem. Ask the groups to present their work.
Tell students that they will come back to this word problem later in the chapter.
Questions are provided for teachers to conduct a class discussion about the task. Students work in groups to recall what they know, discuss what they want to learn and keep track of what they have learned.
This is the main phase of learning for which TM Mathematics principally incorporates three pedagogical approaches to engage students in learning new concepts and skills.
Both concept lessons and formative assessment are centered on the proven activity-based Concrete-Pictorial-Abstract (CPA) approach.
CPA in formative assessment provides feedback to teachers on the level of understanding of students.
CPA in concept lessons consistently and systematically develops deep conceptual understanding in all students.
Concept lessons progress from teacher demonstration and shared demonstration to guided practice, culminating in independent practice and problem solving.
In Let’s Learn, teachers introduce, explain and demonstrate new concepts and skills. They draw connections, pose questions, emphasize key concepts and model thinking.
Students engage in activities to explore and learn mathematical concepts and skills, individually or in groups. They could use manipulatives or other resources to construct meanings and understandings. From concrete manipulatives and experiences, students are guided to uncover abstract mathematical concepts.
Let’s Do is an opportunity for students to work collaboratively on guided practice tasks.
Students work on Let’s Practice tasks individually in class. Teachers assign Exercises in the Practice Book as independent practice for homework.
This approach is about learning through guided enquiry. Instead of giving the answers, teachers lead students to explore, investigate and find answers on their own. Students learn to focus on specific questions and ideas, and are engaged in communicating, explaining and reflecting on their answers. They also lear n to pose questions, process information and data, and seek appropriate methods and solutions.
2.1 Identifying different types of quadrilaterals
Let's Learn Objectives: •State the properties of a rectangle, a square, a parallelogram, a rhombus and a trapezoid •Identify rectangles, squares, trapezoids, parallelograms and
Purposeful questions provided in the Teacher’s Guide help teachers to encourage students to explain and reflect on their thinking.
The three approaches detailed above are not mutually exclusive and are used concurrently in different parts of a lesson. For example, the lesson could start with an activity, followed by teacher-led enquiry and end with direct instruction.
There are multiple opportunities in each lesson for students to consolidate and deepen their learning.
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Practice helps students achieve mastery in mathematics. Let’s Practice in the Coursebook, Exercises in the Practice Book and Digital Practices incorporate systematic variation in the item sets for students to achieve proficiency and flexibility. These exercises provide opportunities for students to strengthen their understanding of concepts at the pictorial and abstract levels and to solve problems at these levels.
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There are a range of activities, from simple recall of facts to application of concepts, for students to deepen their understanding.
Think About It and Math Journal provide opportunities for students to reflect on what they have lear ned, and in doing so, consolidate and deepen their learning.
About It and Math Journal encourage development and communication of mathematical thinking.
Assessment after each chapter and quarterly Reviews provide summative assessment for consolidation of learning throughout the year.
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Summative
Mind Stretcher, Create Your Own, Mission Possible and Mathematical Modeling immerse students in problem solving tasks at various levels of difficulty.
Students
Every lesson is designed to develop deep conceptual
fluency in every student.
Evidenced through its sustained performance on international benchmarking assessments, Singapore’s Mathematics Curriculum Framework (shown in the diagram below) enumerates the critical, inter-related elements of an effective mathematics program and identifies mathematical problem solving as central to mathematics learning.
• Beliefs
• Interest
• Appreciation
• Confidence
• Perseverance
• Numerical calculation
• Algebraic manipulation
• Spatial visualization
• Data analysis
• Measurement
• Use of mathematical tools
• Estimation
• Monitoring of one's own thinking
• Self-regulation of learning
Source: www.moe.gov.sg
• Numerical
• Algebraic
• Geometric
• Statistical
• Probabilistic
• Analytical
• Reasoning, communication, and connections
• Applications and modelling
• Thinking skills and heuristics
TM Mathematics incorporates this framework in its instructional design and develops mathematical problem-solving ability through five-inter-related components: Concepts, Skills, Processes, Metacognition and Attitudes.
In
, problem solving is not only a goal of learning mathematics, it is also a tool of learning.
Suppose the number is 10.
Stage 1: 10 + 20 = 30
Stage 2: 30 – 5 = 25
Stage 3: 25 + 20 = 45
81 – 45 = 36
The number is 36 more than 10.
10 + 36 = 46
At the beginning of each chapter, Explore provides the opportunity to engage prior knowledge in problem solving, leading to independent thinking and greater ownership of learning.
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2.
3.
Throughout the chapter, students revisit the problem and persevere in solving it.
Mathematical problems are used as contexts for introducing concepts and to develop deep conceptual understanding.
Concepts are introduced in Let’s Learn in each unit via problems that the students solve using the Concrete-Pictorial-Abstract approach. Teachers lead students to investigate, explore and find answers on their own. Students are thus guided to uncover abstract mathematical concepts and ideas.
Developing
a problem-solving mindset –students can extrapolate from what they know and apply their knowledge of mathematics in a range of situations, including new and unfamiliar ones.
Students learn to solve problems by applying concepts, skills and processes learned to various problem situations both familiar and non-routine.
Each chapter ends with a problem-solving lesson.
Word problems help students recognize the role that mathematics plays in the world by applying the concepts and skills they have learned within a context.
Word problems assess students’ ability to apply
knowledge learned.
Mind stretchers are specially crafted problems that require students to apply concepts and skills to unusual or complex problem situations and solve the problems using heuristics and higher order thinking skills. Students learn how to select, innovate and compare their strategies.
Teachers will guide students through the worked out examples in the coursebooks. Additional mind stretchers are provided in the Teaching Hub for students to try out such questions on their own.
Create Your Own is a proven problem-posing and problem-solving activity in which students are encouraged to explore, share failures and successes, and question one another In doing so, they become more confident in posing problems and persist with challenging problems.
Students work in pairs or groups to create a word problem, exchange the problem with others, solve the problem and present their work to the class. Students have to explain how they come up with the word problem before presenting the solution.
Building on the mathematics concepts and skills learned, Mission Possible tasks introduce students to computational thinking, an important foundational skill in STEM education.
Prompts are provided in the teacher’s guide for teachers to guide students through the stages of computational thinking (decomposition, pattern recognition, abstraction and algorithms) to solve the problem.
Decomposition
Students break down the problem into smaller and simpler problems.
Pattern recognition
Students analyze the information and look for a pattern.
Abstraction
Students focus on information that will help them solve the problem and ignore the irrelevant details.
Algorithms
Students provide a step-by-step solution for the problem.
MISSION POSSIBLE
Have students complete the task on CB p. 329 independently. Point out to students that the bot is facing the line of symmetry. Go through the task using the prompts given below.
1. Decomposition
Ask: How can we break down the problem into smaller and simpler problems? (Answer varies. Sample: Identify the squares that need to be shaded to complete the figure. Draw a continuous path through the shaded squares. Write down each step to get from the first square to the last square.)
2. Pattern Recognition
Ask: What if the bot is not facing the line of symmetry? Will the first step still require the bot to move forward? (No) What will the first step for the bot be in this case? (To make a turn) When can the first step for the bot to go forward be? (When the bot is in a shaded square and facing the line of symmetry) When will the first step require the bot to make a turn? (When the bot is not facing the line of symmetry)
3. Abstraction Ask: What information will help you solve the problem? (Which grid squares are shaded to form the symmetric figure, where the line of symmetry is, where the bot is, the direction the bot is facing, the restriction that the bot should not return to any grid squares previously colored, the words to use, the steps given, the labels on the grid)
4. Algorithms
Have a student describe the steps he/she used to solve the problem and present the solution. Guide students to generalize the steps needed for the bot to complete a symmetric figure when: a) the bot is in a shaded square facing the line of symmetry. b) the bot is in a shaded square not facing the line of symmetry.
Mathematical Modeling is a way of connecting mathematics with real-world problems. Students represent a real-world problem using mathematics and formulate a model which may describe, explain or predict the real-world problem. The formulated model is thereafter used to obtain a solution to the real-world problem.
Phase 1: Discuss Introduce the real-world problem to the students.
Phase 2: Manipulate Students create a suitable mathematical model or framework for the given problem. They may decide on the variables involved, make sense of data and define terms.
Phase 3: Experiment and Verify Students construct the model. This usually involves the use of concrete materials or pictorial representations.
Phase 4: Present Students present their model with supporting findings and observations.
Phase 5: Reflect Students examines the limitations of the model and extends it to other similar real-world situations.
School Camp
Your school is planning a 3 days 2 nights school camp for all the grade 4 students. The camp site is 235 kilometers away from the school. Everyone will be going to the camp site by rental bus, van or both bus and van.
Work in groups to plan the number of buses and/or vans needed to fit all the people who will be going to the camp site with the least cost.
1. What are some questions you need to ask to complete the task?
Present your findings in a suitable format
a) How many vehicles do you need? b) How much will it cost to rent these vehicles?
TM Mathematics explicitly teaches students to use various thinking skills and heuristics to solve mathematical problems. Thinking skills are skills that can be used in a thinking process, such as classifying, comparing, sequencing, analyzing parts and wholes, and spatial visualization. Heuristics are problem-solving strategies.
TM Mathematics teaches the following heuristics:
Use a representation
Make a calculated guess
Walk through the process
Change the problem
• Draw a picture
• Make a list
• Choose an operation
• Guess and check
• Look for a pattern
• Make a supposition
• Use logical reasoning
• Act it out
• Work backwards
• Restate the problem in another way
• Solve part of the problem
This problem is solved using the guess and check strategy. This strategy provides a starting point for solving problems. Students should modify their subsequent guesses based on the results of the earlier guesses instead of making random guesses.
The bar model method, a key problem-solving strategy in TM Mathematics, helps students understand and draw representations of a problem using mathematical concepts to solve the problem.
In arithmetic word problems, the bar model method helps students visualize the situations involved so that they are able to construct relevant number sentences. In this way, it helps students gain a deeper understanding of the operations they may use to solve problems.
Bottle A contains 75 grams of salt.
Bottle B contains 15 grams more salt than bottle A.
a) What is the mass of salt in bottle B?
b) If Mrs. Chen uses 8 grams of salt from bottle B, what is the mass of salt left in bottle B?
Read the problem. Change the masses in the word problem. How did you decide what masses to use?
Next, solve the word problem. Show your work clearly. What did you learn?
The model method lays the foundation for learning formal algebra because it enables students to understand on a conceptual level what occurs when using complex for mulas and abstract representations. Using the model method to solve algebraic word problems helps students derive algebraic expressions, construct algebraic equations and simplify algebraic equations.
3.3 Mind stretcher
Let's Learn Let's Learn
Brian and his father have a total mass of 90 kilograms. Brian’s father is 50 kilograms heavier than Brian. What is Brian’s mass?
Let the mass of Brian be x
Let the mass of Brian’s father be y
x + y = 90
y = 50 + x
x + y = 90
x + 50 + x = 90
2x + 50 = 90
2x = 40
x = 20
Brian’s mass is 20 kilograms.
model to compare their masses. What is the
Step-by-step guidance in the lesson plans as well as complete worked solutions assist the teachers in teaching students how to solve mathematical problems using the bar model method with confidence.
Develops a growth mindset in every student –the understanding that each effort is instrumental to growth and to be resilient and persevere when initial efforts fail.
A unique 5-step Understand-Plan-Answer-Check-PlusTM (UPAC+TM) problem-solving process that ensures students’ problem-solving efforts are consistently scaffolded and students develop critical and creative thinking skills to not only solve the problem but also to consider alternatives that may be viable.
The “+” in the UPAC+TM problem-solving process, unique to TM Mathematics, is designed to develop “the top skills and skill groups which employers see as rising in prominence … include groups such as analytical thinking and innovation, complex critical thinking and analysis as well as problem-solving” (The Future of Jobs Report 2020, World Economic Forum). It is a crucial step that develops flexible problem solvers who can evaluate information, reason and make sound judgments about the solutions they have crafted, after considering possible alter native solutions. This is critical for solving real world problems.
U Understand the problem.
• If your answer is not correct, go back to Step 1. 1 2 3 4 5
• Can you describe the problem in your own words?
• What information is given?
• What do you need to find?
• Is there information that is missing or not needed?
P Plan what to do.
• What can you do to solve the problem?
• Which strategies/heuristics can you use?
Work out the A Answer
• Solve the problem using your plan in Step 2.
• If you cannot solve the problem, make another plan.
• Show your work clearly.
• Write the answer statement.
C Check if your answer is correct.
• Read the question again. Did you answer the question?
• Does your answer make sense?
• Is your answer correct?
• How can you check if your answer is correct?
• If your answer is not correct, go back to Step 1.
+ Plus
• Is there another way to solve this problem?
• Compare the methods.
• Which is the better method? Why?
Being able to reason is essential in making mathematics meaningful for all students.
Students are provided with opportunities to consolidate and deepen their learning through tasks that allow them to discuss their solutions, to think aloud and reflect on what they are doing, to keep track of how things are going and make changes when necessary, and in doing so, develop independent thinking in problem solving and the application of mathematics.
In Think About It, purposeful questions based on common conceptual misunderstandings or procedural mistakes are posed. Using question prompts as scaffolding, students think about the question, communicate their reasoning and justify their conclusions. Using the graphic organizers in Think About It, teachers act as facilitators to guide students to the correct conclusion, strengthen students’ mathematical knowledge and provide opportunities for students to communicate their reasoning and justify their conclusions.
As students get into the habit of discussing the question, anxieties about mathematical communication are eased, their mathematical knowledge is strengthened and metacognitive skills are honed. Teachers get an insight into students’ understanding and thought processes by observing the discussions.
This question highlights a conceptual misconception about comparison of fractions. Students often compare fractions without realizing that the wholes must be the same for the comparison to be valid.
This question shows a procedural mistake about subtraction of whole numbers. It is common for students to mix up the addition and subtraction algorithms.
Thinking mathematically is developed as a conscious habit.
Math Journal tasks are designed for students to use the prompts to reflect, express and clarify their mathematical thinking, and to allow teachers to observe students’ growth and development in mathematical thinking and reasoning.
There are concept-based and process-based journaling tasks in TM Mathematics Teaching Hub.
Concept-based
Process-based tasks help teachers understand students’ thinking process through a concept.
Let's Practice Practice
Task 1 requires students to count by threes to find the total number of objects and complete the multiplication sentences.
Task 2 requires students to count by threes to complete the patterns.
Students learn through guided enquiry, a process during which instead of giving the answers, teachers lead students to explore, investigate and find answers on their own by posing purposeful questions provided in the Teacher’s Guide. Purposeful questions are used to gather information, probe thinking, make the mathematics visible and encourage reflection and justification. Posing purposeful questions helps to assess and advance students’ reasoning and sense making about important mathematical ideas and relationships.
1.2 Using dot cards
Let's Learn Let's Learn
Objectives:
• Observe the commutative and distributive properties of multiplication
• Relate two multiplication facts using ‘3 more’ or ‘3 less’
• Build up the multiplication table of 3 and commit the multiplication facts to memory
Materials:
• Dot Card F (BM10.1): 1 copy per group, 1 enlarged copy for demonstration
• Dot Card G (BM10.2): 1 copy per group, 1 enlarged copy for demonstration
•Counters
Resources:
There are 18 pears altogether.
Gathering information
• CB: pp. 195–197 • PB: p. 127
(a) Stage: Concrete Experience Draw 6 circles on the board and stick 3 counters in each circle.
Ask: How many counters are there in each group? (3) How many groups are there? (6)
Say: We have 6 groups of 3 counters.
Stage: Pictorial Representation
Say: We can use a dot card to help us find the total number of counters. Have students work in groups. Distribute counters and a copy of Dot Card F (BM10.1) to each group. Stick an enlarged copy of Dot Card F (BM10.1) on the board. Put counters on the three circles in the first row of the dot card.
Say: There is 1 row of counters. There are 3 counters in 1 row. I have shown 1 group of 3.
Ask: How do we show 6 groups of 3 on the dot card? (Put counters on 6 rows of the dot card.)
Demonstrate how the counters are to be placed
Making the mathematics visible
The activity-based Concrete-Pictorial-Abstract (CPA ) approach is a key instructional strategy advocated in the Singapore approach to mathematics learning. In TM Mathematics, the CPA approach is embedded in the learning experiences:
Concept Development
(Objective: Developing deep conceptual understanding): Let’s Learn
Formative Assessment
(Objective: Evaluating levels of understanding): Let’s Do
Summative Assessment
(Objective: Evaluating conceptual mastery and procedural fluency): Let’s Practice, Practice Book Exercises, Digital Practice
Each Let’s Learn segment provides a hands on, teacher-facilitated experience of concepts through the CPA stages.
Concrete
Students use manipulatives or other resources to solve a problem. Through these activities they explore and learn mathematical concepts and skills, individually or in groups, to construct meanings and understandings.
Pictorial
Pictorial representation of the objects used to model the problem in the Concrete stage enables students to see the connections between mathematical ideas and the concrete objects they handled.
Abstract
Once conceptual understanding is developed, students learn to represent the concept using numbers and mathematical symbols.
Throughout the activity, the teacher observes what the students say and do and provides feedback to students.
The CPA approach to mathematics instruction and learning enables students to make and demonstrate mathematical connections, making mathematical understanding deep and long-lasting.
Within each concept lesson, Let’s Do provides vital feedback to the teacher to understand the level of conceptual understanding of each student and to make appropriate instructional decisions for students.
The tasks in Let’s Do are systematically varied so that as students move from one task to the next, the teacher is able to gauge their level of understanding of the concept and if they can progress to independent work.
Task 1(a) requires students to add like fractions within 1 whole with pictorial aid. Task 1(b) is an extension of Task 1(a). It requires students to simplify the answer after adding the fractions.
Let’s Practice, Practice Book Exercises and Digital Practice help students to transition their understanding of concepts from pictorial to abstract levels.
Practices start with pictorial tasks, moving on to abstract tasks with pictorial aids and finally solely abstract tasks to help students make the transition from pictorial to abstract levels.
Coherent framework, spiral curriculum.
Singapore’s Mathematics Curriculum Framework in which mathematical problem solving is the central focus is at the center of the curriculum design of TM Mathematics. The framework stresses conceptual understanding, skills proficiency and mathematical processes and duly emphasizes metacognition and attitudes. It also reflects the 21st century competencies.
Mathematics is hierarchical in nature. TM Mathematics has a focused and coherent content framework and developmental continuum in which higher concepts and skills are built upon the more foundational ones. This spiral approach in the building up of content across the levels is expressed as four Learning Progression Principles that are a composite of the successful practices and lear ning standards of the top performing nations, and, are unique to TM Mathematics.
INESS Metacognition Processes Concepts Skills Attitudes
1 3 MASTERY ENGAGEMENT
2
The
careful spiral sequence of successively more complex ways of reasoning about mathematical concepts – the learning progressions within TM – make the curriculum at the same time, rigorous and effective for all learners.
Learning
Progression Principle 1:
Deep focus on fewer topics builds a strong foundation.
The early learning of mathematics is deeply focused on the major work of each grade— developing concepts underlying arithmetic, the skills of arithmetic computation and the ability to apply arithmetic. This is done to help students gain strong foundation, including a solid understanding of concepts, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the classroom.
Across
Learning Progression Principle 2:
Topics within strands are sequenced to support in-depth and efficient development of mathematics content. New learning is built on prior knowledge. This makes learning efficient, while revisiting concepts and skills at a higher level of difficulty ensures in-depth understanding.
Example
Strand: Numbers and Operations
Grade 1
Topic: Numbers 0 to 10
Development of number sense
• counting
• reading and writing numbers
• comparing numbers
• by matching • by counting
Topic: Number Bonds
Number bonds (part-part-whole relationship):
• 3 and 2 make 5.
• 4 and 1 make 5.
Topic: Addition
Addition (part-part-whole):
• 3 + 2 = 5 part part whole
Topic: Subtraction
Subtraction (part-part-whole):
• 5 – 3 = 2 whole part part
Topic: Numbers to 20
• counting and comparing
• ordering
Topic: Addition and Subtraction
• Addition within 20
• Subtraction within 20
• Students first learn to count, read and write numbers and to compare numbers.
• The concept of number bonds, that the whole is made up of smaller parts, builds on students’ knowledge of counting and comparing.
• The part-part-whole relationship between numbers forms the foundation for understanding addition and subtraction, and the relationship between these operations.
• Counting and comparing are revisited at a higher level of difficulty and are extended to ordering.
• Addition and subtraction are revisited and the concept of regrouping is introduced.
Learning Progression Principle 3:
Sequencing of learning objectives within a topic across grades is based on a mathematically logical progression.
Learning objectives within a topic are sequenced across grades according to a mathematically logical progression.
Example
Strand: Numbers and Operations
Topic: Fractions
Grade 1:
• Halves and quarters
Grade 2:
• Halves, thirds and quarters
• Naming fractions with denominator up to 12
Grade 3:
• Comparison of fractions
• Equivalent fractions
• Addition and subtraction of like and related fractions within 1 whole
Grade 4:
• Mixed numbers and improper fractions
• Fraction and division
• Addition and subtraction of like and related fractions greater than 1 whole
• Multiplication of a fraction and a whole number
Grade 5:
• Addition and subtraction of unlike fractions
• In grades 1 and 2, conceptual understanding of fractions is developed. Students lear n to recognize and name fractions.
• In grade 3, students learn to compare fractions. Equivalent fractions are introduced to help students add and subtract fractions.
• In grades 4 and 5, mixed numbers and improper fractions are introduced. The complexity of operations is also expanded to cover fractional numbers greater than one whole as well as multiplication and division.
Learning Progression Principle 4: Purposeful sequencing of learning objectives across strands deepens links and strengthens conceptual understanding.
The ordering of content for one topic is frequently aligned to reinforce the content of another topic across strands.
Example
Grade 1
Strand: Numbers and Operations
Chapter 16
Topic: Fractions
Learning objective: Recognize and name one half of a whole which is divided into 2 equal halves.
Strand: Measurement
Chapter 18
Topic: Time
Learning objective: Tell time to the half hour
Chapter 16
• Fractions are introduced prior to the lesson on telling time to the half hour so that students will be able to make the connection between the visual representation of halves in fractions and the representation of the half hour on a clock face.
As students lear n to tell time to the half hour, the concept of halves, learned in a prior chapter, is reinforced.
Chapter 18
TM Mathematics covers all the curriculum standards and topics in the curricula of Singapore, Hong Kong and Republic of South Korea. It also completely covers the Cambridge Primary Mathematics curriculum. Additional topics are also available in the Teaching Hub for alignment to different education systems.
TM Mathematics enables every child to succeed by integrating formative and summative assessment with instruction for effective teaching and independent learning.
When instruction is informed by insights from assessment, students are more engaged and take greater ownership of their learning.
Formative assessment is a vital part of the ongoing, interactive process by which teachers gather immediate insight about students’ learning to inform and support their teaching.
Let's Do at each step of concept development are formative and diagnostic assessments. They assess the student’s learning and level of conceptual understanding to provide timely feedback to teachers.
1. Divide. Use the related
Let’s Do enables teachers to immediately assess students’ understanding of the concepts just taught and identify remediation needs.
Task 1 assesses students’ understanding of division by 5 at the pictorial and abstract levels.
Task 2 assesses students’ understanding of division by 5 at the abstract level.
Purposeful Practice tasks in print and digital formats complement and extend learning. They encourage students to develop deep conceptual understanding and confidence to work independently. Practice tasks also serve as for mative and diagnostic assessment providing essential information to students and teachers on learning progress.
5.1
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2.
Recap provides a pictorial and abstract representation of the concrete activity carried out in the class.
Tasks
are ordered by level of difficulty and are systematically varied to gradually deepen the student’s conceptual understanding.
Easy to assign and with instant access, Digital Practice includes hints to support students and provides immediate feedback to teachers on students’ learning.
Students can click on the Hint button if they need help with the practice task.
Summative assessments enable teachers to assess student learning at the end of each chapter and beyond.
Reviews provide summative assessment and enable consolidation of concepts and skills learned across various topics.
There are four reviews per year to consolidate learning across several chapters.
Digital Assessment provides topical, cumulative and progress monitoring assessments for evaluating fluency, proficiency and for benchmarking throughout the year.
There is an assessment at the end of every chapter to consolidate learning for the chapter.
There is an assessment at the end of each quarter of the year to test the topics taught to date.
There are assessments in the middle and end of the year. These assessments can be administered as benchmark tests.
Auto-generated reports for Digital Practice and Assessment make data easily accessible and actionable to support every teacher’s instructional goals. Teachers can review high level reports at class level or dive into the details of each student, chapter, topic, concept and practice or assessment item.
Meaningful actionable insights help teachers easily find learning gaps and gains.
Reports for Practice provide timely formative and diagnostic data on student learning that teachers can act on immediately to adjust instructional practices in an effort to address and maximize individual students’ learning.
Class List by Practice Report shows student performance on each practice.
Teachers can tell at a glance how well students in a class have performed on a practice and determine if remediation is required.
Class List by Learning Objective Report shows student performance against the learning objectives of each practice.
Before proceeding to the next lesson, teachers can review this report to identify the learning objectives that students have difficulty with, reteach these lear ning objectives or pay special attention to the struggling students in class. Doing so will ensure that the next lesson is off to a good start and increase the chances of students keeping up with the lesson.
Reports for Assessments provide in-depth mastery analysis in an easy to access and view format.
Class List by Assessment Report shows student performance on each assessment.
Actionable, real-time reports accessible on the teacher’s dashboard help to monitor student progress and make timely instructional decisions.
This report informs teachers on how well students have learned each chapter.
Class List by Learning Objective Report shows student performance against a topic or learning objective by aggregating the results for it across multiple assessments.
This report helps teachers to identify the strengths and weaknesses of the class as well as individual students and take intervention actions as needed.
All class reports can be drilled down to the individual student level.
All reports in Digital Practice and Assessment can be printed for reporting by school administrators.
A comprehensive range of resources for grades 1 to 6 supports teaching, learning, practice and assessment in a blended, print or digital environment to provide flexibility in planning and instruction, and lear ning.
Serves as a guide for carefully constructed, teacher-facilitated learning experiences for students. This core component provides the content and instruction for all stages of the learning process—readiness, engagement and mastery of concepts and skills.
Correlates to the coursebooks and contains exercises and reviews for independent practice and for mative and summative assessments.
Online opportunities for students to consolidate learning and demonstrate understanding.
Coursebook in online format with embedded videos to ensure that learning never stops.
Comprehensive lesson plans support instruction for each lesson in the Coursebooks.
This one-stop teacher’s resource center provides access to lesson notes, demonstration videos and Coursebook pages for on-screen projection.
A digital component that enables teachers to assign Practice and Assessment tasks to students and provides teachers with meaningful insight into students’ learning through varied, real-time reports.
Video tutorials and related quizzes in this online resource provide anytime, anywhere professional learning to educators.
These posters come with a poster guide to help teachers focus on basic mathematical concepts in class and enhance learning for students.
Every mathematics teacher is a master teacher.
TM Mathematics provides extensive support at point of use to support teacher development along with student lear ning, making teaching mathematics a breeze.
A comprehensive Teacher’s Guide, available in print and digital formats, provides complete program support including:
• developmental continuum,
• Scheme of Work,
• detailed notes for each lesson in the Coursebook,
• answers for practice tasks in the Coursebook and Practice Book, and
• reproducibles for class activities.
This one-stop teacher’s resource center provides resources for planning and teaching. It contains
• all the content from the Coursebook and Practice Book,
• all lesson notes from the Teacher’s Guide,
• lesson demonstration videos embedded at point of use,
• extra lessons addressing learning objectives for regional curricula and
• jour nal tasks.
The Teaching Hub functions as a teacher resource for front-of-class facilitation during lessons. Controlled display of answers in the Coursebook and Practice Book assists teachers in carrying out formative assessment during lessons.
Teachers can view the demonstration video to see and hear a lesson before teaching the lesson to students. The video can even be played during the lesson to help explain the mathematical concept to students.
Teachers can attach content they have created to the Coursebook pages to customize lessons.
Additional lessons and other resources not available in the print Coursebook and Practice Book are downloadable so that teachers can print them for students.
TM Professional Learning Now! provides on-demand professional development for teachers to learn mathematics pedagogy anytime, anywhere — in the convenience and comfort of their home or in-between lessons, or just before teaching a topic. Each learning video is intentionally kept to approximately 5 minutes so that teachers will be able to quickly and effectively learn the pedagogy behind the concept to be taught. With a short quiz of 4 or 5 questions and a performance report, professional development is relevant and effective for teachers at any stage in their teaching career. Teachers can also re-watch learning videos to reinforce their pedagogical content knowledge anytime, anywhere.
TM Mathematics Teacher’s Guides are designed to help teachers implement the program easily and effectively.
The Developmental Continuum provides an overview of prior, current and future learning objectives. Strands are color-coded to help teachers identify the connected topics within a strand.
Numbers and Operations Measurement
Geometry
Data Analysis Algebra
The objectives of each lesson are listed in the Scheme of Work to help teachers establish mathematics goals during lesson planning.
The suggested duration for each lesson is 1 hour. Teachers can adjust the duration based on the school calendar and the pace of individual classes.
Unit 2: Addition and Subtraction Without Regrouping
2.1 Adding a 1-digit number to a 2-digit number
Let's Learn Let's Learn
Objectives:
•Add a 1-digit number and a 2-digit number without regrouping using the ‘counting on’ method, number bonds and place value
•Check the answer to an addition by using a different strategy
Materials:
•2 bundles of 10 straws and 4 loose straws
•Base ten blocks
Resources:
•CB: pp. 27–29
•PB: pp. 23–24
Stage: Concrete Experience
Write: Add 21 and 3. Show students two bundles of 10 straws, and 1 loose straw. Highlight to them that each bundle has 10 straws.
Ask: How many straws are there here? (21) Add another 3 loose straws to the 21 straws.
Ask: How many straws are there now? (24)
Say: When we add 3 straws to 21 straws, we get 24 straws.
(a) Stages: Pictorial and Abstract Representations
Draw a number line with intervals of 1 from 21 to 26 as shown in (a) on CB p. 27 on the board.
Say: We can add by counting on using a number line.
Have students add 21 and 3 by counting on
3 ones from 21. (21, 22, 23, 24) As students count on, draw arrows on the number line as shown on the page.
Ask: Where do we stop? (24)
Say: We stop at 24. When adding a number to 21, we start from 21 and count on because we add. We count on 3 ones because we are adding 3.
Write: 21 + 3 = 24
(b) Stage: Abstract Representation Say: Another way to add is by using number bonds. Show students that 21 can be written as 20 and 1 using number bonds. Write: 21 + 3 = 20 1 Say: First, add the ones. Ask: What do we get when we add 1 and 3? (4) Say: Now, add the tens to the result. We add 20 to 4. Elicit the answer from
Detailed lesson plans explain the pedagogy and methodology for teaching each concept, equipping teachers to teach lessons with confidence.
©
Chapter 2: Addition and Subtraction Within
For each task in Let’s Remember, the objective of the task and the chapter reference to where the skill was taught earlier are listed for teachers to reteach the relevant concepts.
Explore gets students to recall prior knowledge, set learning goals and track their learning as they progress through the chapter. Questions are provided in the Teacher’s Guide to aid class discussion about the context of the task.
Have
to present their work.
Tell students that they will come back to this word problem later in the chapter.
Unit 1: Sum and Difference
1.1 Understanding the meanings of sum and difference
Let's Learn Learn
Objectives:
•Associate the terms ‘sum’ and ‘difference’ with addition and subtraction respectively
•Use a part-whole bar model or a comparison bar model to represent an addition or subtraction problem
Materials:
•Connecting cubes in two colors
•Markers in two colors
Resources:
•CB: pp. 25–26
•PB: p. 22
Vocabulary:
Suggested instructional procedures are provided for the concrete, pictorial and abstract stages of learning.
Let's Do
Task
Task
Let's
Tasks 1 and 3 require students to associate the term ‘sum’ with addition.
Tasks 2 and 4 require students to associate the term ‘difference’ with subtraction.
•difference
•sum
(a) Stage: Concrete Experience
Have students work in pairs. Distribute connecting cubes in two colors, for example, red and blue, to each pair and have students follow each step of your demonstration.
Join 3 red connecting cubes to show 3. Then, join 8 blue connecting cubes to show 8.
Ask: How many red cubes do you see? (3) How many blue cubes do you see? (8)
Join the bar of red cubes and the bar of blue cubes together.
Ask: How many cubes are there altogether? (11)
Stage: Pictorial Representation
Use two markers in different colors to draw a part-whole bar model with 3 equal units and 8 equal units to illustrate the numbers 3 and 8, as shown by the connecting cubes. Relate this model to the earlier connecting cubes activity.
Erase the lines between the units in the bar model to create a simplified version of the model as shown on the right in (a) on CB p. 25.
Say: This is a bar model.
Point out that the length of each part of the model corresponds to the number of connecting cubes of each color.
Say: The two parts form a whole. This model shows the total or the sum of 3 and 8. The sum of two numbers is the total of the two numbers.
We found earlier that the total of 3 cubes and 8 cubes is 11 cubes, so the sum of 3 and 8 is 11.
Separate the bar of connecting cubes into its two parts, 3 and 8, again. Place the bar of 3 cubes above the bar of 8 cubes and left align the bars.
Say: Notice that the total number of cubes has not changed. Let us represent the sum of 3 and 8 in another model.
Draw the comparison bar model as shown in the thought bubble in (a) on the page.
Conclude that we can represent the sum in two types of bar models.
Stage: Abstract Representation
Say: We want to find the sum of 3 and 8. The sum of 3 and 8 is the total of 3 and 8. We find the sum by adding the two numbers.
Write: 3 + 8 = 11
Say: The sum of 3 and 8 is 11.
(b) Stage: Concrete Experience
Have students continue to work in pairs and follow each step of your demonstration.
Reuse the two bars of connecting cubes formed in (a). Place the bar of 3 cubes above the bar of 8 cubes and left align the bars.
Ask: How many red cubes are there? (3) How many blue cubes are there? (8) Which bar is shorter, the bar of red cubes or the bar of blue cubes? (Red cubes) Which is less, 3 or 8? (3)
Say: Let us find out how many more blue cubes than red cubes there are by counting the number of cubes.
For each Let’s Do task, the objective is listed for teachers to reteach the relevant concepts. Answers are provided for all tasks.
For each task in the Practice Book Exercise, the objectives and skills assessed are identified in the Teaching Hub to enable teachers to check learning and address remediation needs. Answers are provided for all tasks.
Digital Practice provides immediate feedback to teachers on students’ learning so that teachers can provide early intervention if required. The items come with hints to promote independent learning for students.
1
2
David is checking his friend's answer to a subtraction.
What did
learn about subtracting a 1-digit number from a 2-digit number with regrouping
Think
Have students work in groups to discuss the tasks. Ask the groups to present their answers.
Point out to students that 12 in the ones column represents 1 ten 2 ones and not 3 ones. David has mixed up addition and subtraction with regrouping in the vertical form. Conclude that David is not correct.
Reiterate that if there are not enough ones to subtract from, we need to first regroup the tens and ones before we subtract.
Make use of the examples presented by the groups to let students understand the importance and usefulness of knowing how to subtract numbers.
Think About It poses purposeful questions to facilitate meaningful mathematical discourse and promote reasoning and communication. Students work in groups to discuss the task and present and justify their answers to the class.
Let's
Objectives:
1. Understand
Have students read the word problem then articulate in their own words what information is given and what is unknown. Pose questions given in the Coursebook to direct students.
2. Plan
Have students plan how to solve the problem. Have them discuss the various strategies they have learned and choose one.
3. Answer
Have students solve the problem using the chosen strategy.
4. Check
Have students check their answer for accuracy or reasonableness.
5. + Plus
Explore other strategies identified in step 2. Compare the different strategies and discuss preferences.
Resources:
Have
1.
2.
3.
have to find.
Ask: How can we find the number of cupcakes left? (Subtract the number of cupcakes given away from the number of cupcakes Emma buys.)
Write: 24 – 16 = Ask a student to work out the subtraction on the board.
Say: Emma has 8 cupcakes left.
For each Let’s Do task, the objective is listed for teachers to reteach the relevant concepts. Answers are provided for all tasks.
For each task in the Practice Book Exercise, the objectives and skills assessed are identified in the Teaching Hub to enable teachers to check learning and address remediation needs. Solutions are provided for all tasks.
Digital Practice provides immediate feedback to teachers on students’ learning so that teachers can provide early intervention if required. The items come with hints to promote independent learning for students.
Solve the word problems. Show your work clearly.
1. There are 82 sandwiches on a table.
25 are egg sandwiches, 34 are tuna sandwiches and the rest are chicken sandwiches.
a) How many egg and tuna sandwiches are there altogether?
b) How many chicken sandwiches are there?
2. Karen had 27 red apples.
She had 18 more green apples than red apples. She used 29 green apples to make some juice.
a) How many green apples did she have at first?
b) How many green apples did she have left after making juice?
3. Vivian has 51 storybooks.
She has 13 more storybooks than Kevin.
2
Students
Students
a) How many storybooks does Kevin have?
b) How many storybooks do they have altogether?
CREATE YOUR OWN
Nathan has 46 stamps. He has 19 more stamps than Tim.
a) How many stamps does Tim have?
b) If Tim gives 8 stamps to Zoe, how many stamps will he have left?
Read the word problem. Replace ‘more’ with ‘fewer’. Next, solve the word problem. Show your work clearly. What did you learn?
To
To solve 1-step word problems involving subtraction with regrouping
Tasks 1 to 3 require students to solve 2-step word problems involving addition and subtraction.
CREATE YOUR OWN
Have students work in groups to create and solve the word problem. Have a few groups present their work.
Students are expected to replace ‘more’ with ‘fewer’ in the word problem. So, they have to add in the first part and subtract in the second part to solve the word problem.
Students are expected to solve a 1-step subtraction word problem involving a comparison situation by finding the difference given the two quantities. They can draw a comparison bar model to help them solve the word problem.
4.2 Mind stretcher
Let's Learn Let's Learn
Objective:
•Solve a non-routine problem involving addition and subtraction using the strategy of working backwards
Resource: •CB: pp. 62–63
Create Your Own tasks facilitate meaningful mathematical discourse and promote reasoning and problem solving. Students work in pairs or groups to discuss the task and present their work to the class.
Have students read the problem on CB p. 62.
1. Understand the problem. Pose the questions in the thought bubble in step 1.
2. Plan what to do.
Say: Since we know the
number, let us
4.2
2.
3.
and Subtraction Within 100 63
Write: Stage 3: + 20 = 81
Say: To find the missing number, we subtract 20 from 81.
Write: 81 – 20 = Elicit the answer from students. (61)
Write ‘61’ in the third box in the diagram.
Write: Stage 2: – 5 = 61
Ask: How do we find the missing number? (Add 5 to 61.)
Write: 61 + 5 = Elicit the answer from students. (66)
Write ‘66’ in the second box in the diagram.
Write: Stage 1: + 20 = 66
Ask: How do we find the missing number?
(Subtract 20 from 66.)
Write: 66 – 20 = Elicit the answer from students. (46)
Write ‘46’ in the first box in the diagram.
Say: Julia starts with the number 46.
4. Check if your answer is correct.
Guide students to check their answer by starting with 46 and going through the three stages in the problem to see if they get 81 in the end.
5. + Plus Solve the problem in another way.
Have students try to solve the problem in a different way.
Have 1 or 2 students share their methods.
If students are unable to solve the problem in a different way, explain the method shown on CB p. 63.
Ask: Which method do you prefer? Why?
(Answers vary.)
EXPLORE
Have students go back to the word problem on
CB p. 24. Get them to write down in column 3 of the table what they have learned that will help them solve the problem, and then solve the problem. Have a student present his/her work to the class.
Mind Stretcher provides opportunities for students to apply concepts and skills learned to unusual or complex problem situations. Encourage students to solve the problem using different strategies.
Chapter 2: Addition and Subtraction Within 100 64
Mission Possible tasks introduce students to computational thinking, an important foundational skill in STEM education. The Teacher’s Guide provides prompts to help teachers facilitate the class discussion.
MISSION POSSIBLE Have students work in groups to complete the task on CB p. 329.
Go through the task using the prompts given below.
1. Decomposition Ask: How can we break down the problem into smaller and simpler problems? (Answer varies. Sample: Find out how much money Miguel has, find all the combinations of two presents Miguel can buy, find the total cost of each combination of presents, find the amount of money left after buying each combination)
2. Pattern Recognition Lead students to say that every time they find the total cost of the combination of presents, they have to check which of the total cost is closest to $78 and less than $78.
3. Abstraction Ask: What information will help you solve the problem? (The notes that Miguel has, the cost of the book, the costs of the presents, and he wants to use up as much of his money as possible)
4. Algorithms Guide students to draw a simple flow chart to show the steps used to solve the problem. Ask a group to write their solution on the board.
MATHEMATICAL MODELING
Duration: 5 h (5 one-hour sessions)
Mathematical Modeling tasks require students to apply mathematics to complex real-world problems. Prompts and rubrics are provided to assist teachers in conducting the lesson and assessing students’ performance.
Material:
•1
Resource:
•CB: pp. 352–353
Topics:
•Multiplication and division of whole numbers (CB5 Chapter 2)
•Line graphs (CB5 Chapter
8.
Digital Chapter Assessment enables consolidation of learning in every chapter. Auto-generated reports provide actionable data for teachers to carry out remediation or extension as required.
Math Journal tasks in the Teaching Hub allows teachers to gain insight into students’ thinking. Rubrics are provided to help teachers give feedback to students.
1To
2To
7To
Digital Quarterly and Half-Yearly Assessments provide opportunities for summative assessment at regular intervals throughout the year. Auto-generated reports help teachers to measure students’ learning.
Practice Book Reviews provide opportunities for summative assessment. They consolidate learning across several chapters. The last review in each grade assesses learning in the entire grade. For each task, the objectives assessed are identified in the Teaching Hub to enable teachers to check learning and address remediation needs.
TM Mathematics can be flexibly used in print, blended or digital formats based on the context to maximize teaching and learning.
Start of school year: Developmental Continuum
Start of chapter: Scheme of Work
Start of lesson: Lesson plan
Lesson demonstration video
Let’s Remember Explore
Teach concepts and skills:
Let’s Learn
Let’s Do Practice Book Exercise
Digital Practice
Think About It
Digital Chapter Assessment
Math Journal
Practice Book Reviews
Digital Quarterly Assessment
Digital Half-Yearly Assessment
Teach problem solving: Let’s Learn (UPAC+™) Let’s Do Practice Book Exercise
Digital Practice
Create Your Own Mind stretcher
Mission Possible Mathematical Modeling
Available
Born and brought up in Mumbai, India, Dr. Aziz has lived in Singapore for more than 30 years now and is the proud grandma of three lovely girls. At Scholastic, she has oversight of Scholastic’s global education business and the research and development of all education products, technology and print.
She completed her doctoral research at the Leeds Metropolitan University and holds a bachelor’s degree in English Language and Literature from the Open University. She holds Graduate degrees in Business Administration and English Studies from the University of Strathclyde and the National University of Singapore, respectively. Inspired by her work in development of materials for teaching and learning, Duriya’s Master’s research was on the role of teacher feedback in materials development, and her doctoral research culminated in the presentation of a framework for development and evaluation of materials in meeting the objectives of stakeholders in education and their impact on teaching and learning.
Though an English language specialist by training, math education found her and Dr. Aziz has spent almost 20 years developing curriculum programs based on Singapore mathematics pedagogical principles and practices for more than 20 countries, in different languages, and worked with ministries, schools and teachers on the implementation of these programs. Dr Aziz’s primary interest is in the development and implementation of programs that incorporate global best practices while remaining culturally and contextually appropriate, to drive sustainable change at a systemic level including development of teacher competence, knowledge and independence.
Duriya has written several textbooks for learners of English and children’s picture books, as well as academic articles on English language teaching and materials development for education.
Kelly Lim holds a Masters in Mathematics Education from The Institute of Education, London, a degree in Mathematics from the National University of Singapore (B. Sc.) and a Post-Graduate Diploma from the National Institute of Education in Singapore (PGDE).
In her current role at Scholastic, she provides program implementation support and professional development to school leaders, educators and parents around the world wherever Scholastic’s acclaimed mathematics programs, particularly PR1ME Mathematics, are in use.
Kelly was the founding headmaster of the third campus of a Singapore international school in Thailand before joining Scholastic. Her stint in Thailand, which lasted for about a decade, was preceded by her time in Singapore, where she taught in government schools for a similar period.
Oscar holds a Master’s degree in Mathematics Education, a specialization on applied ICT in education from Universidad Pedagogica Nacional and a degree in Mathematics from Sergio Arboleda University in Colombia. Currently, he is working on his second Master in Educatronics.
He was a mathematics teacher for 12 years and was head of the mathematics department in a bilingual school, in which he was leading the PR1ME Mathematics program implementation. He has written articles and spoken at different events on Pedagogical Content Knowledge (PCK), B-Learning Flipped Classroom, Statistics teaching and Singapore mathematics.
Oscar was a Calculus and Statistics professor at Sergio Arboleda University in Colombia. Currently, he is the mathematics consultant at Scholastic, offering professional development and supporting the implementation of PR1ME Mathematics and ¡Matemáticas al Máximo! in Latin America, the Caribbean, Middle East, Africa and Europe.
Óscar Mauricio Gómez holds a Master’s degree in Mathematics Education from Francisco José de Caldas District University in Bogotá, Colombia. Óscar has been working in the educational field for the last 14 years as a math teacher in both schools and college classrooms. He worked as a curricular advisor for the Colombian Ministry of Education in the national curricular restructuring in 2016 and the program of Colombia Aprende.
Clara Guerrero’s professional experience combines over 10 years of classroom teaching at the primary level with over 20 years in the field of educational publishing. She has a Master’s degree in Education and a Bachelor’s degree in English Language Teaching.
As a reviewer, Clara focused on the linguistic and cultural aspects of PR1ME Mathematics. Given the fact that a considerable number of students and teachers using the program around the world are not native speakers of English, Clara ensured that the language and contexts used throughout the series were appropriate and did not hinder the lear ning of mathematical concepts.
Teachers can use the Developmental Continuum to understand the links between learning objectives within and across strands and grade levels. It provides a useful overview of prior, current and future learning objectives. Teachers will observe how new learning is built on prior learning across the grades and how each topic forms the foundation for future learning.
Grade 4
Whole Numbers / Place Value
Grade 5 Grade 6
Count within 10 000. Read and write a number within 1 000 000—the numeral and the corresponding number word.
Read and write a number within 10 000—the numeral and the corresponding number word.
Use number notation and place values (thousands, hundreds, tens, ones).
Find the number which is ones, tens, hundreds or thousands more than or less than a given number within 10 000.
*Identify patterns in a hundred chart.
Count on and backwards by ones, tens, hundreds or thousands within 10 000.
Describe, complete and create a number pattern by counting on or backwards by ones, tens, hundreds or thousands within 10 000.
*Describe and complete a number pattern by repeated addition or multiplication.
Compare and order numbers within 10 000.
Use ‘>’ and ‘<’ symbols to compare numbers.
Read and place numbers within 10 000 on a number line.
Give a number between two 4-digit numbers.
Round a 3-digit or 4-digit number to the nearest ten.
Round a 3-digit or 4-digit number to the nearest hundred.
Recognize the historical origins of our number system and begin to understand how it developed.
Identify the values of digits in a 5-digit or 6-digit number. Read and write a number within 10 000 000— the numeral and the corresponding number word.
Compare and order numbers within 1 000 000.
Round a whole number to the nearest ten, hundred or thousand.
Find all the factors of a whole number up to 100.
Find out if a 1-digit number is a factor of a given whole number.
Find the multiples of a whole number up to 10.
Identify the values of digits in a 7-digit number.
Compare and order numbers within 10 000 000.
Round a whole number to the nearest ten thousand, hundred thousand or million.
Find the common factors and greatest common factor of two numbers.
Find out if a number is a common factor of two given numbers.
Relate factors and multiples.Find the common multiples and least common multiple of two numbers.
Find out if a whole number is a multiple of a given whole number up to 10.
Find out if a number is a common multiple of two given numbers.
Identify multiples of 2, 5, 10, 25, 50 and 100 up to 1000. Solve word problems involving common factors and multiples.
Identify prime numbers and composite numbers.
Recognize prime numbers up to 20 and find all prime numbers less than 100.
Identify square numbers.
*Identify triangular numbers.
*Express a number in exponential notation.
*Find the value of a number given in exponential notation.
Whole Numbers / Place Value (continued)
Round a 4-digit number to the nearest thousand.
Addition / Subtraction
Add and subtract within 10 000.
Estimate sums and differences.
Check reasonableness of answers in addition or subtraction using estimation.
Solve 1-step and 2-step word problems involving addition and subtraction.
Find pairs of multiples of 50 with a total of 1000 and write the addition and subtraction facts for each number pair.
Mentally add:
- two 2-digit numbers with regrouping
- a 2-digit, 3-digit or 4-digit number to a 3-digit or 4-digit number with regrouping
- three or four 1-digit or 2-digit numbers
- three 2-digit multiples of 10
Mentally subtract:
- a 2-digit number from another 2-digit number with regrouping
- a 2-digit, 3-digit or 4-digit number from a 3-digit or 4-digit number
*Extend spatial patterns formed from adding and subtracting a constant.
*Extend spatial patterns of square and triangular numbers.
*Identify cube numbers.
Estimate sums and differences.
Check reasonableness of answers in addition or subtraction.
Investigate and generalize the result of adding and subtracting odd and even numbers.
Do mixed operations involving addition and subtraction without parentheses.
Do mixed operations involving the four operations with or without parentheses.
Write simple expressions that record calculations with numbers.
Estimate sums and differences.
Solve multi-step word problems involving the four operations of whole numbers.
Use a calculator to carry out the four basic operations.
Calculate a sum or a difference on a calculator and check the reasonableness of the answer.
Solve multi-step word problems involving the four basic operations using a calculator.
*Solve challenging word problems involving whole numbers.
Interpret numerical expressions without evaluation.
Solve multi-step word problems involving four operations of whole numbers.
Add two numbers up to 4 digits by counting on in thousands, hundreds, tens and ones.
Subtract a number up to 4 digits by counting backwards in thousands, hundreds, tens and ones.
Multiplication / Division
Multiply ones, tens or hundreds by a 1-digit number.
Multiply a 3-digit whole number by a 1-digit number.
Multiply three 1-digit numbers.
Apply the commutative and associative properties of multiplication in computation.
Multiply a whole number up to 3 digits by 10.
Divide hundreds or tens by a 1-digit number.
Divide a 3-digit whole number by a 1-digit number.
Divide a whole number up to 3 digits by 10.
Estimate products and quotients.
Check reasonableness of answers in multiplication or division using estimation.
Solve up to 3-step word problems involving multiplication and division.
Find doubles of 2-digit numbers mentally.
Find doubles of multiples of 10 up to 1000 mentally.
Find doubles of multiples of 100 up to 10 000 mentally.
Find halves of whole numbers up to 200 mentally.
Investigate and generalize the result of multiplying odd and even numbers.
Know and apply tests of divisibility by 2, 3, 4, 5, 10, 25 and 100.
Multiply or divide a whole number by 10, 100 or 1000.
Multiply or divide a whole number by tens, hundreds or thousands.
Multiply pairs of multiples of 10 or multiples of 10 and 100.
Multiply a 4-digit whole number by a 1-digit whole number.
Multiply a 2-digit whole number by a 2-digit whole number.
Divide a 4-digit whole number by a 1-digit whole number.
Divide a 2-digit whole number by a 2-digit whole number.
Estimate products and quotients.
Check reasonableness of answers in multiplication or division.
*Divide a 5-digit whole number by a 1-digit whole number.
Do mixed operations involving multiplication and division without parentheses.
Do mixed operations involving the four operations with or without parentheses.
Write simple expressions that record calculations with numbers.
Find halves of multiples of 20 up to 2000 mentally. Interpret numerical expressions without evaluation.
Find halves of multiples of 200 up to 20 000 mentally.
Solve multi-step word problems involving four operations of whole numbers.
Multiply a 3-digit or 4-digit whole number by a 2-digit whole number.
Divide a 3-digit or 4-digit whole number by a 2-digit whole number.
Estimate products and quotients.
Check reasonableness of answers in multiplication or division.
*Divide a 5-digit whole number by a 2-digit whole number.
Solve multi-step word problems involving the four operations of whole numbers.
Use a calculator to carry out the four basic operations.
Calculate a product or a quotient on a calculator and check the reasonableness of the answer.
Solve multi-step word problems involving the four basic operations using a calculator.
Multiply a 2-digit number by a 1-digit number.
Multiply two 2-digit numbers.
Divide a 2-digit number by a 1-digit number.
*Solve challenging word problems involving whole numbers.
Multiplication / Division (continued)
Fractions / Concepts
Write the sum of a whole number and a proper fraction as a mixed number.
Read and place mixed numbers on a number line.
Compare and order mixed numbers on a number line.
*Decompose a mixed number or a non-unit fraction into a sum of fractions with the same denominator.
*Interpret a non-unit fraction as a multiple of a unit fraction.
Write an improper fraction.
Distinguish among whole numbers, proper fractions, improper fractions and mixed numbers.
Write an improper fraction as a whole number or a mixed number.
Write a mixed number as an improper fraction.
Write a mixed number as another mixed number.
Associate a fraction with division.
Express a whole number as a fraction.
Multiply tens or hundreds by a 1-digit number.
Multiply a 2-digit number close to a multiple of 10 by a 1-digit number.
Multiply a 1-digit or 2-digit number by 25 by multiplying by 100 and dividing by 4.
*Divide tens or hundreds by a 1-digit number.
Find doubles of whole numbers up to 100.
Find halves of whole numbers up to 200.
Read and place positive and negative integers, fractions and decimals on number lines.
Compare and order positive and negative integers, fractions and decimals.
Partition a rectangle into parts with equal areas and express the area of each part as a unit fraction of the whole.
Fractions / Arithmetic Operations
Decimals
Divide a whole number by another whole number and write the quotient as a mixed number.
Add two or three like or related fractions with a sum more than 1 whole.
Subtract one or two fractions from a whole number.
Describe and complete a number pattern involving addition and subtraction of fractions with the same denominator.
Use a fraction to represent a part of a set of objects.
Find the value of a fractional part of a quantity.
Multiply a fraction and a whole number.
Solve 1-step and 2-step word problems involving fractions.
Add and subtract unlike fractions.
*Divide a proper fraction by a proper fraction.
Multiply fractions. *Solve 1-step word problems involving the division of a proper fraction by a proper fraction.
*Add and subtract mixed numbers.
Read and write a decimal up to 2 decimal places.
Express a fraction or mixed number whose denominator is a factor of 100 as a decimal.
*Multiply a whole number by a mixed number.
*Interpret multiplication as scaling.
*Multiply a fraction or mixed number by a mixed number.
*Divide a fraction by a whole number.
*Divide a whole number by a fraction.
*Convert a measurement of length, mass, volume of liquid or time from a larger unit of measure involving a proper fraction or a mixed number to a smaller unit.
*Convert a measurement of length, mass, volume of liquid or time from a larger unit of measure involving a mixed number to compound units.
*Express a measurement of length, mass, volume of liquid or time in the smaller unit as a fraction of a measurement in the larger unit.
Solve multi-step word problems involving fractions.
Read and write a decimal with 3 decimal places.
Interpret a decimal with 3 decimal places in terms of tens, ones, tenths, hundredths and thousandths.
Describe and complete a number pattern involving positive and negative integers, fractions, and decimals by counting on and backwards.
*Solve challenging word problems involving fractions.
Multiply a decimal by 10, 100 or 1000.
Multiply a decimal by tens, hundreds or thousands.
Grade 4
Decimals (continued)
Interpret a decimal up to 2 decimal places in terms of tens, ones, tenths and hundredths.
Identify the values of digits in a decimal up to 2 decimal places.
Express a decimal up to 2 decimal places as a fraction or mixed number in its simplest form.
Write tenths or hundredths as a decimal.
Read and place decimals on a number line with intervals of 0.1 or 0.01.
Compare and order decimals up to 2 decimal places.
Find the number which is 0.1 or 0.01 more than or less than a given number.
Round a decimal to the nearest whole number or 1 decimal place.
Identify the values of digits in a decimal with 3 decimal places.
Express a fraction or mixed number with a denominator of 1000 as a decimal.
Express a decimal with 3 decimal places as a fraction or mixed number in its simplest form.
Read decimals on a number line with intervals of 0.001.
Compare and order decimals up to 3 decimal places.
Compare and order whole numbers, decimals and fractions.
Find the number which is 0.1, 0.01 or 0.001 more than or less than a given number.
Complete a number pattern with decimals involving addition and subtraction.
Round a decimal to 2 decimal places.
Add and subtract decimals up to 2 decimal places.
Multiply and divide decimals up to 2 decimal places by a 1-digit whole number.
Divide a whole number by a 1-digit whole number and give the quotient as a decimal.
Divide a decimal or whole number by 10 or tens.
Divide a whole number by 100 or hundreds.
Divide a whole number by 1000 or thousands.
Multiply a decimal by a 2-digit whole number.
Multiply decimals.
*Express a mixed number as a decimal correct to 2 decimal places.
Divide a decimal by a 2-digit whole number.
Divide a whole number by a decimal.
*Divide a decimal by a decimal.
Estimate products and quotients.
Solve multi-step word problems involving the four operations of decimals.
Read and place positive and negative integers, fractions and decimals on number lines.
Estimate sums, differences, products and quotients. Compare and order positive and negative integers, fractions and decimals.
Check reasonableness of answers in addition, subtraction, multiplication or division.
Solve 1-step and 2-step word problems involving decimals.
Find pairs of decimals with 1 or 2 decimal places with a total of 1.
Describe and complete a number pattern involving positive and negative integers, fractions, and decimals by counting on and backwards.
*Solve challenging word problems involving decimals.
Decimals (continued)
Integers
Rate
Ratio
Find pairs of decimals with 1 decimal place with a total of 10.
Find doubles of decimals with 1 or 2 decimal places.
Find halves of decimals with 1 or 2 decimal places.
Interpret integers in everyday contexts.
Read integers on number lines.
Compare and order integers using number lines.
Describe and complete a number pattern involving positive and negative integers by counting on and backwards by ones, twos, threes, fours, fives or tens.
Read and place positive and negative integers, fractions and decimals on number lines.
Compare and order positive and negative integers, fractions and decimals.
*Understand that absolute value of a number as its distance from zero on the number line.
*Use absolute value to find the magnitude of a positive or negative quantity in a real-world situation.
Add and subtract integers.
Describe and complete a number pattern involving positive and negative integers, fractions, and decimals by counting on and backwards.
*Find the rate by expressing one quantity per unit of another quantity.
*Find a quantity using the given rate.
*Solve word problems involving rate.
Use a ratio to compare two quantities.
Use a ratio to compare a quantity with the total quantity.
Use a comparison bar model to show a ratio.
Use a ratio to compare two quantities given in a comparison bar model.
Write equivalent ratios.
Write a ratio in its simplest form.
Ratio (continued)
Percentage
Length
Recall the units of measurements of length.
Know the meanings of the prefixes ‘kilo’, ‘centi’ and ‘milli’.
Know the relationship between units of length.
*Convert a measurement of length from a larger unit of measure involving a proper fraction or a mixed number to a smaller unit.
*Convert a measurement of length from a larger unit of measure involving a mixed number to compound units.
*Express a measurement of length in the smaller unit as a fraction of a measurement in the larger unit.
Find the missing term in a pair of equivalent ratios.
Write a ratio to compare quantities that are in proportion.
Find missing values in a table of equivalent ratios.
Plot pairs of values in a table of equivalent ratios on a Cartesian plane.
Solve word problems involving ratio and proportion.
*Solve challenging word problems involving ratio.
Read and interpret the percentage of a whole.
Express a fraction as a percentage, and vice versa.
Express a decimal as a percentage, and vice versa.
Interpret and understand that 1 whole is 100%.
Express a part of a whole as a percentage.
Compare fractions, decimals and percentages.
Find the value of a percentage of a quantity.
Solve up to 2-step word problems involving percentage, interest, sales tax and discount.
*Solve challenging word problems involving percentage.
Know imperial units still in common use and approximate metric equivalents.
Choose appropriate units of measure.
Convert a measurement from a larger unit to a smaller unit, and vice versa.
Length (continued)
Perimeter / Area
Measure the length of a line segment in centimeters or centimeters and millimeters.
Draw a line segment given its length in centimeters or centimeters and millimeters.
Convert a measurement of length from compound units to a smaller unit, and vice versa.
Compare and order measurements of length in compound units.
Add and subtract lengths in compound units.
*Measure lengths to the nearest half, quarter or eighth of an inch.
*Measure and compare lengths in feet and inches.
*Express feet and inches in inches, and vice versa.
*Add and subtract lengths in feet and inches.
*Measure and compare lengths in yards and feet.
*Express yards and feet in feet, and vice versa.
*Add and subtract lengths in yards and feet.
*Measure and compare lengths in miles.
*Choose a suitable unit of measure when measuring lengths and distances.
Solve 1-step and 2-step word problems on length.
Measure area in nonstandard units.
Find the area of a figure made up of unit squares and half squares.
Compare areas of figures made up of unit squares and half squares.
Find the perimeter of a figure made up of 1-centimeter or 1-meter squares.
Measure the perimeter of a figure.
Convert a measurement from a larger unit to compound units, and vice versa.
*Solve multi-step word problems involving length.
Compare areas and perimeters of figures made up of 1-centimeter or 1-meter squares.
*Find the area and perimeter of a polygon on a Cartesian plane.
Partition a rectangle into parts with equal areas and express the area of each part as a unit fraction of the whole.
Apply the distributive property to find the area of a rectangle by adding two products.
Perimeter / Area (continued)
Visualize the sizes of 1 square centimeter and 1 square meter.
Find the area of a figure made up of 1-centimeter or 1-meter squares and half squares.
Compare areas of figures made up of 1-centimeter or 1-meter squares and half squares.
*Visualize the sizes of 1 square inch and 1 square foot.
*Find the area of a figure made up of 1-inch or 1-foot squares and half squares.
*Compare areas of figures made up of 1-inch or 1-foot squares and half squares.
Find the perimeter of a rectilinear figure given the lengths of all its sides.
Find the perimeter of a regular polygon given the length of one side.
*Find an unknown side length of a figure given its perimeter and the other side lengths.
Find the area and perimeter of a square given one side.
Find the area and perimeter of a rectangle given its length and width.
Draw a square and a rectangle and measure and calculate their perimeters.
Estimate the area of an irregular shape by counting squares.
Find one side of a rectangle given the other side and its area or perimeter.
Find one side of a square given its area or perimeter.
Solve word problems on areas of squares and rectangles.
Solve word problems on perimeters of polygons.
Find the area and perimeter of a composite figure made up of squares and/or rectangles.
Solve word problems on area and perimeter of composite figures made up of squares and/or rectangles.
*Identify the base and height of a triangle.
*Find the area of a triangle using formula.
*Find the shaded area of a figure related to the area of a triangle.
*Find the area of a parallelogram using formula.
*Find the area of a rhombus using formula.
*Find the area of a trapezoid using formula.
*Find the area of a composite figure made up of squares, rectangles, triangles, parallelograms, rhombuses and trapezoids.
*Solve word problems on area of composite figures made up of squares, rectangles, triangles, parallelograms and/or trapezoids.
*Find the total surface area of prisms and pyramids using formulae.
*Solve word problems involving total surface area or volume of prisms and pyramids.
*Solve challenging word problems involving area.
Volume and Capacity Recall the units of measurements of volume of liquid.
Know the relationship between liter and milliliter.
Know the meanings of the prefixes ‘kilo’, ‘centi’ and ‘milli’.
Compare readings on different scales.
Express liters and milliliters in milliliters, and vice versa.
Compare and order volumes in liters and milliliters.
Add and subtract volumes in liters and milliliters.
Solve 1-step and 2-step word problems on volume.
*Convert a measurement of volume of liquid from a larger unit of measure involving a proper fraction or a mixed number to a smaller unit.
*Convert a measurement of volume of liquid from a larger unit of measure involving a mixed number to compound units.
*Express a measurement of volume of liquid in the smaller unit as a fraction of a measurement in the larger unit.
*Find the volume of a solid made up of unit cubes in cubic units.
*Visualize a solid that is made up of unit cubes and state its volume in cubic units.
*Visualize the sizes of 1 cubic centimeter, 1 cubic meter, 1 cubic inch and 1 cubic foot.
*Find the volume of a solid made up of 1-centimeter, 1-meter, 1-inch or 1-foot cubes.
*Compare the volumes of solids made up of 1-centimeter, 1-meter, 1-inch or 1-foot cubes.
*Find the volume of a rectangular prism, given its length, width and height.
*Find the volume of a rectangular prism, given area of one face and one dimension.
*Find the volume of a solid figure composed of two rectangular prisms.
*Solve word problems involving volume of rectangular prisms.
Mass / Weight Recall the units of measurements of mass.
*Convert a measurement of mass from a larger unit of measure involving a proper fraction or a mixed number to a smaller unit.
Know imperial units still in common use and approximate metric equivalents.
Choose appropriate units of measure.
Convert a measurement from a larger unit to a smaller unit, and vice versa.
Convert a measurement from a larger unit to compound units, and vice versa.
*Solve multi-step word problems involving capacity.
*Find the volume of a rectangular prism, given its length, width and height.
*Find the volume of a rectangular prism, given area of one face and one dimension.
*Solve word problems involving total surface area or volume of prisms and pyramids.
*Solve challenging word problems involving volume.
Know imperial units still in common use and approximate metric equivalents.
Mass / Weight (continued)
Know the relationship between kilogram and gram.
Know the meanings of the prefixes ‘kilo’, ‘centi’ and ‘milli’.
Compare readings on different scales.
Express kilograms and grams in grams, and vice versa.
Compare and order masses in kilograms and grams.
Add and subtract masses in kilograms and grams.
Solve 1-step and 2-step word problems on mass.
*Measure and compare weights in pounds and ounces.
*Express pounds and ounces in ounces, and vice versa.
*Add and subtract weights in pounds and ounces.
*Measure and compare weights in tons.
*Know the relationship between tons, pounds and ounces.
*Choose a suitable unit of measure when measuring weights.
*Solve 1-step and 2-step word problems on weight.
*Convert a measurement of mass from a larger unit of measure involving a mixed number to compound units.
*Express a measurement of mass in the smaller unit as a fraction of a measurement in the larger unit.
Choose appropriate units of measure.
Convert a measurement from a larger unit to a smaller unit, and vice versa.
Convert a measurement from a larger unit to compound units, and vice versa.
*Solve multi-step word problems involving mass.
Time: Clock Tell time to the second.Calculate time intervals in months.
Find the duration of a time interval in seconds.
Measure duration of activities in seconds.
Know the relationship between units of time.
Calculate time intervals in years.
Calculate time intervals in years and months.
*Convert a measurement of time from a larger unit of measure involving a proper fraction or a mixed number to a smaller unit.
Know the relationship between years, decades and centuries.
Convert between years, decades and centuries.
Calculate time in different time zones.
*Understand time intervals less than one second.
Time: Clock (continued)
Choose suitable units to measure time intervals.
Express minutes and seconds in seconds, and vice versa.
Express years and months in months, and vice versa.
Express weeks and days in days, and vice versa.
Tell time using the 24-hour clock notation.
Convert time between the 12-hour and 24-hour clock notations.
Compare times using digital and analog clocks.
Find the duration of a time interval given time in 24-hour clock notation.
Read and interpret timetables in 12-hour and 24-hour clock notations.
Use a timetable to solve problems.
Solve word problems on time.
Speed
Temperature
Money
*Convert a measurement of time from a larger unit of measure involving a mixed number to compound units.
*Express a measurement of time in the smaller unit as a fraction of a measurement in the larger unit.
*Recognize that a time interval can be expressed as a decimal or in compound units.
*Convert between time intervals expressed as a decimal and in compound units.
*Interpret speed as the distance traveled per unit of time.
*Read and write units of speed such as km/h, m/min, m/s and cm/s.
*Understand the relationship between distance, speed and time.
*Calculate speed, distance or time taken given two of the quantities.
*Solve word problems involving speed.
*Solve challenging word problems involving speed.
Calculate a rise or fall in temperature.
Express
Lines and Curves
2D Shapes
Identify open and closed figures.
Differentiate between polygons and non-polygons.
Name polygons according to the number of sides.
Identify regular and irregular polygons.
Find examples of polygons in the environment and in art.
Identify perpendicular and parallel line segments.
Draw perpendicular and parallel line segments.
Recognize that the sum of angle measures in a triangle is 180º.
Identify and describe properties of triangles and classify as isosceles, equilateral or scalene.
Understand the properties of squares and rectangles.
Use properties of squares and rectangles to find unknown angle measures.
Identify right triangles.Use properties of squares and rectangles to find unknown lengths.
Find an unknown angle measure in a triangle. State the properties of a rectangle, a square, a parallelogram, a rhombus and a trapezoid.
Recognize reflective symmetry in regular polygons.
Make polygons on geoboards.
Count the number of lines of symmetry in regular polygons.
Draw polygons on dot grids.Make a symmetric pattern with two lines of symmetry.
Classify polygons using criteria such as the number of right angles, whether or not they are regular and their symmetrical properties.
Identify a symmetric polygon.
Count the number of lines of symmetry in polygons.
Draw lines of symmetry in polygons and patterns.
Find examples of symmetry in the environment and in art.
Recognize rotational symmetry in 2D shapes.
Identify the order of rotational symmetry in 2D shapes.
Identify where a polygon will be after a translation and give instructions for translating the shape.
Predict where a polygon will be after a reflection where the mirror line is parallel to one of the sides.
*Identify the unit shape in a tessellation.
Identify rectangles, squares, trapezoids, parallelograms and rhombuses as examples of quadrilaterals and draw examples of quadrilaterals that do not belong to any of these subcategories.
Classify quadrilaterals using parallel sides, equal sides and equal angles.
Draw polygons on the Cartesian plane given coordinates for the vertices.
Predict where a polygon will be after a translation.
Predict where a polygon will be after a reflection where the sides of the shape are not parallel or perpendicular to the mirror line.
Predict where a polygon will be after a rotation about one of its vertices.
Partition a rectangle into parts with equal areas and express the area of each part as a unit fraction of the whole.
*Identify parts of a circle (radius, diameter, center, circumference).
2D Shapes (continued)
*Make a symmetric pattern with one line of symmetry.
*Determine if a given shape can tessellate.
*Draw a tessellation on dot paper.
*Make different tessellations with a unit shape.
*Make a tessellation with two unit shapes.
*Know the relationship between the radius and diameter of a circle.
*Find the diameter of a circle given its radius, and vice versa.
*Draw a circle with a given radius or diameter.
*Find unknown angle measures involving triangles and quadrilaterals.
3D Shapes Identify and draw different types of prisms and pyramids.
Identify the faces, edges and vertices of prisms and pyramids.
Find examples of prisms and pyramids in the environment and in art.
Understand that cross sections of a prism are of the same shape and size as the parallel faces of the prism.
Understand that cross sections of a pyramid are of the same shape as the base but of different sizes.
Classify prisms and pyramids according to the number and shape of faces, number of vertices and edges.
Identify the nets of prisms and pyramids.
Identify the prism or pyramid which can be formed by a net.
Make prisms and pyramids from nets.
Make nets of prisms and pyramids.
Angles
Name an angle using notations such as ∠ABC and ∠x.
Recognize that the measure of a right angle is 90°.
*Build a solid with unit cubes.
*Visualize a solid drawn on dot paper and state the number of unit cubes used to build the solid.
*Visualize and identify the new solid formed by changing the number of unit cubes of a solid drawn on dot paper.
Recognize that the sum of the angle measures on a straight line is 180°.
Recognize that the sum of the angle measures at a point is 360°.
Understand the properties of squares and rectangles.
Use properties of squares and rectangles to find unknown angle measures.
Angles (continued)
Estimate and measure the size of an angle in degrees and classify the angle as acute, right or obtuse.
Draw acute and obtuse angles using a protractor.
Recognize that vertically opposite angles have equal measures.
Find the unknown measures of angles involving angles on a straight line, angles at a point and vertically opposite angles.
Relate turns to right angles.Recognize that the sum of angle measures in a triangle is 180º.
Relate a 1 4 -turn to 90°, a 1 2 -turn to 180°, a 3 4 -turn to 270° and a complete turn to 360°.
Position and Movement
State the properties of a rectangle, a square, a parallelogram, a rhombus and a trapezoid.
Classify quadrilaterals using parallel sides, equal sides and equal angles.
*Find unknown angle measures involving triangles and quadrilaterals.
Find an unknown angle measure in a triangle.
Relate turns to right angles.Read and plot coordinates in the 1st quadrant of the Cartesian plane.
Relate a 1 4 -tur n to 90°, a 1 2 -tur n to 180°, a 3 4 -tur n to 270° and a complete turn to 360°.
Tell direction using the 8-point compass.
Give and follow directions to a place on a grid.
*Plot corresponding terms from two patterns on a Cartesian plane.
*Solve word problems involving the Cartesian plane.
Identify where a polygon will be after a translation and give instructions for translating the shape.
Predict where a polygon will be after a reflection where the mirror line is parallel to one of the sides.
Recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
Read and plot coordinates in all four quadrants of the Cartesian plane.
Draw polygons on the Cartesian plane given coordinates for the vertices.
*Find the distance between two points with the same first or second coordinate.
*Find the length of a side joining points with the same first or second coordinate.
*Find the area and perimeter of a polygon on a Cartesian plane.
Predict where a polygon will be after a translation.
Predict where a polygon will be after a reflection where the sides of the shape are not parallel or perpendicular to the mirror line.
Predict where a polygon will be after a rotation about one of its vertices.
*Solve word problems involving the Cartesian plane.
Plot pairs of values in a table of equivalent ratios on a Cartesian plane.
Data Collection Collect and present data in a graph.
Identify the data to collect to answer a set of related questions.
Distinguish between statistical questions and those that are not.
Collect and present data in an appropriate data display. Write a statistical question and explain what data could be collected to answer the question.
Draw conclusions from data and identify further questions to ask.
Collect and present data in a bar line chart.
Tables
Graphs Collect and present data in a graph.
Interpret a graph.
Compare the impact of representations where scales have different intervals.
*Make, read and interpret a line plot with a scale marked in whole numbers, halves, quarters or eighths.
*Make, read and interpret a line plot.
Collect and present data in a bar line chart.
Consider the effect of changing the scale on the axis.
Read and interpret a bar line chart.
Complete, read and interpret a line graph.
Distinguish between categorical data and numerical data.
Collect and record data in a frequency table.
*Represent the relationship between two variables using equations, tables and graphs.
*Represent the relationship between two variables using equations, tables and graphs.
Read and interpret a conversion graph.
Present data in a pie chart.
Read and interpret a pie chart.
Make, read and interpret a dot plot.
Describe the distribution of data in a dot plot.
Make, read and interpret a histogram.
Describe the distribution of data in a histogram.
Recognize that the number of intervals may affect the shape of the histogram.
*Present data in a waffle diagram.
*Read and interpret a waffle diagram.
*Present data in a scatter graph.
*Read and interpret a scatter graph.
*Describe the center, variability and shape of a data distribution in a dot plot or histogram.
Venn Diagrams Sort data in a Venn diagram with 2 or 3 criteria.
Data Analysis
Find the mean, median and mode of a set of data.
Find the mean given the total amount and the number of items.
Find the total amount given the mean and the number of items.
Know that mean, median and mode are measures of center of a set of data.
Describe a distribution using mean, mode and median.
Compare the mean, median and mode of a set of data and discuss which one best describes the set of data.
Solve word problems involving mean, median and mode.
*Find the interquartile range of a set of data.
*Find the mean absolute deviation of a set of data.
*Describe the variability in a data set using the interquartile range or mean absolute deviation.
*Know that interquartile range and mean absolute deviation are measures of variability of a set of data.
*Describe a distribution using interquartile range and mean absolute deviation.
*Compare the interquartile range and mean absolute deviation of a set of data and discuss which one best describes the set of data.
*Describe the center, variability and shape of a data distribution in a dot plot or histogram.
*Make a box plot.
*Describe the distribution of data in a box plot.
*Compare the distribution of data in two box plots.
*Solve challenging word problems involving statistics.
Probability
*Identify events that will happen, will not happen or might happen.
Identify events as being ‘certain’ or ‘uncertain’ to happen.
Identify events as being ‘possible’ or ‘impossible’ to happen.
Identify events as being ‘likely’ or ‘unlikely’ to happen.
List all the possible outcomes in a chance experiment or situation.
Describe events as being ‘equally likely’, ‘more likely’, ‘less likely’, ‘most likely’ or ‘least likely’ to occur.
Compare and order chances of events occurring from least likely to most likely to occur, and vice versa.
*Identify when two events can happen at the same time and when they cannot, and know that the latter are called ‘mutually exclusive’.
Find the probability of an event and express it as a fraction or a decimal.
Recognize that probabilities range from 0 to 1 and relate it to their likelihood of happening.
Understand the difference between theoretical and experimental probabilities.
Find the theoretical and experimental probabilities of an event.
*Conduct chance experiments, using small and large numbers of trials, and present and describe the results using the language of probability.
Patterns
*Identify patterns in a hundred chart.
Complete a number pattern with decimals involving addition and subtraction.
Describe, complete and create a number pattern by counting on or backwards by ones, tens, hundreds or thousands within 10 000.
*Describe and complete a number pattern by repeated addition or multiplication.
Describe and complete a number pattern involving positive and negative integers by counting on and backwards by ones, twos, threes, fours, fives or tens.
*Generate two number patterns from given rules and identify the relationships between corresponding terms.
Describe and complete a number pattern involving positive and negative integers, fractions, and decimals by counting on and backwards.
Patterns (continued)
Describe and complete a number pattern involving addition and subtraction of fractions with the same denominator.
Expressions
Equations
Write simple expressions that record calculations with numbers.
Interpret numerical expressions without evaluation.
Use a letter to represent an unknown number.
Write an algebraic expression in one variable.
Find the value of an algebraic expression by substitution.
*Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient) and view one or more parts of an expression as a single entity.
*Find the value of an algebraic expression given in exponential notation.
*Simplify an algebraic expression in one variable.
*Add or subtract two terms with a common factor using distributive property.
*Write equivalent expressions for algebraic expressions made up of terms with common factors.
*Identify equivalent expressions.
Solve word problems using algebraic expressions.
*Understand what an equation is.
*Identify an algebraic equation.
*Use substitution to determine whether a given number makes an equation true or false.
*Solve an algebraic equation.
*Analyze the relationship between dependent and independent variables using a table and write an equation to express the dependent variable in terms of the independent variable.
Equations (continued)
Inequalities
*Lessons are available in PR1ME Mathematics Teaching Hub.
*Represent the relationship between two variables using equations, tables and graphs.
*Solve word problems by forming an algebraic equation or inequality.
*Identify an inequality.
*Use substitution to determine whether a given number makes an inequality true or false.
*Represent the solutions of an inequality of the form x > c or x < c on a number line diagram.
*Solve an inequality and represent the solutions on a number line diagram.
*Solve word problems by forming an algebraic equation or inequality.
Strand: Numbers and Operations
• CB: pp. 1–2
• Count and write a number within 10 000—the numeral and the corresponding number word
• Identify the values of digits in a 4-digit number
• Compare numbers within 10 000
• Compare and order numbers within 10 000
• Round a 4-digit number to the nearest ten, hundred or thousand
• Identify odd and even numbers
Let’s Remember
• hundred thousand
• million • ten thousand
• CB: pp. 3–5
• PB: p. 9
• Digital Practice
Unit 1: Numbers to 1 000 000
• Read and write a number within 1 000 000—the numeral and the corresponding number word
1.1 Reading and writing numbers
• CB: p. 5
• PB: p. 10
• Digital Practice
• CB: p. 6
• PB: p. 11
• Digital Practice
• Identify the values of digits in a 5-digit or 6-digit number
1.2 Identifying values of digits
• Write a 5-digit or 6-digit number in terms of hundred thousands, ten thousands, thousands, hundreds, tens and ones
• Compare and order numbers within 1 000 000
1.3 Comparing and ordering numbers
Unit 2: Estimation
• CB: pp. 7–8
• PB: p. 12
• Digital Practice
• CB: pp. 9–11
• PB: p. 13
• Digital Practice
• Round a whole number to the nearest ten, hundred or thousand
2.1 Rounding whole numbers to the near est ten, hundred or thousand
• Estimate an answer in addition or subtraction using rounding and front-end estimation
2.2 Estimating sums and differences
• Check reasonableness of an answer in addition or subtraction
Unit 3: Odd and Even Numbers
• CB: pp. 12–14
• PB: pp. 14–15
• Digital Practice
• Connecting cubes
• Investigate and generalize the result of adding and subtracting odd and even numbers
3.1 Adding and subtracting odd and even numbers
• CB: pp. 15–16
• PB: p. 16
• Digital Practice
• factor
• CB: pp. 17–18
• PB: p. 17
• Digital Practice
• CB: pp. 18–19
• PB: p. 18
• Digital Practice
• Connecting cubes
• Investigate and generalize the result of multiplying odd and even numbers
3.2 Multiplying odd and even numbers
Unit 4: Factors
• Unit cubes
• Find all the factors of a whole number up to 100
4.1 Finding factors of a whole number
• Find out if a 1-digit number is a factor of a given whole number
4.2 Finding out if a number is a factor of another number
Unit
• multiple
Unit 5: Multiples
• CB: p. 20
• PB: p. 19
• Digital Practice
• CB: p. 21
• PB: p. 20
• Digital Practice
• CB: pp. 22–23
• PB: p. 21
• Digital Practice
• CB: pp. 23–25
• PB: p. 22
• Digital Practice
• composite number
• CB: pp. 26–29
• PB: p. 23
• prime number
• Digital Practice
• square number
• CB: p. 30–31
• PB: p. 24
• Digital Practice
• CB: pp. 32–33
• Counters
• Find the multiples of a whole number up to 10
5.1 Finding multiples of a whole number
• Relate factors and multiples
• Find out if a whole number is a multiple of a given whole number up to 10
5.2 Relating factors and multiples
• Identify multiples of 2, 5, 10, 25, 50 and 100 up to 1000
5.3 Identifying multiples of 2, 5, 10, 25, 50 and 100
• Know and apply tests of divisibility by 2, 3, 4, 5, 10, 25 and 100
5.4 Using divisibility rules
• 1 copy of Number Chart (BM1.1) per student
• Identify prime and composite numbers
• Recognize prime numbers up to 20 and find all prime numbers less than 100
6.1 Identifying prime and composite numbers
• Counters
• Identify square numbers
6.2 Identifying square numbers
• Solve a non-routine problem involving factors and multiples using the strategy of logical reasoning
Mind stretcher
7.1
Digital Practice and Assessment
Digital Chapter Assessment — Available in PR1ME Mathematics
The suggested duration for each lesson is 1 hour.
1. Count and write the number in numerals
4. Arrange the numbers in order. Begin with the least. 3072 3720
(least)
5. Round 2195 to the a) nearest ten.
b) nearest hundred. c) nearest thousand.
6. Write in the blanks.
a) The digit in the ones place of an odd number is or
b) The digit in the ones place of an even number is , , , or
ABC
How can we solve this problem? Discuss in your group and fill in columns 1 and 2. EXPLORE
Find factors
Chapter Overview
Let’s Remember
Unit 1: Numbers to 1 000 000
Unit 2: Estimation
Unit 3: Odd and Even Numbers
Unit 4: Factors
Unit 5: Multiples
Unit 6: Prime, Composite and Square Numbers
Unit 7: Problem Solving
Let's Remember Let's Remember
Recall:
1. Counting and writing a number within 10 000— the numeral and the corresponding number word (CB4 Chapter 1)
2. Identifying the values of digits in a 4-digit number (CB4 Chapter 1)
3. Comparing numbers within 10 000 (CB4 Chapter 1)
4. Comparing and ordering numbers within 10 000 (CB4 Chapter 1)
5. Rounding a 4-digit number to the nearest ten, hundred or thousand (CB4 Chapter 1)
6. Identifying odd and even numbers (CB3 Chapter 4)
Have students read the word problem on CB p. 2. Discuss with students the following questions:
•Have you participated in a food donation drive?
•Why do you think ABC Grocery Store donated food to the food donation drive?
•What are some other ways to give to charity?
Have students form groups to complete the tasks in columns 1 and 2 of the table. Let students know that they do not have to solve the word problem. Ask the groups to present their work.
Tell students that they will come back to this word problem later in the chapter.
1.1 Reading and writing numbers
Let's Learn Let's Learn
Objective:
• Read and write a number within 1 000 000— the numeral and the corresponding number word
Resources:
•CB: pp. 3–5
•PB: p. 9
Vocabulary:
•hundred thousand
•million
•ten thousand
(a) Stages: Pictorial and Abstract Representations
Have students look at the thousand-blocks in (a) on CB p. 3.
Say: Each thousand-block is made up of 1000 unit cubes. Let us count the unit cubes in the first row of thousand-blocks.
Count aloud with students in steps of 1000 from 1000 to 10 000.
Write: 10 000
Say: We read this number as ten thousand. Let us count the unit cubes in the second row of thousand-blocks.
Count aloud with students in steps of 1000 from 1000 to 5000.
Write: 10 000 + 5000
Continue to count the hundred-squares, tenrods and unit cubes on the page and write the corresponding numbers on the board to get the expression ‘10 000 + 5000 + 200 + 60 + 3’. Have students look at the place value cards on the page. Point out that each place value card corresponds to a number on the board. Have students observe that the place value cards form the number 15 263 when they are overlapped.
Write: 10 000 + 5000 + 200 + 60 + 3 = 15 263
Point to the number 10 000 on the board.
Ask: How do we read this number? (Ten thousand)
Point to the number 5000 on the board. Ask: How do we read this number? (Five thousand) What is 10 000 plus 5000? (15 000)
Write: 15 000 fifteen thousand
Point to the expression ‘200 + 60 + 3’ on the board.
Ask: What is 200 plus 60 plus 3? (263)
Write: 15 000 + 263 = 15 263 fifteen thousand, two hundred and sixty-three
Say: We read 15 263 as fifteen thousand, two hundred and sixty-three.
(b) Stage: Pictorial Representation
Have students look at the base ten blocks in (b) on CB p. 3.
Ask: How many thousand-blocks are there in 1 row? (10)
Say: There are 10 thousand-blocks in 1 row. So, there are 10 000 unit cubes in 1 row. Let us now count on in steps of 10 000.
Count aloud with students in steps of 10 000 from 10 000 to 100 000. Have students point to each row of thousand-blocks as they count.
Stage: Abstract Representation
Write: 100 000 one hundred thousand
Say: We read this number as one hundred thousand.
(c) Stage: Abstract Representation
Have students look at the dark green place value card in (c) on CB p. 4.
Ask: How do we read this number? (Two hundred thousand)
Have students look at the light green place value card on the page.
Ask: How do we read this number? (Thirty thousand)
Have students look at the orange place value card on the page.
Ask: How do we read this number? (Five thousand) What is 200 000 plus 30 000 plus 5000? (235 000)
Write: 235 000 two hundred and thirty-five thousand
Repeat the above procedure for the remaining place value cards to teach students to read and write 471.
Write: 235 000 + 471 = 235 471 two hundred and thirty-five thousand, four hundred and seventy-one
Say: We read 235 471 as two hundred and thirty-five thousand, four hundred and seventy-one.
(d) Stage: Abstract Representation
Write: 100 000
Say: We read this number as one hundred thousand.
Have students count on in steps of 100 thousand.
Say: 100 thousand, 200 thousand, 300 thousand, …, 1000 thousand.
Write: 1 000 000
Say: 1000 thousands is equal to 1 million. We read this number as one million.
Write: 1 000 000 = 1 million
Let's Do Let's
Task 1 requires students to write 5-digit and 6-digit numbers in numerals, given the numbers in words.
Task 2 requires students to write 5-digit and 6-digit numbers in words, given the numerals.
d) Count on in steps of
Do
1. Write the numerals.
a) fifty-four thousand, one hundred and one
b) two hundred and six thousand, four hundred and three
2. Write the numbers in words.
a) 10 299
ten thousand, two hundred and ninety-nine
five hundred and fifty-nine thousand, three hundred and eleven
b) 559 311
Let's Practice
1. Write the numerals.
a) fourteen thousand, two hundred and ninety
b) four hundred and thirty-five thousand, one hundred and seventeen
c) eight hundred and eight thousand, nine hundred and sixty-five
d) seven hundred and twenty thousand and twelve
Practice
Practice
Task 1 requires students to write 5-digit and 6-digit numbers in numerals, given the numbers in words.
Task 2 requires students to write 5-digit and 6-digit numbers in words, given the numerals.
Let's Learn Let's Learn
Objectives:
•Identify the values of digits in a 5-digit or 6-digit number
•Write a 5-digit or 6-digit number in terms of hundred thousands, ten thousands, thousands, hundreds, tens and ones
Resources:
•CB: p. 5
•PB: p. 10
Stage: Abstract Representation
Copy the place value table on CB p. 5 on the board without filling in the numbers.
Write: 318 526
Say: Let us write this number in the place value chart.
Write 318 526 in the place value chart. Point to the digit 3 in the place value chart. Say: We can identify the value of each digit by looking at its place value. In this number, the digit 3 is in the hundred thousands place. So, there are 3 hundred thousands. The value of the digit 3 is 300 000.
Ask: Which place is the digit 1 in? (Ten thousands) What is its value? (10 000)
Ask similar questions about the other digits. Point to the digits in the place value chart. Say: 300 000, 10 000, 8000, 500, 20 and 6 make 318 526.
Write: 318 526 = 300 000 + 10 000 + 8000 + 500 + 20 + 6
In 318 526, the digit 3 is in the hundred thousands place and its value is
the digit 1 is in the place and its value is the digit 2 is in the tens place and its value is 20.
1. Write the missing numbers or words.
a) In 750 168, the digit 7 is in the place and its value is
b) In 98
the
1. Write the missing numbers or words.
a) In 601 325, the digit 0 is in the place.
b) In 713 950, the digit is in the hundred thousands place and its value is c) 90 310 = 90 000 + + 10 eighteen thousand and thirty six hundred thousand, five hundred and seventy-two nine hundred and seventy-six thousand, one hundred and three
Task 1 requires students to identify the values and place values of digits in 5-digit and 6-digit numbers.
Let's Learn Let's Learn
Objective:
• Compare and order numbers within 1 000 000
Resources:
•CB: p. 6
•PB: p. 11
Stage: Abstract Representation
Write: 730 650, 720 650, 72 650
Copy the place value chart on CB p. 6 on the board without filling in the numbers.
Say: Let us compare these three numbers using a place value chart.
Invite three students to fill in the place value chart to show the numbers 730 650, 720 650 and 72 650.
Say: When we compare numbers, we begin by comparing the digits in the highest place value.
Ask: What is the highest place value in this place value chart? (Hundred thousands)
Say: Look at the three numbers in the place value chart.
Ask: Which number does not have hundred thousands? (72 650) So, what can we say about this number? (It is the least.)
Say: Let us now compare the two remaining numbers, 730 650 and 720 650. The number of hundred thousands is the same so we compare the digits in the next place value, the ten thousands place.
Ask: What is the digit in the ten thousands place in 730 650? (3) What is the digit in the ten thousands place in 720 650? (2) Which is greater, 3 ten thousands or 2 ten thousands? (3 ten thousands)
Say: Since 3 ten thousands is greater than 2 ten thousands, 730 650 is greater than 720 650.
Write: 730 650 > 720 650
Ask: So, which number is the greatest? (730 650)
Guide students to arrange the numbers in order, beginning with the least.
Write: 72 650, 720 650, 730 650 (least)
Let's Do Let's Do
Task 1 requires students to compare two numbers within 1 000 000 using the symbols ‘>’ and ‘<’.
Let's Practice Let's Practice
Task 1 requires students to compare three numbers within 1 000 000.
Task 2 requires students to compare and order three numbers within 1 000 000.
1. Round each number.
2.1 Rounding whole numbers to the nearest ten, hundred or thousand
a)
In 7343, the digit in the ones place is 3. 7343 is nearer to 7340 than to 7350. So, we round down.
7343 is 7340 when rounded to the nearest ten. 7343 ≈ 7340
b)
In 48 776, the digit in the tens place is 7. 48 776 is nearer to 48 800 than to 48 700. So, we round up. 48 776 is 48 800 when rounded to the nearest hundred.
500 is when rounded to the nearest thousand.
To round a number to a place value, look at the digit on the right of the place value. If the digit is 5 or greater, round up. If the digit is less than 5, round down.
2.1 Rounding whole numbers to the nearest ten, hundred or thousand
Let's Learn Let's Learn
Objective:
•Round a whole number to the nearest ten, hundred or thousand
Resources:
•CB: pp. 7–8
•PB: p. 12
(a) Stage: Pictorial Representation
Draw the number line in (a) on CB p. 7 on the board but do not label ‘7343’.
Guide students to see that there are 10 equal intervals between 7340 and 7350 and each interval stands for 1.
Invite a student to mark 7343 on the number line.
Say: 7343 is between two tens, 7340 and 7350.
Ask: How many intervals are there from 7340 to 7343? (3) How many intervals are there from 7343 to 7350? (7) Is 7343 nearer to 7340 or to 7350? (Nearer to 7340)
Stage: Abstract Representation
Say: 7343 is nearer to 7340 than to 7350. So, we round down. 7343 is 7340 when rounded to the nearest ten.
Write: 7343 ≈ 7340
1. Round each number to the nearest ten.
a) 4138 ≈ b) 1652 ≈
c) 2237 ≈ d) 7695 ≈
a) $4303 ≈ b) $35 043 ≈
c) $27 533 ≈ d) $11 753 ≈
e) $135 942 ≈ f) $478 284 ≈
Round each amount to the nearest thousand dollars.
a) $7455 ≈
c) $30 628 ≈
e) $144 550 ≈ f)
Say: We read this statement as ‘7343 is approximately 7340’.
Explain that ‘≈’ is the approximation sign and means ‘approximately’.
(b) Stages: Pictorial and Abstract Representations Follow the procedure in (a).
(c) Stages: Pictorial and Abstract Representations Follow the procedure in (a). Point out that when a number is halfway between two thousands, the greater thousand is to be taken as the nearest thousand. In this case, the greater thousand is 122 000.
Conclude that to round a number to a place value, look at the digit on the right of the place value. If the digit is 5 or greater, round up. If the digit is less than 5, round down.
Task 1 requires students to round whole numbers to the nearest ten, hundred or thousand.
Task 1 requires students to round whole numbers to the nearest ten.
Task 2 requires students to round amounts of money to the nearest hundred dollars.
Task 3 requires students to round amounts of money to the nearest thousand dollars.
Let's Learn Let's Learn
Objectives:
• Estimate an answer in addition or subtraction using rounding and front-end estimation
•Check reasonableness of an answer in addition or subtraction
Resources:
•CB: pp. 9–11
•PB: p. 13
(a) Stage: Abstract Representation
Write: Find the sum of 4593 and 23 862. Say: First, let us find the sum of 4593 and 23 862. Have students add 4593 and 23 862.
Ask: What is the value of 4593 plus 23 862? (28 455)
Write: 4593 + 23 862 = 28 455
Method 1
Say: Let us check the answer using rounding. Ask: What do we get when we round 4593 to the nearest thousand? (5000) What do we get when we round 23 862 to the nearest thousand? (24 000)
Write: 4593 + 23 862 ≈ 5000 + 24 000
Say: In the addition sentence shown, we use the approximation sign to show that we are finding the estimated value of 4593 + 23 862, not the exact value of 4593 + 23 862.
Ask: What do we get when we add 5000 and 24 000? (29 000)
Write: 4593 + 23 862 ≈ 5000 + 24 000 = 29 000
Explain to students that the use of the equal sign is to indicate that we are finding the exact value of the sum of 5000 and 24 000.
Say: 28 455 is about 29 000. Since the exact value of the sum is close to the estimated value of the sum, we can say that our answer of 28 455 is reasonable. A reasonable answer is an answer that makes sense.
Method 2
Say: Let us check the answer using front-end estimation.
Ask: What is the leftmost digit of 4593? (4) Say: When we use front-end estimation, we keep the leftmost digit and change the other digits to zeros. So, 4593 is estimated as 4000.
2.2 Estimating sums and differences
Let's Learn
a) Find the sum of 4593 and 23 862.
4593 + 23 862 = 28 455
We can use estimation to check if the answer is reasonable.
Method 1: Using rounding
4593 + 23 862 ≈ 5000 + 24 000 = 29 000
28 455 is about 29 000. The answer is reasonable.
Method 2: Using front-end estimation
4593 + 23 862 ≈ 4000 + 20 000 = 24 000
28 455 is about 24 000. The answer is reasonable.
b) Find the difference between 5812 and 32 248.
32 248 – 5812 = 26 436
We can use estimation to check if the answer is reasonable.
32 248 – 5812 ≈ 32 000 – 6000 =
Is the answer 26 436 reasonable?
Method 2: Using front-end estimation
32 248 – 5812 ≈ 30 000 – 5000 =
Method 1: Using rounding 26 000 25 000 Yes Yes
Is the answer 26 436 reasonable?
Ask: Using front-end estimation, what is the estimated value of 23 862? (20 000)
Write: 4593 + 23 862 ≈ 4000 + 20 000
Ask: What is the estimated value of 4593 + 23 862? (24 000)
Say: 28 455 is about 24 000. Since the exact value is close to the estimated value, we can say that our answer of 28 455 is reasonable.
(b) Stage: Abstract Representation
Follow the procedure in (a). Point out that students can use rounding or front-end estimation to check the reasonableness of their answers in subtraction.
1. Find the sum or difference. Then, use estimation to check your answer.
a) 365 + 709 = 365 + 709 ≈ + = Is your answer reasonable?
b) 6740 – 1826 = 6740 – 1826 ≈ – = Is your answer reasonable?
1. Find the sum or difference. Then,
Let's Do Let's and Let's Practice Let's Practice
Task 1 requires students to find the sum or difference of two numbers before using estimation to check the reasonableness of their answers.
Sarah and David get different estimated sums of 341 and 2138.
Sarah
341 + 2138 ≈ 340 + 2140 = 2480 341 + 2138 ≈ 300 + 2100 = 2400
David
Who is correct? Why do you say so? Who is wrong? Why do you say so?
Both are correct. Sarah estimates the sum by rounding the numbers to the nearest hundred. David estimates the sum by rounding the numbers to the nearest ten.
No one is wrong. Sarah and David round the numbers to different place values so they get different estimated sums.
We can estimate sums by rounding numbers to the different place values. We can round the numbers to nearest ten, hundred or thousand before adding.
Think of a time in your daily life when estimation is useful.
It takes 12 minutes to walk from my house to the supermarket and 19 minutes to walk from the supermarket to the library. The library is 26 minutes away from my house.
12 + 19 + 26 ≈ 10 + 20 + 30 = 60
The journey would take about 1 hour if I visit the supermarket and the library.
I have learned to... round a whole number to the nearest ten, hundred or thousand estimate sums and differences
Have students work in groups to discuss the tasks. Ask the groups to present their answers.
Guide students to observe that Sarah estimates the sum by rounding the numbers to the nearest hundred while David estimates the sum by rounding the numbers to the nearest ten. Have students work out the exact value of the sum and observe that the two estimates are close to the exact value. Conclude that both Sarah and David are correct.
Reiterate to students that we can estimate sums by rounding numbers to different place values. Point out that when a question asks for the estimated sum, students can choose to round the numbers to their preferred place value if it is not specified in the question.
Make use of the examples presented by the groups to let students understand the importance and usefulness of estimation.
3.1 Adding and subtracting odd and even numbers
Let's Learn Let's Learn
Objective:
•Investigate and generalize the result of adding and subtracting odd and even numbers
Materials:
•Connecting cubes
Resources:
•CB: pp. 12–14
•PB: pp. 14–15
(a) Stage: Concrete Experience
Have students work in groups. Distribute connecting cubes to each group. Write the even numbers and the odd numbers from 1 to 10 on the board. Have a student pick two of the even numbers, for example, 4 and 2. Change the numbers in the following steps according to the numbers chosen by the student. Have the groups take out 4 connecting cubes and join the cubes to make 2 pairs.
Ask: Is 4 an odd number or an even number? (Even)
Say: Since there are no cubes left unpaired, 4 is an even number.
Have the groups take out 2 more connecting cubes and join the cubes to make a pair.
Ask: Are there any cubes left unpaired when you join the 2 cubes? (No) Is 2 an even number? (Yes)
How many cubes are there altogether? (6)
Are all the cubes paired? (Yes) Is 6 an odd number or an even number? (Even number)
Write: 4 + 2 = 6
Ask: Is each number in the addition sentence odd or even? (Even)
Write ‘even number’ below each number in the addition sentence. Guide students to conclude that they added an even number, 4, to an even number, 2, to get an even number, 6.
Stage: Pictorial Representation
Have students look at the first addition of the connecting cubes in (a) on CB p. 12. Relate the picture to the earlier activity and go through the example.
Stage: Abstract Representation
Ask: What pattern do you notice? (The sum of two even numbers is an even number.)
Say: We always get an even number when we add two even numbers because when all pairs are joined together, they still remain in pairs.
Let's Learn
Stages: Concrete Experience and Pictorial Representation
Repeat the above procedure to guide students to add an odd number and an even number.
Stage: Abstract Representation
Ask: What pattern do you notice when we add an odd number and an even number? (The sum is always an odd number.)
Say: We always get an odd number when we add an odd number and an even number because there is one object left unpaired from the odd number.
Remind students of the commutative property of addition and point out that adding an even number and an odd number will also result in an odd number.
Stages: Concrete Experience, and Pictorial and Abstract Representations
Guide students to add two odd numbers using connecting cubes. Explain that when we add two odd numbers, there is one object left unpaired in each odd number. The two objects that are left unpaired can be paired. So, there are no objects left unpaired and the total is an even number.
Say: We always get an even number when we add two odd numbers.
(b) Stage: Concrete Experience
Have students continue to work in groups. Have a student pick two even numbers from 1 to 10, for example, 8 and 2. Change the numbers in the following steps according to the numbers chosen by the student. Have the groups take out 8 connecting cubes and join the cubes to make 4 pairs.
Say: Since there are no cubes left unpaired, 8 is an even number.
Have the groups remove 2 connecting cubes. Students should remove a pair of connecting cubes.
Ask: Is 2 an even number? (Yes)
Have students count the number of cubes left.
Ask: How many cubes are left? (6) Are all the cubes paired? (Yes) Is 6 an odd number or an even number? (Even number)
Write: 8 – 2 = 6
Ask: Is each number in the subtraction sentence odd or even? (Even)
Write ‘even number’ below each number in the subtraction sentence. Guide students to conclude that they subtracted an even number, 2, from an even number, 8, to get an even number, 6.
Stage: Pictorial Representation
Have students look at the picture of the first subtraction in (b) on CB p. 13. Relate the picture to the earlier activity and go through the example.
Stage: Abstract Representation
Ask: What pattern do you notice? (The difference between two even numbers is an even number.)
Say: We always get an even number when we subtract an even number from an even number because when we take away pairs from pairs, we will have pairs left over.
Stages: Concrete Experience and Pictorial Representation
Repeat the above procedure to guide students to subtract an even number from an odd number.
Stage: Abstract Representation
Ask: What pattern do you notice when we subtract an even number from an odd number? (The difference is an odd number.)
Say: We always get an odd number when we subtract an even number from an odd number because after taking away pairs, the unpaired object from the odd number still remains.
Stages: Concrete Experience and Pictorial Representation
Repeat the above procedure to guide students to subtract an odd number from an even number.
Stage: Abstract Representation
Ask: What pattern do you notice when we subtract an odd number from an even number? (The difference is an odd number.)
Say: We always get an odd number when we subtract an odd number from an even number because we remove an unpaired object from pairs.
Stages: Concrete Experience, and Pictorial and Abstract Representations Guide students to subtract an odd number from an odd number using connecting cubes. Explain that when we subtract an odd number from another odd number, we remove the unpaired object from the greater odd number. So, there are no objects left unpaired.
Say: We always get an even number when we subtract an odd number from an odd number.
Task 1 requires students to investigate and generalize the result of adding two even numbers.
Task 2 requires students to investigate and generalize the result of adding two odd numbers.
Task 3 requires students to investigate and generalize the result of adding odd and even numbers.
Task 4 requires students to investigate and generalize the result of subtracting an even number from another even number.
Task 5 requires students to investigate and generalize the result of subtracting an odd number from another odd number.
Task 6 requires students to investigate and generalize the result of subtracting an even number from an odd number.
Task 7 requires students to investigate and generalize the result of subtracting an odd number from an even number.
Task 1 requires students to use the properties of odd and even numbers in addition. Students are required to use the properties to identify if the sum is even or odd without adding the numbers.
Task 2 requires students to use the properties of odd and even numbers in subtraction. Students are required to use the properties to identify if the difference is even or odd without subtracting the numbers.
Let's Learn
Let's Learn
Objective:
•Investigate and generalize the result of multiplying odd and even numbers
Materials:
•Connecting cubes
Resources:
•CB: pp. 15–16
•PB: p. 16
Stage: Concrete Experience
Have students work in groups. Distribute connecting cubes to each group. Have a student pick two even numbers from 1 to 10, for example, 2 and 4. Change the numbers in the following steps according to the numbers chosen by the student.
Say: 2 × 4 is 2 groups of 4.
Have the groups make 2 groups of 4 connecting cubes and count the total number of cubes.
Ask: Is 2 an odd number or an even number? (Even number) Is 4 an odd number or an even number? (Even number) What is 2 × 4? (8) Is 8 an odd number or an even number?
(Even number)
Write: 2 × 4 = 8
Write ‘even number’ below each number in the multiplication sentence. Guide students to conclude that they have multiplied two even numbers, 2 and 4, to get an even number, 8.
Stage: Pictorial Representation
Have students look at the first picture of the connecting cubes on CB p. 15. Relate the picture to the earlier activity and go through the example.
Stage: Abstract Representation
Explain to students that when we multiply two even numbers, we are putting together even groups of objects and the number of objects in each group is also even. So, all objects remain in pairs and the product is an even number. Say: We always get an even number when we multiply two even numbers.
Stages: Concrete Experience and Pictorial Representation
Repeat the above procedure to guide students to multiply an odd number and an even number.
Stage: Abstract Representation
Explain to students that when we multiply an odd number by an even number, we are putting together odd groups of objects but the number
of objects in each group is even. So, they are still in pairs and the product is an even number. Say: We will always get an even number when we multiply an odd number by an even number.
Stages: Concrete Experience, and Pictorial and Abstract Representations
Guide students to multiply an even number and an odd number using connecting cubes. Guide them to see the number of objects in each group is odd. So, there is an object left unpaired in each group. Since we are putting together even groups of objects, we can pair the objects left in every two groups to make a pair. Hence, the product is even.
Say: We always get an even number when we multiply an even number and an odd number.
Stages: Concrete Experience, and Pictorial and Abstract Representations
Guide students to multiply an odd number by an odd number using connecting cubes. Guide them to see that both the number of objects in each group and the number of groups is odd. So, even after pairing the unpaired objects from each group, there is an object left unpaired. Hence, the product is odd.
Say: We always get an odd number when we multiply two odd numbers.
Task 1 requires students to investigate and generalize the result of multiplying two even numbers.
Task 2 requires students to investigate and generalize the result of multiplying two odd numbers.
Task 3 requires students to investigate and generalize the result of multiplying odd and even numbers.
Task 1 requires students to use the properties of odd and even numbers in multiplication. Students are expected to use the properties of odd and even numbers to identify if the product is odd or even, without multiplying the numbers.
4.1 Finding factors of a
Let's Learn Let's Learn
Objective:
•Find all the factors of a whole number up to 100
Materials:
•Unit cubes
Resources:
•CB: pp. 17–18
•PB: p. 17
Vocabulary:
•factor
(a) Stage: Concrete Experience
Have students work in groups. Distribute unit cubes to each group.
Ask students to arrange 10 cubes in rows and ensure that every row has the same number of cubes. Have students make as many such arrangements as possible.
Ask students the following questions for each arrangement of cubes:
Ask: How many rows of cubes are there? (Answers vary.) How many cubes are there in each row? (Answers vary.) How many cubes are there altogether? (10)
Conclude that we can arrange 10 cubes in rows and columns in different ways.
Stages: Pictorial and Abstract Representations
Have students look at the first row of flowers in (a) on CB p. 17. Relate the flowers to the unit cubes.
Get students to write a multiplication sentence to find the total number of flowers.
Write: 1 × 10 = 10
Say: 10 is the product of 1 and 10. We call 1 and 10 the factors of 10.
Repeat the above procedure for the next group of flowers.
Ask: What are the factors of 10? (1, 2, 5 and 10) How many factors does 10 have? (4)
Lead students to see that a number can be written as a product of two factors in different ways and it can have more than two factors.
(b) Stage: Abstract Representation
Say: Let us find the factors of 60. First, we find the pairs of numbers that can be multiplied to get 60.
Ask students which pairs of numbers can be multiplied to get 60 and write the multiplication sentences on the board.
4.1 Finding factors of a whole number Unit 4 Factors Let's Learn
Ask: Have you found all the pairs of numbers that can be multiplied to get 60? (Yes or no.)
How do you know? (Answers vary.)
Guide students to write the multiplication sentences with products of 60 systematically. Ask: What whole number and 1 can be multiplied to get 60? (60)
Write: 1 × 60 = 60
Ask: What whole number and 2 can be multiplied to get 60? (30)
Write: 2 × 30 = 60
Repeat the above procedure until you reach 7. Say: We cannot find a whole number that gives a product of 60 when multiplied by 7. So, 7 is not a factor of 60.
Continue to write multiplication sentences with products of 60. When you reach 10 × 6 = 60, point out that 10 × 6 = 6 × 10 and we write each pair of factors only once. So, we do not write 10 × 6 = 60. Stop writing multiplication sentences after 10 × 6 = 60 because all the factors are accounted for in the earlier multiplication sentences.
Ask: From the multiplication sentences, what are the factors of 60? (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60)
Let's Do Let's
Task 1 requires students to find all the factors of a whole number up to 100.
Let's Practice Let's Practice
Task 1 requires students to find the unknown factor of a number up to 100, given another factor of the number.
Task 2 requires students to find all the factors of a whole number up to 100.
Have students go back to the word problem on CB p. 2.
Ask: Can you solve the problem now? (Answer varies.) What else do you need to know? (Answer varies.)
Students are not expected to be able to solve the problem now. They will learn more skills in subsequent lessons and revisit this problem at the end of the chapter.
4.2 Finding out if a number is a factor of another number
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Objective:
•Find out if a 1-digit number is a factor of a given whole number
Resources:
•CB: pp. 18–19
•PB: p. 18
(a) Stage: Abstract Representation
Write: Is 5 a factor of 30?
Say: The numbers that are multiplied to form the product are the factors of the number. So, a product can be divided by any of its factors and there will be no remainder.
Have a student divide 30 by 5 in the vertical form on the board.
Ask: Is there any remainder when 30 is divided by 5? (No)
Say: So, 30 can be divided by 5 exactly. This means 5 is a factor of 30.
(b) Stage: Abstract Representation
Follow the procedure in (a).
Ask: Is there any remainder when 53 is divided by 5? (Yes) What does this mean? (53 cannot be divided exactly by 5.) So, is 5 a factor of 53? (No)
Let's Do Do and Let's Practice Let's Practice
Task 1 requires students to find out if a 1-digit number is a factor of a given whole number.
Task 2 requires students to find out if a 1-digit number is a factor of the given whole numbers.
5.1 Finding multiples of a whole number
Let's Learn Let's Learn
Objective:
• Find the multiples of a whole number up to 10
Materials: •Counters
Resources:
•CB: p. 20
•PB: p. 19
Vocabulary: •multiple
Stage: Concrete Experience
Stick 4 counters in a row on the board.
Ask: How many rows of counters are there? (1)
How many counters are there in each row? (4)
How many counters are there altogether? (4)
Repeat the above procedure with 8, 12 and 16 counters with 4 counters in each row.
Stages: Pictorial and Abstract Representations
Have students look at the dot cards on CB p. 20.
Relate the dot cards to the counters.
Ask students to look at the first dot card.
Say: There is 1 row of dots and there are 4 dots in a row.
Write: 1 × 4 = 4
Repeat the above procedure for the remaining dot cards.
Ask: What is the pattern you notice in the dot cards? (Each dot card has one more row of 4 dots than the previous dot card.)
Have students look at the multiplication sentences. Guide them to see that these are multiplication facts in the multiplication table of 4.
Say: The products of the multiplication sentences are called multiples. 4, 8, 12 and 16 are multiples of 4.
Explain to students that a multiple of a number is the product of that number and another number. In this case, a multiple of 4 is the product of 4 and another number.
Say: Look at the multiplication sentences again.
Ask: How are the multiplication sentences alike? (Every multiplication sentence has 4 as a factor.) Explain that a number is a factor of all its multiples. In this example, 4 is a factor of the multiples of 4. So, 4 is a factor of 4, 8, 12 and 16.
Reinforce students’ understanding of multiples by getting them to find the next six multiples of 4.
Let's Do Let's Do and Let's Practice Let's Practice
Task 1 requires students to find the multiples of a whole number up to 10.
Let's Learn Let's Learn
Objectives:
•Relate factors and multiples
•Find out if a whole number is a multiple of a given whole number up to 10
Resources:
•CB: p. 21
•PB: p. 20
(a) Stage: Abstract Representation
Ask a student to work out the division of 18 by 2 on the board.
Ask: Can 18 be divided by 2 exactly? (Yes) Is 18 a multiple of 2? (Yes) Is 2 a factor of 18? (Yes)
Say: 18 is a multiple of 2. This also means that 2 is a factor of 18. Multiples of 2 can be divided by 2 exactly and have 2 as a factor.
(b) Stage: Abstract Representation
Follow the procedure in (a).
Lead students to see how factors and multiples are related. If a number can be divided exactly by a second number, the first number is a multiple of the second number. This also means that the second number is a factor of the first number.
Let's Do Let's Do
Task 1 requires students to find out if a whole number is a multiple of a given whole number up to 10, and relate factors and multiples.
Let's Practice Let's Practice
Task 1 requires students to find out if a whole number is a multiple of a given whole number up to 10.
Task 2 requires students to relate factors and multiples.
Objective:
•Identify multiples of 2, 5, 10, 25, 50 and 100 up to 1000
Resources:
•CB: pp. 22–23
•PB: p. 21
Stage: Abstract Representation
Have three students write the multiplication tables of 2, 5 and 10 on the board.
Ask: Look at the digits in the ones place of the multiples in multiplication table of 2. What pattern do you observe? (Multiples of 2 have the digit 0, 2, 4, 6 or 8 in the ones place.)
Say: A multiple of 2 has 0, 2, 4, 6 or 8 in the ones place.
Ask: Look at the digits in the ones place of the multiples in multiplication table of 5. What pattern do you observe? (Multiples of 5 have the digit 0 or 5 in the ones place.)
Say: A multiple of 5 has 0 or 5 in the ones place.
Ask: Look at the digits in the ones place of the multiples in multiplication table of 10. What pattern do you observe? (Multiples of 10 have the digit 0 in the ones place.)
Say: A multiple of 10 has 0 in the ones place. Repeat the above procedure with multiplication tables of 25, 50 and 100.
Guide students to observe the patterns in the tens and ones places of the multiples in multiplication tables of 25, 50 and 100.
Point out to students that a multiple of 25 ends with ‘00’, ‘25’, ‘50’ or ‘75’, a multiple of 50 ends with ‘00’ or ‘50’ and a multiple of 100 ends with ‘00’.
Multiply by 2 Multiply by 5 Multiply by 10
Multiples of 100 end with ‘00’. Let's Learn 5.3 Identifying multiples of 2, 5, 10, 25, 50 and 100
Multiply by 25 Multiply by 50 Multiply by 100
Multiples of 2 have the digit 0, 2, 4, 6 or 8 in the ones place.
Multiples of 5 have the digit 0 or 5 in the ones place.
Multiples of 10 have the digit 0 in the ones place.
Multiples of 25 end with ‘00’, ‘25’, ‘50’ or ‘75’.
Multiples of 50 end with ‘00’ or ‘50’.
Let's Do Let's Do
Task 1 requires students to determine if a whole number is a multiple of 2, 5, 10, 25, 50 and 100.
Let's Practice Let's Practice
Task 1 requires students to determine if a whole number is a multiple of 2, 5, 10, 25, 50 and 100.
Task 2 requires students to identify multiples of 10.
Task 3 requires students to identify multiples of 5 and 50.
Let's Learn Let's Learn
Objective:
•Know and apply tests of divisibility by 2, 3, 4, 5, 10, 25 and 100
Resources:
•CB: pp. 23–25
•PB: p. 22
Stage: Abstract Representation
Say: We can use divisibility rules to quickly tell if a large number can be divided by another number exactly.
Say: To determine if a number is divisible by 2, we look at its last digit. If the last digit of a number is even, that is, if it is 0, 2, 4, 6 or 8, we can conclude that the number is divisible by 2.
Write: 176
Ask: What is the last digit in 176? (6) Is 6 an even number? (Yes)
Say: Since the last digit in 176 is even, we can say that 176 is divisible by 2. Have students divide 176 by 2 to check. (176 ÷ 2 = 88)
Say: To determine if a number is divisible by 3, we look at the sum of its digits. If the sum of its digits is divisible by 3, we can conclude that the number is divisible by 3.
Write: 543
Ask: What are the digits in 543? (5, 4 and 3)
What is the sum of the digits in 543? (12) Is 12 divisible by 3? (Yes)
Say: Since 12 is divisible by 3, we can say that 543 is divisible by 3.
Have students divide 543 by 3 to check.
(543 ÷ 3 = 181)
Say: To determine if a number is divisible by 4, we look at the number formed by its last two digits. If the number formed by its last two digits is divisible by 4, we can conclude that the number is divisible by 4.
Write: 324
Ask: What is the number formed by the last two digits in 324? (24) Is 24 divisible by 4? (Yes)
Say: The number formed by the last two digits in 324 is 24, which is divisible by 4. So, 324 is divisible by 4.
Have students divide 324 by 4 to check. (324 ÷ 4 = 81)
Say: To determine if a number is divisible by 5, look at its last digit. If the last digit is 0 or 5, we can conclude that the number is divisible by 5.
Write: 495
Ask: What is the last digit in 495? (5)
Say: Since the last digit in 495 is 5, we can say that 495 is divisible by 5.
Have students divide 495 by 5 to check.
(495 ÷ 5 = 99)
Say: To determine if a number is divisible by 10, look at its last digit. If the last digit is 0, we can conclude that the number is divisible by 10.
Write: 540
Ask: What is the last digit in 540? (0)
Say: Since the last digit in 540 is 0, we can say that 540 is divisible by 10.
Have students divide 540 by 10 to check.
(540 ÷ 10 = 54)
Say: To determine if a number is divisible by 25, look at its last two digits. If the last two digits are ‘00’, ‘25’, ‘50’ or ‘75’, we can conclude that the number is divisible by 25.
Write: 575
3If the sum of its digits is divisible by 3. Is 543 divisible by 3?
Check: 5 + 4 + 3 = 12 12 is divisible by 3.
So, 543 is divisible by 3.
4If the number formed by its last two digits is divisible by 4. Is 324 divisible by 4?
Check: 324
24 is divisible by 4.
So, 324 is divisible by 4.
5If the last digit is 0 or 5. Is 495 divisible by 5?
Check: 495
The last digit is 5.
So, 495 is divisible by 5.
10If the last digit is 0. Is 540 divisible by 10?
Check: 540
The last digit is 0.
So, 540 is divisible by 10.
25If its last two digits are ‘00’, ‘25’, ‘50’ or ‘75’. Is 575 divisible by 25?
Check: 575
The last two digits are ‘75’.
So, 575 is divisible by 25.
100If its last two digits are ‘00’. Is 6700 divisible by 100?
Check: 6700
The last two digits are ‘00’. So, 6700 is divisible by 100.
Ask: What are the last two digits in 575? (75)
Say: Since the last two digits in 575 are ‘75’, we can say that 575 is divisible by 25.
Have students divide 575 by 25 to check.
(575 ÷ 25 = 23)
Say: To determine if a number is divisible by 100, look at its last two digits. If the last two digits are ‘00’, we can conclude that the number is divisible by 100.
Write: 6700
Ask: What are the last two digits in 6700? (00)
Say: Since the last two digits in 6700 is ‘00’, we can say that 6700 is divisible by 100.
Have students divide 6700 by 100 to check.
(6700 ÷ 100 = 67)
Let's Do Let's Do
Task 1 requires students to know and apply tests of divisibility by 2, 3, 4, 5, 10, 25 and 100.
Let's Practice Let's Practice
Task 1 requires students to know and apply test of divisibility by 2.
Task 2 requires students to know and apply test of divisibility by 3.
Task 3 requires students to know and apply test of divisibility by 4.
Task 4 requires students to know and apply test of divisibility by 25.
Task 5 requires students to know and apply tests of divisibility by 10 and 100.
1. Complete the table. Write Yes or No Number Is the number divisible by …? 23451025100
6.1 Identifying prime and composite numbers
Let's Learn Let's Learn
Objectives:
•Identify prime and composite numbers
• Recognize prime numbers up to 20 and find all prime numbers less than 100
Materials:
• 1 copy of Number Chart (BM1.1) per student
Resources:
•CB: pp. 26–29
•PB: p. 23
Vocabulary:
•composite number
•prime number
(a) Stage: Pictorial Representation
Draw a row of 7 circles on the board.
Ask: How many rows of circles are there? (1)
How many circles are there in each row? (7)
How many circles are there altogether? (7)
Stage: Abstract Representation
Say: Let us write a multiplication sentence to find the total number of circles.
Write: 1 × 7 = 7
Say: 1 and 7 are factors of 7.
Ask: Can we arrange 7 circles in other ways so that every row has the same number of circles? (No)
Write: The factors of 7 are 1 and 7.
Say: The factors of 7 are 1 and 7. A number that has only two factors, 1 and the number itself, is a prime number. Since 7 has only two factors, 1 and 7, 7 is a prime number.
Write: 7 is a prime number.
(b) Stages: Pictorial and Abstract Representations
Draw a row of 6 circles on the board.
Ask: How many rows of circles are there? (1)
How many circles are there in each row? (6)
How many circles are there altogether? (6)
Say: Let us write a multiplication sentence to find the total number of circles.
Write: 1 × 6 = 6
Say: 1 and 6 are factors of 6.
Ask: Can we arrange 6 circles in other ways so that every row has the same number of circles? (Yes)
Invite a student to draw 6 circles in another way so that every row has the same number of circles.
Ask questions similar to those above to derive the multiplication sentence 2 × 3 = 6 or 3 × 2 = 6.
Say: 2 and 3 are also factors of 6. The factors of 6 are 1, 2, 3 and 6.
Write: The factors of 6 are 1, 2, 3 and 6.
Ask: How many factors does 6 have? (4)
Say: 6 has more than 2 factors, so 6 is not a prime number. A number that has more than two factors is a composite number. So, 6 is a composite number.
Write: 6 is a composite number.
Ask: Is 1 a prime number or a composite number? (Answers vary.)
Have students discuss in pairs why they think 1 is a prime number or a composite number. Guide them to see that we can write 1 as a product of 1 and 1, and 1 = 1 × 1. This means that 1 has only one factor, 1.
Say: So, 1 is neither a prime number nor a composite number.
(c) Stages: Concrete Experience, and Pictorial and Abstract Representations
Distribute a copy of Number Chart (BM1.1) to each student.
Say: Let us use the number chart to find all the prime numbers within 100. First, cross out 1 as it is neither a prime number nor a composite number.
Have students cross out the number 1 on the number chart.
Ask: What are the factors of 2? (1 and 2) Is 2 a prime number? (Yes)
Say: Circle the number 2.
Lead students to see that multiples of 2 greater than 2 have at least three factors, 1, 2 and the number itself. So, multiples of 2 that are greater than 2 are not prime numbers.
Say: Cross out all the multiples of 2 that are greater than 2.
Ask: What are the factors of 3? (1 and 3) Is 3 a prime number? (Yes)
Say: Circle the number 3 and cross out the multiples of 3 that are greater than 3.
Explain to students that multiples of 3 greater than 3 have at least three factors, 1, 3 and the number itself. So, multiples of 3 that are greater than 3 are not prime numbers.
Ask: Which is the next prime number? (5) Why? (It has only two factors, 1 and 5.)
Say: Circle the number 5 and cross out the multiples of 5 that are greater than 5.
Continue this process until all the numbers in the number chart are circled or crossed out.
Say: All the numbers that are circled are the prime numbers between 1 and 100.
Ask: What type of numbers are the numbers that are crossed out? (Composite numbers except for the number 1.)
Say: All the numbers greater than 1 that are crossed out are the composite numbers between 1 and 100.
Guide students to use their number chart to identify if 39 is a prime number.
Ask: Is 39 a prime number? (No)
Have students observe that 39 is crossed out in the number chart. Lead them to see that the factors of 39 are 1, 3, 13 and 39, so 39 is not a prime number.
12345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899100
Follow these steps to find all the prime numbers within 100.
Step 1: Cross out 1 in the number chart as it is neither a prime nor a composite number.
Step 2: 2 is the first prime number. Circle it and cross out all multiples of 2 that are greater than 2.
Step 3: 3 is the next prime number. Circle it and cross out all multiples of 3 that are greater than 3.
Step 4: 5 is the next prime number. Circle it and cross out all multiples of 5 that are greater than 5.
Step 5: Continue this process until all the numbers are circled or crossed out.
All the numbers that are circled are the prime numbers between 1 and 100.
Is 39 a prime number? Is 59 a prime number? Is 96 a prime number? No Yes No
Say: 39 is not a prime number because it has factors other than 1 and 39 itself.
Ask: Is 59 a prime number? (Yes)
Have students observe that 59 is circled in the number chart.
Say: 59 is a prime number because it only has factors 1 and 59 itself.
Ask: Is 96 a prime number? (No)
Have students observe that 96 is crossed out in the number chart.
Have them list the factors of 96. (1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48 and 96)
Say: 96 is not a prime number because it has factors other than 1 and 96 itself.
1. Write the factors of each of the following numbers. Then, write Yes or No
a) Factors of 11: b) Factors of 25: Is 11 a prime number? Is 25 a prime number?
2. Circle the prime numbers. Cross out the composite numbers.
7 18 23 35 41 57
3. Write in the blanks. Use the number chart on page 27 to help you.
a) What are the prime numbers from 1 to 10?
b) What are the composite numbers from 11 to 20?
c) What is the least prime number?
d) What is the first odd composite number?
Let's Practice Let's Yes 1, 11 1, 5, 25 No 2, 3, 5, 7 12, 14, 15, 16, 18, 20 2 9
1. Write all the prime numbers from 1 to 20.
2, 3, 5, 7, 11, 13, 17, 19
2. Write all the composite numbers from 31 to 50.
32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50
3. Complete the table with the following numbers.
5 9 17 40 42 49 55 64 73 97
5, 17, 73, 97 9, 40, 42, 49, 55, 64
4. Cross out the prime numbers. 17 8 31 24 67 29 45 51
5. Circle the numbers that are both composite and odd.
2 3 5 9 11 12 13 15
Let's Do Let's
Task 1 requires students to identify prime numbers.
Tasks 2 and 3 require students to identify prime and composite numbers.
Let's Practice Let's Practice
Task 1 requires students to recognize prime numbers up to 20.
Task 2 requires students to find all composite numbers from 31 to 50.
Task 3 requires students to identify prime and composite numbers.
Task 4 requires students to identify prime numbers.
Task 5 requires students to identify composite odd numbers.
I want to bring some beads to school to share equally among my friends. The number of beads cannot be a prime number as I need to divide the number of beads equally among my friends. The number of beads needs to be a composite number. Prime numbers have only two factors, 1 and the number itself. All prime numbers, except 2, are odd numbers.
Have students work in groups to discuss the tasks. Ask the groups to present their answers.
Ask students to write the prime numbers that are less than 20. Have students observe that 2 is a prime number but it is not an odd number. Conclude that Sarah is not correct.
Reiterate to students that prime numbers have only two factors, 1 and the number itself. All prime numbers, except 2, are odd numbers.
Make use of the examples presented by the groups to let students understand the importance and usefulness of knowing prime numbers.
Let's Learn Let's Learn
Objective:
•Identify square numbers
Materials: •Counters
Resources:
•CB: pp. 30–31
•PB: p. 24
Vocabulary: •square number
Stages: Concrete Experience and Pictorial Representation
Have students work in groups. Distribute counters to each group. Have students follow each step of your demonstration. Stick 1 counter on the board.
Ask: How many rows of counters are there? (1) How many counters are there in the row? (1)
What is the multiplication sentence we can write to find the number of counters? (1 × 1 = 1)
Write the number sentence below the counter to show the number of counters.
Write: 1 × 1 = 1
Draw a box around the counter on the board.
Point out to students that the shape of the box around the counter is a square. Have students relate this to the first picture of counters from the left on CB p. 30.
Next, stick 2 rows of 2 counters beside the first counter on the board as shown in the second picture of counters from the left on the page. Point out that there are 2 rows of 2 counters.
Write: 2 × 2 = 4
Draw a box around the 2 rows of 2 counters on the board.
Ask: What is the shape of the box around the second group of counters? (Square)
Have students relate this to the second picture of counters from the left on the page. Follow the same procedure for the third and fourth pictures of counters from the left on the page. Lead students to see that the boxes drawn around the group of 9 counters and the group of 16 counters are also squares.
Ask: What is the multiplication sentence we can write to find the number of counters in the third picture of counters? (3 × 3 = 9) What is the multiplication sentence we can write to find the number of counters in the fourth picture of counters? (4 × 4 = 16)
Write: 3 × 3 = 9 4 × 4 = 16
6.2 Identifying square numbers
Stage: Abstract Representation
Point to the single counter on the board.
Say: When we multiply a number by itself, we get the square of the number. Since 1 × 1 = 1, the square of 1 is 1. The square of a whole number is called a square number.
Write: square number
Highlight to students that all square numbers can be arranged in equal groups to form a square.
Write: 12 = 1
Explain to students that we use the index notation ‘2’ to represent 1 multiplied by itself, that is 1 × 1. Tell them that 12 is read as 1 squared. Point to the second group of counters on the board.
Say: Since 2 × 2 = 4, the square of 2 is 4. 2 squared is 4.
Write: 22 = 4
Follow the same procedure for ‘3 × 3 = 9’ and ‘4 × 4 = 16’ and lead students to see that the square of 3 is 9 and the square of 4 is 16.
Say: 1, 4, 9 and 16 are square numbers.
Let's Do Let's Do
Task 1 requires students to model square numbers. Students are required to interpret the diagrams and determine the square numbers represented by the diagrams.
Task 2 requires students to understand the index notation of square numbers.
Task 3 requires students to identify square numbers.
Let's Practice Let's Practice
Task 1 requires students to model square numbers. Students are required to color the small squares to represent a square number.
Task 2 requires students to understand the index notation of square numbers.
Task 3 requires students to identify square numbers.
7.1 Mind stretcher
Let's Learn Let's Learn
Objective:
•Solve a non-routine problem involving factors and multiples using the strategy of logical reasoning
Resource: •CB: pp. 32–33
Have students read the problem on CB p. 32.
1. Understand the problem. Pose the questions in the thought bubble in step 1.
2. Plan what to do.
Point out to students that they can use logical reasoning to help them solve the problem.
3. Work out the Answer.
Have students read the first clue.
Ask: How can we tell if a number is even? (The last digit is 0, 2, 4, 6 or 8.)
Have students read the second clue. Ask: What is the range of numbers that could be the mystery number? (Numbers from 201 to 499.)
Have students read the third clue.
Ask: How do you identify multiples of 25? (Multiples of 25 end with ‘00’, ‘25’, ‘50’ or ‘75’.) What are the multiples of 25 that are between 200 and 500? (225, 250, 275, 300, 325, 350, 375, 400, 425, 450, 475)
Write: 225, 250, 275, 300, 325, 350, 375, 400, 425, 450, 475
You will learn to...
• solve a non-routine problem involving factors and multiples
Let's Learn
Anne writes a mystery number in her notebook and asks Marcus to find out what the number is. She gives him these clues:
• The number is even.
• The number is between 200 and 500.
• The number is a multiple of 25 and has 7 as a factor. What is the mystery number?
Is the number odd or even?
How many digits does the number have? What can the number be divided by? What do I have to find?
Plan what to do.
Work out the Answer 3
I can use logical reasoning to solve the problem.
The mystery number is a multiple of 25 so it ends with ‘00’, ‘25’, ‘50’ or ‘75’.
The mystery number is between 200 and 500 so it is one of the following:
225, 250, 275, 300, 325, 350, 375, 400, 425, 450, 475
The mystery number is an even number. Cross out the odd numbers.
225, 250, 275, 300, 325, 350, 375, 400, 425, 450, 475
Ask: Since the mystery number is an even number, what are the numbers that should be crossed out? (225, 275, 325, 375, 425 and 475)
Cross out 225, 275, 325, 375, 425 and 475.
Ask: The mystery number also has 7 as a factor. How can you use this clue to identify the mystery number? (Divide the remaining numbers by 7 to find the number which gives a quotient without any remainder.)
Ask five students to divide 250, 300, 350, 400 and 450 by 7 on the board.
Ask: What number can be divided by 7 exactly? (350) What is the mystery number? (350)
Say: The mystery number is 350.
4. Check if your answer is correct. Have students check the answer by matching each clue to the mystery number, 350.
5. + Plus Solve the problem in another way. Have students try to solve the problem in a different way.
Have 1 or 2 students share their methods. If students are unable to solve the problem in a different way, explain the method shown on CB p. 33.
Ask: Which method do you prefer? Why? (Answers vary.)
Have students go back to the word problem on CB p. 2. Get them to write down in column 3 of the table what they have learned that will help them solve the problem, and then solve the problem.
Have a student present his/her work to the class.
Exercise 1.1
1. a) 23 704 b) 543 319 c) 925 602 d) 700 083
2. a) fifty thousand, nine hundred and one b) six hundred and fifty-nine thousand, five hundred and forty-seven c) two hundred thousand, three hundred and ninety d) nine hundred and eight thousand and four
Exercise 1.2
1. a) ten thousands
b) hundred thousands; 600 000
c) 7; 7000
d) 0; 0
2. a) 30 000 b) 500 c) 20 d) 9000 e) 200 000 f) 50 000
Exercise 1.3
1. a) > b) < c) > d) <
2. a) 30 458, 304 508, 304 580
b) 123 406, 123 456, 132 056, 132 456
3. a) 944 700, 904 779, 94 797
b) 332 220, 332 000, 222 200, 92 222
Exercise 2.1
1. a) 3540 b) 67 220 c) 850 600 d) 215 000
2. a) $8800 b) $9500 c) $56 500
d) $37 100 e) $974 400 f) $350 000
3. a) $9000 b) $4000 c) $73 000
d) $97 000 e) $533 000 f) $400 000
Exercise 2.2
1. Estimates vary. Sample:
a) 803; 200; 600; 800; Yes b) 1938; 6000; 4000; 2000; Yes
2. Estimates vary. Sample: a) 960; 900 b) 44 572; 45 000 c) 6217; 6300 d) 21 523; 21 000
Exercise 3.1
1. a) 15; odd; odd b) 79, even; odd c) 466; even; even d) 600; odd; even
2. a) odd b) even c) even d) even e) odd f) odd g) even h) odd i) even j) odd k) odd l) even 3. a) 13; even; odd b) 15; odd; odd c) 272; even; even d) 208; odd; even
4. a) even b) odd c) even d) even e) odd f) odd g) even h) even i) odd j) even
Exercise 3.2
1. a) 30 b) 40 c) 70 d) 60 even; even 2. a) 21; odd; odd b) 48; even; even 3. a) even b) even c) odd d) even 4. even
5. odd
Exercise 4.1
1. a) 8 b) 5 c) 6 d) 9 2. a) 1, 2, 4, 5, 10 and 20 b) 1, 3, 7, 9, 21 and 63 c) 1, 3, 5, 9, 15 and 45
Exercise 4.2
1. a) Yes b) No c) No d) Yes 2. 48, 36, 24
Exercise 5.1
1. 4; 8; 12; 16; 20 2. a) 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 b) 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 c) 7, 14, 21, 28, 35, 42, 49, 56, 63, 70 d) 8, 16, 24, 32, 40, 48, 56, 64, 72, 80 e) 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 f) 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
Exercise 5.2
1. a) Yes; Yes; No b) Yes; No; Yes 2. a) factor b) factor c) multiple d) multiple 3. a) factor b) factor c) multiple d) multiple
Exercise 5.3
1. a) Yes b) No c) No d) Yes e) Yes f) No
2. Circle: 325; 55; 510; 500; 170; 485
3. Circle: 8; 80; 340; 24; 60; 600 Cross out: 80; 340; 60; 600
4. Circle: 300; 225; 450; 475; 400; 850 Cross out: 300; 450; 400; 850
5. Circle: 750; 700; 450; 500; 650 Cross out: 700; 500
5. even
6. odd
7. odd
Number Is the number divisible by …?
23451025100
640 Yes No YesYesYes NoNo
3575 NoNoNo Yes No Yes No
93 400 Yes No YesYesYesYesYes
2. Circle:
656; 42 358; 59 000
3. Circle:
748; 5340; 813 456; 639 484
4. Circle:
630; 455 370; 7505; 855
5. Circle:
6370; 23 150; 7490; 905 050
6. Circle:
300; 125 600; 4550; 85 675; 219 900
Cross out:
300; 125 600; 219 900
Exercise 6.1
1. 2, 3, 5, 7, 11, 13, 17, 19
2. 4, 6, 8, 9, 10, 12, 14, 15, 16, 18
3. a) Yes b) No c) Yes
d) Yes e) No f) No
4. Circle:
33; 72; 57; 65; 18; 27; 55; 49; 87; 30; 51; 63; 39; 93; 77
Cross out:
2; 5; 89; 47; 97; 13; 31; 71; 61; 23; 83; 53; 19; 11; 43
3. a) 7; 7 b) 36; 36 c) 15; 15 d) 45; 45
4. Circle:
49; 9; 4; 64
5. 25, 36, 49, 64, 81
• composite number A composite number is a number that has more than two factors.
1 × 8 = 8 2 × 4 = 8
The factors of 8 are 1, 2, 4 and 8. So, 8 is a composite number.
• coordinate plane See Cartesian plane.
• coordinates Coordinates are a set of values that shows an exact position on a Cartesian plane. When writing the coordinates of a point, we write the x-coordinate followed by the y-coordinate in the form (x y). We write coordinates in brackets and separate the two numbers with a comma.
(2, 3) and (5, 0) are coordinates.
E • equilateral triangle
An equilateral triangle is a triangle with three equal sides and three equal angles.
A B C AB = BC = AC
/BAC = m/ACB = m/ABC = 60° ABC is an equilateral triangle.
L • likely
A likely event is an event that has a high chance of happening.
• line graph
A line graph is a graph used to present data that changes over time. M • million 1 million = 1000 thousands = 1 000 000
• mirror line
A mirror line is a line of reflection. It is also a line of symmetry.
• multiple The multiple of a number is the product of the number and any other whole number except zero.
F
• factor
Factors are whole numbers that are multiplied to get another number.
3 × 6 = 18 3 and 6 are factors of 18.
A factor is a whole number that divides another whole number exactly, without any remainder.
18 ÷ 6 = 3 6 is a factor of 18.
H
• hundred thousand
1 hundred thousand = 100 thousands = 100 000
I
• impossible
An impossible event is an event that definitely will not happen.
• integers
An integer is one of the numbers ..., –3, –2, –1, 0, 1, 2, 3, ... –5, 0 and 2 are integers.
3 4 and 0.6 are not integers.
• isosceles triangle
An isosceles triangle is a triangle with two equal sides and two equal angles.
A B C
AB = BC m/BAC = m/ACB
ABC is an isosceles triangle.
1 × 8 = 8 2 × 8 = 16 3 × 8 = 24 8, 16 and 24 are multiples of 8. N • negative integers
Integers less than 0 are called negative integers ..., –5, –4, –3, –2, –1 are negative integers.
• negative numbers
Numbers less than 0 are called negative numbers –20, –8.5, –5 and –1.2 are negative numbers.
• numerical expression
A numerical expression contains only numbers and operation signs. 6 – 2 and 4 ÷ 5 are numerical expressions.
O • order of rotational symmetry
The number of times a shape fits into its original position during a complete turn about its center is called the order of rotational symmetry
This shape fits into its original position 3 times during the complete turn.
The order of rotational symmetry of this shape is 3.
• ordered pair
An ordered pair, also known as coordinates, is used to describe the location of a point on a Cartesian plane. An ordered pair is made up of the x-coordinate and the y-coordinate of that point.
(4, 2) is an ordered pair on the Cartesian plane. It means 4 units to the right and 2 units up from the origin.
• origin The point where the x-axis and the y-axis cross at a right angle is called the origin. The coordinates of the origin is (0, 0).
P • parallel (//)
These two line segments are parallel They are always the same distance apart and will never meet.
• perimeter The perimeter of a figure is the length around the figure. A B D C Perimeter of ABCD = AB + BC + CD +
• positive integers
Integers greater than 0 are called positive integers 1, 2, 3, 4, 5, ... are positive integers.
• possible A possible event is an event that may happen.
• prime number A prime number is a number that has only two factors, 1 and the number itself.
The factors of 3 are 1 and 3. So, 3 is a prime number.
Q • quadrants (Cartesian plane)
The x-axis and the y-axis divide the Cartesian plane into four quadrants
Point A is in the 1st quadrant.
Point B is in the
two line segments are perpendicular. They cross at a right angle.
R • reflection
A reflection is a flip along the mirror line. In a reflection, the size and shape do not change but the orientation and position change.
• reflective symmetry
When a shape has two matching halves folded about a line of symmetry, we say that the shape is symmetric or has reflective symmetry
• right triangle
A right triangle is a triangle with a right angle (90°).
A B C /BAC is a right angle.
ABC is a right triangle.
• rotational symmetry
When a shape fits into its original position during a turn less than a complete turn, it has rotational symmetry
S • scalene triangle
6 cm
A scalene triangle is a triangle with no equal sides. K 9 cm L 7 cm J
LJK is a scalene triangle.
• square number
When a number is multiplied by itself, the product is a square number
5 × 5 = 25
5 multiplied by itself is 25. So, 25 is a square number.
T
• ten thousand
1 ten thousand = 10 thousands = 10 000
• thousandth
1 thousandth is 1 out of 1000 equal parts.
1 1000 = 0.001
• transformation A transformation happens when a shape undergoes a change. The shape can change its position, size, shape or orientation after a transformation. Translations and reflections are types of transformations.
• translation 7 units
A translation is a movement along a straight line. This shape shows a translation. The triangle has moved 7 units to the right. In a translation, the size, shape and orientation of the shape do not change but its position does.
U
• uncertain
An uncertain event is an event that may or may not happen.
• unlike fractions Unlike fractions are fractions that do not have the same denominator.
2 3 and 2 5 are unlike fractions because they have different denominators, 3 and 5.
• unlikely An unlikely event is an event that has a low chance of happening.
V • vertically opposite angles
Vertically opposite angles are the angles opposite to each other when two lines cross. Vertically opposite angles have equal measures. W Y X Z d b c a /a and /c are a pair of vertically opposite angles.
m/a = m/c /b and /d are also a pair of vertically opposite angles.
m/b = m/d X • x-axis
The horizontal axis on the Cartesian plane is called the x-axis
• x-coordinate
The x-coordinate measures the distance of a point from the origin along the x-axis. The x-coordinate of (3, 5) is 3.
Y • y-axis
The vertical axis on the Cartesian plane is called the y-axis
• y-coordinate
The y-coordinate measures the distance of a point from the origin along the y-axis.
The y-coordinate of (3, 5) is 5.
A world-class program incorporating the highly effective Readiness-Engagement-Mastery model of instructional design
100% coverage of Cambridge Primary Mathematics Curriculum Framework Incorporates Computational Thinking and Math Journaling Builds a Strong Foundation for STEM
Scholastic TM Mathematics (New Edition) covers five strands of mathematics across six grades: Numbers and Operations, Measurement, Geometry, Data Analysis, and Algebra.
The instructional design of the program incorporates the Readiness-Engagement-Mastery process of learning mathematics, making learning meaningful, and lesson delivery easy and effective.
In Coursebook In Practice Book Digital
Think About It Practice
Create Your Own Assessment
Mind Stretcher Math Jour nal
Mission Possible Exercises Reviews
Each chapter of the coursebook starts with Let’s Remember and Explore to ready students for learning new content and comprises units of study developed on carefully grouped learning objectives. Each unit is delivered through specially crafted daily lessons that focus on a concept or an aspect of it. Concepts and skills are introduced in Let’s Learn Let’s Do and Let’s Practice provide opportunities for immediate formative assessment and practice.
Let’s Remember offers an opportunity for systematic recall and assessment of prior knowledge in preparation for new learning.
Explore encourages mathematical curiosity and a positive learning attitude. It gets students to recall prior knowledge, set targeted learning goals for themselves and track their learning as they progress through the unit, seeking to solve the problem.
In Let’s Learn, concepts and skills are introduced and developed to mastery using the concrete-pictorialabstract approach. This proven, research-based approach develops deep conceptual understanding.
Systematic variation of tasks in Let's Do and Let's Practice reinforces students’ understanding and enables teachers to check learning and identify remediation needs.
Practice Book links lead to exercises in the Practice Book to further reinforce understanding of the concepts and skills learnt.
Think About It develops metacognition by providing opportunities for mathematical communication, reasoning and justification. Question prompts take students through the mathematical reasoning process, helping teachers identify misconceptions.
A Problem Solving lesson concludes each chapter. With a focus on both the strategies and the process of problem solving, these word problems provide a meaningful context for students to apply mathematical knowledge and skills.
A 5-step process guides students to systematically solve problems by applying appropriate strategies and to reflect on their problemsolving approach.
Create Your Own and Mind Stretcher develop higher-order thinking skills and metacognitive ability.
Mission Possible develops computational thinking through a scaffolded approach to solving complex problems with newly learnt skills.
Mathematical Modeling provides opportunities for students to model solutions to real-world problem situations.
To make learning and teaching fun and engaging, digital components are available with
Mathematics (New Edition).
Digital practice and assessment further strengthen understanding of key concepts and provide diagnostic insight in students' capabilities and gaps in understanding.
In addition to the course materials for in-class projection, the Hub offers valuable resources including videos, lesson notes, and additional content at point of use.
Chapter 5 Perpendicular and
Line Segments
Chapter 6 Triangles Let’s
Chapter 7 Symmetry
Chapter 9 Addition and Subtraction of Decimals
1. Count and write the number in numerals and in words.
Numeral:
Words:
2. Complete the sentences.
a) In 3518, the value of the digit 5 is .
b) In 6042, the digit 6 is in the place.
3. Compare 1354, 1340 and 2098. Then, write in the blanks.
a) is the greatest number.
b) 1354 is than 1340.
c) is the least number.
Compare the digits in the place values from left to right.
4. Arrange the numbers in order. Begin with the least. 3072 3720 , , (least)
5. Round 2195 to the a) nearest ten.
b) nearest hundred. c) nearest thousand.
6. Write in the blanks.
a) The digit in the ones place of an odd number is , , , or .
b) The digit in the ones place of an even number is , , , or .
ABC Grocery Store donated 100 cans of food during a food donation drive. The store owner wants to pack the cans of food equally into bags. How many bags with an equal number of cans of food can there possibly be?
How can we solve this problem? Discuss in your group and fill in columns 1 and 2.
You will learn to...
• read and write numbers within 1 000 000
• identify the values of digits in 5-digit and 6-digit numbers
• compare and order numbers within 1 000 000
Let's Learn Let's Learn
We read 10 000 as ten thousand. 15 263 is read as fifteen thousand, two hundred and sixty-three.
b) These are 100 000 unit cubes.
Count on in steps of 10 000 from 10 000.
10 000, 20 000, 30 000, 40 000, 50 000, 60 000, 70 000, 80 000, 90 000, 100 000
100 000 comes after 90 000. We read 100 000 as one hundred thousand.
200 000 + 30 000 + 5000 + 400 + 70 + 1 = 235 471 235 000 + 471 = 235 471
235 471 is read as two hundred and thirty-five thousand, four hundred and seventy-one.
d) Count on in steps of 100 000 from 100 000.
100 000, 200 000, 300 000, 400 000, 500 000, 600 000, 700 000, 800 000, 900 000, 1 000 000
1 000 000 comes after 900 000. We read 1 000 000 as one million.
1. Write the numerals.
a) fifty-four thousand, one hundred and one
b) two hundred and six thousand, four hundred and three
2. Write the numbers in words.
a) 10 299
b) 559 311
1. Write the numerals.
a) fourteen thousand, two hundred and ninety
b) four hundred and thirty-five thousand, one hundred and seventeen
c) eight hundred and eight thousand, nine hundred and sixty-five
d) seven hundred and twenty thousand and twelve
2. Write the numbers in words.
a) 18 030
b) 600 572
c) 976 103
In 318 526, the digit 3 is in the hundred thousands place and its value is 300 000. the digit 1 is in the place and its value is the digit 2 is in the tens place and its value is 20.
318 526 = 300 000 + + 8000 + 500 + 20 + 6
Let's Do Let's Do
1. Write the missing numbers or words.
a) In 750 168, the digit 7 is in the place and its value is .
b) In 98 132, the digit is in the thousands place.
c) 654 927 = 600 000 + + 4000 + 900 + 20 + 7
Let's Practice
1. Write the missing numbers or words.
a) In 601 325, the digit 0 is in the place.
b) In 713 950, the digit is in the hundred thousands place and its value is
c) 90 310 = 90 000 + + 10
Let's Learn Let's Learn
Compare 730 650, 720 650 and 72 650.
First, compare the hundred thousands. There are no hundred thousands in 72 650. 72 650 is the least number.
Then, compare the ten thousands in 730 650 and 720 650.
3 ten thousands is greater than 2 ten thousands.
730 650 > 720 650
730 650 is the greatest number.
Arrange the numbers in order. Begin with the least. , , (least)
Let's Do Let's Do
1. Write > or <. a) 340 238 341 002 b) 970 331 970 329
Let's Practice
1. Compare 537 041, 54 359 and 536 302.
a) Which is the greatest number?
b) Which is the least number?
2. Arrange the numbers in order. Begin with the greatest. 680 386, 608 863, 668 308 , , I have learned to... read and write numbers within 1 000 000 identify the values of digits in 5-digit and 6-digit numbers compare and order numbers within 1 000 000
You will learn to...
• round a whole number to the nearest ten, hundred or thousand
• estimate sums and differences
Let's Learn Let's Learn
a)
In 7343, the digit in the ones place is 3. 7343 is nearer to 7340 than to 7350. So, we round down.
7343 is 7340 when rounded to the nearest ten.
7343 ≈ 7340
b)
48 700
776
750
800
In 48 776, the digit in the tens place is 7. 48 776 is nearer to 48 800 than to 48 700. So, we round up.
48 776 is 48 800 when rounded to the nearest hundred.
48 776 ≈ 48 800
c)
In 121 500, the digit in the hundreds place is 5. So, we round up.
121 500 is when rounded to the nearest thousand.
121 500 ≈
To round a number to a place value, look at the digit on the right of the place value. If the digit is 5 or greater, round up. If the digit is less than 5, round down.
1. Round each number.
2558 is when rounded to the nearest ten.
c) 94 445 is when rounded to the nearest hundred.
531 634 is when rounded to the nearest thousand.
Let's Practice
1. Round each number to the nearest ten.
a) 4138 ≈ b) 1652 ≈
c) 2237 ≈ d) 7695 ≈
2. Round each amount to the nearest hundred dollars.
a) $4303 ≈ b) $35 043 ≈
c) $27 533 ≈ d) $11 753 ≈
e) $135 942 ≈ f) $478 284 ≈
3. Round each amount to the nearest thousand dollars.
a) $7455 ≈ b) $46 013 ≈
c) $30 628 ≈ d) $68 140 ≈
e) $144 550 ≈ f) $248 742 ≈
Let's Learn
a) Find the sum of 4593 and 23 862.
4593 + 23 862 = 28 455
We can use estimation to check if the answer is reasonable.
4593 + 23 862 ≈ 5000 + 24 000 = 29 000
28 455 is about 29 000. The answer is reasonable.
Method
4593 + 23 862 ≈ 4000 + 20 000 = 24 000
28 455 is about 24 000. The answer is reasonable.
b) Find the difference between 5812 and 32 248.
32 248 – 5812 = 26 436
We can use estimation to check if the answer is reasonable.
Method
32 248 – 5812 ≈ 32 000 – 6000 = Is the answer 26 436 reasonable?
Method
32 248 – 5812 ≈ 30 000 – 5000 = Is the answer 26 436 reasonable?
1. Find the sum or difference. Then, use estimation to check your answer.
a) 365 + 709 = 365 + 709 ≈ + = Is your answer reasonable?
b) 6740 – 1826 = 6740 – 1826 ≈ – = Is your answer reasonable?
1. Find the sum or difference. Then, use estimation to check your answer.
a)
+ 316 =
+ 316 ≈
b) 9383 + 21 475 = 9383 + 21 475 ≈ c) 7239 – 447 = 7239 – 447 ≈
d) 45 290 – 23 811 = 45 290 – 23 811 ≈
Sarah and David get different estimated sums of 341 and 2138.
Who is correct? Why do you say so? Who is wrong? Why do you say so?
Think of a time in your daily life when estimation is useful.
I have learned to... round a whole number to the nearest ten, hundred or thousand estimate sums and differences
You will learn to...
• tell if the sum or difference of odd and/or even numbers is odd or even • tell if the product of odd and/or even numbers is odd or even
Let's Learn
a) We can tell if a whole number is odd or even by pairing objects. An even number of objects can be grouped in pairs. An odd number of objects has 1 object left over after the other objects are grouped in pairs.
Do we get an odd or even number when we add two odd numbers?
Can the total number of objects be grouped in pairs without any left over?
even number + even number = even
odd number + odd number = even number
odd number + even number = odd number even number + odd number = odd number
8 – 2 = 6 7 – 4 = 3 8 – 3 = 5
number number number
even number – even number = even number odd number – odd number = even number odd number – even number = odd number even number – odd number = odd number
Let's Do Let's Do
1. Add. Then complete the rule.
Do we get an odd or even number when we subtract an odd number from an odd number? 7 – 3
Can the remaining objects be grouped in pairs without any left over?
a) 8 + 6 = b) 24 + 24 = c) 32 + 40 =
even number + even number = number
2. Add. Then complete the rule.
a) 7 + 7 = b) 35 + 13 = c) 29 + 51 =
odd number + odd number = number
3. Add. Then complete the rule.
a) 8 + 9 = b) 39 + 58 = c) 56 + 27 =
odd number + even number = number even number + odd number = number
4. Subtract. Then complete the rule. a) 12 – 4 = b) 56 – 20 = c) 64 – 36 =
even number – even number = number
5. Subtract. Then complete the rule. a) 29 – 7 = b) 57 – 23 =
odd number – odd number = number
6. Subtract. Then complete the rule.
19 – 6 = b) 45 – 30 =
odd number – even number = number
75 – 39 =
51 – 28 =
7. Subtract. Then complete the rule. a) 28 – 7 = b) 46 – 13 = c) 60 – 21 = even number – odd number = number
Let's Practice
1. Are the sums odd or even? Write odd or even. a) 25 + 19 b) 63 + 48 c) 58 + 204 d) 136 + 67 e) 38 + 111
587 + 292
2. Are the differences odd or even? Write odd or even
25 – 13
× 4 = 8 3 × 4 = 12 4 × 3 = 12
Do we get an odd or even number when we multiply two odd numbers?
Can the total number of objects be rearranged and put in pairs without any left over? 5 × 3 = 5 groups of 3
even number × even number = even number
odd number × odd number = odd number
odd number × even number = even number even number × odd number = even number
1. Multiply. Then complete the rule. a) 6 × 2 = b) 4 × 4 = c) 8 × 6 = d) 4 × 8 = e) 4 × 10 = f) 2 × 12 = even number × even number = number
2. Multiply. Then complete the rule.
a) 7 × 1 =
9 × 3 = c) 3 × 5 =
7 × 7 = e) 5 × 9 =
9 × 11 = odd number × odd number = number
3. Multiply. Then complete the rule. a) 9 × 4 = b) 2 × 5 = c) 7 × 6 = d) 4 × 7 = e) 3 × 8 = f) 8 × 9 = odd number × even number = number even number × odd number = number
Let's Practice
1. Are the products odd or even? Write odd or even.
I have learned to... tell if the sum or difference of odd and/or even numbers is odd or even tell if the product of odd and/or even numbers is odd or even
You will learn to...
• list all factors of a whole number up to 100
• find out if a 1-digit number is a factor of a given whole number
Let's Learn
a)
1 × 10 = 10
10 is a product of 1 and 10. 1 and 10 are factors of 10.
2 × 5 = 10
2 and 5 are also factors of 10.
factor × factor = product
A number can have more than two factors. So, 1, 2, 5 and 10 are factors of 10.
b) List all the factors of 60.
1 × 60 = 60
2 × 30 = 60
3 × 20 = 60
4 × 15 = 60
5 × 12 = 60
6 × 10 = 60
factors
10 × 6 is the same as 6 × 10.
The factors of 60 are .
1. Write the factors of 14.
The factors of 14 are .
Let's Practice
1. Write the missing factors.
a) × 9 = 27
b) 10 × = 90
c) 7 × = 49 d) 3 × = 24
2. Find the factors of 32.
32 = 1 × 32 32 = 2 ×
The factors of 32 are
= 4 ×
>> Look at EXPLORE on page 2 again. Can you solve the problem now? What else do you need to know?
Let's Learn
a) Is 5 a factor of 30? 30 ÷ 5 = 6
30 can be divided by 5 exactly. So, 5 is a factor of 30.
b) Is 5 a factor of 53?
53 ÷ 5 = 10 R3
53 be divided by 5 exactly. So, 5 a factor of 53.
Let's Do Let's Do
1. Write Yes or No. Show your work clearly.
a) Is 4 a factor of 28? b) Is 5 a factor of 43?
2. Which of the following numbers have 2 as a factor?
10, 13, 16, 20
Let's Practice Let's Practice
1. Write Yes or No. Show your work clearly.
a) Is 4 a factor of 46?
b) Is 2 a factor of 50?
c) Is 6 a factor of 62? d) Is 3 a factor of 78?
2. Which of the following numbers have 3 as a factor?
8, 11, 14, 18, 21, 30
I have learned to... list all factors of a whole number up to 100 find out if a 1-digit number is a factor of a given whole number
B Chapter 1: Exercise 4.2
You will learn to...
• list the multiples of a whole number up to 10
• relate factors and multiples
• find out if a whole number is a multiple of a given whole number up to 10
• identify multiples of 2, 5, 10, 25, 50 and 100 up to 1000
• apply rules of divisibility by 2, 3, 4, 5, 10, 25 and 100
Let's Learn Let's Learn
The first four multiples of 4 are 4, 8, 12 and 16.
The multiples of 4 have 4 as a factor.
Let's Do Let's Do
1. Write the first four multiples of 8.
The next two multiples of 4 are and . The first four multiples of 8 are .
Let's Practice Let's
1. Write the next five multiples.
a) 2, 4, , , , , b) 5, 10, , , , ,
c) 7, 14, , , , ,
d) 9, 18, , , , ,
a) 18 ÷ 2 = 9
18 can be divided by 2 exactly.
18 is a multiple of 2.
2 is a factor of 18.
Multiples of 2 can be divided by 2 exactly. Multiples of 2 have 2 as a factor.
b) 45 ÷ 2 = 22 R1
45 cannot be divided by 2 exactly.
45 a multiple of 2. 2 a factor of 45.
Let's Do Let's Do
1. Write Yes or No. Show your work clearly.
a) Is 54 a multiple of 9?
b) Is 39 a multiple of 9? Is 9 a factor of 54? Is 9 a factor of 39?
Let's Practice Let's Practice
1. Write Yes or No.
a) Is 64 a multiple of 3?
b) Is 80 a multiple of 2? Is 64 a multiple of 4? Is 80 a multiple of 6? Is 64 a multiple of 5? Is 80 a multiple of 10?
2. Write factor or multiple 8 × 3 = 24
a) 3 is a of 24. b) 8 is a of 24.
c) 24 is a of 3. d) 24 is a of 8.
Let's Learn
Look for patter ns in the multiples of 2, 5, 10, 25, 50 and 10.
Multiply by 2
1 × 2 = 2
2 × 2 = 4
3 × 2 = 6
4 × 2 = 8
5 × 2 = 10
6 × 2 = 12
7 × 2 = 14
8 × 2 = 16
9 × 2 = 18
10 × 2 = 20
Multiply by 25
1 × 25 = 25
2 × 25 = 50
3 × 25 = 75
4 × 25 = 100
5 × 25 = 125
6 × 25 = 150
7 × 25 = 175
8 × 25 = 200
9 × 25 = 225
10 × 25 = 250
Multiply by 5
1 × 5 = 5
2 × 5 = 10
3 × 5 = 15
4 × 5 = 20
5 × 5 = 25
6 × 5 = 30
7 × 5 = 35
8 × 5 = 40
9 × 5 = 45
10 × 5 = 50
Multiply by 50
1 × 50 = 50
2 × 50 = 100
3 × 50 = 150
4 × 50 = 200
5 × 50 = 250
6 × 50 = 300
7 × 50 = 350
8 × 50 = 400
9 × 50 = 450
10 × 50 = 500
Multiply by 10
1 × 10 = 10
2 × 10 = 20
3 × 10 = 30
4 × 10 = 40
5 × 10 = 50 6 × 10 = 60
7 × 10 = 70
8 × 10 = 80
9 × 10 = 90
10 × 10 = 100
Multiply by 100
1 × 100 = 100
2 × 100 = 200
3 × 100 = 300
4 × 100 = 400
5 × 100 = 500
6 × 100 = 600
7 × 100 = 700
8 × 100 = 800
9 × 100 = 900
10 × 100 = 1000
Multiples of 2 have the digit 0, 2, 4, 6 or 8 in the ones place.
Multiples of 5 have the digit 0 or 5 in the ones place.
Multiples of 10 have the digit 0 in the ones place.
Multiples of 25 end with ‘00’, ‘25’, ‘50’ or ‘75’.
Multiples of 50 end with ‘00’ or ‘50’.
Multiples of 100 end with ‘00’.
Let's Do Let's Do
1. Is 125 a multiple of the following numbers? Write Yes or No.
a) 2 b) 5
c) 10 d) 25
e) 50 f) 100
Let's Practice Let's Practice
1. Write Yes or No.
a) Is 52 a multiple of 2? b) Is 89 a multiple of 5?
c) Is 62 a multiple of 10?
d) Is 75 a multiple of 25?
e) Is 300 a multiple of 50? f) Is 111 a multiple of 100?
2. Circle the numbers that are multiples of 10.
3. Circle the numbers that are multiples of 5. Cross out the numbers that are multiples of 50.
32 87 19 6
Let's Learn Let's
We can use divisibility rules to quickly tell if a large number can be divided by another number exactly.
A number is divisible by Rule Example
2If the last digit is even (0, 2, 4, 6 or 8). Is 176 divisible by 2?
Check: 176 6 is even. So, 176 is divisible by 2.
A number is divisible by Rule Example
3If the sum of its digits is divisible by 3.
4If the number for med by its last two digits is divisible by 4.
Is 543 divisible by 3?
Check: 5 + 4 + 3 = 12 12 is divisible by 3.
So, 543 is divisible by 3.
Is 324 divisible by 4?
Check: 324
24 is divisible by 4.
So, 324 is divisible by 4.
5If the last digit is 0 or 5.
Is 495 divisible by 5?
Check: 495
The last digit is 5.
So, 495 is divisible by 5.
10If the last digit is 0. Is 540 divisible by 10?
Check: 540
The last digit is 0.
So, 540 is divisible by 10.
25If its last two digits are ‘00’, ‘25’, ‘50’ or ‘75’.
100If its last two digits are ‘00’.
Is 575 divisible by 25?
Check: 575
The last two digits are ‘75’.
So, 575 is divisible by 25.
Is 6700 divisible by 100?
Check: 6700
The last two digits are ‘00’.
So, 6700 is divisible by 100.
1. Complete the table. Write Yes or No.
by …?
1. Circle the numbers that are divisible by 2.
2. Circle the numbers that are divisible by 3.
3. Circle the numbers that are divisible by 4.
4. Circle the numbers that are divisible by 25.
5. Circle the numbers that are divisible by 10. Cross out the numbers that are divisible by 100.
I have learned to... list the multiples of a whole number up to 10 relate factors and multiples find out if a whole number is a multiple of a given whole number up to 10 identify multiples of 2, 5, 10, 25, 50 and 100 up to 1000 apply rules of divisibility by 2, 3, 4, 5, 10, 25 and 100
You will learn to...
• identify prime and composite numbers
• identify square numbers
Let's Learn
a)
1 × 7 = 7
The factors of 7 are 1 and 7.
Numbers that have only two factors, 1 and the number itself, are called prime numbers.
So, 7 is a prime number.
b) 1 × 6 = 6 2 × 3 = 6
The factors of 6 are 1, 2, 3 and 6.
Numbers that have more than two factors are called composite numbers.
So, 6 is a composite number
The number 1 has only one factor. 1 × 1 = 1
It is neither a prime number nor a composite number.
21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960
61626364656667686970 71727374757677787980
81828384858687888990 919293949596979899100
Follow these steps to find all the prime numbers within 100.
Step 1: Cross out 1 in the number chart as it is neither a prime nor a composite number.
Step 2: 2 is the first prime number. Circle it and cross out all multiples of 2 that are greater than 2.
Step 3: 3 is the next prime number. Circle it and cross out all multiples of 3 that are greater than 3.
Step 4: 5 is the next prime number. Circle it and cross out all multiples of 5 that are greater than 5.
Step 5: Continue this process until all the numbers are circled or crossed out.
All the numbers greater than 1 that are crossed out are the composite numbers between 1 and 100. 12345678910 11121314151617181920
A multiple of 2 has 2 as a factor. This means that aside from the number 2, any multiple of 2 has other factors besides 1 and itself. So, it cannot be a prime number.
All the numbers that are circled are the prime numbers between 1 and 100. Is 39 a prime number?
Is 59 a prime number?
Is 96 a prime number?
1. Write the factors of each of the following numbers. Then, write Yes or No.
a) Factors of 11: b) Factors of 25: Is 11 a prime number? Is 25 a prime number?
2. Circle the prime numbers. Cross out the composite numbers. 7 18 23 35 41
3. Write in the blanks. Use the number chart on page 27 to help you.
a) What are the prime numbers from 1 to 10?
b) What are the composite numbers from 11 to 20?
c) What is the least prime number?
d) What is the first odd composite number?
1. Write all the prime numbers from 1 to 20.
2. Write all the composite numbers from 31 to 50.
3. Complete the table with the following numbers.
4. Cross out the prime numbers.
5. Circle the numbers that are both composite and odd.
Sarah relates prime numbers to odd numbers.
All prime numbers are odd numbers.
Is Sarah correct? Why do you say so?
What did you learn about prime numbers?
Think of a time in your daily life when knowing prime numbers can be useful.
Let's Learn 1 × 1 = 1 2 × 2 = 4 3 × 3 = 9 4 × 4 = 16 12 = 1 22 = 4 32 = 9 42 = 16
We can arrange 1, 4, 9 and 16 counters into squares.
When we multiply a number by itself, we get the square of the number. 3 × 3 = 9 32 = 9
The square of 3 is 9. We read 32 as 3 squared. It means 3 × 3.
The square of a whole number is called a square number.
The numbers 1, 4, 9 and 16 are square numbers.
Let's Do Let's Do
1. Write the square number for each diagram.
2. Complete the multiplication sentences.
a) 92 = × b) 212 = ×
3. Circle the square numbers.
9 46 25 100 63 58 81
1. Color the small squares to show a square number that is greater than 20. What square number have you shown?
2. Complete the multiplication sentences.
a) 82 = × b) 142 = × c) 252 = × d) 312 = ×
3. Write a square number that is a) odd. b) even.
I have learned to... identify prime and composite numbers identify square numbers
You will learn to...
Anne writes a mystery number in her notebook and asks Marcus to find out what the number is. She gives him these clues:
• The number is even.
• The number is between 200 and 500.
• The number is a multiple of 25 and has 7 as a factor. What is the mystery number?
1
Understand the problem.
• solve a non-routine problem involving factors and multiples Plan what to do.
2 Work out the Answer. 3
Is the number odd or even?
How many digits does the number have?
What can the number be divided by?
What do I have to find?
I can use logical reasoning to solve the problem.
The mystery number is a multiple of 25 so it ends with ‘00’, ‘25’, ‘50’ or ‘75’.
The mystery number is between 200 and 500 so it is one of the following:
225, 250, 275, 300, 325, 350, 375, 400, 425, 450, 475
The mystery number is an even number. Cross out the odd numbers.
225, 250, 275, 300, 325, 350, 375, 400, 425, 450, 475
5
Check if your answer is correct.
The mystery number has 7 as a factor. So, divide the remaining numbers by 7 to find the mystery number.
250, 300, 400 and 450 cannot be divided by 7 exactly. Only 350 can be divided by 7 exactly.
So, the mystery number is 350.
4 I have learned to... solve a non-routine problem involving factors and multiples 1. Understand 2. Plan 3. Answer 4. Check 5. Plus
+ Plus Solve the problem in another way.
350 is an even number. It is between 200 and 500. It is a multiple of 25 and has 7 as a factor.
My answer is correct.
The mystery number can be divided by 25 and 7 exactly. 25 × 7 = 175
The mystery number is a multiple of 175.
Find the multiples of 175.
1 × 175 = 175
2 × 175 = 350
3 × 175 = 525
Only 350 is between 200 and 500. 350 is an even number.
So, the mystery number is 350.
Compare the methods in steps 3 and 5. Which method do you prefer? Why?
>> Look at EXPLORE on page 2 again. Fill in column 3. Can you solve the problem now?
• angles at a point
The sum of angle measures at a point is 360°.
/x, /y and /z are angles at point O.
They are angles at a point. m/x + m/y + m/z = 360°
• angles on a straight line
The sum of angle measures on a straight line is 180°.
• bar line chart
A bar line chart uses thin bars that look like lines to show comparisons between categories of data. 0 readingcraftsgamessports Number of students Hobbies Favorite hobbies of students 2 4 6 8
• Cartesian plane
The Cartesian plane is a grid with a system of coordinates for locating or plotting a point on the plane. A Cartesian plane is also called a coordinate plane.
AOB is a straight line.
/p, /q and /r are angles on a straight line. m/p + m/q + m/r = 180°
• certain
A certain event is an event that will definitely happen.
• composite number
A composite number is a number that has more than two factors.
1 × 8 = 8
2 × 4 = 8
The factors of 8 are 1, 2, 4 and 8. So, 8 is a composite number.
• coordinate plane See Cartesian plane.
• coordinates
Coordinates are a set of values that shows an exact position on a Cartesian plane. When writing the coordinates of a point, we write the x-coordinate followed by the y-coordinate in the form (x, y). We write coordinates in brackets and separate the two numbers with a comma. (2, 3) and (5, 0) are coordinates.
• equilateral triangle
An equilateral triangle is a triangle with three equal sides and three equal angles.
A B C
AB = BC = AC
m/BAC = m/ACB = m/ABC = 60° ABC is an equilateral triangle.
• factor
Factors are whole numbers that are multiplied to get another number.
3 × 6 = 18
3 and 6 are factors of 18.
A factor is a whole number that divides another whole number exactly, without any remainder.
18 ÷ 6 = 3
6 is a factor of 18. H
• hundred thousand
1 hundred thousand = 100 thousands = 100 000
• impossible
An impossible event is an event that definitely will not happen.
• integers
An integer is one of the numbers ..., –3, –2, –1, 0, 1, 2, 3, ... –5, 0 and 2 are integers. 3 4 and 0.6 are not integers.
• isosceles triangle
An isosceles triangle is a triangle with two equal sides and two equal angles.
B C
AB = BC m/BAC = m/ACB ABC is an isosceles triangle.
• likely
A likely event is an event that has a high chance of happening.
• line graph
A line graph is a graph used to present data that changes over time.
• negative integers
Integers less than 0 are called negative integers ..., –5, –4, –3, –2, –1 are negative integers.
• negative numbers
Numbers less than 0 are called negative numbers –20, –8.5, –5 and –1.2 are negative numbers.
• numerical expression
A numerical expression contains only numbers and operation signs. 6 – 2 and 4 ÷ 5 are numerical expressions.
• order of rotational symmetry
• million
1 million = 1000 thousands = 1 000 000
• mirror line
A mirror line is a line of reflection. It is also a line of symmetry.
The number of times a shape fits into its original position during a complete tur n about its center is called the order of rotational symmetry.
mirror line
• multiple
The multiple of a number is the product of the number and any other whole number except zero. 1 × 8 = 8 2 × 8 = 16 3 × 8 = 24 8, 16 and 24 are multiples of 8.
This shape fits into its original position 3 times during the complete turn. The order of rotational symmetry of this shape is 3.
• ordered pair
An ordered pair, also known as coordinates, is used to describe the location of a point on a Cartesian plane. An ordered pair is made up of the x-coordinate and the y-coordinate of that point. (4, 2) is an ordered pair on the Cartesian plane. It means 4 units to the right and 2 units up from the origin.
• origin
The point where the x-axis and the y-axis cross at a right angle is called the origin. The coordinates of the origin is (0, 0).
• parallel (//)
These two line segments are parallel. They are always the same distance apart and will never meet.
• perimeter
The perimeter of a figure is the length around the figure.
B D C
Perimeter of ABCD = AB + BC + CD + DA
• perpendicular (⊥)
These two line segments are perpendicular. They cross at a right angle.
• positive integers
Integers greater than 0 are called positive integers 1, 2, 3, 4, 5, ... are positive integers.
• possible
A possible event is an event that may happen.
• prime number
A prime number is a number that has only two factors, 1 and the number itself.
The factors of 3 are 1 and 3.
So, 3 is a prime number. Q
• quadrants (Cartesian plane)
The x-axis and the y-axis divide the Cartesian plane into four quadrants.
quadrant
quadrant
quadrant
quadrant
Point A is in the 1st quadrant. Point B is in the 2nd quadrant. Point C is in the 3rd quadrant. Point D is in the 4th quadrant.
• reflection
A reflection is a flip along the mirror line. In a reflection, the size and shape do not change but the orientation and position change.
• reflective symmetry
When a shape has two matching halves folded about a line of symmetry, we say that the shape is symmetric or has reflective symmetry.
• right triangle
A right triangle is a triangle with a right angle (90°).
• scalene triangle
A scalene triangle is a triangle with no equal sides. K 9 cm L 7 cm J 6 cm
LJK is a scalene triangle.
• square number
When a number is multiplied by itself, the product is a square number. 5 × 5 = 25
5 multiplied by itself is 25. So, 25 is a square number.
T• ten thousand
1 ten thousand = 10 thousands = 10 000
A
B C
/BAC is a right angle. ABC is a right triangle.
• rotational symmetry
When a shape fits into its original position during a turn less than a complete turn, it has rotational symmetry
• thousandth
1 thousandth is 1 out of 1000 equal parts.
1 1000 = 0.001
• transformation
A transformation happens when a shape undergoes a change. The shape can change its position, size, shape or orientation after a transformation.
Translations and reflections are types of transformations.
• translation
7 units
• vertically opposite angles
Vertically opposite angles are the angles opposite to each other when two lines cross. Vertically opposite angles have equal measures.
A translation is a movement along a straight line. This shape shows a translation. The triangle has moved 7 units to the right. In a translation, the size, shape and orientation of the shape do not change but its position does.
• uncertain
An uncertain event is an event that may or may not happen.
• unlike fractions
Unlike fractions are fractions that do not have the same denominator. 2 3 and 2 5 are unlike fractions because they have different denominators, 3 and 5.
• unlikely
An unlikely event is an event that has a low chance of happening.
/a and /c are a pair of vertically opposite angles.
m/a = m/c
/b and /d are also a pair of vertically opposite angles.
m/b = m/d
• x-axis
The horizontal axis on the Cartesian plane is called the x-axis.
• x-coordinate
The x-coordinate measures the distance of a point from the origin along the x-axis.
The x-coordinate of (3, 5) is 3.
• y-axis
The vertical axis on the Cartesian plane is called the y-axis.
• y-coordinate
The y-coordinate measures the distance of a point from the origin along the y-axis.
The y-coordinate of (3, 5) is 5.
A world-class program incorporating the highly effective
Readiness-Engagement-Mastery model of instructional design
Name
PR1ME Mathematics Digital Practice and Assessment provides individualized learning support and diagnostic performance reports
Scholastic TM Mathematics (New Edition) covers five strands of mathematics across six grades: Numbers and Operations, Measurement, Geometry, Data Analysis, and Algebra.
Each Practice Book comprises chapters with several Exercises. Chapters end with Problem Solving exercises. A Review follows after every four or five chapters.
Exercises provide comprehensive practice for students to attain fluency and mastery of topics.
Recap helps students to recall what was taught in the coursebook and assist them with the exercise.
Tasks in each exercise are systematically varied to provide comprehensive practice and formative assessment.
Reviews provide summative assessment and enable consolidation of concepts and skills learnt across various topics.
Review 3
Chapter 1 Whole Numbers
Exercise 1.1 Reading and writing numbers .........................
Exercise 1.2 Identifying values of digits ..............................
Exercise 1.3 Comparing and ordering numbers ................... 11
Exercise 2.1 Rounding numbers to the nearest ten, hundred or thousand ................................................
Exercise 2.2 Estimating sums and differences ......................
Exercise 3.1 Adding and subtracting odd and even numbers .............................................
Exercise 3.2 Multiplying odd and even numbers
Exercise 4.1 Finding factors of a whole number ...................
Exercise 4.2 Finding out if a number is a factor of another number ..........................................
Exercise 5.1 Finding multiples of a whole number ................
Exercise 5.2 Relating factors and multiples .........................
Exercise 5.3 Identifying multiples of 2, 5, 10, 25, 50 and 100 ....
Exercise 5.4 Using divisibility rules ......................................
Exercise 6.1 Identifying prime and composite numbers
Exercise 6.2 Identifying square numbers ............................
Chapter 2 Multiplication and Division of Whole Numbers
Exercise 1.1 Multiplying by 10, 100 or 1000 ..........................
Exercise 1.2 Multiplying by tens, hundreds or thousands ........
Chapter 3 Fractions
Exercise
Exercise
Exercise
Exercise
Chapter 4 Angles
Exercise
Chapter 5 Perpendicular and Parallel Line Segments
Exercise 1.1 Identifying perpendicular line segments ............
Exercise
Exercise 1.3 Using a protractor and a ruler, or a set square to draw perpendicular line segments ...............
Exercise 2.1 Identifying parallel line segments
Exercise 2.2 Drawing parallel line segments on a square grid .................................................
Exercise 2.3 Using a set square and a ruler to draw parallel line segments ..............................................
Chapter 6 Triangles
Exercise 1.1 Sum of angle measures in a triangle ................. 76
Exercise 1.2 Types of triangles ......................................... 77
Exercise 1.3 Finding unknown angle measures in triangles ..... 78
Chapter 7 Symmetry
Exercise 1.1 Reflective symmetry ..................................... 79
Exercise 1.2 Making symmetric patter ns ............................
Exercise 1.3 Rotational symmetry .....................................
Chapter 8 Decimals
Exercise 1.1 Reading and writing decimals ........................ 82
Exercise 1.2 Interpreting decimals in terms of tens, ones, tenths, hundredths and thousandths ................ 83
Exercise 1.3 Identifying values of digits .............................. 84
Exercise 1.4 Expressing fractions and mixed numbers as decimals ................................................ 85
Exercise 1.5 Expressing decimals as fractions or mixed numbers in their simplest for m ......................... 86
Exercise 1.6 Reading number lines ................................... 87
Exercise 1.7 Comparing and ordering numbers ................... 88
Exercise 1.8 Finding ‘more than’ and ‘less than’ .................. 89
Exercise 1.9 Completing number patter ns ......................... 90
Exercise 1.10 Rounding decimals to 2 decimal places ........... 91
2 .................................................................. 92
Exercise 1.1 Adding tenths or hundredths without regrouping ................................................. 98
Exercise 1.2 Adding tenths or hundredths with regrouping ..... 99
Exercise 1.3 Adding decimals with 1 decimal place with regrouping ................................................. 100
Exercise 1.4 Adding decimals with 2 decimal places with regrouping ................................................. 101
Exercise 1.5 Estimating sums ........................................... 102
Exercise 1.6 Solving word problems .................................. 103
Exercise 2.1 Subtracting tenths from decimals less than 1 or from whole numbers ..................................... 105
Exercise 2.2 Subtracting tenths from decimals greater than 1 ....................................................... 106
Exercise 2.3 Subtracting hundredths from decimals less than 1 or from whole numbers ................... 107
Exercise 2.4 Subtracting decimals with 2 decimal places from whole numbers ..................................... 108
Exercise 2.5 Subtracting hundredths from decimals greater than 1 ............................................. 109
Exercise 2.6 Subtracting decimals with 1 decimal place ....... 110
Exercise 2.7 Subtracting decimals up to 2 decimal places ..... 111
Exercise 2.8 Adding and subtracting decimals close to whole numbers ............................................ 112
Exercise 2.9 Estimating differences ................................... 113
Exercise 2.10 Solving word problems .................................. 114
Exercise 3.1 Word problems ............................................ 116
10 Mental Strategies
Exercise 1.1 Making 1 .................................................... 119
Exercise 1.2 Making 10 .................................................. 120
Exercise 2.1 Adding using the jump strategy ....................... 121
Exercise 2.2 Subtracting using the jump strategy ................. 122
Exercise 3.1 Multiplying tens or hundreds by a 1-digit number ..................................................... 123
Exercise 3.2 Multiplying tens and hundreds ........................ 124
Exercise 3.3 Multiplying numbers near tens by 1-digit numbers .................................................... 125
Exercise 3.4 Multiplying by 25 .......................................... 126
Exercise 4.1 Finding doubles of numbers ............................ 127
Exercise 4.2 Finding halves of numbers .............................. 128
Chapter 11 Multiplication and Division of Decimals
Exercise 1.1 Multiplying tenths or hundredths without regrouping ................................................. 129
Exercise 1.2 Multiplying tenths or hundredths with regrouping ................................................. 130
Exercise 1.3 Multiplying decimals with 1 decimal place by 1-digit whole numbers ................................... 131
Exercise 1.4 Multiplying decimals with 2 decimal places by 1-digit whole numbers ................................... 132
Exercise 1.5 Estimating products ...................................... 133
Exercise 1.6 Solving word problems .................................. 134
Exercise 2.1 Dividing tenths or hundredths without regrouping ................................................. 136
Exercise 2.2 Dividing decimals and whole numbers by 1-digit whole numbers to get quotients in tenths .......... 137
Exercise 2.3 Dividing decimals by 1-digit whole numbers to get quotients in hundredths ............................ 138
Exercise 2.4 Dividing decimals with 2 decimal places by 1-digit whole numbers ................................... 139
Exercise 2.5 Dividing decimals and whole numbers with zero as a placeholder ................................... 140
Exercise 2.6 Estimating quotients ..................................... 141
Exercise 2.7 Rounding quotients ...................................... 142
Exercise 2.8 Solving word problems . ................................. 144
Exercise 3.1 Word problems ............................................ 146
Review 3 .................................................................. 149
Chapter 12 Integers
Exercise 1.1 Reading and placing integers on number lines ... 156
Exercise 1.2 Comparing integers on number lines ................ 157
Exercise 1.3 Completing number patter ns ......................... 158
Chapter 13 Probability
Exercise 1.1 Certain or uncertain, possible or impossible ........ 159
Exercise 1.2 Likely or unlikely ........................................... 161
Chapter 14 Handling Data
Exercise 1.1 Collecting and presenting data ...................... 163
Exercise 2.1 Making bar line charts ................................... 164
Exercise 2.2 Reading and interpreting bar line charts ........... 166
Exercise 3.1 Completing, reading and interpreting broken line graphs ................................................. 169
Exercise 3.2 Reading and interpreting straight line graphs ..... 172
Chapter 15 Position and Movement
Exercise 1.1 Reading and plotting points in the first quadrant ................................................... 175
Exercise 2.1 Translations ................................................. 177
Exercise 2.2 Reflections ................................................. 179
Chapter 16 Area and Perimeter
Exercise 1.1 Measuring perimeter .................................... 181
Exercise 1.2 Comparing areas and perimeters .................... 183
Exercise 1.3 Finding the perimeter of a figure ...................... 185
Exercise 1.4 Finding perimeters of squares and rectangles ..... 186
Exercise 2.1 Estimating area ........................................... 188
Exercise 2.2 Finding areas of rectangles and squares ........... 189
Exercise 3.1 Finding an unknown side of a rectangle or square given its perimeter .............................. 191
Exercise 3.2 Finding an unknown side of a rectangle or square given its area .................................... 193
Exercise 4.1 Word problems ............................................ 195
Chapter 17 Time
Exercise 1.1 Duration of time in months or years .................. 198
Exercise 1.2 Duration of time in years and months ............... 199
Recap
463 653 is read as four hundred and sixty-three thousand, six hundred and fifty-three.
Five hundred and twenty thousand, seven hundred and thirty is written as the numeral 520 730.
1. Write the numerals.
a) twenty-three thousand, seven hundred and four
b) five hundred and forty-three thousand, three hundred and nineteen
c) nine hundred and twenty-five thousand, six hundred and two
d) seven hundred thousand and eighty-three
2. Write the numbers in words.
a) 50 901
b) 659 547
c) 200 390
d) 908 004
In 756 321, the digit 7 is in the hundred thousands place and its value is 700 000.
756 321 = 700 000 + 50 000 + 6000 + 300 + 20 + 1
1. Write the missing numbers or words.
a) In 423 546, the digit 2 is in the place.
b) In 634 543, the digit 6 is in the place and its value is .
c) In 547 893, the digit is in the thousands place and its value is .
d) In 906 428, the digit is in the ten thousands place and its value is .
2. Write the missing numbers.
a) 834 765 = 800 000 + + 4000 + 700 + 60 + 5
b) 327 504 = 300 000 + 20 000 + 7000 + + 4
c) 104 023 = 100 000 + 4000 + + 3
d) 809 702 = 800 000 + + 700 + 2
e) 200 398 = + 300 + 90 + 8
f) 656 565 = 600 000 + + 6000 + 500 + 60 + 5
Recap
Compare 530 602 and 530 623.
The digits in the hundred thousands, ten thousands, thousands and hundreds places are the same.
Compare the digits in the tens place. 0 tens is less than 2 tens.
530 602 < 530 623
1. Write > or <.
a)
2. Arrange the numbers in order. Begin with the least.
a) 304 580, 30 458, 304 508
b) 123 456, 132 456, 123 406, 132 056
3. Arrange the numbers in order. Begin with the greatest.
a) 94 797, 944 700, 904 779
b) 222 200, 332 220, 92 222, 332 000
Recap
To round a number to a place value, look at the digit on the right of the place value. If the digit is 5 or greater, round up. If the digit is less than 5, round down.
25 535
25 500
25 550
25 535 is nearer to 25 550 than to 25 600.
25 600
25 535 is 25 500 when rounded to the nearest hundred.
25 535 ≈ 25 500
1. Write in the blanks.
a) 3542 is when rounded to the nearest ten.
b) 67 218 is when rounded to the nearest ten.
c) 850 565 is when rounded to the nearest hundred.
d) 215 094 is when rounded to the nearest thousand.
2. Round each amount to the nearest hundred dollars.
a) $8762
c) $56 506
e) $974 435
b) $9542
d) $37 091
f) $349 957
3. Round each amount to the nearest thousand dollars.
a) $9459
c) $73 231
e) $532 652
b) $3740
d) $96 602
f) $399 559
Recap
Estimate the sum of 34 876 and 9591.
34 876 + 9591 = 44 467
Method 1:
34 876 + 9591 ≈ 35 000 + 10 000 = 45 000
44 467 is about 45 000.
The answer is reasonable.
Method 2:
34 876 + 9591 ≈ 30 000 + 9000 = 39 000
44 467 is about 39 000.
The answer is reasonable.
1. Find the sum or difference. Then, use estimation to check your answer.
a) 235 + 568 = 235 + 568 ≈ + = Is your answer reasonable?
b) 5882 – 3944 = 5882 – 3944 ≈ –= Is your answer reasonable?
2. Find the sum or difference. Then, use estimation to check your answer.
Exercise 3.1 Adding and subtracting odd and even numbers
Recap + 5 + 3
The sum of 5 and 3 can be grouped in pairs without any left over. The sum of 5 and 3 is an even number.
1. Add. Then, complete each rule.
a) 6 + 9 = even number + number = number
b) 43 + 36 = odd number + number = number
c) 142 + 324 = number + even number = number
d) 537 + 63 = number + odd number = number
2. Are the sums odd or even? Write odd or even.
a) 17 + 8 b) 7 + 25
c) 35 + 25 d) 90 + 82
e) 78 + 433 f) 931 + 42
g) 108 + 126 h) 294 + 129
i) 143 + 357 j) 474 + 103
k) 418 + 227 l) 531 + 179
3. Subtract. Then, complete each rule.
a) 17 – 4 = odd number – number = number
b) 68 – 53 = even number – number = number
c) 368 – 96 = number – even number = number
d) 423 – 215 = number – odd number = number
4. Are the differences odd or even? Write odd or even.
a) 40 – 8 b) 36 – 15
c) 76 – 38 d) 261 – 93
e) 393 – 48 f) 402 – 99
g) 254 – 146 h) 273 – 179
i) 347 – 252 j) 921 – 877
5. I am an odd number. I am the sum of two numbers and . is an odd number. Is an odd or even number?
6. The difference between two numbers is an even number. One of the numbers is an odd number. Is the other number an odd or even number?
7. The difference between number X and number Y is an odd number. If number X is an odd number, is the sum of number X and number Y an odd or even number?
Recap 2 × 3 = 6 even number odd number even number
1. Multiply. Then, complete the rules.
a) 6 × 5 = b) 5 × 8 = c) 10 × 7 = d) 5 × 12 = even number × odd number = number odd number × even number = number
2. Multiply. Then, complete each rule.
a) 3 × 7 = odd number × number = number
b) 4 × 12 = number × even number = number
3. Are the products odd or even? Write odd or even a) 33 × 2 b) 16 × 8 c) 245 × 5 d) 104 × 3
4. The product of two numbers is an even number. One of the numbers is an odd number. Is the other number an even or odd number?
5. Is the product of two odd numbers an odd number or an even number?
Recap
1. Write the missing factors.
3 × 6 = 18 3 and 6 are factors of 18.
factor × factor = product
a) 4 × = 32 b) × 6 = 30 c) 7 × = 42 d) × 9 = 81
2. a) Find the factors of 20.
The factors of 20 are .
b) Find the factors of 63.
The factors of 63 are .
c) Find the factors of 45.
The factors of 45 are .
Exercise 4.2 Finding out if a number is a factor of another number
Recap Is 3 a factor of 15?
15 ÷ 3 = 5 15 can be divided by 3 exactly. So, 3 is a factor of 15. 5
1. Answer the questions. Show your work clearly.
a) Is 7 a factor of 98?
b) Is 4 a factor of 77?
c) Is 8 a factor of 131?
d) Is 9 a factor of 243?
2. Which of the following numbers have 6 as a factor?
48, 27, 36, 13, 24, 51
Recap
The first eight multiples of 3 are 3, 6, 9, 12, 15, 18, 21 and 24.
1. Write the first five multiples of 4.
The first five multiples of 4 are , , , and .
2. Write the first ten multiples of each number.
Recap
24 ÷ 4 = 6
24 can be divided by 4 exactly.
24 is a multiple of 4.
4 is a factor of 24.
1. Answer the questions using Yes or No
a) Is 42 a multiple of 3?
Is 42 a multiple of 6?
Is 42 a multiple of 9?
b) Is 63 a multiple of 7?
Is 63 a multiple of 8?
Is 63 a multiple of 3?
2. Write factor or multiple. 5 × 7 = 35
a) 7 is a of 35.
b) 5 is a of 35.
c) 35 is a of 7.
d) 35 is a of 5.
3. Write factor or multiple 8 × 9 = 72
a) 9 is a of 72.
b) 8 is a of 72.
c) 72 is a of 9.
d) 72 is a of 8.
30 ÷ 4 = 7 R2
30 cannot be divided by 4 exactly.
30 is not a multiple of 4.
4 is not a factor of 30.
Recap
Multiples of 2 have the digit 0, 2, 4, 6 or 8 in the ones place.
Multiples of 5 have the digit 0 or 5 in the ones place.
Multiples of 10 have the digit 0 in the ones place.
Multiples of 25 end with ‘00’, ‘25’, ‘50’ or ‘75’.
Multiples of 50 end with ‘00’ or ‘50’.
Multiples of 100 end with ‘00’.
1. Answer the questions using Yes or No
a) Is 96 a multiple of 2?
b) Is 57 a multiple of 5?
c) Is 23 a multiple of 10?
d) Is 125 a multiple of 25?
e) Is 250 a multiple of 50?
f) Is 365 a multiple of 100?
2. Circle the numbers that are multiples of 5.
3. Circle the numbers that are multiples of 2. Cross out the numbers that are multiples of 10.
4. Circle the numbers that are multiples of 25. Cross out the numbers that are multiples of 50.
5. Circle the numbers that are multiples of 50. Cross out the numbers that are multiples of 100.
Recap
A number is divisible by:
• 2 if the last digit is even. Examples: 438, 76, 808.
• 3 if the sum of its digits is divisible by 3. Example: 87, 267, 804.
• 4 if the number formed by its last two digits is divisible by 4. Examples: 124, 560, 344.
• 5 if the last digit is 0 or 5. Examples: 35, 455, 960.
• 10 if the last digit is 0. Examples: 80, 640, 320.
• 25 if its last two digits are ‘00’, ‘25’, ‘50’ or ‘75’. Examples: 275, 950, 500.
• 100 if its last two digits are ‘00’. Examples: 600, 100, 400.
1. Write Yes or No.
2. Circle the numbers that are divisible by 2.
3. Circle the numbers that are divisible by 4.
4. Circle the numbers that are divisible by 5.
5. Circle the numbers that are divisible by 10.
6. Circle the numbers that are divisible by 25. Cross out the numbers that are divisible by 100.
Recap
Prime numbers are numbers that have only two factors, 1 and the number itself. Examples: 3 and 5
Composite numbers are numbers that have more than two factors. Examples: 6 and 10
1. List the first eight prime numbers.
2. List the first ten composite numbers.
3. Answer the questions using Yes or No.
a) Is 23 a prime number?
b) Is 37 a composite number?
c) Is 15 a composite number?
d) Is 67 a prime number?
e) Is 91 a prime number? f) Is 73 a composite number?
4. Circle the composite numbers. Cross out the prime numbers.
Recap
5 × 5 = 25
52 = 25
The square of 5 is 25.
1. Draw a diagram to show that 9 is a square number.
2. Draw a diagram to show that 16 is a square number.
3. Complete the multiplication sentences. a) 72 = ×
4. Circle the square numbers.
5. Write the square numbers that are between 21 and 85.