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.
Let's Remember
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.
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.
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.
2.
1.
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.
1.
2.
3.
4.
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6.
Summative
Mind Stretcher, Create Your Own, Mission Possible and Mathematical Modeling immerse students in problem solving tasks at various levels of difficulty.
Students
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 TM , 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 =
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.
Compare the methods in steps 3 and 5. Which method do you prefer? Why?
Look
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
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
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.
4.2 Mind stretcher Let's Learn
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.
CREATE YOUR OWN
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
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 and his father have a total mass of 90 kilograms. Brian’s father is 50 kilograms heavier than Brian. What is Brian’s mass? I can draw a
2
model to compare their masses. What is the total mass of Brian and his father? Who is heavier? How many kilograms heavier? What do I have to find? Understand the problem. 1 Plan what to do.
Brian’s mass is 20 kilograms.
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.
• Can you describe the problem in your own words?
• What information is given?
• What do you need to find?
• If your answer is not correct, go back to Step 1. 1 2 3 4 5
• 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.
Process-based tasks
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’
cards
• 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:
• 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 on the dot card to show 6 groups of 3.
Have
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
The careful spiral sequence of successively more complex ways of reasoning about mathematical concepts – the learning progressions within – 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
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
Let's
1.
Recap provides a pictorial and abstract representation of the concrete activity carried out
2.
1.
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.
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
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.
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.
2. Adding and subtracting within
using number
(CB1 Chapter 7) 3. Adding and subtracting within 20 using the ‘counting on’ or ‘counting backwards’ method (CB1 Chapter 7)
EXPLORE
Have
•What will happen if we do not protect such sites?
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.
Unit 1: Sum and Difference
1.1 Understanding the meanings of sum and difference
Let's Learn Let's 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 1 requires students to associate the terms ‘sum’ and ‘difference’ with addition and subtraction respectively.
Task 2 requires students to associate the
‘sum’ with addition.
Let's Practice
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.
Chapter 2: Addition and Subtraction Within 100 30
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 To add tens and a 2-digit number without regrouping using the ‘counting on’ method
2 To add tens and a 2-digit number without regrouping using number bonds
To add tens and a 2-digit number without regrouping using place value
Students are expected to add by counting on tens using the number lines provided.
Students are expected to split the 2-digit number into tens and ones. They are required to first add the tens, then the ones.
Students are expected to first add the ones, then the tens. They are required to find each sum and color the corresponding boxes at the bottom of the page. The colored path serves as a check for students’ answers. In the first three tasks, students are provided the addition in the vertical form. In the other tasks, students are required to write the addition in the
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.
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:
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?
1, 3 To solve 1-step word problems involving addition with regrouping
2 To solve 1-step word problems involving subtraction with regrouping
4 To solve 1-step word problems involving subtraction with regrouping
Students are expected to solve 1-step word problems involving addition with regrouping. They can draw a partwhole bar model to help them solve each word problem.
Students are expected to solve a 1-step subtraction word problem involving a comparison situation by finding the smaller quantity given the larger quantity and the difference. They can draw a comparison bar model to help them solve the word problem.
3. Vivian has 51 storybooks. She has 13 more storybooks than Kevin.
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 solve 1-step word problems involving addition with regrouping
To solve 1-step word problems involving subtraction with regrouping
Let's Practice
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 word problem involving subtraction with regrouping. They can draw a part-whole bar model to help them solve the word problem.
Students are expected to solve a 1-step addition word problem involving a comparison situation by finding the larger quantity given the smaller quantity and the difference. They can draw a comparison bar model to help them 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 final
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 63
MISSION POSSIBLE
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.
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:
Topics:
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
6To
7To
8To
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
Teach problem solving: Let’s Learn (UPAC+™) Let’s Do Practice Book Exercise
Digital Practice
Create Your Own
Mind stretcher
Mission Possible Mathematical Modeling
Digital Chapter Assessment
Math Journal
Practice Book Reviews
Digital Quarterly Assessment
Digital Half-Yearly Assessment
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.
5
Whole Numbers / Place Value
Read and write a number within 1 000 000—the numeral and the corresponding number word.
Identify the values of digits in a 5-digit or 6-digit number.
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.
6
Recognize the historical origins of our number system and begin to understand how it developed.
Read and write a number within 10 000 000—the numeral and the corresponding number word.
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.
Identify multiples of 2, 5, 10, 25, 50 and 100 up to 1000.
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.
*Extend spatial patterns formed from adding and subtracting a constant.
*Extend spatial patterns of square and triangular numbers.
*Identify cube numbers.
Find out if a number is a common multiple of two given numbers.
Solve word problems involving common factors and multiples.
*Express a number in exponential notation.
*Find the value of a number given in exponential notation.
Grade 5
Addition / Subtraction
Multiplication / Division
Estimate sums and differences.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.
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.
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.
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.
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.
Estimate products and quotients.Multiply a 2-digit number by a 1-digit number.
Multiplication / Division (continued)
Fractions / Concepts
Fractions / Arithmetic Operations
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.
Interpret numerical expressions without evaluation.
Solve multi-step word problems involving four operations of whole numbers.
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.
Multiply two 2-digit numbers.
Divide a 2-digit number by a 1-digit number.
*Solve challenging word problems involving whole numbers.
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.
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.
Fractions / Arithmetic Operations (continued)
Decimals
*Add and subtract mixed numbers.Describe and complete a number pattern involving positive and negative integers, fractions, and decimals by counting on and backwards.
*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.
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.
*Solve challenging word problems involving fractions.
Multiply a decimal by 10, 100 or 1000.
Multiply a decimal by tens, hundreds or thousands.
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.
Decimals (continued)
Integers
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.
Estimate sums, differences, products and quotients.
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.
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.
*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.
Compare and order positive and negative integers, fractions and decimals.
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.
Read and place positive and negative integers, fractions and decimals on number lines.
Read integers on number lines.Compare and order positive and negative integers, fractions and decimals.
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.
*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.
Integers (continued)
Rate
Ratio
Percentage
*Find the rate by expressing one quantity per unit of another quantity.
*Find a quantity using the given rate.
*Solve word problems involving rate.
Add and subtract integers.
Describe and complete a number pattern involving positive and negative integers, fractions, and decimals by counting on and backwards.
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.
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.
Percentage (continued)
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 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.
Perimeter / Area
Find the perimeter of a figure made up of 1-centimeter or 1-meter squares.
Measure the perimeter of a figure.
Compare areas and perimeters of figures made up of 1-centimeter or 1-meter 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.
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 length.
*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.
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.
Perimeter / Area (continued)
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 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
*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.
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.
Volume and Capacity (continued)
Mass / Weight
Time: Clock
Speed
*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.
*Convert a measurement of mass from a larger unit of measure involving a proper fraction or a mixed number to a smaller unit.
*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.
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 mass.
Calculate time intervals in months.Know the relationship between years, decades and centuries.
Calculate time intervals in years.Convert between years, decades and centuries.
Calculate time intervals in years and months. Calculate time in different time zones.
*Convert a measurement of time from a larger unit of measure involving a proper fraction or a mixed number to a smaller unit.
*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.
*Understand time intervals less than one second.
*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.
Speed (continued)
Temperature
Lines and Curves
2D Shapes
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.
Identify right triangles.
Find an unknown angle measure in a triangle.
Recognize reflective symmetry in regular polygons.
*Solve word problems involving speed.
*Solve challenging word problems involving speed.
Calculate a rise or fall in temperature.
Count the number of lines of symmetry in regular polygons.
Make a symmetric pattern with two lines of symmetry.
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.
*Determine if a given shape can tessellate.
Understand the properties of squares and rectangles.
Use properties of squares and rectangles to find unknown angle measures.
Use properties of squares and rectangles to find unknown lengths.
State the properties of a rectangle, a square, a parallelogram, a rhombus and a trapezoid.
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).
*Know the relationship between the radius and diameter of a circle.
*Draw a tessellation on dot paper.*Find the diameter of a circle given its radius, and vice versa.
2D Shapes (continued)
3D Shapes
Angles
Position and Movement
*Make different tessellations with a unit shape.
*Make a tessellation with two unit shapes.
*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°.
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.
Recognize that the sum of angle measures in a triangle is 180º.
Find an unknown angle measure in a triangle.
Read and plot coordinates in the 1st quadrant of the Cartesian plane.
*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.
*Draw a circle with a given radius or diameter.
*Find unknown angle measures involving triangles and quadrilaterals.
Understand the properties of squares and rectangles.
Use properties of squares and rectangles to find unknown angle measures.
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.
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.
Position and Movement (continued)
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
Identify the data to collect to answer a set of related questions.
Collect and present data in an appropriate data display.
Draw conclusions from data and identify further questions to ask.
Collect and present data in a bar line chart.
Distinguish between statistical questions and those that are not.
Write a statistical question and explain what data could be collected to answer the question.
Distinguish between categorical data and numerical data.
Collect and record data in a frequency table.
Tables
Graphs
*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.
*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 bar line chart.Read and interpret a pie chart.
Complete, read and interpret a line graph.
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.
Graphs (continued)
Data Analysis
*Read and interpret a scatter graph.
*Describe the center, variability and shape of a data distribution in a dot plot or histogram.
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
Expressions
Complete a number pattern with decimals involving addition and subtraction.
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.
Write simple expressions that record calculations with numbers.
Interpret numerical expressions without evaluation.
Describe and complete a number pattern involving positive and negative integers, fractions, and decimals by counting on and backwards.
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.
Expressions (continued)
Equations
Inequalities
*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.
*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.
*Lessons are available in PR1ME Mathematics Teaching Hub.
Strand: Numbers and Operations
• CB: p. 1
Objectives
• Read and write a number within 1 000 000 — the numeral and the corresponding number word
• Identify the values of digits in a 6-digit number
• Compare numbers within 1 000 000
• Find all the factors of a whole number up to 100
• Find the multiples of a whole number up to 10
Unit
Let’s Remember
• ten million
• CB: pp. 2–4
• PB: p. 9
• Digital Practice
• Hieroglyphic Symbols (BM1.1): 1 copy per group, 1 enlarged copy for demonstration
• CB: p. 5
• PB: p. 10
• Digital Practice
Unit 1: Whole Numbers to 10 000 000
• Recognize the historical origins of our number system and begin to understand how it developed
• Read and write a number within 10 000 000 — the numeral and the corresponding number word
1.1 Reading and writing numbers
• Identify the values of digits in a 7-digit number
1.2 Identifying values of digits
• CB: p. 6
• PB: p. 11
• Digital Practice
• Compare and order numbers within 10 000 000
1.3 Comparing and ordering numbers
• CB: pp. 7–8
• PB: p. 12
• Digital Practice
• CB: p. 9
• PB: p. 13
• Digital Practice
• CB: pp. 9–10
• PB: p. 14
• Digital Practice
• common factor
• greatest common factor (GCF)
• CB: pp. 11–12
• PB: p. 15
• Digital Practice
• CB: pp. 12–13
• PB: p. 16
• Digital Practice
• Round a whole number to the nearest ten thousand, hundred thousand or million
1.4 Rounding whole numbers to the nearest ten thousand, hundred thousand or million
• common multiple
• least common multiple (LCM)
• CB: pp. 14–16
• PB: p. 17
• Digital Practice
• CB: pp. 16–17
• PB: p. 18
• Digital Practice
• Estimate sums and differences
1.5 Estimating sums and differences
• Estimate products and quotients
1.6 Estimating products and quotients
Unit 2: Factors
• Find the common factors of two numbers
• Find the greatest common factor of two numbers
2.1 Finding common factors of two numbers
• Find out if a number is a common factor of two given numbers
2.2 Finding out if a number is a common factor of two given numbers
• Find the common multiples of two numbers
• Find the least common multiple of two numbers
Unit 3: Multiples
3.1 Finding common multiples of two numbers
• Find out if a number is a common multiple of two given numbers
3.2 Finding out if a number is a common multiple of two given numbers
Unit 4: Problem Solving
• CB: pp. 18–22
• PB: pp. 19–21
• Digital Practice
• CB: pp. 22–23
• Solve word problems involving common factors and multiples
4.1 W ord problems
• Solve a non-routine problem involving common multiples using the strategy of making a list
4.2 Mind stretcher
Digital Chapter Assessment — Available in PR1ME Mathematics Digital Practice and Assessment
The suggested duration for each lesson is 1 hour.
Chapter Overview
Let’s Remember
Unit 1: Whole Numbers to 10 000 000
Unit 2: Factors
Unit 3: Multiples
Unit 4: Problem Solving
Let's Remember
Recall:
1. Reading and writing a number within 1 000 000 — the numeral and the corresponding number word (CB5 Chapter 1)
2. Identifying the values of digits in a 6-digit number (CB5 Chapter 1)
3. Comparing numbers within 1 000 000 (CB5 Chapter 1)
4. Finding all the factors of a whole number up to 100 (CB5 Chapter 1)
5. Finding the multiples of a whole number up to 10 (CB5 Chapter 1)
EXPLORE
Have students read the word problem on CB p. 1. Discuss with students the following questions:
•Have you ever taken part in competitions together with other boys and girls?
•Why do you think that there is a need to have equal number of boys and girls in each team?
•What other sports can boys and girls take part together?
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.
2. Write the missing numbers or words.
In 320 678, a) the digit 3 is in the place. b) the digit 0 is in the place. c) the value of the digit 2 is d) the digit is in the hundreds place.
3. Compare 704 513, 740 053 and 704 053. Which is the least number?
4. Write all the factors of 81.
5. Write the first 10 multiples of 7.
1, 3, 9, 27, 81
7, 14, 21, 28, 35, 42, 49, 56, 63, 70
1.1 Reading and writing numbers
Let's Learn Let's Learn
Objectives:
• Recognize the historical origins of our number system and begin to understand how it developed
• Read and write a number within 10 000 000— the numeral and the corresponding number word
Materials:
• Hieroglyphic Symbols (BM1.1): 1 copy per group, 1 enlarged copy for demonstration
Resources:
• CB: pp. 2–4
• PB: p. 9
Vocabulary:
• ten million
(a) Before the lesson, cut out the hieroglyphic symbols in BM1.1.
Stage: Pictorial Representation
Have students look at the first diagram in (a) on CB p. 2.
Say: The diagram shows some key developments in number systems. You will learn about these number systems.
Ask: During primitive times, how do people keep count? (They used tally marks.)
Say: Let us use tally marks to represent 15 items. Draw a rectangle with 15 tally marks as shown below.
Say: Each tally mark represents 1 item. There are 15 tally marks to represent 15 items.
Ask: What is the disadvantage of this system? (It takes up a lot of space and is difficult to read if there are too many tally marks.)
Stage: Abstract Representation
Say: The Egyptians and Romans came up with better ways to represent numbers using symbols.
Have students look at the table on the page. Say: These hieroglyphic symbols are used in the Egyptian number system. Each symbol represents a number.
You will learn to...
• understand how our number system is developed
• read and write numbers within 10 000 000
• identify the values of digits in 7-digit numbers
• compare and order numbers within 10 000 000
• round whole numbers to the nearest ten thousand, hundred thousand or million
estimate sums, differences, products and quotients
Let's Learn Let's Learn
a) Number systems have evolved throughout history. The diagram below shows some key developments in number systems.
In primitive times, number systems were developed when there was a need to keep track of or count things. For example, a shepherd could use 1 tally mark to represent 1 animal and keep track of the animals.
Writing larger numbers would take up a lot of space if tally
numbers by using symbols.
2
Use the hieroglyph cut-outs from Hieroglyphic Symbols (BM1.1) to explain what number each symbol represents. For example, to represent the number 100, we draw a coil of rope. Have students work in groups. Distribute a copy of Hieroglyphic Symbols (BM1.1) to each group.
Ask students to cut out the symbols along the gridlines of the table.
Write: 235
Say: Let us use hieroglyphic symbols to represent the number 235. Stick hieroglyph cut-outs on the board to represent the number 235. Call out a few numbers and have students use their cut-outs to represent the numbers.
Ask: Is this number system better than the primitive one shown earlier? (Yes) Why? (It is easier to represent large numbers using this system.)
Say: The Roman number system uses these letters to represent numbers. Write the numbers that the symbols represent below the symbols.
Say: Roman numerals are still used today.
Ask: Where do you see Roman numerals? (Answer varies. Samples: clocks, names of sports events)
Guide students to use Roman numerals to represent the numbers 11 and 9. (XI and IX)
Ask: What are the disadvantages of the Egyptian and Roman number systems? (More symbols needed for larger numbers. Not easy to do operations like addition or subtraction.)
Say: The Babylonians used a place value number system. The symbols and the position of symbols contribute to the value of the number.
Write: 1 10
Point to the symbols and explain what number each symbol represents.
Have students look at the Babylonian number symbols on CB p. 3 and observe that it is easy to decipher numbers written in this system.
(b) Stage: Abstract Representation
Have students look at the dark blue place value card in (b) on CB p. 3.
Say: We read this number as eight million
Write: 8 000 000 eight million
Have students look at the next place value card on the page.
Ask: How do we read this number? (7 hundred thousand)
Have students look at the next place value card on the page.
Ask: How do we read this number? (60 thousand)
Have students look at the next place value card on the page.
Ask: How do we read this number? (4 thousand) What is 7 hundred thousand plus 60 thousand plus 4 thousand? (764 thousand)
Write: 8 000 000 + 764 000 eight million, seven hundred and sixtyfour thousand
Repeat the above procedure for the remaining place value cards to teach students to read and write 291.
Write: 8 000 000 + 764 000 + 291 = 8 764 291 eight million, seven hundred and sixty-four thousand, two hundred and ninety-one
While the Egyptian and Roman number systems were a big improvement, to write larger numbers you have to keep making more symbols. It was also not easy to do operations like addition or subtraction.
People found a clever way to represent numbers using place value. The Babylonians developed the earliest place value system. A place value system uses symbols and the position of the symbol contributes to the value of the number.
The number system we are learning today is called the Hindu-Arabic number system. It is the most widely used number system in the world. It is a powerful number system that can be used to represent any number using ten symbols:
Say: We read 8 764 291 as eight million, seven hundred and sixty-four thousand, two hundred and ninety-one.
c) Count on in steps of 1 000 000 from 1 000 000. 1 000 000, 2
Let's Do Do
1. Write the numerals.
a) four million, five hundred and forty-four thousand, one hundred and eleven
b) nine million, four hundred and seventy-two thousand, one hundred and twenty-three
2. Write the numbers in words.
a) 1 089 657
one million, eighty-nine thousand, six hundred and fifty-seven
b) 5 239 500
five million, two hundred and thirty-nine thousand and five hundred
Let's Practice
1. Write the numerals.
a) three million, three hundred thousand, one hundred and one
b) two million, six hundred and fourteen thousand and ninety
c) nine million, five hundred and thirty-two thousand, nine hundred and fifty-three
2. Write the numbers in words.
a) 6 030 584
six million, thirty thousand, five hundred and eighty-four
b) 1 043 495
one million, forty-three thousand, four hundred and ninety-five
c) 8 395 200
eight million, three hundred and ninety-five thousand and two hundred
4
(c) Stage: Abstract Representation
Write: 1 000 000
Say: We read this number as one million. Have students count on in steps of 1 million.
Say: 1 million, 2 million, 3 million, …, 10 million
Write: 10 000 000
Say: We read this number as ten million.
Write: ten million = 10 000 000
Let's Do Do and Let's Practice Let's
Task 1 requires students to write 7-digit numbers in numerals, given the numbers in words.
Task 2 requires students to write 7-digit numbers in words, given the numerals.
1.2 Identifying values of digits
Let's Learn Let's Learn
Objective:
• Identify the values of digits in a 7-digit number
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.
1.2 Identifying values of digits
1. Write the missing numbers or words.
In
Write: 2 539 764
Say: Let us write this number in the place value chart.
Write 2 539 764 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 2 is in the millions place so there are 2 millions. The value of the digit 2 is 2 000 000. Ask: Which digit is in the hundred thousands place? (5) How many hundred thousands are there? (5) What is the value of the digit 5? (500 000)
Ask similar questions about the other digits. Say: 2 000 000, 500 000, 30 000, 9000, 700, 60 and 4 make 2 539 764.
Write: 2 539 764 = 2 000 000 + 500 000 + 30 000 + 9000 + 700 + 60 + 4
Let's Do Let's and Let's Practice Practice
Task 1 requires students to identify the values and place values of digits in 7-digit numbers.
Let's Learn Let's Learn
Objective:
•Compare and order numbers within 10 000 000
Resources:
•CB: p. 6
•PB: p. 11
Stage: Abstract Representation
Write: 7 350 119, 7 250 210 and 765 005
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 7 350 119, 7 250 210 and 765 005.
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? (Millions)
Say: Look at the three numbers in the place value chart.
Ask: Which number does not have millions? (765 005) So, what can we say about the number? (It is the least number.)
Say: Let us now compare the two remaining numbers, 7 350 119 and 7 250 210. The number of millions is the same, so we compare the digits in the next place value, the hundred thousands place.
Ask: What is the digit in the hundred thousands place in 7 350 119? (3) What is the digit in the hundred thousands place in 7 250 210? (2) Which is greater, 3 hundred thousands or 2 hundred thousands? (3 hundred thousands)
Say: Since 3 hundred thousands is greater than 2 hundred thousands, 7 350 119 is greater than 7 250 210.
Write: 7 350 119 > 7 250 210
Ask: Which number is the greatest? (7 350 119)
Have students arrange the three numbers in order, beginning with the least. Invite a student to write the answer on the board. (765 005, 7 250 210, 7 350 119)
Reinforce students’ understanding by asking them to rearrange the numbers in order, beginning with the greatest.
Task 1 requires students to compare numbers within 10 000 000.
Task 1 requires students to compare and order numbers within 10 000 000.
1.4 Rounding whole numbers to the nearest ten thousand, hundred thousand or million
Objective:
•Round a whole number to the nearest ten thousand, hundred thousand or million
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 ‘187 325’.
Guide students to see that there are 10 equal intervals between 180 000 and 190 000 and each interval stands for 1000.
Ask a student to mark 187 325 on the number line.
Say: 187 325 is between two ten thousands, 180 000 and 190 000.
Ask: Is 187 325 nearer to 180 000 or to 190 000?
(Nearer to 190 000)
Stage: Abstract Representation
Say: 187 325 is nearer to 190 000 than to 180 000. So, 187 325 is 190 000 when rounded to the nearest ten thousand.
Write: 187 325 ≈ 190 000
Say: We read this statement as ‘187 325 is approximately 190 000’.
(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 millions, the greater million is to be taken as the nearest million. In this case, the greater million is 9 000 000. 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 a whole number to the nearest ten thousand or hundred thousand.
Task 2 requires students to round a whole number to the nearest million.
Task 3 requires students to round a whole number to the nearest ten thousand, hundred thousand or million.
Task 1 requires students to round whole numbers to the nearest ten thousand, hundred thousand or million.
Let's Learn Let's Learn
Objective:
•Estimate sums and differences
Resources:
•CB: p. 9
•PB: p. 13
(a) Stage: Abstract Representation
Write: Estimate the value of 45 943 + 2380.
Say: Let us round 45 943 to the nearest ten thousand.
Ask: What do we get when we round 45 943 to the nearest ten thousand? (50 000)
Say: Let us round 2380 to the nearest thousand.
Ask: What do we get when we round 2380 to the nearest thousand? (2000)
Write: 45 943 + 2380 ≈ 50 000 + 2000
Ask: What do we get when we add 50 000 and 2000? (52 000)
Write: 45 943 + 2380 ≈ 50 000 + 2000 = 52 000
Say: The estimated value of 45 943 + 2380 is 52 000.
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.
(b) Stage: Abstract Representation
Follow the procedure in (a).
Let's Do Do and Let's Practice Let's
Task 1 requires students to estimate sums and differences.
1.6 Estimating products and quotients
Let's Learn Let's Learn
Objective:
•Estimate products and quotients
Resources:
•CB: pp. 9–10
•PB: p. 14
(a) Stage: Abstract Representation
Write: Estimate the value of 24 836 × 5.
Say: To estimate the product, we first round 24 836 to the nearest ten thousand.
Ask: What is 24 836 rounded to the nearest ten thousand? (20 000)
1.5 Estimating sums and differences
a) Estimate the sum of 45 943 and 2380.
45 943 + 2380 ≈ 50 000 + 2000 =
b) Estimate the difference between 732 946 and 29 682.
732 946 – 29 682 ≈ 700 000 – 30 000 =
1. Estimate the value of each of the following. a) 261 380 + 2998 ≈ + =
Estimates vary. Sample:
b) 1 394 188 – 423 782 ≈ –=
1. Estimate the value of each of the following. a) 14 034 + 49 558 b) 211 495 + 2 376 342 c) 69 853 – 45 914 d) 9 275 439 – 349 845
Estimates vary. Sample:
≈ 10 000 + 50 000 = 60 000
≈ 200 000 + 2 000 000 = 2 200 000
≈ 70 000 – 50 000 = 20 000
≈ 9 000 000 – 300 000 = 8 700 000
1.6 Estimating products and quotients
Learn Let's Do Let's
a) Estimate the product of 24 836 and 5. 24 836 × 5 ≈ 20 000 × 5 = 100 000
Write: 24 836 × 5 ≈ 20 000 × 5
Point out that it is not necessary to round the number 5 to the nearest ten as it is easy to calculate 20 000 × 5 mentally.
Ask: What is the product of 20 000 and 5? (100 000)
Write: 24 836 × 5 ≈ 20 000 × 5 = 100 000
Say: The estimated value of 24 836 × 5 is 100 000.
(b) Stage: Abstract Representation
Write: Estimate the value of 5 541 238 ÷ 6.
Say: To estimate the quotient, we look for a multiple of 6 that is close to 5 541 238. Have students recite the first ten multiples of 6. As they recite the multiples, write them on the board: 6, 12, 18, …, 48, 54, 60.
Say: Since 48, 54 and 60 are multiples of 6, 4 800 000, 5 400 000 and 6 000 000 are also multiples of 6.
Ask: Between which two multiples is 5 541 238? (5 400 000 and 6 000 000) Which multiple is 5 541 238 nearer to? (5 400 000) If necessary, draw a number line to help students see that 5 541 238 is nearer to 5 400 000 than to 6 000 000.
Say: 5 541 238 is approximately 5 400 000
Write: 5 541 238 ÷ 6 ≈ 5 400 000 ÷ 6
Say: 5 541 238 ÷ 6 is approximately 5 400 000 ÷ 6. 5 400 000 is 54 hundred thousands.
Ask: What do we get when we divide 54 hundred thousands by 6? (9 hundred thousands) What do we get when we divide 5 400 000 by 6? (900 000)
Write: 5 541 238 ÷ 6 ≈ 5 400 000 ÷ 6 = 900 000
Say: The estimated value of 5 541 238 ÷ 6 is 900 000.
Let's Do and Let's Practice
Task 1 requires students to estimate products and quotients.
b) Estimate the quotient. 5 541 238 ÷ 6 ≈ 5 400 000 ÷ 6 = 900 000
Let's Do Let's Do
1. Estimate the value of each of the following. a) 58 409 × 6 ≈ × = b) 349 881 ÷ 7 ≈ ÷ =
Let's Practice Practice
1. Estimate the value of each of the following.
I have learned to... understand how our number system is developed read and write numbers within 10 000 000 identify the values of digits in 7-digit numbers compare and order numbers within 10 000 000 round whole numbers to the nearest ten thousand, hundred thousand or million estimate sums, differences, products and quotients
2.1 Finding common factors of two numbers
Let's Learn Let's Learn
Objectives:
•Find the common factors of two numbers
•Find the greatest common factor of two numbers
Resources:
•CB: pp. 11–12
•PB: p. 15
Vocabulary:
•common factor
•greatest common factor (GCF)
Stage: Abstract Representation
Say: Let us find the factors of 30.
Ask a student to write multiplication sentences with products of 30 on the board.
Ask: What are the factors of 30? (1, 2, 3, 5, 6, 10, 15, 30)
Have a student write the factors of 30 on the board.
Repeat the above procedure to find the factors of 42.
Ask: Compare the factors of 30 and 42. Which factors are the same? (1, 2, 3, 6)
Circle the common factors of 30 and 42 on the board.
Say: 1, 2, 3 and 6 are factors of 30 and 42. We call them common factors of 30 and 42.
Write: Common factors of 30 and 42 = 1, 2, 3 and 6
Ask: Compare the common factors of 30 and 42. Which common factor is the greatest? (6) Say: We say that the greatest common factor or the GCF of 30 and 42 is 6.
Write: GCF of 30 and 42 = 6
Let's Do Do
Task 1 requires students to find the common factors and greatest common factor of two numbers.
You will learn to... • find the common factors and the greatest common factor of two numbers • find out if a number is a common factor of two given numbers
2.1 Finding common factors of two numbers
Let's Learn Let's Learn
Find the common factors and the greatest common factor of 30 and 42. We write the factors of 30 and 42 first.
1, 2, 3 and 6 are factors of 30 and 42. 1, 2, 3 and 6 are common factors of 30 and 42.
Compare the common factors. The greatest common factor (GCF) of 30 and 42 is 6.
Let's Do Let's
2.2
Practice
Let's Learn
a)
Let's Practice Let's Practice
Task 1 requires students to find the common factors and greatest common factor of two numbers.
Have students go back to the word problem on CB p. 1.
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.
2.2 Finding out if a number is a common factor of two given numbers
Let's Learn Let's Learn
Objective:
•Find out if a number is a common factor of two given numbers
Resources:
•CB: pp. 12–13
•PB: p. 16
(a) Stage: Abstract Representation
Write: Is 3 a common factor of 36 and 54?
Say: If 3 is a common factor of 36 and 54, it is a factor of 36 and 54. If 3 is a factor of 36 and 54, 36 and 54 can be divided by 3 exactly and there will be no remainder.
Have a student divide 36 by 3 on the board.
Ask: Can 36 be divided by 3 exactly? (Yes) Is 3 a factor of 36? (Yes)
Repeat the above procedure to determine that 3 is also a factor of 54.
Say: 3 is a factor of 36 and 3 is also a factor of 54. So, 3 is a common factor of 36 and 54.
(b) Stage: Abstract Representation
Repeat the procedure in (a). Conclude that 3 is not a common factor of 72 and 49.
Task 1 requires students to find out if a number is a common factor of two given whole numbers.
Practice Let's Practice
Tasks 1 and 2 require students to find out if a number is a common factor of two given whole numbers.
You will learn to...
• find the common multiples and the least common multiple of two numbers
• find out if a number is a common multiple of two given numbers
3.1 Finding common multiples of two numbers
Find the common multiples and the least common multiple of 3 and 4.
Sarah and David are discussing whether they can find the greatest possible number of candies Joe has.
Joe has some candies. He can pack them equally into bags of 3 or bags of 5.
The
of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36
1 × 4 = 4 2 × 4 = 8 3 × 4 = 12
4 × 4 = 16 5 × 4 = 20 6 × 4 = 24
7 × 4 = 28 8 × 4 = 32 9 × 4 = 36
10 × 4 = 40 11 × 4 = 44 12 × 4 = 48
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48 . . .
Then, we compare the multiples of 3 and 4.
12 is a multiple of 3 and 4.
12 is a common multiple of 3 and 4.
The next two common multiples of 3 and 4 are 24 and 36.
The least common multiple (LCM) of 3 and 4 is 12.
Let's Do Let's
1. Find the first two common multiples and the least common multiple of 2 and 6. Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 … Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 …
a) The first two common multiples of 2 and 6 are
First, we list some of the multiples of 3 and 4. 6 and 12 6
b) The least common multiple of 2 and 6 is
3.1 Finding common multiples of two numbers
Let's Learn Let's Learn
Objectives:
•Find the common multiples of two numbers
•Find the least common multiple of two numbers
Resources:
•CB: pp. 14–16
Vocabulary:
•common multiple
•PB: p. 17
•least common multiple (LCM)
Stage: Abstract Representation
Have two students write the multiplication tables of 3 and 4 until 12 × 3 and 12 × 4 on the board.
Ask: What are the first twelve multiples of 3?
(3, 6, 9, …, 36) What are the first twelve multiples of 4? (4, 8, 12, …, 48) Which multiples are the same? (12, 24, 36)
Circle the common multiples of 3 and 4.
Say: 12, 24 and 36 are multiples of 3 and 4. We call them common multiples of 3 and 4.
Write: Common multiples of 3 and 4 = 12, 24, 36, …
Can you find the greatest possible number of candies Joe has? Yes, I can.
David Sarah
Is David correct? Why do you say so?
David is not correct. To find the greatest possible number of candies, he has to find the greatest common multiple of 3 and 5, which is not possible. He can only find the least possible number of candies.
What did you learn about the common multiples of two numbers?
We can find the least common multiple but not the greatest common multiple of two numbers.
Think of a time in your daily life when you need to use the least common multiple of two numbers.
I wanted to know how many packs of sausages and buns I should buy at the supermarket to serve hotdogs at a party. Sausages come in packs of 6 and buns come in packs of 8. I found the least common multiple of 6 and 8 which is 24. So, I needed to buy 4 packs of sausages and 3 packs of buns so that I could make 24 hotdogs, with no leftover of buns or sausages.
Ask: Which common multiple is the least? (12) Say: We say that the least common multiple or the LCM of 3 and 4 is 12.
Write: LCM of 3 and 4 = 12
Task 1 requires students to find the common multiples and the least common multiple of two numbers.
Have students work in groups to discuss the tasks. Ask the groups to present their answers.
Have students find common multiples of 3 and 5. Lead them to observe that they can only find the least common multiple and not the greatest common multiple of 3 and 5. Conclude that David is not correct.
Reiterate to students that we can find the least common multiple but not the greatest common multiple of two numbers.
Make use of the examples presented by the groups to let students understand the importance and usefulness of knowing how to find the least common multiple of two numbers.
1.
9, 18, 27, 36
b) What is the least common multiple of 9 and 6?
2. Find the least common multiple of 3 and 5.
Multiples of 3: 3, 6, 9, 12, 15, 18, …
Multiples of 5: 5, 10, 15, 20, 25, …
The least common multiple of 3 and 5 is 15.
3.2 Finding out if a number is a common multiple of two given numbers
Let's Learn Let's Learn
a) Is 72 a common multiple of 2 and 4?
72 can be divided by 2 and 4 exactly.
b) Is 104 a common multiple of 4 and 6?
104
16
Let's Practice Let's Practice
Task 1 requires students to find the common multiples and the least common multiple of two numbers.
Task 2 requires students to find the least common multiple of two numbers.
3.2 Finding out if a number is a common multiple of two given numbers
Let's Learn Let's Learn
Objective:
•Find out if a number is a common multiple of two given numbers
Resources:
•CB: pp. 16–17
•PB: p. 18
(a) Stage: Abstract Representation
Write: Is 72 a common multiple of 2 and 4?
Say: If 72 is a common multiple of 2 and 4, it is a multiple of 2 and 4. If 72 is a multiple of 2 and 4, 72 can be divided by 2 and 4 exactly and there will be no remainder.
Let's Do Let's 1. Is 216 a common multiple of 3 and 8?
be
be divided by 3 exactly. divided by 8 exactly. So, 216 a common multiple of 3 and 8.
Let's Practice Let's
1. a) Is 252 a common multiple of 6 and 7?
Is 378 a common multiple of 5 and 9?
have learned to... find the common multiples and the least common multiple of two numbers find out if a number is a common multiple of two given numbers
Have a student divide 72 by 2 on the board. Ask: Is 72 a multiple of 2? (Yes) Repeat the above procedure to determine that 72 is also a multiple of 4.
Say: 72 is a multiple of 2 and 72 is also a multiple of 4. So, 72 is a common multiple of 2 and 4.
(b) Stage: Abstract Representation
Repeat the procedure in (a). Conclude that 104 is not a common multiple of 4 and 6.
Task 1 requires students to find out if a number is a common multiple of two given numbers.
Tasks 1 and 2 require students to find out if a number is a common multiple of two given numbers.
4.1 Word problems
Let's Learn Let's Learn
Objective:
•Solve word problems involving common factors and multiples
Resources:
•CB: pp. 18–22
•PB: pp. 19–21
1. Have students read the word problem on CB p. 18.
1. Understand the problem. Pose the questions in the thought bubble in step 1.
2. Plan what to do.
Say: The number of apples on each platter must be the same. This is also true for oranges and cherries.
Point out that students can make a list to find the number of fruit platters Nicole can make.
3. Work out the Answer.
Say: First, we find the number of platters of apples Nicole can make. There are 24 apples, so we write the multiplication sentences with products of 24.
Invite a student to write the multiplication sentences with products of 24.
Point to the multiplication sentence 1 × 24 = 24.
Explain that one factor represents the number of apples on a platter and the other factor represents the number of platters.
Say: Nicole can make 1 platter of 24 apples or 24 platters of 1 apple. Similarly, use the remaining multiplication sentences to describe the number of platters of apples that can be made.
Say: The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. Nicole can make 1, 2, 3, 4, 6, 8, 12 or 24 platters of apples. Repeat the above procedure to find the number of platters of oranges and cherries.
Ask: What are the common factors of 24, 36 and 30? (1, 2, 3, 6) What is the greatest common factor of 24, 36 and 30? (6) Say: The greatest number of platters Nicole can make is 6.
You will learn to... • solve word problems involving common factors and multiples • solve a non-routine problem involving common multiples
Let's Learn Let's
1. Nicole has 24 apples, 36 oranges and 30 cherries. She wants to make identical fruit platters using all of the fruits. What is the greatest number of platters she can make?
1
Understand the problem.
2
Plan what to do.
3
Work out the Answer
What types of fruits does Nicole have? How many fruits of each type are there? What does Nicole want to do? What do I have to find?
The number of apples on each platter must be the same. This is also true for oranges and cherries. I can make a list to find the number of platters I can make with each type of fruit.
Apples: 1 × 24 = 24 2 × 12 = 24
3 × 8 = 24 4 × 6 = 24
Oranges: Cherries:
1 × 36 = 36 1 × 30 = 30
2 × 18 = 36 2 × 15 = 30
3 × 12 = 36 3 × 10 = 30
4 × 9 = 36 5 × 6 = 30 6 × 6 = 36
Nicole can make 1 platter of 24 apples, 24 platters of 1 apple, 2 platters of 12 apples, 12 platters of 2 apples, …
The common factors of 24, 36 and 30 are 1, 2, 3 and 6. The greatest common factor of 24, 36 and 30 is 6. The greatest number of platters Nicole can make is 6.
4. Check if your answer is correct. Guide students to check their answer by checking if Nicole will use all the fruits to make 6 platters of fruit. Circle 4 × 6 = 24 on the board. Say: There are 4 apples on each platter. Repeat the above procedure for the oranges and cherries.
Ask: Will Nicole use all the fruits? (Yes) How many apples, oranges and cherries are there on each platter? (4 apples, 6 oranges and 5 cherries) Are the platters identical? (Yes)
Say: The answer is correct.
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. 19.
Ask: Which method do you prefer? Why? (Answers vary.)
2. Have students read the word problem on CB p. 19.
1. Understand the problem. Pose the questions in the thought bubble in step 1.
Work out the Answer 3
Since the buttons are sold in packs, I can make a list of the possible number of buttons can buy.
The multiples of 12 are 12, 24, 36, 48, 60, 72, 84, …
The multiples of 18 are 18, 36, 54, 72, 90, …
The multiples of 24 are 24, 48, 72, 96, …
The least common multiple of 12, 18 and 24 is 72. 72 buttons of each color are needed.
12 × 6 = 72 18 × 4 = 72 24 × 3 = 72
I need to buy 6 packs of gold buttons, 4 packs of silver buttons and 3 packs of black buttons.
72 ÷ 6 = 12 72 ÷ 4 = 18 72 ÷ 3 = 24
My answer is correct.
The greatest number among 12, 18 and 24 is 24 so the number of buttons will be a multiple of 24. The multiples of 24 are 24, 48, 72, 96, … Find the least multiple of 24 that can be divided by 12 and 18 exactly. Check if your answer is correct. 4 + Plus Solve the problem in another way. 5
NumberDivisible by 12? Divisible by 18?
24 Yes No
48 Yes No
72 Yes Yes
72 ÷ 12 = 6 72 ÷ 18 = 4 72 ÷ 24 = 3
I need to buy 6 packs of gold buttons, 4 packs of silver buttons and 3 packs of black buttons.
Compare the methods in steps 3 and 5. Which method do you prefer? Why?
1. Understand 2. Plan 3. Answer 4. Check 5. Plus
2. Plan what to do.
Say: Before we can find the number of packs of buttons in each color, we have to find the number of buttons in each color. Point out that students can make a list of the possible number of buttons to buy.
3. Work out the Answer
Say: First, we find the number of gold buttons. There are 12 gold buttons in each pack.
Ask a student to write the first ten multiples of 12 on the board.
Repeat the above procedure for the silver and black buttons.
Ask: What is the least common multiple of 12, 18 and 24? (72)
Say: 72 buttons of each color are needed. Circle the multiplication facts 12 × 6 = 72, 18 × 4 = 72 and 24 × 3 = 72 on the board. Explain that we need to buy 6 packs of gold buttons, 4 packs of silver buttons and 3 packs of black buttons to have 72 buttons in each color.
4. Check if your answer is correct. Guide students to check their answer by dividing 72 by the number of packs of buttons to get the number of buttons in each pack.
Let's Do Let's
Solve the word problems. Show your work clearly.
Next, try solving each problem in a different way. Which method do you prefer? Why?
1. Understand 2. Plan 3. Answer 4. Check 5. Plus
1. Mike has two rods of lengths 18 centimeters and 42 centimeters. He wishes to cut the two rods into shorter pieces of equal length with no remainder. What is the greatest possible length of each of the shorter pieces of rod?
Find the greatest common factor of 18 and 42.
2. Oranges are sold in packs of 8, lemons in packs of 10 and plums in packs of 20. What is the least number of packs of each type of fruits that Lisa needs to buy if she wants to make a bag having an equal number of each type of fruits in it, with none left over?
Let's Practice Let's
Solve the word problems. Show your work clearly.
1. Ralph makes snack bags using 12 granola bars, 18 pretzels and 30 cheese sticks. Each snack bag has the same mix of snacks in the same quantities. What is the greatest number of snack bags Ralph makes?
2. Mrs. Sanders has 36 walnuts, 54 pecans and 90 almonds. She wants to place an equal number of nuts in bowls, without mixing the different types. What is the greatest number of nuts she can have in each bowl?
3. A red lamp flashes every 10 seconds, a yellow lamp flashes every 24 seconds and a blue lamp flashes every 30 seconds. How many times will the three lamps flash at the same time in 10 minutes?
4. A radio station is having a contest in which every 12th caller receives a voucher, every 15th caller receives a concert ticket and every 18th caller receives a limousine ride. Which caller will be the first to win all three prizes?
Write: 72 ÷ 6 = ______
Elicit the answer from students. (12)
Say: We get 12 gold buttons in each pack. Repeat the above procedure for the packs of silver and black buttons.
Ask: Is our answer correct? (Yes)
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. 20.
Ask: Which method do you prefer? Why?
(Answers vary.)
Task 1 requires students to solve a word problem involving common factors.
Task 2 requires students to solve a word problem involving common multiples.
Let's Practice Let's Practice
Tasks 1 and 2 require students to solve word problems involving common factors.
Tasks 3 and 4 require students to solve word problems involving common multiples.
Have students work in pairs. Get students to create a word problem and exchange the word problem with their partner. Ask students to solve the word problem from their partner. Have a few pairs of students present their work. They should first explain how they decide what numbers to use and their partner has to explain the solution.
Students are expected to change the bus frequencies. They should find the least common multiple of the two bus frequencies to solve the word problem.
Let's Learn Let's Learn
Objective:
•Solve a non-routine problem involving common multiples using the strategy of making a list
Resource:
•CB: pp. 22–23
Have students read the problem on CB p. 22.
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 make a list to find the common number of candies if Kelly gives each friend 8 or 9 candies.
3. Work out the Answer
Draw the table shown in step 3 on the page on the board without filling in the numbers in the green boxes of rows 2 to 5. Point to the second row of the table. Say: Let us fill in this row by finding the number of candies given if each friend gets 8 candies. Ask: If 8 candies are given to 1 friend, how many candies are given away? (8) Write ‘8’ in the second row of the table. Ask similar questions to fill in the entire second row. Point to the third row of the table. Say: If Kelly gives each friend 8 candies, she will have 5 candies left. We fill in this row by finding the number of candies she has. Ask: If she gives away 8 candies and has 5 candies left, how many candies does she have at first? (13)
CREATE YOUR OWN
Route A bus arrives at a bus stop every 25 minutes. Route B bus arrives at the same bus stop every 10 minutes. Both buses are at the bus stop now. In how many minutes will the two buses be at the bus stop again?
Read the word problem. Change the numbers in the word problem. How did you decide what numbers to use?
Answer varies. Sample: Change 25 to 20 and 10 to 15.
Next, solve the word problem. Show your work clearly. What did you learn?
The multiples of 20 are 20, 40, 60, 80, …
The multiples of 15 are 15, 30, 45, 60, …
The least common multiple of 20 and 15 is 60.
4.2 Mind stretcher
Let's Learn Let's
The two buses will be at the bus stop again in 60 minutes.
Kelly wants to give some candies to her friends. If she gives each friend 8 candies, she will have 5 candies left. If she gives each friend 9 candies, the last friend will receive 8 candies. How many candies does Kelly have?
1
Understand the problem.
Plan what to do. 2
Work out the Answer 3
How many candies does Kelly want to give to her friends? How many candies will there be left? How many candies will the last friend receive? Does the number of friends change? What do I have to find?
should make a list. Then, find the common number.
Number of friends 123456
8 candies to each friend 8 1624324048
5 candies left 1321293745 53
9 candies to each friend 91827364554
1 fewer candy 817263544 53
Kelly has 53 candies.
Write ‘13’ in the third row of the table. Ask similar questions to fill in the entire third row. Repeat the above procedure to fill in the next two rows.
Say: We want to find the number of candies Kelly has, so we compare the third and last rows.
Ask: What is the common number? (53)
Say: The least number of candies Kelly can have is 53 candies.
4. Check if your answer is correct.
Ask: How can we check our answer? (Divide 53 by 8 and 9 and check the remainders.)
Have students divide 53 by 8.
Ask: What is the remainder when we divide 53 by 8? (5)
Say: There are 5 candies left if she gives 8 candies to each of her 6 friends. Have students divide 53 by 9 and guide them to see the remainder is 8.
Say: The last friend gets 8 candies if she gives 9 candies to each of her 6 friends. Our answer is correct.
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. 23.
Ask: Which method do you prefer? Why? (Answers vary.)
Have students go back to the word problem on CB p. 1. 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) 1 250 497 b) 4 050 014
c) 5 302 888 d) 2 000 060
2. a) nine million, six hundred and fifty-two thousand and thirty-nine
b) six million, seven hundred and thirty-one thousand, five hundred and forty-six
c) three million, four hundred thousand, eight hundred and eight
d) one million, ninety thousand and forty
Exercise 1.2
1. a) millions; 7 000 000
b) hundred thousands; 0
c) 3000
d) 5; 5 000 000
e) 1; 10 000
2. a) 2 000 000 b) 200 000
c) 7000 d) 90 000
Exercise 1.3
1. a) > b) < c) > d) <
2. a) 5 690 800, 4 690 800, 969 800
b) 2 345 665, 1 345 660, 1 345 600, 1 234 560
3. a) 500 555, 5 500 505, 5 500 555
b) 3 154 234, 3 154 324, 3 452 896, 4 872 435
Exercise 1.4
1. a) 250 000 b) 440 000 c) 8 050 000
d) 6 280 000 e) 5 730 000 f) 1 000 000
2. a) 500 000 b) 900 000 c) 6 300 000
d) 1 900 000 e) 4 200 000 f) 4 000 000
3. a) 3 000 000 b) 2 000 000 c) 5 000 000
d) 5 000 000 e) 6 000 000 f) 10 000 000
Exercise 1.5
1. Estimates vary. Sample:
a) 900 000; 60 000; 960 000
b) 600 000; 400 000; 1 000 000
c) 8 000 000; 90 000; 8 090 000
d) 100 000; 6000; 94 000
e) 1 000 000; 200 000; 800 000
f) 7 000 000; 4 000 000; 3 000 000
2. Estimates vary. Sample:
a) 37 000 b) 6 900 000 c) 5 000 000
d) 4 500 000 e) 48 000 f) 1 000 000
Exercise 1.6
1. Estimates vary. Sample:
a) 20 000; 3; 60 000
b) 1 000 000; 7; 7 000 000
c) 2 000 000; 5; 10 000 000
d) 48 000; 4; 12 000
e) 240 000; 8; 30 000
f) 1 800 000; 6; 300 000
2. Estimates vary. Sample:
a) 2 700 000 b) 8 000 000 c) 480 000
d) 15 000 e) 30 000 f) 1 200 000
Exercise 2.1
1. a) 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48
b) 1, 2, 4, 7, 8, 14, 28 and 56
c) 1, 2, 4 and 8
d) 8
2. a) 1, 5 and 25; 25
b) 1, 2, 4, 8, 16 and 32; 32 c) 1, 2 and 4; 4
Exercise 2.2
1. a) Yes b) No c) Yes
2. Circle: 4; 9; 12
Exercise 3.1
1. a) 7, 14, 21, 28, …
b) 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28,
c) 14; 28
d) 14
2. a) 5, 10, 15, 20, 25, 30, 35, 40, …
b) 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …
c) 20; 40
d) 20
3. a) 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, … b) 8, 16, 24, 32, 40, 48, …
c) 24; 48
d) 24
4. 18
Exercise 3.2
1. No
2. No
3. Yes
4. Circle: 216; 372; 468
Exercise 4.1
1. 7 cm
2. 4 packs of green pens, 3 packs of blue pens
3. 4 4. 1
5. 18
6. 5 bags of dark chocolate bars, 6 bags of almond chocolate bars, 4 bags of milk chocolate bars
A
• algebraic expression
An algebraic expression contains operation signs and letters which are known as variables. The letters can represent any number.
n + 5 and 4x are algebraic expressions.
• average See mean
C
• categorical data
When we can group data into categories, we call this type of data categorical data
• center of rotation
The center of rotation is the point about which a shape rotates.
• century
A century is a period of 100 years.
• circle graph See pie chart
• common factor
A common factor is a factor of two or more numbers.
The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30.
The factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42.
The common factors of 30 and 42 are 1, 2, 3 and 6.
• common multiple A common multiple is a multiple of two or more numbers.
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, …
The multiples of 6 are 6, 12, 18, 24, 30, 36, …
The first three common multiples of 4 and 6 are 12, 24 and 36.
• conversion graph
A conversion graph is a line graph that shows the relationship between two units.
Conversion between yards and feet
• discount Discount is the amount of money you save when buying things at a reduced selling price.
Discount = usual price – selling price
• discounted price Discounted price is the selling price after discount.
Discounted price = usual price – discount
• dot plot
A dot plot uses dots to represent numerical data. Each dot represents 1 value.
• equivalent ratios
Equivalent ratios are ratios that represent the same relationship between quantities.
The ratio of the number of triangles to the number of circles is 8 : 6 or 4 : 3. 8 : 6 and 4 : 3 are equivalent ratios.
• event An event is an outcome or group of outcomes of an experiment. When we toss a coin, getting a head is an example of an event.
• experimental probability
Experimental probability is calculated from the actual outcomes of an experiment. It is different from the theoretical probability.
Experimental probability
= Number of favorable outcomes
F
• Coordinated Universal Time (UTC)
The Coordinated Universal Time (UTC) is a global time standard which is based on the Greenwich Mean Time (GMT).
D
• decade A decade is a period of 10 years.
• Greenwich Mean Time (GMT)
The Greenwich Mean Time (GMT) is the time measured at the Royal Observatory in Greenwich, England.
L • least common multiple (LCM)
The least common multiple is the least of all the common multiples of two or more numbers.
The first three common multiples of 4 and
6 are 12, 24 and 36.
The least common multiple of 4 and 6 is 12.
• least likely
When an event has the lowest chance of happening among all the other events, the event is least likely to occur. Jane has 3 red dresses, 2 blue dresses and 1 green dress. It is least likely for Jane to select a green dress to wear.
• less likely
When an event has a lower chance of happening than another event, this event is less likely to occur. There are 4 girls and 1 boy. It is less likely that a boy will be selected from the five children.
• likelihood Likelihood is the probability that an event will happen.
I • interest Interest is the amount of money the bank pays you for
money with them.
M
• mean
The mean of a set of data is often called the average of the data. It is the sum of the set of data divided by the number of data.
Mean Sum of data Number of data
E
• equally likely
When two events each have the same chance of happening, the events are equally likely to occur.
It is equally likely for a coin to land on heads or tails when it is tossed.
Total number of trials
• favorable outcome
A favorable outcome is the outcome you are looking for among all the possible outcomes of an event.
G
• greatest common factor (GCF)
The greatest common factor is the greatest of all the common factors of two or more numbers.
The common factors of 30 and 42 are 1, 2, 3 and 6. The greatest common factor of 30 and 42 is 6.
• measure of center A measure of center uses a single value to summarize a set of data. Mean, median and mode are measures of center.
• median
The median is the middle number in an ordered list of a set of data.
• mode The mode is the value that appears the most frequently in a set of data.
• more likely
When an event has a greater chance of happening than another event, this event is more likely to occur. There are 4 girls and 1 boy. It is more likely that a girl will be selected from the five children.
• most likely
When an event has the greatest chance of happening among all the other events, the event is most likely to occur. Jane has 3 red dresses, 2 blue dresses and 1 green dress. It is most likely for Jane to select a red dress to wear.
N
• numerical data
When we can group data into numbers, we call this type of data numerical data
O
• outcome An outcome is the possible result of a chance experiment.
Kate rolls the die. The outcome can be any number from 1 to 6.
• outlier An outlier is an extreme or unusual occurrence of values.
P
• parallelogram A parallelogram is a four-sided figure with opposite sides that are parallel and equal in length.
• pie chart
A pie chart is a circle graph divided into parts, with each part showing the relative size of a quantity of the whole.
The pie chart below shows the quantities of four types of fruit sold at a supermarket in a day.
A pie chart is also called a circle graph.
• probability The probability of an event happening is the chance or likelihood that the event will happen.
Probability of an event = Number of favorable outcomes
Total number of possible outcomes
• proportion When two or more ratios are equivalent to one another, we say that they are in proportion.
These ratios are in proportion:
1 : 2 = 2 : 4 = 3 : 6
• random To pick randomly means to pick without deciding or knowing the outcome.
Ben randomly picks 1 ball from the bag. It is equally likely that he picks any ball.
• range A range is the difference between the greatest and the least values in a set of data.
Given set of data: 4, 5, 7, 8, 10, 15 Range = 15 – 4 = 11
• ratio A ratio is a comparison of quantities. It does not have units.
The ratio of the number of triangles to the number of circles is 4 : 3.
• rhombus A rhombus is a four-sided figure with four equal sides and its opposite sides are parallel.
A C D AB = BC = CD = DA AB // DC and AD // BC • rotation
The shapes show a 90° rotation clockwise about point O. In a rotation, the size and shape do not change but the orientation changes.
S • sales tax Sales tax is the tax we pay when we buy goods and services. It is also known as Value Added Tax (VAT).
• skewed A skewed data distribution is not symmetrical. It has a ‘tail’ to the left or to the right. In a skewed distribution, the mean, median and mode are not equal. This dot plot has a ‘tail’ to the right. The shape is right skewed.
Countries visited by students
Number of countries 012345
• statistical question
A statistical question is a question that can be answered by collecting data and the data collected will vary.
• statistics Statistics is about the collection, presentation and analysis of data.
• symmetrical
The left and right sides of a symmetrical data distribution are reflections of each other. In a symmetrical distribution, the mean, median and mode are equal. This dot plot is symmetrical.
Number of pens students have
Number of pens 1023456 358
T
• ten million
1 ten million = 10 millions = 10 000 000
• term (ratio)
The terms of a ratio are the numbers in the ratio.
2 : 3 is a ratio.
first term second term
• theoretical probability
Theoretical probability is about what you expect to happen. It may not be what actually happens. It can be obtained using the formula:
Theoretical probability
= Number of favorable outcomes
Total number of possible outcomes
• time zone A time zone is a part of the world with the same local time. The world is divided into 24 time zones.
• trapezoid A trapezoid is a four-sided figure with only one pair of parallel sides.
B A C D AB // DC
• trial A trial is a round of experiment. In an experiment where a coin is tossed 5 times, each toss is known as a trial.
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 TM Mathematics (New Edition).
PR1ME Mathematics Digital Practice and Assessment
Digital practice and assessment further strengthen understanding of key concepts and provide diagnostic insight in students' capabilities and gaps in understanding.
PR1ME Mathematics Teaching Hub
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 2 Multiplication and Division of Whole Numbers
Chapter
Chapter 4 Introduction to Algebra
Chapter 5 Quadrilaterals
Let’s
Chapter 11 Ratio
Let’s
Chapter 12 Percentage
Let’s
Chapter 15
Chapter 16 Graphs
Chapter 17 Introduction to Statistics
Chapter
1. Write five hundred and fourteen thousand and nine, in numerals.
2. Write the missing numbers or words.
In 320 678,
a) the digit 3 is in the place.
b) the digit 0 is in the place.
c) the value of the digit 2 is .
d) the digit is in the hundreds place.
3. Compare 704 513, 740 053 and 704 053. Which is the least number?
4. Write all the factors of 81.
5. Write the first 10 multiples of 7.
40 girls and 32 boys want to participate in a team race during Sports Day. To ensure that there is fair competition among the teams, each team must have the same number of girls and boys. How many girls and boys will there be on each team?
How can we solve this problem? Discuss in your group and fill in columns 1 and 2.
1. What I already know that will help me solve the problem 2. What I need to find out and learn 3. What I have learned
will learn to...
• understand how our number system is developed
• read and write numbers within 10 000 000
• identify the values of digits in 7-digit numbers
• compare and order numbers within 10 000 000
• round whole numbers to the nearest ten thousand, hundred thousand or million
• estimate sums, differences, products and quotients
a) Number systems have evolved throughout history. The diagram below shows some key developments in number systems.
Primitive times
Egyptian and Roman number systems Babylonian number system Hindu-Arabic number system
tally marks on counting devices use of symbols to represent numbers place value systemnumber system used in modern times
In primitive times, number systems were developed when there was a need to keep track of or count things. For example, a shepherd could use 1 tally mark to represent 1 animal and keep track of the animals.
Primitive people etched tally marks on counting devices, like the Ishango bone, to help them keep track of things more accurately.
Writing larger numbers would take up a lot of space if tally marks were used. The Egyptian and Roman number systems have a better way to represent numbers by using symbols.
Roman number system
While the Egyptian and Roman number systems were a big improvement, to write larger numbers you have to keep making more symbols. It was also not easy to do operations like addition or subtraction.
People found a clever way to represent numbers using place value. The Babylonians developed the earliest place value system. A place value system uses symbols and the position of the symbol contributes to the value of the number.
The number system we are learning today is called the Hindu-Arabic number system. It is the most widely used number system in the world. It is a powerful number system that can be used to represent any number using ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
c) Count on in steps of 1 000 000 from 1 000 000.
1 000 000, 2 000 000, 3 000 000, 4 000 000, 5 000 000, 6 000 000, 7 000 000, 8 000 000, 9 000 000, 10 000 000
10 000 000 comes after 9 000 000. We read 10 000 000 as ten million
Let's Do Let's Do
1. Write the numerals.
a) four million, five hundred and forty-four thousand, one hundred and eleven
b) nine million, four hundred and seventy-two thousand, one hundred and twenty-three
2. Write the numbers in words.
a) 1 089 657
b) 5 239 500
Let's Practice Let's Practice
1. Write the numerals.
a) three million, three hundred thousand, one hundred and one
b) two million, six hundred and fourteen thousand and ninety
c) nine million, five hundred and thirty-two thousand, nine hundred and fifty-three
2. Write the numbers in words.
a) 6 030 584
b) 1 043 495
c) 8 395 200
Let's Learn Let's Learn
In 2 539 764, the digit 2 is in the millions place and its value is 2 000 000. the digit 3 is in the place and its value is . the digit 5 stands for 500 000.
2 539 764 = 2 000 000 + 500 000 + + 9000 + 700 + 60 + 4
Let's Do Let's Do Let's Practice Let's Practice
1. Write the missing numbers or words.
In 3 920 168,
a) the digit 3 is in the place and its value is .
b) the digit is in the thousands place.
c) the digit 9 stands for
1. Write the missing numbers or words.
a) In 8 145 023, the digit 8 is in the place and its value is .
b) In 6 132 599, the digit 3 is in the place.
c) In 2 713 950, the digit is in the hundred thousands place and its value is .
d) In 7 029 512, the digit 7 stands for .
e) 4 510 703 = + 500 000 + 10 000 + 700 + 3
f) 9 085 046 = 9 000 000 + + 5000 + 40 + 6
Compare 7 350 119, 7 250 210 and 765 005.
First, compare the millions.
There are no millions in 765 005. 765 005 is the least number.
Then, compare the hundred thousands in 7 350 119 and 7 250 210. 3 hundred thousands is greater than 2 hundred thousands.
7 350 119 > 7 250 210
7 350 119 is the greatest number.
Arrange the numbers in order. Begin with the least. , , (least)
1. Write > or <. a) 2 320 105
1. Compare 5 704 100, 5 741 100 and 5 714 100.
a) Which is the greatest number?
b) Which is the least number?
c) Arrange the numbers in order. Begin with the greatest. , , Let's Learn Let's Learn
whole numbers to the nearest ten thousand, hundred thousand or million
Let's Learn Let's Learn
a) The number of visitors to a state park in July was 187 325.
187 325
180 000 185 000 190 000
187 325 is between 180 000 and 190 000. It is nearer to 190 000 than to 180 000. So, we round up.
187 325 is 190 000 when rounded to the nearest ten thousand.
187 325 ≈ 190 000
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.
b) Round 4 325 680 to the nearest hundred thousand.
4 300 000 4 350 000 4 400 000 4 325 680
4 325 680 is 4 300 000 when rounded to the nearest hundred thousand.
4 325 680 ≈ 4 300 000
c) Round 8 500 000 to the nearest million.
8 500 000 ≈ 1.4
8 500 000
The digit in the ten thousands place is 2. So, we round down.
The digit in the hundred thousands place is 5. So, we round up.
8 000 000 9 000 000
8 500 000 is 9 000 000 when rounded to the nearest million.
Let's Do Let's Do
1. Round 183 260 to the nearest ten thousand and to the nearest hundred thousand.
a) 183 260
180 000
000
000
183 260 is when rounded to the nearest ten thousand. b)
183 260 is when rounded to the nearest hundred thousand.
2. Round 5 540 986 to the nearest million.
5 540 986 is when rounded to the nearest million.
3. Round 6 375 819 to the nearest a) ten thousand. b) hundred thousand.
c) million.
Let's Practice Let's Practice
1. Complete the table.
a) Estimate the sum of 45 943 and 2380.
45 943 + 2380 ≈ 50 000 + 2000 =
b) Estimate the difference between 732 946 and 29 682. 732 946 – 29 682 ≈ 700 000 – 30 000 =
1. Estimate the value of each of the following.
a) 261 380 + 2998 ≈ + =
b) 1 394 188 – 423 782 ≈ –=
Let's Practice Let's Practice
1. Estimate the value of each of the following.
a) 14 034 + 49 558
b) 211 495 + 2 376 342
c) 69 853 – 45 914
d) 9 275 439 – 349 845
a) Estimate the product of 24 836 and 5. 24 836 × 5 ≈ 20 000 × 5 = 100 000
2 ten thousands × 5 = 10 ten thousands
b) Estimate the quotient.
5 541 238 ÷ 6 ≈ 5 400 000 ÷ 6 = 900 000
4 800 000, 5 400 000, 6 000 000 5 541 238 is nearer to 5 400 000 than to 6 000 000. 54 hundred thousands ÷ 6 = 9 hundred thousands
Let's Do Let's Do
1. Estimate the value of each of the following.
a) 58 409 × 6 ≈ × =
b) 349 881 ÷ 7 ≈ ÷ =
Let's Practice Let's Practice
1. Estimate the value of each of the following.
a) 81 429 × 4
b) 634 959 × 3
c) 77 239 ÷ 5
d) 7 985 691 ÷ 8
I have learned to... understand how our number system is developed read and write numbers within 10 000 000 identify the values of digits in 7-digit numbers compare and order numbers within 10 000 000 round whole numbers to the nearest ten thousand, hundred thousand or million estimate sums, differences, products and quotients
You will learn to...
• find the common factors and the greatest common factor of two numbers
• find out if a number is a common factor of two given numbers
Let's Learn Let's Learn
Find the common factors and the greatest common factor of 30 and 42. We write the factors of 30 and 42 first.
1 × 30 = 30
1 × 42 = 42
2 × 15 = 30 2 × 21 = 42
3 × 10 = 30 3 × 14 = 42
5 × 6 = 30 6 × 7 = 42
1, 2, 3, 5, 6, 10, 15 and 30
1, 2, 3, 6, 7, 14, 21 and 42 are factors of 30. are factors of 42.
1, 2, 3 and 6 are factors of 30 and 42.
1, 2, 3 and 6 are common factors of 30 and 42.
Compare the common factors. The greatest common factor (GCF) of 30 and 42 is 6.
Let's Do Let's Do
1. Find the common factors and the greatest common factor of 50 and 40.
1 × 50 = 50 1 × 40 = 40
2 × = 50 2 × = 40 × = 50 4 × = 40 × = 40
a) The common factors of 50 and 40 are .
b) The greatest common factor of 50 and 40 is .
1. Complete the table.
NumbersCommon factorsGreatest common factor 15 and 45 16 and 56 a) b)
P B Chapter 1: Exercise 2.1
>> Look at EXPLORE on page 1 again. Can you solve the problem now? What else do you need to know?
2.2 Finding out if a number is a common factor of two given numbers
Let's Learn Let's Learn
a) Is 3 a common factor of 36 and 54?
36 and 54 can be divided by 3 exactly.
3 is a factor of 36; 3 is also a factor of 54.
3 is a common factor of 36 and 54.
b) Is 3 a common factor of 72 and 49?
72 can be divided by 3 exactly. 49 cannot be divided by 3 exactly. 3 is a factor of 72 but not a factor of 49.
3 a common factor of 72 and 49.
Let's Do Let's Do
1. Is 8 a common factor of 104 and 128? 8104
104 be 128 be divided by 8 exactly. divided by 8 exactly. So, 8 a common factor of 104 and 128.
Let's Practice Let's Practice
1. a) Is 9 a common factor of 117 and 135?
b) Is 6 a common factor of 72 and 140?
2. Circle the numbers that are common factors of 48 and 96. 3 4 5 9 12
P B Chapter 1: Exercise 2.2
I have learned to... find the common factors and the greatest common factors of two numbers find out if a number is a common factor of two given numbers
You will learn to...
• find the common multiples and the least common multiple of two numbers
• find out if a number is a common multiple of two given numbers
Let's Learn Let's Learn
Find the common multiples and the least common multiple of 3 and 4.
First, we list some of the multiples of 3 and 4.
1 × 3 = 3 2 × 3 = 6 3 × 3 = 9
4 × 3 = 12 5 × 3 = 15 6 × 3 = 18
7 × 3 = 21 8 × 3 = 24 9 × 3 = 27
10 × 3 = 30 11 × 3 = 33 12 × 3 = 36
The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 . . .
1 × 4 = 4 2 × 4 = 8 3 × 4 = 12
4 × 4 = 16 5 × 4 = 20 6 × 4 = 24
7 × 4 = 28 8 × 4 = 32 9 × 4 = 36
10 × 4 = 40 11 × 4 = 44 12 × 4 = 48
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48 . . .
Then, we compare the multiples of 3 and 4.
12 is a multiple of 3 and 4.
12 is a common multiple of 3 and 4.
The next two common multiples of 3 and 4 are 24 and 36.
The least common multiple (LCM) of 3 and 4 is 12.
Let's Do Let's Do
1. Find the first two common multiples and the least common multiple of 2 and 6.
Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 …
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 …
a) The first two common multiples of 2 and 6 are
b) The least common multiple of 2 and 6 is .
Sarah and David are discussing whether they can find the greatest possible number of candies Joe has.
Joe has some candies. He can pack them equally into bags of 3 or bags of 5. Is David correct? Why do you say so?
Can you find the greatest possible number of candies Joe has?
Yes, I can.
What did you learn about the common multiples of two numbers?
Think of a time in your daily life when you need to use the least common multiple of two numbers.
1. a) Write the first four multiples of 9 and 6.
Multiples of 9: Multiples of 6: b) What is the least common multiple of 9 and 6?
2. Find the least common multiple of 3 and 5.
a) Is 72 a common multiple of 2 and 4?
72 can be divided by 2 and 4 exactly.
72 is a multiple of 2; 72 is also a multiple of 4.
72 is a common multiple of 2 and 4.
b) Is 104 a common multiple of 4 and 6?
104 can be divided by 4 exactly. 104 cannot be divided by 6 exactly.
104 is a multiple of 4 but not a multiple of 6.
104 a common multiple of 4 and 6.
Let's Do Let's Do
1. Is 216 a common multiple of 3 and 8?
216 be 216 be divided by 3 exactly. divided by 8 exactly.
So, 216 a common multiple of 3 and 8.
Let's Practice Let's Practice
1. a) Is 252 a common multiple of 6 and 7?
b) Is 378 a common multiple of 5 and 9?
2. Circle the numbers that are common multiples of 5 and 10.
I have learned to... find the common multiples and the least common multiple of two numbers find out if a number is a common multiple of two given numbers
You will learn to...
• solve word problems involving common factors and multiples
• solve a non-routine problem involving common multiples
Let's Learn Let's Learn Understand the problem. 1 Plan what to do. 2
1. Nicole has 24 apples, 36 oranges and 30 cherries. She wants to make identical fruit platters using all of the fruits. What is the greatest number of platters she can make?
What types of fruits does Nicole have? How many fruits of each type are there?
What does Nicole want to do? What do I have to find?
The number of apples on each platter must be the same. This is also true for oranges and cherries. I can make a list to find the number of platters I can make with each type of fruit.
Work out the Answer. 3
Apples:
1 × 24 = 24 2 × 12 = 24
3 × 8 = 24 4 × 6 = 24
Oranges: Cherries:
1 × 36 = 36 1 × 30 = 30
2 × 18 = 36 2 × 15 = 30
3 × 12 = 36 3 × 10 = 30
4 × 9 = 36 5 × 6 = 30
6 × 6 = 36
Nicole can make 1 platter of 24 apples, 24 platters of 1 apple, 2 platters of 12 apples, 12 platters of 2 apples, …
The common factors of 24, 36 and 30 are 1, 2, 3 and 6.
The greatest common factor of 24, 36 and 30 is 6.
The greatest number of platters Nicole can make is 6.
4
Check if your answer is correct.
5
+ Plus Solve the problem in another way.
4 × 6 = 24 6 × 6 = 36 5 × 6 = 30
There are 4 apples, 6 oranges and 5 cherries on each platter. The platters are identical.
My answer is correct.
1 × 24 = 24 2 × 12 = 24
3 × 8 = 24 4 × 6 = 24
The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
Nicole can make 1, 2, 3, 4, 6, 8, 12 or 24 platters with 24 apples.
We want to find the greatest number of platters that Nicole can make. So, we start with the greatest factor of 24.
The greatest number of platters Nicole can make is 6.
Compare the methods in steps 3 and 5. Which method do you prefer? Why?
1. Understand 2. Plan 3. Answer 4. Check 5. Plus
2. Gold buttons are sold in packs of 12, silver buttons are sold in packs of 18 and black buttons are sold in packs of 24. If you want to have the same number of buttons in each color, what is the least number of packs of buttons of each color you need to buy? Understand the problem.
1
What colors are the buttons? How many buttons are there in each pack? What do I have to find?
3
Plan what to do.
2 Work out the Answer
The multiples of 12 are 12, 24, 36, 48, 60, 72, 84, …
The multiples of 18 are 18, 36, 54, 72, 90, …
The multiples of 24 are 24, 48, 72, 96, …
The least common multiple of 12, 18 and 24 is 72. 72 buttons of each color are needed.
12 × 6 = 72 18 × 4 = 72 24 × 3 = 72
I need to buy 6 packs of gold buttons, 4 packs of silver buttons and 3 packs of black buttons.
4 + Plus Solve the problem in another way.
Check if your answer is correct.
Since the buttons are sold in packs, I can make a list of the possible number of buttons I can buy. 72 ÷ 6 = 12
÷ 4 = 18
My answer is correct.
5 72 ÷ 12 = 6
÷ 3 = 24
The greatest number among 12, 18 and 24 is 24 so the number of buttons will be a multiple of 24. The multiples of 24 are 24, 48, 72, 96, …
Find the least multiple of 24 that can be divided by 12 and 18 exactly.
÷ 18 = 4 72 ÷ 24 = 3
I need to buy 6 packs of gold buttons, 4 packs of silver buttons and 3 packs of black buttons.
Compare the methods in steps 3 and 5. Which method do you prefer? Why?
Solve the word problems. Show your work clearly.
Next, try solving each problem in a different way. Which method do you prefer? Why?
1. Understand 2. Plan 3. Answer 4. Check 5. Plus
1. Mike has two rods of lengths 18 centimeters and 42 centimeters. He wishes to cut the two rods into shorter pieces of equal length with no remainder. What is the greatest possible length of each of the shorter pieces of rod?
Find the greatest common factor of 18 and 42.
2. Oranges are sold in packs of 8, lemons in packs of 10 and plums in packs of 20. What is the least number of packs of each type of fruits that Lisa needs to buy if she wants to make a bag having an equal number of each type of fruits in it, with none left over?
Let's Practice Let's Practice
Solve the word problems. Show your work clearly.
1. Ralph makes snack bags using 12 granola bars, 18 pretzels and 30 cheese sticks. Each snack bag has the same mix of snacks in the same quantities. What is the greatest number of snack bags Ralph makes?
2. Mrs. Sanders has 36 walnuts, 54 pecans and 90 almonds. She wants to place an equal number of nuts in bowls, without mixing the different types. What is the greatest number of nuts she can have in each bowl?
3. A red lamp flashes every 10 seconds, a yellow lamp flashes every 24 seconds and a blue lamp flashes every 30 seconds. How many times will the three lamps flash at the same time in 10 minutes?
4. A radio station is having a contest in which every 12th caller receives a voucher, every 15th caller receives a concert ticket and every 18th caller receives a limousine ride. Which caller will be the first to win all three prizes?
P B Chapter 1: Exercise 4.1
Route A bus arrives at a bus stop every 25 minutes. Route B bus arrives at the same bus stop every 10 minutes. Both buses are at the bus stop now. In how many minutes will the two buses be at the bus stop again?
Read the word problem. Change the numbers in the word problem. How did you decide what numbers to use?
Next, solve the word problem. Show your work clearly. What did you learn?
4.2 Mind stretcher
Kelly wants to give some candies to her friends. If she gives each friend 8 candies, she will have 5 candies left. If she gives each friend 9 candies, the last friend will receive 8 candies. How many candies does Kelly have?
How many candies does Kelly want to give to her friends?
How many candies will there be left? How many candies will the last friend receive? Does the number of friends change? What do I have to find?
I should make a list. Then, find the common number.
has 53 candies.
5
4 + Plus Solve the problem in another way.
1.Understand 2.Plan 3.Answer 4.Check 5.Plus Check if your answer is correct.
53 ÷ 8 = 6 R5
53 ÷ 9 = 5 R8
My answer is correct.
If Kelly gives each friend 8 candies, she will have 5 candies left. If she gives the remaining 5 candies to her friends so that each friend has 9 candies, she will be short of 1 candy. This means that there are 6 friends.
6 × 8 = 48
48 + 5 = 53
Kelly has 53 candies.
Compare the methods in steps 3 and 5. Which method do you prefer? Why?
I have learned to... solve word problems involving common factors and multiples solve a non-routine problem involving common multiples
>> Look at EXPLORE on page 1 again. Fill in column 3. Can you solve the problem now?
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.
1. Write five million, four hundred and seven thousand and eleven as a numeral.
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 ...................
Exercise 1.4 Rounding whole numbers to the nearest ten thousand, hundred thousand or million .............
Exercise 1.5
Exercise 1.6 Estimating products and quotients
Exercise 2.1 Finding common factors of two numbers
Exercise 2.2 Finding out if a number is a common factor of two given numbers .......................................
Exercise 3.1 Finding common multiples of two numbers ........
Exercise 3.2 Finding out if a number is a common multiple of two given numbers .......................................
Exercise
Chapter 2 Multiplication and Division of Whole Numbers
Exercise
Exercise
Exercise
Exercise
Exercise
Exercise 3.1
Chapter 3 Basic Operations Using Calculators
Exercise
Exercise
Exercise
Chapter 4 Introduction to Algebra
Exercise
Exercise 1.2 Finding the value of an algebraic expression involving addition or subtraction ......................
Exercise 1.3 Writing and evaluating algebraic expressions involving multiplication ..................................
Exercise 1.4 Writing and evaluating algebraic expressions involving division ..........................................
Exercise 1.5 Writing and evaluating algebraic expressions involving more than one operation ..................
Chapter 5 Quadrilaterals
Exercise 1.1 Properties of squares and rectangles ................
Exercise 1.2 Finding unknown angle measures ....................
Exercise 1.3 Finding unknown lengths ................................
Exercise 2.1 Identifying different types of quadrilaterals .........
Exercise 2.2 Classifying quadrilaterals ................................
Chapter 6 Decimals
Exercise 1.1 Multiplying tenths, hundredths or thousandths by 10 ......................................................... 63
Exercise 1.2 Multiplying decimals by 10 ............................. 64
Exercise 1.3 Multiplying decimals by tens ...........................
Exercise 1.4 Multiplying decimals by 100 ............................ 66
Exercise 1.5 Multiplying decimals by 1000 ........................... 67
Exercise 1.6 Multiplying decimals by hundreds or thousands ...
Exercise 2.1 Dividing ones or tenths by 10 ..........................
Exercise 2.2 Dividing whole numbers or decimals by 10
Exercise 2.3 Dividing whole numbers or decimals by tens ......
Exercise 2.4 Dividing whole numbers by 100 .......................
Exercise 2.5 Dividing whole numbers by 1000 ...................... 73
Exercise 2.6 Dividing whole numbers by hundreds or thousands................................................ 74
Exercise 3.1 Multiplying decimals by 2-digit whole numbers ....
Exercise 3.2 Solving word problems ..................................
Exercise 4.1 Multiplying tenths by tenths or hundredths
Exercise 4.2 Multiplying decimals .....................................
Exercise 4.3 Solving word problems ..................................
Exercise 5.1 Dividing decimals by 2-digit whole numbers .......
Exercise 5.2 Dividing whole numbers by decimals ................
Exercise 5.3 Solving word problems ..................................
Exercise 6.1 Word problems ............................................
Chapter 7 Length, Mass and Capacity
Exercise 1.1 Imperial units of measurement ........................ 90
Exercise 1.2 Converting a measurement from a larger unit to a smaller unit .............................................. 91
Exercise 1.3 Converting a measurement from a larger unit to compound units .......................................... 92
Exercise 1.4 Converting a measurement from a smaller unit to a larger unit ................................................ 93
Exercise 1.5 Converting a measurement from compound units to a larger unit ............................................ 94
Chapter 8 Mental Strategies
Exercise 1.1 Multiplying 2-digit numbers by 1-digit numbers .... 95
Exercise 1.2 Multiplying two 2-digit numbers using factors ...... 96
Exercise 1.3 Multiplying two 2-digit numbers by halving and doubling .............................................. 97
Exercise 1.4 Dividing 2-digit numbers by 1-digit numbers ........ 98
Review 2 .................................................................. 99
Chapter 9 Positive and Negative Numbers
Exercise 1.1 Reading and placing positive and negative numbers on number lines ............................... 105
Exercise 1.2 Comparing positive and negative numbers on number lines ............................................... 106
Exercise 2.1 Adding integers ........................................... 107
Exercise 2.2 Subtracting integers ...................................... 108
Exercise 2.3 Completing number patter ns .......................... 109
Chapter 10 Position and Movement
Exercise 1.1 Reading and plotting points in all four quadrants ................................................... 110
Exercise 1.2 Drawing polygons on a Cartesian plane ............ 113
Exercise 2.1 Translations ................................................. 114
Exercise 2.2 Reflections .................................................. 116
Exercise 2.3 Rotations .................................................... 118
Chapter 11 Ratio
Exercise 1.1 Using a ratio to compare two quantities ............ 120
Exercise 1.2 Using a ratio to compare a quantity with the total quantity ............................................... 121
Exercise 1.3 Using a bar model to show a ratio ................... 122
Exercise 2.1 Writing equivalent ratios ................................ 123
Exercise 2.2 Writing a ratio in its simplest form ...................... 124
Exercise 2.3 Finding a missing ter m in a pair of equivalent ratios ......................................................... 125
Exercise 3.1 Using ratios to show quantities in proportion ........ 126
Exercise 4.1 Word problems ............................................ 128
Chapter 12 Percentage
Exercise 1.1 Understanding percentage ............................ 131
Exercise 1.2 Expressing fractions as percentages .................. 132
Exercise 1.3 Expressing decimals as percentages ................. 133
Exercise 1.4 Expressing percentages as decimals ................. 134
Exercise 1.5 Expressing percentages as fractions .................. 135
Exercise 2.1 Relating 1 whole to 100% ................................ 136
Exercise 2.2 Expressing fractions with denominators less than 100 as percentages ...................................... 137
Exercise 2.3 Expressing parts of a whole as percentages ....... 138
Exercise 2.4 Expressing fractions with denominators greater than 100 as percentages ............................... 139
Exercise 2.5 Comparing fractions, decimals and percentages ............................................... 140
Exercise 2.6 Solving 2-step word problems .......................... 141
Exercise 3.1 Finding the value of a percentage of a quantity .. 144
Exercise 4.1 Word problems involving percentage of a quantity ................................................... 145
Exercise 4.2 Word problems involving interest, tax and discount ..................................................... 148
Exercise 4.3 Word problems involving increase and decrease .................................................... 151 Review 3 .................................................................. 154
Chapter 13 Probability
Exercise 1.1 Listing possible outcomes of events .................. 162
Exercise 1.2 Describing and comparing likelihood of events ... 163
Exercise 2.1 Finding the probability of an event ................... 164
Exercise 2.2 Understanding probabilities from 0 to 1 .............. 166
Exercise 2.3 Understanding the difference between theoretical and experimental probabilities ......... 168
Chapter 14 Area and Perimeter
Exercise 1.1 Breaking apart rectangles .............................. 169
Exercise 1.2 Finding perimeters of composite figures ............ 170
Exercise 1.3 Finding area of a rectangle by adding areas of two smaller rectangles .................................. 171
Exercise 1.4 Finding areas of composite figures by adding areas ............................................... 172
Exercise 1.5 Finding areas of composite figures by subtracting areas ......................................... 174
Exercise 2.1 Word problems ............................................ 176
Chapter 15 Time
Exercise 1.1 Converting decades and centuries ................. 179
Exercise 2.1 Calculating time in different time zones ............ 180
Chapter 16 Graphs
Exercise 1.1 Reading and interpreting conversion graphs ...... 181
Exercise 2.1 Reading and interpreting pie charts which display whole numbers .................................. 183
Exercise 2.2 Reading and interpreting pie charts which display fractions ........................................... 186
Exercise 2.3 Reading and interpreting pie charts which display percentages ..................................... 189
Chapter 17 Introduction to Statistics
Exercise 1.1 Posing statistical questions .............................. 192
Exercise 1.2 Collecting data ........................................... 193
Exercise 2.1 Making, reading and interpreting a dot plot ....... 195
Exercise 2.2 Describing distribution of data in a dot plot ........ 197
Exercise 3.1 Making, reading and interpreting a histogram .... 199
Exercise 3.2 Describing distribution of data in a histogram ..... 202
Chapter 18 Data Distribution
Exercise 1.1 Finding the mean of a set of data .................... 204
Exercise 1.2 Finding the mean given the total amount and the number of data ...................................... 206
Exercise 1.3 Finding the total amount given the mean and the number of data ...................................... 207
Exercise 1.4 Finding the median of a set of data .................. 208
Exercise 1.5 Finding the mode of a set of data .................... 209
Exercise 1.6 Describing distribution using mean, mode and median ................................................ 210
Exercise 2.1 Word problems ............................................ 211
Recap
5 768 905 is read as five million, seven hundred and sixty-eight thousand, nine hundred and five.
Six million, nine hundred and five thousand, three hundred and twenty is written as the numeral 6 905 320.
1. Write the numerals.
a) one million, two hundred and fifty thousand, four hundred and ninety-seven
b) four million, fifty thousand and fourteen
c) five million, three hundred and two thousand, eight hundred and eighty-eight
d) two million and sixty
2. Write the numbers in words.
a) 9 652 039
b) 6 731 546
c) 3 400 808
d) 1 090 040
In 6 041 729, the digit 6 is in the millions place. It stands for 6 000 000 and its value is 6 000 000.
6 041 729 = 6 000 000 + 40 000 + 1000 + 700 + 20 + 9
1. Write the missing numbers or words.
a) In 7 564 328, the digit 7 is in the place and its value is
b) In 9 021 635, the digit 0 is in the place and its value is .
c) In 2 803 157, the digit 3 stands for .
d) In 5 042 198, the digit is in the millions place and its value is .
e) In 4 310 247, the digit is in the ten thousands place and its value is .
2. Write the missing numbers.
a) 2 903 507 = + 900 000 + 3000 + 500 + 7
b) 3 210 058 = 3 000 000 + + 10 000 + 50 + 8
c) 8 027 013 = 8 000 000 + 20 000 + + 10 + 3
d) 7 490 900 = 7 000 000 + 400 000 + + 900
Recap
Compare 4 550 100 and 4 437 904.
First, compare the millions. They are the same.
4 550 100 > 4 437 904
Then, compare the hundred thousands. 5 hundred thousands is greater than 4 hundred thousands.
1. Write > or <. a) 4 300 520 4 000 520 b) 1 208 765 1 280 765 c) 7 100 321 7 010 321 d) 8 231 580 8 231 805
2. Arrange the numbers in order. Begin with the greatest. a) 4 690 800, 5 690 800, 969 800 b) 1 345 600, 1 234 560, 1 345 660, 2 345 665
3. Arrange the numbers in order. Begin with the least.
a) 5 500 555, 500 555, 5 500 505 b) 3 154 234, 3 452 896, 4 872 435, 3 154 324
Recap
Rounding whole numbers to the nearest ten thousand, hundred thousand or million
Round 6 374 092 to the nearest hundred thousand.
6 300 000 6 400 000 6 374 092
6 374 092 is between 6 300 000 and 6 400 000. It is nearer to 6 400 000 than to 6 300 000. So, we round up.
6 374 092 is 6 400 000 when rounded to the nearest hundred thousand.
6 374 092 ≈ 6 400 000
1. Round each number to the nearest ten thousand.
a) 251 312 ≈ b) 437 543 ≈
c) 8 054 631 ≈ d) 6 278 914 ≈
e) 5 725 097 ≈ f) 1 000 986 ≈
2. Round each number to the nearest hundred thousand.
a) 541 321 ≈ b) 850 987 ≈ c) 6 304 092 ≈ d) 1 867 234 ≈
e) 4 214 361 ≈ f) 3 954 364 ≈
3. Round each number to the nearest million.
a) 2 532 070 ≈ b) 1 964 376 ≈
c) 4 603 986 ≈ d) 5 407 092 ≈
e) 6 093 179 ≈ f) 9 804 387 ≈ Chapter 1: Practice 1.4
Recap
Estimate the sum of 246 703 and 45 290.
246 703 + 45 290 ≈ 200 000 + 50 000 = 250 000
Estimate the difference between 4 821 590 and 704 228.
4 821 590 − 704 228 ≈ 5 000 000 − 700 000 = 4 300 000
1. Estimate the value of each of the following.
a) 850 701 + 56 641 ≈ + =
b) 605 302 + 424 991 ≈ + =
c) 8 089 729 + 92 045 ≈ + =
d) 108 046 – 5538 ≈ − =
e) 1 492 111 – 239 817 ≈ − =
f) 7 047 643 – 3 590 399 ≈ − =
2. Estimate the value of each of the following.
a) 27 240 + 6684 ≈
b) 899 668 + 6 259 153 ≈
c) 2 379 001 + 3 463 568 ≈
d) 4 574 294 – 472 377 ≈
e) 47 858 – 1756 ≈
f) 5 347 620 – 3 774 175 ≈
Recap
Estimate the product of 49 831 and 3.
49 831 × 3 ≈ 50 000 × 3 = 150 000
Estimate the value of 308 627 ÷ 8.
308 627 ÷ 8 ≈ 320 000 ÷ 8 = 40 000
240 000, 320 000, 400 000 308 627 is nearer to 320 000 than to 240 000.
320 thousands ÷ 8 = 40 thousands
1. Estimate the value of each of the following.
a) 22 529 × 3 ≈ × =
b) 959 224 × 7 ≈ × =
c) 1 898 910 × 5 ≈ × =
d) 49 206 ÷ 4 ≈ ÷ =
e) 247 460 ÷ 8 ≈ ÷ = f) 1 665 113 ÷ 6 ≈ ÷ =
2. Estimate the value of each of the following.
a) 301 782 × 9 ≈
b) 2 202 427 × 4 ≈
c) 79 099 × 6 ≈
d) 46 992 ÷ 3 ≈
e) 235 652 ÷ 7 ≈
f) 5 966 069 ÷ 5 ≈
Recap
Find the common factors and the greatest common factor of 36 and 54.
1, 2, 3, 4, 6, 9, 12, 18 and 36 are factors of 36.
1, 2, 3, 6, 9, 18, 27 and 54 are factors of 54.
The common factors of 36 and 54 are 1, 2, 3, 6, 9 and 18.
The greatest common factor of 36 and 54 is 18.
1. a) Find the factors of 48.
The factors of 48 are .
b) Find the factors of 56.
The factors of 56 are
c) The common factors of 48 and 56 are .
d) The greatest common factor of 48 and 56 is .
2. Complete the table.
a) b) c) NumbersCommon
25 and 75 32 and 96 28 and 100
Exercise 2.2 Finding out if a number is a common factor of two given numbers
Recap
Is 5 a common factor of 50 and 75?
50 and 75 can be divided by 5 exactly. 5 is a common factor of 50 and 75.
1. a) Is 3 a common factor of 42 and 90?
b) Is 7 a common factor of 63 and 104?
c) Is 9 a common factor of 54 and 99?
2. Circle the numbers that are common factors of 36 and 72.
Recap
Find the common multiples and the least common multiple of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …
The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, …
12 is a common multiple of 4 and 6.
The next two common multiples of 4 and 6 are 24 and 36.
The least common multiple of 4 and 6 is 12.
1. a) List the multiples of 7.
b) List the multiples of 2.
c) The first two common multiples of 7 and 2 are and .
d) The least common multiple of 7 and 2 is .
2.
a) List the multiples of 5.
b) List the multiples of 4.
c) The first two common multiples of 5 and 4 are and .
d) The least common multiple of 5 and 4 is .
3.
a) List the multiples of 3.
b) List the multiples of 8.
c) The first two common multiples of 3 and 8 are and .
d) The least common multiple of 3 and 8 is .
4. Find the least common multiple of 9 and 6.
Exercise 3.2 Finding out if a number is a common multiple of two given numbers
Recap
Is 36 a common multiple of 4 and 8?
36 can be divided by 4 exactly.
36 cannot be divided by 8 exactly.
36 is a multiple of 4 but not a multiple of 8.
36 is not a common multiple of 4 and 8.
1. Is 315 a common multiple of 3 and 6?
2. Is 265 a common multiple of 4 and 5?
3. Is 448 a common multiple of 7 and 8?
4. Circle the numbers that are common multiples of 4 and 6.
Solve the word problems. Show your work clearly.
1. Understand 2. Plan 3. Answer 4. Check 5. Plus
1. Davis has two strips of paper measuring 21 centimeters and 49 centimeters. He wants to cut the strips of paper into shorter pieces of equal length with no remainders. What is the greatest possible length of each of the shorter pieces of paper?
2. Green pens are sold in packs of 3 and blue pens are sold in packs of 4. If Jess wants to have the same number of pens in each color, what is the least number of packs of pens of each color she has to buy?
3. Zac has 12 cookies, 20 sweets and 32 stickers. He wants to distribute all the items to his friends. Each friend should get the same set of items. What is the greatest number of friends to whom he can distribute the items?
4. Alarm A rings every 5 minutes, alarm B rings every 7 minutes and alarm C rings every 10 minutes. How many times will the three alarms ring at the same time in 2 hours?
5. Sandra has 36 sheets of green paper, 54 sheets of blue paper and 72 sheets of white paper. She wants to make identical notebooks using all the colored paper. What is the greatest number of notebooks Sandra can make?
6. Dark chocolate bars are sold in bags of 12, almond chocolate bars in bags of 10 and milk chocolate bars in bags of 15. What is the least number of bags of each type of chocolate bar Aaron needs to buy if he wants to have an equal number of each type of chocolate bar?