for Teaching Portfolio: During this math class, we went over the
Mathematics many different strands within
the Ontario math curriculum. We went over the five strands within the curriculum; Number Sense and Numeration, Measurement, Geometry and Spatial Sense, Patterning and algebra, and Data Management and Probability. Each strand was then reinforced by our colleagues; going over fractions & decimals, integers, proportional thinking, patterning & algebra, geometry, measurement, data management & probability, and technology. All of these topics are covered within the five strands of mathematics respectively, and the resources were collected from our colleagues provided within their presentations regarding their topics. The first strand within the Ontario Math curriculum is Number Sense and Numeration, which is overviewed within the grade levels 1-8. This strand includes fractions, decimals and whole numbers, multiplication of fractions and decimals, integers, and proportional relationships. Number Sense and numeration requires understanding relationships, having an operational sense, and knowing proportional relationships. The second strand within the Ontario Math curriculum is measurement, which can be encapsulated by the presented topic of measurement. This strand includes metric units of area, volume, capacity, estimation, and the use of various different tools. The main topic within this strand is measuring attributes, including the area of an abject, and the volume of a prism.
The third strand within this Ontario math curriculum is Geometry and Spatial Sense. This strand is concerned with understanding geometric properties, relationships involving lines, triangles, polygons, and relating them to the Cartesian planes. The most important topic I found was being able to make connections between transformations and the real world. Geometric properties, relationships and location and movement were the topics that our colleagues went over. The fourth strand was patterning and algebra, representing linear growth patterns, using graphs, algebraic expressions, and equations.
The first brave presenters for this class were Julia, Zach, and Kevon. All three of these teacher candidates presented their topic of integers. Below is Julia’s presentation on Integers.
LEARNING ACTIVITY PRESENTATION Topic: Fractions Grade Level: Grade 7 and up Mathematics curriculum strand: Number sense, Numeration and Patterning. Content Expectation: Adding and subtraction of simple fractions and representing the growing pattern relationship (Page no. 97) Process Expectation: (Page no. 98) 1. Problem solving- Develop, select, apply, and compare a variety of problem-solving strategies as they pose and solve problems and conduct investigations, to help deepen their mathematical understanding. 2. Reasoning and Proving: develop and apply reasoning skills (e.g., recognition of relationships, generalization through inductive reasoning, use of counter-examples) to make mathematical conjectures, assess conjectures and justify conclusions, and plan and construct organized mathematical arguments. Source: Nelson- Making Math Meaningful to Canadian Students, K-8. Chapter 11, Activity 11.14 Date: October 2nd. 2015 Name: Anjali Sharma
This activity is designed to explore sums and differences of fraction that form a pattern. I worked with different types of neighboring and related fraction and found that, they were forming patterns with numerators and denominators. 1 1 1 1 1 1 1 , , , , , , . Addition of these 2 3 4 5 6 7 8 fractions creates a pattern where, numerator and denominator increase by 2. Type 1: Neighboring fraction-
Type 2: Fractions with common denominator -
2 , 8,
3 4 5 6 7 , , , , On Addition, 8, 8, 8 8 8
numerator increases by 2. Type 3: Denominator of 1st fraction is numerator of next fraction-
1 2 3 4 , , , , 2 3 4 5
5 6 7 , , . On adding, numerator remains same and denominator increases by 2. 6 7 8 3 4 5 6 7 , , , , . Addition shows interesting results. 2 3 4 5 8 Here, numerator increases by 4 and denominator by 2. Type 4: Improper fractions:
Decimals Making Math Meaningful to Canadian Students, K-8: Marian Small October 2nd, 2015, Mariska Ceci
Target Grade Level: Grade 4/5 Curriculum Strand: Number Sense & Numeration Activity 12.4 (pg. 285, text by Marian Small) Students can colour designs on a decimal grid and give the design a decimal value. Students can also be given a value and asked to draw something to match it. Today we will be using 3 colours to create our designs from the initial decimal value and each colour needs to be given its own decimal value. 100 square grid = 1 whole For example: Pumpkin Drawing in 0.72 of a whole - Orange = 0.60, Green = 0.02, Black = 0.10 ďƒ Total = 0.72 of a whole Expectations: Gr. 4 (pg. 66- 67, Curriculum)- decimal numbers to 10ths, demonstrate understanding of magnitude by counting forward & backward by 0.1, addition & subtraction of decimal numbers to 10ths
- Demonstrate an understanding of place value in whole numbers & decimal numbers for 0.1 – 10 000, represent, compare & order decimal numbers to 10ths using a variety of tools Gr. 5 (pg. 78 -79, Curriculum) – decimal numbers to 100ths, counting backward and forward by 0.01 - Demonstrate & explain equivalent representation of decimal numbers using concrete materials & drawings (0.3 = 0.30)
Integers Julia Chamberlain October 9, 2015
Minds On: Introduction • • •
Use a large-scale deck of cards to introduce integers and represent and order integers by comparing them to real life tools/manipulatives. Deck of cards: Black cards are negative and reds are positive Ace is low and equal to 1 and Jokers counts as 0. The line simulates the number scale we use for integers and helps for visual learning and hands on, minds on involvement activity.
Activity: Integro (Activity 14.6 in Making Math Meaningful. Ch.14, pg • • • • • •
327) Rules: In groups of 2 or 4, a student shuffles and deals cards equally to their group (Using only numbers 2-10 and Aces -- Reds cards are positives, Black cards are negative, Aces are 1, Remove face cards and jokers) In a round, each player places one card face up on the table. The first person to call out the sum of the cards wins all the cards in the turn. These cards go into the players bank pile. Tied players play additional rounds until someone wins. When a player runs out of cards, the player shuffles his or her bank pile and continues playing. If the player’s bank is empty the player is out. The game ends when one player has won all the cards.
Consolidation: Integers start to show up in the Ontario curriculum in Grade 7 and are a part of the Number Sense and Numeration stream. By the end of grade 7 students have the overall expectation to, “represent, compare, and order numbers, including integers,” and also, “demonstrate an understanding of addition and subtraction of fractions and integers, and apply a variety of computational strategies to solve problems involving whole numbers and decimal numbers.” Their specific expectations are to, “represent and order integers, using a variety of tools (e.g., two-colour counters, virtual
manipulatives, number lines)” as well as, “add and subtract integers, using a variety of tools (e.g., two-colour counters, virtual manipulatives, number lines).” This activity would ideally be used in grade 7 classrooms, where they are first being introduced to integers and how they can be represented in addition and subtraction. Matt presented next:
Proportional Reasoning
October 23rd, 2015 Mathieu Carrière Activity Target Grade: 4-6 Source of activity: Making Math Meaningful to Canadian Students, K-8 A couple points on proportional reasoning: -The essence of proportional reasoning is the consideration of number in relative terms, rather than absolute terms. -Ratios are not introduced until grade 6, although they are introduced in informal ways earlier on. Example 1: Kindergarten teachers will say there are 2 eyes for every person. They are using the ratio 2:1 Example 2: A grade 2 or 3 teacher might ask how many wheels are on 5 bicycles. The students will use the ratio of 2:1 to solve the problem. Curriculum expectations for Grade 4 Number Sense and Numeration: Compare and order fractions (i.e., halves, thirds, fourths, fifths, tenths) by considering the size and the number of fractional p.66 Demonstrate an understanding of simple multiplicative relationships involving unit rates, through investigation using concrete materials and drawings p.68 Describe relationships that involve simple whole-number multiplication (e.g.,“If you have 2 marbles and I have 6 marbles, I can say that I have three times the number of marbles you have.”) p.68 Determine and explain, through investigation, the relationship between fractions (i.e., halves, fifths, tenths) and decimals to tenths, using a variety of tools p.68 Activity 13.7, p.311 -For a grade 4 class, I would tell students to enlarge the picture so that it is twice as high and twice as wide.
-For a grade 6 class, I could ask questions such as ‘What is the ratio of the pumpkin’s eyes?’ and ‘What is the new ratio of the pumpkin, if you enlarge it by half of its original
ratio? Kelsey presented next;
Patterning
Activity Target: Grade 4 Source of Activity: Making Math Meaningful to Canadian Students, K-8 Curriculum Expectations for Grade 4 Patterning and Algebra: Overall Expectations: • •
Describe extend and create a variety of numeric and geometric patterns make predictions related to the patterns, and investigate repeating patterns involving reflections; Demonstrate and understanding of equality between pairs of expressions, using addition, subtraction and multiplication (73)
Introduction to Patterning: Core: the shortest part of the pattern that repeats itself
Core Repeating Patterns are also sometimes described using a letter code ie. AAB Multi-Attribute Patterns: patterns that contain more than a single attribute ie. color, shape, size etc. Color Pattern: ABC Shape Pattern: ABB
Activity 22.5: Ask Students to choose a criterion from the list below for creating a pattern: • Use three colors of counters to create a pattern • Create a repeating pattern that has a core of three elements • Create a growing pattern where the 10th term is 100 • Create a pattern that grows but not by the same amount each time • Create a shrinking pattern where the 4th number is 16 Julian Foglia
Geometry- 2-D shapes intro
1) what is a polygon? Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up). Give examples: 2) Properties of polygons and different types: Triangles- they are classified in terms of their relationship to their sides... The length of the size, the angles of each side etc.--> in the text, they mention the types of triangles:
Quadrilaterals- they are 4-sided polygons. Most common are squares and rectangles however there are many other types. -Students must understand what properties are in polygons. They are, the traits and characteristics of each shape; angle, straight sided, curves etc. Students must list the properties of the triangles and the other shapes given. From there we can teach them what different types of lines/ segments there are in geometry. Parallel lines- Lines that do not meet and run in the same direction.
Intersection- the lines meet at a single point. Perpendicular- The lines intersect, but they only meet at a right angle.
Reflections - Geometry and Spatial Sense By: Marissa Di Camillo Activity Target Grades: 6 and 7 Curriculum Expectations: • Grade 6 – Create and analyse designs made by reflecting, translating, and/or rotating a shape, or shapes, by 90 degrees or 180 degrees (Pg. 93) • Grade 7 – Create and analyse designs involving translations, reflections, dilatations, and/or simple rotations of two-dimensional shapes, using a variety of tools (e.g., concrete materials, Mira, drawings, dynamic geometry software) and strategies (e.g., paper folding) (pg. 104) Source of Activity: Making Math Meaningful to Canadian Students, K-8 – Page 397 What is a flip/ Reflection? • A flip (or reflection) can be thought of as the result of picking up a shape and turning it over.
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A flip is always made over a line called the flip line or line of reflection. This line can be place horizontal, vertical or slanted. What is a transparent Mirror? • A transparent mirror is a useful tool for performing reflections. By placing the mirror in front of a shape, you can see the flip image when you look through the plastic at the other side. Activity 16.8 – Flips (Reflections) Ask students where to put the mirror on the original shape to create the two images. • First practice flips and reflections using the provided shapes and graph paper o The first flip will be vertical – leave 3 boxes between the shape and the line of reflection o The second flip will be horizontal – leave 5 boxes between the shape and the line of reflection o The third flip will be slanted – leave 2 boxes between the shape and the line of reflection • Using the provided shapes with the graph paper, figure out where the line of reflection would be
Original
3D
Geometry: Representing Shapes Nicole Horlings November 6, 2015
Activity Target Grade: 4 Source of Activity: Making Math Meaningful to Canadian Students, K-8 What does it mean to be able to represent a shape? - By demonstrating that they are able to create or draw a shape, students show that they have visualizations skills and a good grasp of spatial sense. - Being able to conceptualize a shape and accurately draw it is important for students to understand the relationship between 2-D and 3-D objects. - A real life example that demonstrates the importance of conceptualizing and representing shapes is an architect who makes blue prints for buildings and needs to understand what those 2-D blue prints will look like when they become 3-D buildings.
Curriculum expectations for Grade 4 Geometry and Spatial Sense: Overall expectation: - construct three-dimensional figures, using two-dimensional shapes (p. 71) Specific expectation: - construct skeletons of three-dimensional figures, using a variety of tools (e.g., straws and modelling clay, toothpicks and marshmallows, Polydrons), and sketch the skeletons (p. 71) Activity 15.13, p. 360 - “Use balls of clay for vertices and sticks for edges to build the skeleton of a 3-D shape� (p. 360). - Instead of clay, I will be using mini marshmallows for this activity - I will make the students create a cube using their tooth picks and marshmallows -
Once the students have created their cubes, I will ask them to record how many edges and vertices there are I will also ask the students what the angles that the cube has are called As an extra challenge if there is time, I will hand the students a sheet of isometric paper, and have them draw an image of the cube where 3 faces of the cube are visible.
Tim’s Presentation: My Instructions:
1) Introduce topic (measurement of length). Length is one-dimensional. 2) Length is: assigning a qualitative or quantitative description of size to an object based on particular attributes. In simple terms - measurements are markers that we use every day to help describe the dimensions of a particular thing (i.e. CN Tower is 553M, Toronto is about 110KM from St. Kits) 3) You’ll notice something about the measurements I just gave you: they’re widely known. They’re like benchmarks. Sometimes what we use to measure are standard measurement units, other times they are contextual (i.e. that truck weighs as much as a whale). Not perfect, but still get a sense of how much the truck weighs. 4) Standard measurements are used to simplify and clarify communication of size of objects and simplifies measurements. In other words – if we didn’t have CMs, Ms, KMs, etc… it would be difficult for us to tell others how far we are talking about. 5) You’ll notice one more thing: we are dealing in metres and centimetres; imperial (US, Liberia, Myanmar) (Wikipedia) metric (everywhere else). Who here thinks of their weight in pounds? Their size in feet & inches? (2m) Task: 1) Have five students come to the front and stand against the board (30s) 2) Line up from shortest to tallest, then estimate how tall they are in CM or M/CM (gave you a hint at the bottom of the page with inch conversion). Draw a line on the board with names while people estimate. (2m) 3) Have all students sit down; give students who volunteered a chance to estimate heights just from line on board and without a human to help them visualize); in meantime ask students to explain how they got their estimates (visualization, prior knowledge?) or ask them to describe what they see. (3m) 4) Ask for five volunteers to measure the lengths in CMs, write it on the corresponding line with their name (2m) • • •
5) Who thinks their estimates were close? Not close? Why do you think I wanted you to estimate first? (in the real world you don’t have a tape measure, can be quicker, what’s more important: the time it takes to measure or exact measurement?). Do you think it was more difficult for volunteers to estimate the lengths based on the lines on the board rather than seeing a person?
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Do you think it would have been more difficult to estimate height if we didn’t line them up shortest to tallest? Why?
Grade 4 Math Problem: Estimating and Measuring Length Activity 17.2 with Modifications
Corresponding Strands: Measurement and Geometry and Spatial Sense (pg. 8/9) (measuring using a ruler and visualizing lengths) Grade 4 Measurement Overall Expectation: - Estimate, measure, and record length, perimeter, area, mass, capacity, volume, and elapsed time, using a variety of strategies (pg. 69) Specific Expectation: - estimate, measure, and record length, height, and distance, using standard units (i.e., millimetre, centimetre, metre, kilometre) (e.g., a pencil that is 75 mm long) (pg. 69) Volunteers (shortest to Estimates (in M/CM) tallest) 1. 2. 3. 4. The mathematical processes • problem solving • reasoning and proving • reflecting • selecting tools and computational strategies • connecting • representing • communicating
Actual Height (in M/CM)
30CM = 11.8 Inches
Victoria Medeiros Friday, November 27, 2015 Technology Grades 4-8 (Grade 5) Geometry and Spatial Sense
Using technology to teach Geometry: Kahoot! Process Expectations • Selecting tools and computational strategies: select and use a variety of concrete, visual, and electronic learning tools and appropriate computational strategies to investigate mathematical ideas and to solve problem •
Communicating: communicate mathematical thinking orally, visually, and in writing, using everyday language, a basic mathematical vocabulary, and a variety of representations, and observing basic mathematical conventions
Overall Expectations • identify and classify two-dimensional shapes by side and angle properties and compare and sort three-dimensional figure Specific Expectations • Geometric Properties o distinguish among polygons, regular polygons, and other two-dimensional shapes o distinguish among prisms, right prisms, pyramids, and other threedimensional figures o identify and classify acute, right, obtuse, and straight angles o identify triangles (i.e., acute, right, obtuse, scalene, isosceles, equilateral), and classify them according to angle and side properties Kahoot! Kahoot! is an online tool that teachers can use to create online quizzes, discussions and surveys in order to assess student learning. It is a more fun and interactive way to assess learning than the traditional method of handing out a quiz. This would be best used at the end of a unit for Assessment of Learning. A teacher can design the questions, how many answers there are, how much time there is to answer and also if it is worth points. The teacher can make this game into a challenge like I will show today or simply use it as an assessment tool. Another great aspect is the website is free to use. Visit it at: https://getkahoot.com/ Data Management and Probability Bar Graphs Activity: 19.4 (modified), page 527 of Making Math Meaningful to Canadian Student, K8 textbook Grade: 4/5 Overall expectations: - collect and organize discrete primary data and display the data using charts and graphs, including double bar graphs
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read, describe, and interpret primary data and secondary data presented in charts and graphs
What is a bar graph? A bar graph is a diagram in which the numerical values of variables are represented by the height or length of lines or rectangles (bars) of equal width and equal space between them. Single bar graph: Double bar graph:
Activity: Choose a partner and each person roll a dice 10 times. Record your data and create a double bar graph using the data collected. Step 1: Each person take turns to roll the dice 10 times. Step 2: Record each person’s dice outcomes (i.e. Student 1 may roll a 3 and Student 2 may roll a 5) in the chart below. # of Rolls 1 2 3 4 5 6 7 8 9 10
Student 1
Student 2
Step 3: Create a double bar graph using the data collected on the graph paper provided. (Do not forget to label the axes and have a legend).
References:
Grade 5: Making a Double Bar Graph: Introducing the Concept. (n.d.). Retrieved November 18, 2015, from http://www.eduplace.com/math/mw/background/5/06a/te_5_06a_graphs_ideas.html Small, M. (2013) 2nd Edition. Making Math Meaningful to Canadian Students, K-8. 2nd Edition, Toronto, Nelson. Maddison Furtado 27th November 2015
Proportional Thinking: Ratios + Equivalent Ratio’s Integrating Technology into the Classroom Target Grade Level: Grade 6 & 7 Overall Expectations: Grade 6 & 7 Number Sense and Numeration Pg. 88, 99 "Demonstrate an understanding of relationships involving percent, ratio and unit rate." Specific Expectations: Proportional Relationships "Represent ratios found in real-life contexts, using concrete materials, drawings, and standard fractional notation” (89) "Determine and explain, through investigation using concrete materials, drawings, and calculators, the relationships among fractions, decimal numbers, and percents” (89) •
“Determine, through investigation, the relationships among fractions, decimals, percents, and ratios” (100)
Source of Activity: Math Play Ground: An Educational Website that Includes a Variety of Math Activities and Videos http://www.mathplayground.com Activity: Ratio Stadium
There will be a ratio presented at the bottom center of the screen. One needs to identify the equivalent ratio, from the options presented, in order to increase the speed of the bike. If the wrong answer is chosen, the speed will decrease. Answer as many questions as you can to win the race!
What is a ratio? A ratio is a way to compare quantities Example:
Part 1: Pineapples
Part 2: Apples
-----------------Total: All the Fruit Together----------------Ratio of Pineapples to apples: 2 to 3 , 2:3 ,
2 3
Ratio of Apples to Pineapples: 3 to 2 , 3:2 ,
3 2
Ratio of Pineapples to total amount of fruit: 2 to 5 , 2:5 , Ratio of Apples to total amount of fruit: 3 to 5 , 3:5 ,
3 5
2 5