Math

Introduction

I first became intrigued by math manipulatives and concrete materials when in teachers’ college and I learned about Piaget’s experiment. One of the discoveries this psychologist made is that if you place two identical sticks a couple of centimeters apart, young children will always think one is longer than the other. This concept of length and measurement, which seems as natural to us as breathing, is something we all had to learn. The time children spend playing with blocks and other toys helps them learn important concepts they will need in mathematics. When you realize that even concepts as simple as length have to be learned by handling physical objects, you have to wonder why this approach is not used more often to explain more complicated concepts.

Manipulatives also force us to focus on concepts. I believe that drill and repetition are important parts of mathematics and the basics such as multiplication tables should be taught. But I also believe that students too often waste their time responding to instruction such as, “Turn to page 17 and do numbers 1 to 38.” This seems uncanny: if they can do the first ten problems, we know they can do the rest. If they cannot do the first ten, forcing them through the rest is unlikely to help.

So what are math manipulatives? Manipulatives are brightly coloured, mathematically provocative materials that have an allure that even the most reluctant cannot overcome. With “hands-on-learning” becoming almost synonymous with good teaching, manipulatives have garnered a heightened interest all across the country.

Why the enthusiasm? There is a good reason. Manipulatives enhance student understanding, enable students and teachers to have a conversation that is grounded in a common model, and help students recognize and correct their own errors in thinking. Throughout their schooling, students need to be actively engaged in doing mathematics and need to be provided with many and varied rich opportunities to explore mathematical concepts and to develop a rich understanding of the fundamental mathematical concepts. The language of mathematics is pivotal to the development of this understanding. Working in small groups, either in pairs or in groups consisting of two or more students, will enable the students to exchange ideas, observe and compare data and results.

These experiences are enhanced and enriched through the use of concrete materials/manipulatives. This booklet describes and provides tips/examples for how to use the following materials in a mathematical classroom for junior students: hundreds board, geoboards, interlocking cubes, pattern blocks, Cuisenaire rods, base ten, tangrams, polydrons, and logic blocks.

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Table of Contents

Title Page....................................................................................1

Introduction................................................................................2

Table of Contents........................................................................3

Hundreds Boards....................................................................4-6

Geoboards...............................................................................7-9

Interlocking Cubes.............................................................10-12

Pattern Blocks....................................................................13-15

Cuisenaire Rods................................................................16-17

Base Ten...........................................................................18-20

Tangrams..........................................................................21-22

Polydrons..........................................................................23-25

Logic Blocks....................................................................26-28

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What is it?

A hundreds board is a 10-by-10 grid with the numbers one to one hundred printed in the squares. A hundreds chart can be sized so that each student has her or his own, or it can be poster-size for use with the whole class. Some hundreds charts have clear pockets stitched onto vinyl or fabric while others contain magnetic squares.

Why use it?

The purpose of a hundreds chart is to provide a framework for students to think about our base ten number system and to allow students to build a mental model of the mathematical structure of our number system. Hundreds boards allow children to explore concepts from counting to adding, to investigating prime numbers and patterns.

The standards for mathematical practice note that “mathematically proficient students look closely to discern a pattern or structure. Implementing the use of a hundreds board into lessons helps students look for and make sense of the pattern and structure of the board to become computationally flexible and fluent.

The hundreds board is one of the most versatile manipulative devices available for teaching mathematics. You can certainly count on the chart. The chart can be used for teaching number patterns and number relationships, operations, and problem solving. Hundreds board activities are designed to help elementary students develop number sense and number relationships and the activities can be used with individual students, small groups or with an entire class.

Internet Resources

Teaching Tables: http://www.teachingtables.co.uk/

Learning about Number Relationships and Properties of Numbers Using Calculators and Hundred Boards: Displaying Number Patterns: http://standards.nctm.org/document/eexamples/chap4/4.5/index.htm

Things to do with a hundreds board http://letsplaymath.net/2008/09/22/things-to-do-hundred-chart/

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Activities/Exercises

1. Counting and Number Awarenessa. Distribute the cards or tiles

numbered 1 to 20 (2 or more to each student). Ask: Who has 1? Put it on the board. Who has 2? Put it on the board. What is the next number?

b. Have students cover numbers as you give the directions

i. 22 and count on 3 more

ii. 37 and count on 6 more

iii. 63 and count on 2 more

2. Subtractiona. Have students cover numbers

as you give the directions:i. 35 and count back 2

ii. 47 and count back 4iii. 58 and count back 3

3. Ten More or Lessa. Have students place a marker

on the number that is 10 more than or less than:

i. 10 more than 2ii. 10 more than 55

iii. 10 less than 484. Race to 100 Game

a. Provide students a hundreds board and dice. In pairs students will take turns rolling the dice and moving forward that many spaces on their hundreds board. If you correctly predict your landing space before you move (without counting squares!), then you can go one extra space as a bonus. The first student to reach 100 wins!

5. Patterns with Multiplesa. Cover all the even numbers

with your counters. These are the multiples of 2.

i. What patterns do you observe?

b. Clear your hundreds board of counters. This time cover every third number. These are the multiples of 3.

i. What are some of the patterns that you noticed?

c. Repeat the process for the multiples of 4 and 5. What do you notice?

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6. Introduction to Prime Numbers (The Sieve of Eratosthenes)

a. Have students predict how many prime numbers are between 0 and 99. Have students cross out all the multiples of 2 (except 2)

b. Now cross out all the multiples of 3 (some will have been crossed out already)

c. The next number that is not crossed out is 5. Leave that one and cross out any multiples of 55 that are not already crossed out.

d. Look at the next number that is not crossed out. It should be a 7. Leave it but cross out all the other multiples of 7.

e. Identify the remaining numbers (2,3,5,7,11,13,17,19,23,29,31,41,43,47,53,59,61,67,71,73,83,89,97) and inform the

students that these are the prime numbers.

f. Explain that prime numbers only have two factors, 1 and the number itself. Ask students if they see any pattern with the prime numbers? Inform them that no one has found the pattern and whoever finds it will be given $5 million! (This will keep them interested and looking for that pattern)

7. Investigating Diagonal Sums

Investigate the sums of numbers that are diagonally opposite of each other. If you add the numbers diagonally opposite each other, what happens? Why?

32+43 = 75 and 21+54=75

33+42=75 and 24+51 =75

8. Place Value (Arrows)

→ to mean move across 1 (to add 1)↑ to mean move up 1 row (to add 10)← to mean move back 1 (to subtract 1)↓ to mean move down 1 row (to subtract 10)

a. 32↑↑↑b. 44←↑↑c. 89↑↑→→↓

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Geoboards

What is it?

A geoboard is a mathematical manipulative used to explore basic concepts in plane geometry such as perimeter, area and the characteristics of triangles and polygons. It consists of a physical board with a certain number of nails half driven in, around which are wrapped rubber bands. Geoboards are now available in a variety of sizes, styles, and colours. A preferred model is the transparent geoboard that can be placed on an overhead projector to facilitate sharing of student observations and conclusions.

Why use it?

Geoboards are particularly useful in developing conceptual understanding of area and perimeter. However, they can be used to explore mathematics from any of the curriculum strands. A wide array of resources including the Ontario Grade 7/8 Exemplars, are available to support the use of this manipulative. Resources included engaging activities involving fractions, the Pythagorean theorem, tessellations, transformations and patterning. When

students work together using geoboards, they have opportunities to improve communication skills, share ideas, and use mathematical vocabulary.

Internet Resources

Interactive geoboard activities http://standards.nctm.org/document/eexamples/chap4/4.2/

Geoboards in the classroom http://mathforum.org/trscavo/geoboards/contents/html

Dot paper http://mathforum.org/trscavo/geoboards/dotpaper.html

15 Questions http://math.about.com/od/manipulatives/a/Using-A-Geo-Board-In-Math.htm

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Activities/Exercises

1. Introductory Tasks

A geoboard shape can be described by a number pair which gives the number of pegs on its boundary and the number of pegs inside...

For example, the shape (3,0) means there are 3 dots on the boundaries, and 0 dots on the inside of the shape.

Have students investigate the shapes (3,0) (3,1) (4,0) (4,1) (5,0) (6,5) (7,5) (8,7)

How many different figures can you make for each of these?

How many different three-sided, four-sided, five-sided... shapes can you make on the geoboard?

How many different shapes can you make that have just one peg on the inside of the figure?

How many different shapes can you make that have just four pegs on the figure?

2. Constructing Quadrilaterals (Geometry and Spatial Sense)

Invite students to make each of the following quadrilaterals on the geoboard. If

you believe it is impossible to make any of these, provide reasons why you believe it is not possible to make it.

- 4 equal sides, 4 non-equal sides- 3 equal sides, 3 non-equal sides- 2 equal sides, 2 non-equal sides- One pair of perpendicular sides- On pair of parallel sides- Two pairs of perpendicular/parallel

3. Pick’s Theorem

When the dots on square dotty paper are joined by straight lines the resulting figures have dots on their perimeter (p) and often internal (i) ones as well.

Each figure can be described as (p,i). Each figure always encloses an area (A)

Pick’s Theorem computes the area of an arbitrary polygon simply by counting pegs.

The polygon in the above figure touches ten pegs and surrounds six pegs. Now if we take the number of boundary pegs (which is 10 in this case) and divide it in half, add the number of interior pegs (which is 6) and subtract one, we get...

10/2 + 6 – 1 = 10

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Have students create a table chart that consists of pegs on, pegs inside, and area. They will then use investigate and try to solve the area for each and determine the relationship between the three variables (p, i, A) and then introduce them to Pick’s Theorem.

Can they find the formula without you?

Pegs on Pegs Inside Area3 0 0.54 0 15 0 1.53 1 24 1 25 1 2.5

4. Finding Area

Many students may already know how to calculate the area of squares and rectangles; possibly even that of parallelograms and triangles. Using the geoboard can introduce or strengthen a student’s understanding of area, providing the framework that builds the formula.

Through the use of geoboards students will realize that to find the area of a parallelogram they can divide it into shapes they already know how to compute.

Implementing geoboards allows “hands-on” experiences that engage children and can lead into the formula from a textbook,

building off the knowledge from their investigations.

5. Line Symmetry

Geoboards allow students to practice constructing a symmetrical design. Ask students how many symmetrical shapes they can design on their geoboards.

Provide a handout that contains shapes after they have practiced with their geoboards. The handout can contain shapes such as these below – ask students which shapes have a line of symmetry? How many?

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Interlocking Cubes

What is it?

Interlocking cubes are mathematical manipulatives that are available in different sizes and different colours. Generally used are the 2cm interlocking cubes that connect on all six faces, available in 10 colours.

Why use it?

Interlocking cubes help students develop spatial sense. They are also used to develop understanding of number and measurement concepts. Students can use cubes to create, identify, and extend patterns. The patterns can be used to develop algebraic models. The variety of colours also allows cubes to be used in probability experiments.

Interlocking Cubes help students within their recommended capacity based purely on their ability to connect to each other.  However, be mindful that if quick rearrangements are required for a task, connecting the Cubes

and disconnecting them can take a bit of time overall. 

Internet Resources

Interactive Isometric Drawing Tool http://illuminations.nctm.org/Activities.aspx?grade=all&srchstr=isometric

Space Blocks, Algebra, 6-8 http://nlvm.usu.edu/en/nav/search.html?qt=blocks

Grade 8 Exemplar, Patterning and Algebra http://www.edu.gov.on.ca/eng/curriculum/elelementary/math8ex/pattern.pdf

101 Activities Using Cubes http://www.tsusmell.org/downloads/Conferences/2008/Barnard_2008.pdf

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Activities/Exercises

1. Patterns

Interlocking Cubes allows students to determine patterns through a hands-on method. Visually seeing the patterns and determining what should come next. Examples like the picture below are great to have students determine the pattern and then determine what would be the 10th, 15th, 30th cube in the pattern.

2. Spatial Sense

Task One – for this you will need one yellow cube and four blue cubes

For this task all four blue cubes should be touching the surface and the yellow cub should be on top of the four blue cubes. How many different arrangements can you

make with the four blue cubes on the surface and the yellow cube on top?

Task Two - Building Problems

Have students construct buildings based on the clues given for each building; ensure that students meet all the specifications.

One – need six cubes (2 green, 1 red, 2 white, and 1 brown)

Clue #1 The two green cubes do not touch each other

Clue #2 The red cube shares one face with each of the other cubes

Clue #3 The two white cubes are opposite each other

Clue #4 One of the cubes is brown

Two – need five cubes (red, blue, white, green and yellow)

Clue #1 Build a tower five cubes high

Clue #2 The red cube is above the yellow

Clue #3 The middle cube is not white

Clue #4 The bottom cube is not green

Three – need six cubes (2 yellow, 1 red, 1 brown, 1 green, and 1 white)

Clue #1 The brown cube shares a face with three cubes

Clue #2 The red cube shares a face with two cubes

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Clue #3 The white cube is between the two yellow cubes

Clue #4 The green cube is on the bottom level

3. Volume and Area

Build each figure below with interlocking cubes. Determine the surface area and volume for each figure.

4. Cubes and Fractions

Build a cube such that one-half is made with yellow cubes, one-eighth with brown cubes and three-eighths with green cubes. How many different cubes can you build?

Build a structure such that two-thirds is made with red cubes and one-third with brown cubes. How many different structures can you build?

Build a design such that one-half is made with green cubes, one-tenth is made with yellow cubes, two-tenths is made with green cubes and the rest is made with black cubes.

Students can also be provided with examples and asked to replicate then record the fractions and convert to equivalent percents and decimals.

5. Probability

A big cube is made up of 64 smaller cubes. All of the faces of this big cube are painted red. The cube is now taken apart and the unit cubes are put in a bag.

You pick a cube out of the bag. What’s the probability that the cube you choose has:

- No faces painted red?- One face painted red?- Two faces painted red?- Three faces painted red?

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Pattern Blocks

What is it?

Pattern blocks are a set of blocks that contain six colour-coded geometric solids. The top and bottom surfaces of these solides are geometric shapes: hexagon, trapezoid, square, triangle, parallelogram (2). Except for the trapezoid, the lengths of all the sides of the shapes are the same. This allows students to form a variety of patterns with these solids.

Why use it?

As the name suggests, pattern blocks are used to create, identify, and extend patterns. However, pattern blocks can be used for many more mathematical concepts. Students can use the many relationships among the pieces to explore fraction, angles,

transformations, patterning, symmetry, and measurement.

Internet Resources

Virtual pattern blocks http://arcytech.org/java/patterns/patters_j.shtml

Exploring fractions http://math.rice.edu/~lanius/Patterns/

Working with pattern blocks http://fcit.usf.edu/math/resource/manips/pattern.pdf

Math forum – Investigating Tessellations http://mathforum.org/sum95/suzanne/active/html

Polygon playground http://www.mathcats.com/explore/polygons/html

Interactive manipulatives and activities http://matti.usu.edu/nlvm/nav/category_g_4_t_3.html

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Activities/Exercises

1. Patterns

Provide students with a pattern, such as the ones below, and have them continue it. Ask ‘What will be the 10th, 15th, 30th, block in the pattern?’

Students must gain proficiency in recognizing patterns since it is essential in the study of mathematics. Pattern blocks allow the beginner to copy, continue, and

create patterns. Since each pattern block has two distinct attributes, shape and colour, they are perfect for pattern exploration.

2. Angles

Use three blocks to make a pentagon. How many different ways can you do this? What is the sum of the interior angles in each case.

Allow students to investigate and use the pattern blocks to identify the interior angles of each shape.

- What are the angles in the hexagon?- What are the angles in the triangle?- What is the measurement of each

angle in your set of pattern blocks?

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3. Fractions

How many fractions can you find?

What fraction does each pattern block show? Using pattern blocks, demonstrated below, introduces students to the concepts of fractions through a hands-on method.

4. Pattern Blocks and Algebraic Thinking

Provide students with the following questions to introduce algebraic thinking through the use of concrete materials.

- Determine the angles of each the Pattern Blocks

- Use Pattern Blocks to make a polygon such that the sum of its angles is the same as the sum of the hexagon.

- Use Pattern Blocks to make a polygon such that the sum of its angles is less than the sum for the hexagon but more than the sum for the square.

- Use pattern blocks to make a polygon such that the sum of its angles is greater than the sum for the hexagon. Determine the sum of the angles.

- If the sum of the angles of a “Pattern Block” is 1620°, how many sides does it have?

- What is the relationship between the number of sides of a Pattern Block polygon and the sum of its angles?

5. Tessellations

Pattern blocks can also be used to introduce students to tessellation – a tessellation of a flat surface is the tiling of a plane using one or more geometric shapes (in this case pattern blocks) with no overlaps and no gaps.

Introduce the concept and have students use pattern blocks, one kind at a time, or multiples, to create a tessellation.

Will triangles work?

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Squares?

Hexagons?

Ask students why they think that we do not use hexagons but rather squares and rectangles in bricklaying, floor tiles, ceilings, etc. when hexagons is what appears throughout nature (honeycombs)?

Cuisenaire Rods

What is it?

Cuisenaire, or relational rods, are rectangular solids of related lengths. A set usually contains between 70 and 80 rods. In a set, all rods of the same length are the same colour. The smallest rod is a 1-cm cube. The largest of the 10 rods has a volume of 10 cm³ and it is ten times as long as the small cube. The lengths of the different coloured rods increase incrementally from the smallest size to the largest size (shown in the picture above).

Why use it?

Cuisenaire rods help students visualize mathematical concepts. They are primarily used to help students develop understanding of fractions and proportional reasoning. However, there are many ways they can be used in all of the curriculum strands.

Cuisenaire rods were invented by Georges Cuisenaire as a means to teach him pupils the study of arithmetic. He made then a discovery now established as a component

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in mathematics teaching today. He found that by making use of children’s inclination to play, and giving them an appealing material which demonstrated the relationships on which mathematics is based, it was possible to provide understanding for them all.

Internet Resources

Cuisenaire Rods http://cuisenaire.co.uk

Learning Fractions http://teachertech.rice.edu/Participants/silha/Lessons/cuisen2.html

Fractions http://www.learner.org/channel/courses/learningmath/number/session8/part_b/

Communication http://www.learner.org/channel/courses/teachingmath/grades6_8/session_02/section_02_h.html

Activities/Exercises

1. Representing Fractions

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How many ways can you show each of the following fractions?

- ½- ¼- 2/3- ¾- 1/3

2. Equivalent Fractions

Write some fractions that are equivalent to each of the following:

- ½- 1/3- 2/3- ¾

3. Comparing Fractions

Use Cuisenaire Rods to determine which of the two fractions is bigger.

- ½ or 1/3- ½ or ¾- 2/3 or ¾- 3/10 or ½

4. Adding Fractions

Use Cuisenaire Rods to add each of the following:

- ¼ + ½- 3/10 + ½- ¾ + 5/8

Base Ten

What is it?

There are four different sizes of base 10 blocks. The smallest blocks, called units, are 1 cm³. The next largest block is a long narrow block that measures 10cm by 1 cm

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by 1 cm. These 10-unit pieces are called rods. The flat square blocks are 10 cm by 10 cm by 1 cm and are called flats. The largest blocks are 10 cm by 10 cm by 10 cm and are called cubes. These terms are used to signify the interchangeableness of the pieces in place value.

Why use it?

The size relationships of the blocks can be used to explore number concepts. Students can explore place value concepts such as addition, subtraction, multiplication, and division with both whole and decimal numbers. These blocks provide a visual representation and foundation for understanding traditional algorithms. They can also be used to explore perimeter, area, and volume concepts.

Although algebra tiles are a better manipulative to explore algebra, base 10 blocks can be used in single-variable activities. The unit would represent the number. The rod would represent the single variable such as x. The flat would represent the square of the variable such as x².

Internet Resources

Using base 10 blocks to represent numbers http://nlvm.usu.edu/en/nav/frames_asid_152_g_1_t_1.html?from=topic_t_1.html

Addition http://nlvm.usu.edu/en/nav/frames_asid_154_g_1_t_1.html?from=topic_t_1.html

Subtraction http://nlvm.usu.edu/en/nav/frams_asid_155_g_1_t_1.html?from=topic_t_1.html

Adding and Subtracting Decimals http://www.folksemantic.com/visits/104455

Base 10 Activities http://www.susancanthony.com/Resources/base10ideas.html

Activities/Exercises

1. Represent and determine the value

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2. Addition

Addition is fairly straight forward with base 10 blocks. By allowing students to use the blocks they can see and understand larger numbers when added together, doing it on their desk with their hands to deepen their understanding.

3. Subtraction

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4. Other Sample Questions

Solve this problem, using base 10 blocks: A video game company wants to pack their games to send out to stores. The game is the same size as a flat. They have decided to fit 12 games in a box. What are their box size options? Which box would be the most cost efficient box (use the least amount of packaging)?

Show how to model 4 x 22 with the base 10 block. Is there another way to arrange the blocks for the same answer?

Solve this problem, using base 10 blocks: It cost $120 for 6 people to enter an amusement park. Model how you would determine how much each person would pay to get into the park.

Solve this problem, using base 10 blocks: Josie, Christina, Audrey, and Manny go shopping. Josie spends 4/5 of her money, Christina spends 75% of her money, Audrey spends 0.68 of her money and Manny spends 17/20 of his money. Who has the largest percentage of money left?

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Tangrams

What is it?

One tangram set consists of seven shapes that can be arranged to form a square. The square tangram puzzle was invented in China and is still being used to challenge individuals to create different shapes using the seven pieces.

A tangram contains 7 shapes

- Two large right triangles- One medium sized right triangle- Two small right triangles- One small square - One parallelogram

Why use it?

Tangrams are particularly useful in problem-solving activities. Frequently, tangrams are also used for exploring geometry, proportional reasoning, area, and algebra. Tangrams encourage students to visualize spatial relationships between shapes. Allowing students to learn through the use of tangrams may help them:

- Classify shapes- Develop positive feelings about

geometry- Gain a stronger grasp of spatial

relationships- Hone spatial rotation skills- Acquire a precise vocabulary for

manipulating shapes (e.g., “flip,” “rotate”)

- Learn the meaning of congruent

Internet Resources

Exploring the Pythagorean Theorem with tangram pieces http://members.aol.com/sth777/page24.html

Construct a tangram set http://mathforum.org/trscavo/tangrams/construct.html

Interactive tangram set http://standards.nctm.org/documents/eexamples/chap4/4.4/part2.htm

Sample puzzles http://members.aol.com/sth777/page3.html

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Activities/Exercises

1. Primary Activities

Assume that each tangram has a value of 1 whole. Find the value of each piece (based on surface area of one face) stated as a fraction. (Do the same activity but state the value as a decimal or percent).

Assume a whole tangram set costs $1.60. Determine the value for each piece of the set.

2. Discovering the Pythagorean Theorem

Step 1 – place one of the small triangles in the center of you paper and trace around it. Label the longest side of the triangle “C” (hypotenuse) and the other two sides “A” and “B”. Discuss the 90° angle and the two acute angles and the relationship among the sides (c ˃ a; c > b; a + b > c)

Step 2 – Use tangram pieces to form a perfect square along each side of the triangle you drew. Trace around the squares, making sure they are attached to the sides of the triangle. How many small triangles were used on sides “A” and “B”? (two) How many small triangles are needed to make the

square on side “C”? (four) Discuss how the perfect squares of “A” and “B” combined make a perfect square on side “C”.

Step 3 – Repeat steps 1 and 2 using the medium triangle. Can the perfect squares be made by using only the small triangles? How many triangles are used on sides “A” and “B”? (four) How many small triangles would be need for side “C”? (eight)

Step 4 – repeat the activity using the large triangle. Determine how many triangles would be needed for sides “A” and “B”, (Two triangles or five of the smaller pieces) and for side “C”. (All seven tangram pieces.)

Step 5 – compare the three drawings. Discuss the relationship of the areas of the squares along each leg of the right triangle to the area of the square along the hypotenuse. The sum of the squares on the legs is equal to the square of the hypotenuse.

Step 6 – Use the formula for area (A = l x w) to find the areas of the squares on all three sides. Have students try to write an equation to represent the relationship (a² + b² = c²)

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Polydrons

What is it?

Polydrons are an assortment of regular polygons with the ability to connect to one another to form a hinge. These hinges are the edges of the three-dimensional solids they can form. Once all of the necessary pieces are connected and in a viable net for a solid the faces come together on their hinges and for the remaining connections. The result is a shell (the proper term for a hollow three-dimensional figure). There are also open face pieces that allow for the creation of open boxes and shapes. The focus is on the student’s ability to design and manipulate the pieces into the shapes.

Why use it?

Polydrons help students to transcend the rote memorization of nets to conceptually seeing any net come together in their mind’s eye. Take a cube for example, it has six faces. How many nets are there for a cube? Students can actively and dynamically work through this problem recording ones that work and even ones that do not work. Polydrons encourage and teach students geometry, spatial sense, and can be used to introduce Euler’s Rule.

Internet Resources

Using Polydrons in the Classroom http://shop.polydron.co.uk/images/manuals/polydronbooklet.pdf

Polydron Fun! http://www.shawnee.edu/acad/ms/ENABLdoc/Summer08pdfs/Polydrons%20Lesson%20Plan.pdf

Geometry and Spatial Sense, Grades 4 to 6 http://www.edugains.ca/resourcesLNS/GuidestoEffectiveInstruction/GEI_Math_K-6_GeomSpatialSense_Gr4-6/Guide_Geometry_Spatial_Sense_456.pdf

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Activities/Exercises

1. Rich tasks and questions for students- What is the fewest number of faces that a polyhedron can have?- How many different nets for a tetrahedron are there?- How many different polyhedral can you make from five pieces? How many of these

polyhedral can you name?- How many different nets for a cube can you make? How do you know when you have

made them all?- How many different regular polyhedral can you make? How do you know when you have

made them all?- In any simple polyhedron, there is a connection between the number of edges, vertices

and faces. What is the connection?- How many different pyramids can you make?- How many different prisms can you make?

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2. Introducing Euler’s Rulea) How many different nets for a cube can you make with your polydron pieces? Sketch the

nets on graph paper.b) How many different nets for a tetrahedron (triangular pyramid) can you makes? Sketch

the nets.c) What are some of the polyhedral that you can make with five and only five pieces? Name

the polyhedral you can make.d) The mathematician Euler made a wonderful discovery when he counted the number of

faces, edges, and vertices of polyhedrons. Count the number of faces, edges and vertices of the following polyhedral to find out what he noticed.

# of faces # of vertices # of edges Euler’s RuleTetrahedronHexahedron (cube)

Triangular PrismSquare-based Pyramid

Pentagonal Pyramid

Logic/Attribute Blocks

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What is it?

Attribute (Logic) blocks are math manipulatives that have four different features. These are shape, color, size and thickness. The shapes are circles, hexagons, squares, rectangles and triangles. The colours are red, blue, and yellow. The two sizes are big and small and the shapes are either thick or thin.

Why use it?

Attribute blocks are a good introduction to logical thinking for young learners. Students will learn colour and shape concepts by sorting and investigating with these blocks. Students learn one attribute of the blocks at a time; most will recognize the colour differences without any need of explanation. Students will learn to differentiate and sort the shapes based on the different attributes and can place them into Venn diagrams.

By combining four attributes, students can learn shapes, classification (sorting) skills, congruent vs. similar, fractions, proportions, patterns, comparison/contrast, patterning, and many other mathematical concepts and thinking skills.

Internet Resources

Attributes http://nlvm.usu.edu/en/nav/frames_asid_270_g_2_t_3.html?open=instructions

Activities/Exercises

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1. Attribute Block Trains

Attribute Block Trains is an activity designed to help students recognize attributes and patterns of attributes by placing pieces along a trail. The blocks placed along the trail must follow the rule for number of differences. For example, a large, blue square and a large, red square differ in only one way (shape). A large, blue hexagon and a large red circle differ in two ways (shape and colour)

2. Logical Relationships with Venn Diagrams

Students can use attribute/logic blocks to learn how to use a Venn Diagram and to show the logical relationships between a set of objects.

The regions in a Venn diagram define logical relationships. In this diagram circle A and circle B intersect. The region formed by the intersection is used to represent the logical relationship between objects in set A and the objects in Set B.

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For example if the rule for Set A is ‘circle’ and the rule for set B is ‘blue’, any Attribute Block that is a circle belongs in circle A and any Attribute Block that is blue belongs in circle B. Blue circles belong in the intersection. Have students develop a set of rules without explaining it and see what they come up with.

Later students should be able to do this activity with three intersecting circles (shown below).

Ask students to try and sort out their Attribute Blocks according to a rule to complete a three intersection Venn diagram.

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What is it?

The Geometer’s Sketchpad® is the world’s leading software for teaching mathematics.Sketchpad® gives students at all levels—from third grade through college—a tangible, visual way to learn mathematics that increases their engagement, understanding, and achievement. Make math more meaningful and memorable using Sketchpad.

Why use it?

Elementary students can manipulate dynamic models of fractions, number lines, and geometric patterns. Middle school students can build their readiness for algebra by exploring ratio and proportion, rate of change, and functional relationships through numeric, tabular, and graphical representations. And high school students can use Sketchpad to construct and transform geometric shapes and functions—from linear to trigonometric—promoting deep understanding.

With Sketchpad, students are able to measure lengths of segments, measures of angles, area, perimeter, and more. They can also use tools that allow them to create objects in relation to selected objects. The transform function allows the user to create points in relation to objects, which include distance, angle, ratio, and others. With these tools, one can create numerous different objects, measure them, and potentially figure

out hard-to-solve math problems.Sketchpad is the optimal tool for interactive whiteboards. Teachers can use it daily to illustrate and illuminate mathematical ideas. Classroom-tested activities are accompanied by presentation sketches and detailed teacher notes, which provide suggestions for use by teachers as a demonstration tool or for use by students in a computer lab or on laptops.

Internet Resources

Free sketchpad resources: http://www.keycurriculum.com/sketchpad-resources

Downloadable Version: http://www.chartwellyorke.com/sketchpad/x24795.html

Sample Activities: http://learningcenter.dynamicgeometry.com/x20.xml

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Activities/Exercises

Task 1In the figure below, the circle and the triangle are inscribed in the square. If you find the area of each of the three figures and add them up, you get 28 units2. What's the edge length of the square, rounded to the nearest tenth?

Task 2We have a triangle ABC, and it has an area of 24 units2. Point A is at (6,0) and point B is at (10,0).

1. Where can point C be? (Read the given parts carefully.)

2. What if ABC is isosceles? Now where can C be?

Be thorough in your explanation!

Task 3

ABCD is a rectangle. Find the area of ABCD, and be sure to explain how you did it!

If you're feeling really creative, you might try to find two completely separate ways of solving this problem, and we might give you bonus credit.

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Task 3

In the picture below, if the measure of ∠1 is 2x + 10, the measure of ∠2 is 5x - 40, and the measure of ∠4 is 600, find the actual angle measures of ∠1, ∠2, and ∠3.

Extra: If we don't know that the measure of ∠4 is 600, can we still determine anything about the measure of ∠3? Why or why not?

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