USING MATH MANIPULATIVES

TO

BUILD UNDERSTANDING

Table of Contents Algebra Tiles..........................................................................................................................1

Relational/Cuisenaire Rods...................................................................................................2

Five & Ten Frames.............................................................................................................3-4

Rekenreks...........................................................................................................................5-6

Base 10 Blocks...................................................................................................................7-8

Colour Tiles...........................................................................................................................9

Pattern Blocks......................................................................................................................10

Tangrams..............................................................................................................................11

Connecting/Unifix Cubes.....................................................................................................12

Fraction Strips & Towers................................................................................................13-16

Fraction Circles...............................................................................................................17-18

Geoboards............................................................................................................................19

Other Recommended Manipulatives...................................................................................20

Useful Math Websites..........................................................................................................21

GAINS: Tips for Manipulatives

Algebra Tiles

What are Algebra Tiles? Algebra tiles are rectangular shapes that provide area models of variables and integers. They usually consist of x sets and y sets. Different pieces are used to model 1, x, x2, y, y2, and xy. Sets consist of two different colours to represent both positive and negative terms. Overhead versions are used for whole class learning opportunities. A clear plastic organizer prevents tiles from moving around. How do Algebra Tiles help students? Algebra tiles are used to build concrete area representations of abstract algebraic concepts. The concrete representations help students become comfortable with using symbols to represent algebraic concepts. Algebra tiles are typically used to explore integers, algebraic expressions, equations, factoring, and expanding. They can also be used to explore fractions and ratios.

How many are recommended? Students usually work in pairs or small groups when using algebra tiles. Each pair of students needs an x set, a y set, and a plastic organizer. Students can use card stock to create algebra tile sets. Other representations can also be created using card stock, e.g., z sets. A transparent set of tiles is useful for overhead demonstrations by students and/or teachers. When students first use algebra tiles, allow for exploration time. Sample Activities 1. Determine the number of different ways that zero (0) can be represented using tiles from a set of 3 blue

one-tiles and 2 red one-tiles. 2. Use the one-tiles to model different integer values, e.g., a loss of $4; 2 metres above sea level. 3. Create models for integer operations, e.g., show that (-4) + (+1) = -3; show that 2(-3) = -6 4. Build an algebra tile model to show that 2x + 3 – 4x – 2 + 5x – 1 = 3x 5. Build an algebra tile model to show that (2x + 3) + (-5x –3) = -3x 6. Build an algebra tile representation of 2(3x + 1). Use the model to show that 2(3x + 1) = 6x + 2. 7. Make two different models of the ratio 3:2. 8. Build algebra tile models for (x + 1)2 and x2 + 1. Use your models to explain why these expressions are not

equivalent. 9. Try to arrange two red x-tiles and three red one-tiles into one rectangular arrangement. (Note: This activity

builds understanding of factoring.) Is it always possible to make a rectangular arrangement? 10. Solve this problem:Jen used a set of x tiles to model 2x2 – 3x + 4. Can the same model be used to represent 2a 2 – 3a + 4 11. Use the red one-tiles to show all possible factors of 12. 12. Build a tile train. What is the colour and

shape of the 200th cube in the train?

Recommended Websites http://matti.usu.edu/nlvm/nav/category_g_4_t_2.html Virtual Algebra Activities http://www.pbs.org/teachers/connect/resources/4450/preview/ Yo-yo Activity http://www.uen.org/Lessonplan/preview.cgi?LPid=6393 Evaluating Expressions using Algebra Tiles http://illuminations.nctm.org/swr/review.asp?SWR=227 Understanding Algebraic Factoring

GAINS: Tips for Manipulatives

Relational Rods

What are Relational Rods? 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. How do Relational Rods help students? Relational 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 relational rods can be used in all of the mathematics curriculum strands. Since the rods’ attributes can be used for length, area and volume, ensure that students understand which attribute is being used in the problem.

How many are recommended? One set of rods per pair of students is ideal; however students can make effective use of the rods in groups of four. Allow students time to explore their attributes and the relationships between rods. A transparent set of relational rods is useful for overhead demonstrations by students and/or teachers. Students can use any square grid paper to record solutions but 1-cm grid paper is particularly useful. Many activities can also be done with trains of 1 to 10 connecting cubes. Paper versions can also be made. Sample Activities 1. Build a staircase. What is the total volume? Determine the volume if there are 100 steps. 2. Create a structure or design. Hide it from your partner’s view and describe it so your partner can build it. 3. How many different ways can you make “trains” that have the same length as one yellow rod? (e.g., one

purple and one white, or two reds and one white) 4. Describe the relationship between one purple rod and one dark green rod. 5. Solve this problem: Jake has one white rod and one red rod. Taz has one orange rod and one yellow rod.

Make a list of comparisons between their two sets. 6. Solve this problem: Jasmine has a train of four white rods and a train of two red rods.

She writes the equation 4w = 2r to algebraically model what she sees. Explain the connection between the train model and the algebraic model.

7. What does one green rod represent, if one red rod represents one-half? (or vice-versa). 8. Solve this problem: Terry represents the fraction two-thirds by placing a red rod on top of a light green rod.

Use the relational rods and Terry’s method to build different models of two-thirds. 9. Use the length of the yellow rod as a unit of measure. Measure the width of this page. Measure the width of

this page using the red rod. (Were you able to determine the answer without actually measuring?) 10. Use five different rods and 1-cm grid paper. Place the rods to create a shape that can be cut from the paper.

Challenge a classmate to determine how you created the shape. Can other shapes be created using the same rods so that the shapes have the same perimeter? same area? Which shape has the greatest perimeter?

11. Create a pattern. Extend the pattern. Develop a rule for the pattern. 12. Create a pattern. Draw a reflection of the pattern. Recommended Websites http://teachertech.rice.edu/Participants/silha/Lessons/cuisen2.html Learning Fractions with Relational Rods http://www.learner.org/channel/courses/learningmath/number/session8/part_b/ Fractions with Relational Rods http://www.learner.org/channel/courses/teachingmath/grades6_8/session_02/section_02_h.html Communication http://mason.gmu.edu/~mmankus/Handson/crods.htm Template for Making Relational Rods

Five and Ten Frames

GAINS: Tips for Manipulatives  1 

What are Five and Ten Frames? Five and ten frames are equal-sized rectangular boxes in a row where each box is large enough to hold a counter. The five frame is arranged in a 1-by-5 array. A ten frame is a set of two five frames or a 2-by-5 array. How do Five and Ten Frames help students? Five and ten frames help students to relate given numbers to 5 and 10 by providing a visual image. The frames may be filled in from left to right so that students can learn to subitize. Their use encourages counting strategies beyond counting by one or counting on each time they are asked to identify a number or work on an addition or subtraction problem. Students think about combinations of number that make other numbers, e.g., 7 is two more than 5, or 9 is one less than 10. These number relationships help build the foundation for the development of more complex mental computations. Students start with the five frames before moving on to ten frames and may explore double ten frames later to develop a better understanding of place value. How many are recommended? It is recommended that every child have a five frame to begin and, when developmentally ready, they should also have a ten frame. Blackline masters of frames can be mounted on cardboard. Students also need counters (at least 10 per child) to place in and beside the frames for counting. Non-permanent markers can be used with laminated frames. It is possible to use stickers in the frames or have the students colour the frames, although these options make for a one-time use of the frames. Sample Activities 1. Ask students to only put one counter in each space on a five frame to

show 3. Ask them to explain ways they have displayed 3. Continue for 0-5. Once the students have displayed a number, ask “How many more counters are needed to make 5?” to continually reference 5. As a next step, call out numbers greater than 5 and have students place those additional counters outside the frame so they see that 7 is two more than 5.

2. Once students have had experience with five frames, repeat the above activity with the ten frame cards. Note: When using the ten frames, ask students to fill the top row up first, before moving on to the second row, as this will provide a “standard” way to show numbers and reinforce the concept of 5s and 10s as anchors.

3. Using two-sided counters, find all the ways to make 5 (or 10).

Five Frames

Ten Frames

Five and Ten Frames

GAINS: Tips for Manipulatives  2 

4. When students have had experience with five or ten frames, play a game by quickly flashing a filled frame and ask how many dots there were. Encourage students to share strategies of how they could tell without counting.

5. Have five (or ten) frames pre-coloured and ask students to match them to pre-made expressions, such as 5 + 3, 2 + 2 + 2, etc.

6. Call out numbers as a shared class experience and students build that number on their frames. Note if students clear their frame each time. If this happens, encourage a volunteer to call out what they do to the previous number to make the new number, e.g., The first number called out is 8. If the second number called out is 12, students call out “add 4.” If the third number called is 6, students call out “subtract 6.”

7. Hold up a frame with some frames already marked and say “I wish I had 5 (or 10).” Students figure out how many more counters are needed to make that number.

8. One student arranges counters on the ten frame and hides it from a partner. The partner can ask Yes or No questions to figure out the hidden number, e.g., Is the top row full? or Are there more than 3 spaces empty?

9. Once students have had experience with the frames, they could try visualizing the counters. Ask students to imagine 6 counters in a ten frame and adding 7 more counters. Ask: What do the frames look like now?

10. Use the frames to prove which number is greater 6 or 9. 11. Wayne has 3 more toy trucks than Craig. Craig has 4 trucks. How many does Wayne have? 12. Mary had 16 silly bands but she gave 7 away to her friends. How many are left? Adapted from Teaching Student-Centered Mathematics: Volume One, Grades, K-3. John Van de Walle, Boston: Pearson, 2006. ISBN 205-40843-6 Recommended Websites http://illuminations.nctm.org/activitydetail.aspx?id=74 – five frames interactive website http://illuminations.nctm.org/activitydetail.aspx?ID=75 – ten frames interactive website http://sites.google.com/site/angelatuell/usingtenframestosupportmathlearning – using five and ten frames http://nrich.maths.org/2479 – explanation and activities for ten frames

Rekenreks

GAINS: Tips for Manipulatives  1 

What are Rekenreks? Rekenreks are arithmetic racks, developed by Adrian Treffers, a mathematics curriculum researcher at the Freudenthal Institute in Holland. There are two rods of 10 beads. Each rod has 5 beads of one colour followed by 5 of another colour. The

order and colours on the top rod are repeated on the lower rod. The colours most often used are red and white. The starting position should show all the beads pushed to the far right. The student enters a number by sliding the beads to the left in a one-push motion. It is important that everyone in the class is visualizing and communicating the patterns in the same way. How do Rekenreks help students? Rekenreks are used to help develop addition and subtraction strategies, such as doubling or finding near doubles as well as thinking in terms of 5s and 10s, instead of counting from one each time or counting on in addition and subtraction. Students improve their ability to regroup numbers when solving addition and subtraction problems. How many are recommended? It is recommended that each child have a Rekenrek to represent/ visualize mathematical thinking. It is also recommended that the teacher have a large demonstration Rekenrek which is visible to the whole class for shared activities. Sample Activities 1. To introduce the Rekenreks, push all ten beads from the top row to the left and cover the

bottom row(s). Students do the same on their individual Rekenreks. Ask the students what they notice.

2. Push various numbers to the left and ask the students to quickly tell how many beads they see. Start with 1, then 5, 7, 9, 12, 16, etc. Ask students how they know how many they see and listen for answers that involve visualizing 5 and 10, or seeing doubles, as opposed to counting individual beads.

3. Reinforce the idea of showing a number on the Rekenrek in “one push.” Ask students to explain how they knew they were pushing the right number. Notice reasoning that involves visualization of 5s and 10s, as well as doubles.

4. Model mathematical situations such as: 8 birds were eating crumbs in the park and 4 more came to eat. How many birds were there altogether?

5. Use the Rekenreks to model student thinking, e.g., Did you think of 7 as two more than 5? Show that 5 red beads pushed together with two additional white ones does make 7.

6. Use the Rekenrek as a tool for problem solving, e.g., When you add 6 and 5, you can see it as 5 and 5 with one more. Show 5 red on the top rod and 5 red on the bottom rod with one additional white bead, making the 5-5 pattern explicit.

7. Play team class games by pushing some of the top rod of beads and the class pushes the bottom set of beads to make the chosen number, e.g., To make 9, push 5 red beads to the left from the top row. Students push 4 on the bottom rod. Look and listen for strategies. Once shared a few times, students could be encouraged to play the game with a partner.

Rekenreks

GAINS: Tips for Manipulatives  2 

8. Make numeral cards from 1-20. Hold up a card at random and ask students to show that number on their individual Rekenreks. Allow one push for numbers up to 10 and two pushes for any number larger. Debrief various solutions with and how students arrived at the position of beads. To further this activity in pairs, students have a barrier between them. One student draws a card from the pile, saying the number out loud and making that number on their Rekenrek in any manner they choose. The first player provides a clue as to how many beads they pushed on the top rod but the partner must figure out how many are pushed on the bottom rod to replicate the partner’s solution.

9. Determine all the ways to make 10, using beads from each row. 10. Use the Rekenrek to prove that 3 + 2 = 1 + 4 11. Ask: Is 7 + 8 = 8 + 7? How do you know? 12. Once familiar with the Rekenreks and the number of beads on each row, show some beads

to the class and ask the class to figure out the number of hidden beads to make a certain number.

13. There were 12 students playing on the play structure in the playground. Four were on the top level. How many were on the bottom level?

14. Six students were on the stage in the gym practising for the school play, while four were on the floor setting up chairs. Three more students came to help. How many students were in the gym?

15. Out of the 20 cupcakes brought into class for Owen’s birthday only 3 were left. How many were eaten?

Recommended Websites http://therekenrek.com/about_the_rekenrek.html – about the Rekenrek http://therekenrek.com/sample_lesson_plan.pdf – sample lesson with the Rekenrek http://www.mathlearningcenter.org/media/Rekenrek_0308.pdf – using Rekenreks http://www.ronblond.com/MathGlossary/Division01/Rekenrek/REKENREK/index.html – virtual Rekenrek http://www.k-5mathteachingresources.com/Rekenrek.html – Rekenrek activities http://www.mefeedia.com/watch/26880976 – video of students using Rekenreks http://www.mathematicallyminded.com/downloads/Rekenrek%20Activities_Directions.pdf – activities and instruction on how to make a Rekenrek

Base 10 Blocks

GAINS: Tips for Manipulatives 1

What are Base 10 Blocks? 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 10 cm by 1 cm 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.

How do Base 10 Blocks help Students? The size relationships of the blocks can be used to explore number concepts. Students can explore place value concepts as well 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.

(x + 1) (2x + 2) = 2x² + 4x + 2

How many are recommended? Students can use the base 10 blocks individually, in pairs, or small groups depending on the activity as well as the concept being explored. A class set of 1000 unit cubes, 200 rods, 120 flats and 10 cubes will allow students to perform a variety of activities. When students are first learning to use base 10 blocks, allow for exploration time. A transparent set is useful for overhead or document camera demonstrations by students(s) or teacher(s). Sample Activities 1. Use the base 10 blocks to represent the following numbers: 1342, 211.1, 13.28, 2.524 2. How many ways can you represent 43.21, using the blocks? 3. Use any combination of blocks to represent 258. Place your blocks on centimetre grid paper to make a

polygon so that there are no empty spaces in the middle. Record the shape and perimeter of the shape. Rearrange the blocks and find the new perimeter. How can you show the shortest perimeter? the longest perimeter?

4. The object of the game is to get closest to one whole after 10 rounds. For this game, a flat is equal to one whole. Students take turns rolling two numbered cubes in each round and choose how to arrange the digits to make a number less than one. Students then decide whether they add or subtract that number from 1 and can trade blocks for their flat, if necessary. After 10 rounds see which player is closest to one whole. Discuss strategies.

5. 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)?

6. If there are no cubes available, how else can you represent 1000? How many tens is a thousand worth? How do you know?

7. How do you know that 16 hundreds are more than 1000? If you know a number is 36 hundreds, how do you know how many tens it is?

Base 10 Blocks

GAINS: Tips for Manipulatives 2

8. Show how to model 4 × 22 with the base 10 blocks. Is there another way to arrange the blocks for the same answer? What do you notice about the two numbers you multiply in your new arrangement? Try it again for 16 × 23.

9. 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.

10. Use the base 10 blocks to prove that 0.4 and 0.40 are the same 11. Put the amounts in order from least to greatest: 14.2, 1.42, 0.14, 12.4, 1.24 12. If a cube is a whole, how much is a flat worth? a rod? a unit? two rods? a flat and 4 units? 13. Which is larger 4.2 or 4.12? How do you know? 14. Use the blocks to model the rules as to why (n +1) (n + 1) = n² + 2n + 1. 15. Solve this problem, using base 10 blocks (from EQAO 2007-2008): Josie, Christina, Audrey, and Manny go

shopping: Josie spends 45

of her money, Christina spends 75% of her money, Audrey spends 0.68 of her

money and Manny spends 1720

of his money. Who has the largest percentage of money left?

Recommended Websites http://nlvm.usu.edu/en/nav/frames_asid_152_g_1_t_1.html?from=topic_t_1.html using base 10 blocks to represent numbers http://nlvm.usu.edu/en/nav/frames_asid_154_g_1_t_1.html?from=topic_t_1.html addition http://nlvm.usu.edu/en/nav/frames_asid_155_g_1_t_1.html?from=topic_t_1.html subtraction http://nlvm.usu.edu/en/nav/frames_asid_264_g_1_t_1.html?from=topic_t_1.html adding and subtracting decimals http://www.folksemantic.com/visits/104455 adding and subtracting decimals http://www.susancanthony.com/Resources/base10ideas.html base 10 activities http://ejad.best.vwh.net/java/b10blocks/description.html#algebra base 10 activities

GAINS: Tips for Manipulatives

Colour Tiles

What are Colour Tiles? Colour tiles have two square surfaces. They are usually referred to as “square” colour tiles even though they are 3-D objects. Sets usually come with four colours of tiles. How do Colour Tiles help students? Colour tiles can be used for explorations, investigations, or games in any of the mathematics curriculum strands. The variety of colours allows the tiles to be used for probability, as well as proportional reasoning. Students can use colour tiles to create, identify, and extend patterns. The patterns can be used to develop algebraic models.

How many are recommended? Students usually work in pairs or small groups, when using colour tiles. A class set of about 700 to 1000 pieces allows students to do a variety of activities. Students can make paper versions. Many colour-tile activities can be done with connecting cubes. When students first use colour tiles, allow for exploration time. A transparent set is useful for overhead demonstrations by students and/or teachers. Sample Activities 1. Build a tile train. What colour is the 200th cube in the train? 2. How many different ways can you use tiles to represent 3

4 (or a decimal, or a percent)?

3. Use two different colours of tiles to model integer questions. (Note: The number of tiles represents size and the colour of tiles represents sign.)

4. Design a sequence of patterns. Analyse the pattern and determine an attribute of the 100th term in the sequence (connect to algebraic modelling).

5. Explore relationships between perimeter and area. 6. Put different coloured tiles into a paper bag. Determine the probability of choosing a yellow tile. 7. Create a pattern. Draw its reflection and check your answer using a mirror. 8. Pick any number. Determine if the number is prime by using colour tiles. (Note: If the number is prime, there

will be only one possible rectangular arrangement of the tiles – a single row.) 9. Use the colour tiles to show all possible factors of 24. 10. Model the ratio 4:1 using four red tiles and one yellow tile. Place the tiles in a row. Add a second identical

row and discuss similarities and differences. Continue adding rows until there are 100 tiles in total. How does this illustrate that 4

5is the same as 80%?

11. Let r dollars represent the value of one red tile. Let y dollars represent the value of one yellow tile, and so on. Determine an expression that represents the total value of a collection of tiles. Combine two different collections and determine an expression for the total value of the new combined collection. Assign a dollar value to each different coloured tile and use the algebraic expression to determine the total value.

12. Use patterns to develop algebraic models. Develop understanding that two different algebraic models can be simplified to show equivalence.

Recommended Websites http://matti.usu.edu/nlvm/nav/grade_g_3.html Interactive Tiles (go to Algebra – Gr. 6-8, Polyominoes) http://mathcentral.uregina.ca/RR/database/RR.09.98/loewen2.4.html Selection of Activities http://math.rice.edu/~lanius//Lessons/Patterns/rect.html Pattern Challenge

n + n(n - 1) is equivalent to (n - 2)2 + 2n +2(n-2) and to (2n - 1) + (n - 1)2

Total number of tiles = (number of light tiles) + (number of dark tiles), so:

light tiles: 2n - 1dark tiles: (n - 1)2

light tiles: (n - 2)2dark tiles: 2n + 2(n-2)

light tiles: ndark tiles: n(n-1)

GAINS: Tips for Manipulatives

Pattern Blocks

What are Pattern Blocks? One set of pattern blocks has six colour-coded geometric solids. The top and bottom surfaces of these solids are geometric shapes: hexagon, trapezoid, square, triangle, parallelogram (2). Except for the trapezoid, the lengths of all sides of the shapes are the same. This allows students to form a variety of patterns with these solids. How do Pattern Blocks help students? As their name suggests, pattern blocks are used to create, identify, and extend patterns. Students use the many relationships among the pieces to explore fractions, angles, transformations, patterning, symmetry, and measurement.

How many are recommended? Students usually work in pairs or small groups, when using pattern blocks. A class set of about 700 to 1000 pieces allows students enough pieces to do a variety of activities. Sometimes a single set of six pieces per pair is sufficient but larger amounts are often required. Allow time for students to explore the blocks and to become familiar with their attributes. Discuss the variety of names that can be used for each piece, e.g., the two parallelogram faces are also rhombi. The triangle face can also be called an equilateral triangle, an acute triangle, as well as a regular three-sided polygon. Note: Ensure that students understand that blocks are named for the large faces although each block is actually a 3-D geometric solid. For example, instead of properly naming the yellow block as a hexagonal prism, it is usually called a hexagon. Sample Activities 1. How many different ways can you name the orange (square) block? 2. How many different ways can you cover the hexagon with other shapes? 3. 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? 4. Design a tessellating floor pattern. 5. How many lines of symmetry are there for each block? 6. Create a symmetrical design. Describe the design to a partner. 7. How many different angles can you create by placing two or more blocks together so they meet at one

vertex? 8. Determine the size of each different face as a fraction of the size of the hexagon. 9. If the hexagon represents 5

6, what fraction does the triangle represent?

10. Build a shape with a perimeter of 10 and an area of 5. 11. Design a sequence of patterns. Analyse the pattern and determine an attribute of the 100th term in the

sequence. 12. Put a variety of pieces into a paper bag. Determine the probability of choosing one type of block. 13. Let a represent the area of a hexagon. Determine a representation for the area of each other block. 14. Create a shape with three or more pattern blocks. Choose a variable to represent the area of each block in

the shape. Create an expression for the total area. Make several copies of the shape. Create an algebraic expression for the total area of all of the shapes.

Recommended Websites http://nlvm.usu.edu/en/nav/frames_asid_169_g_1_t_2.html Virtual Pattern Blocks http://matti.usu.edu/nlvm/nav/category_g_4_t_3.html Interactive Manipulatives and activities http://math.rice.edu/~lanius/Patterns/ Exploring Fractions http://fcit.usf.edu/math/resource/manips/pattern.pdf Explorations with Pattern Blocks http://mathforum.org/sum95/suzanne/active.html Investigating Tessellations using Pattern Blocks http://www.mathcats.com/explore/polygons.html Polygon Playground (interactive)

GAINS: Tips for Manipulatives

Tangrams

What are Tangrams? 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. How do Tangrams help students? Tangrams are particularly useful in problem-solving activities. Frequently, tangrams are also used for exploring geometry, proportional reasoning, area, and algebra.

How many are recommended? Tangram activities are often done with pairs of students so, one tangram set per pair of students is sufficient. Students can make their own set from a template. When tangrams are introduced give students time to experiment and explore. Reassemble the tangram square before storing in a small zip-lock bag. Sample Activities 1. Assume that each tangram has a value of one whole. Find the value of each piece (based on surface area of

one face) stated as a fraction. (Repeat the activity but state the value as a decimal or percent.) 2. Assume a whole tangram set costs $1.60. Determine a value for each piece of the set. 3. If the largest triangle represents 5

8 then what fraction does the smallest triangle represent?

(As an alternative to 58

use an integer, decimal or percent.)

4. Use four tangram pieces to make a parallelogram. 5. How many ways can right isosceles triangles be formed with the tangram pieces? (As an alternative, form

squares, rectangles, or parallelograms.) 6. Find the perimeter/area of each piece. (This is an opportunity to use the Pythagorean theorem). 7. Use the smallest triangle and the largest triangle to explore what happens to the area of a triangle when the

lengths of both height and base are doubled. 8. Choose one of the triangles to represent a loading ramp. Calculate the slope of ramp. 9. Stack the right triangles so that the right angles are aligned. Make an observation about the hypotenuses. 10. Let a represent the area of the smallest triangle. What algebraic expression would represent the area of each

other piece? 11. Create a tangram design using two or more pieces. Then create an algebraic expression to represent the area

of the design. 12. Create convex polygons using tangram pieces. Investigate the sum of the interior angles. 13. Sort and classify the tangram pieces. 14. Create a shape using tangram pieces. Give instruction so your partner can build the same shape (sight

unseen). 15. Create a “spinner” using the names of the tangram pieces that meet at one vertex. Determine the probability

that the spinner will land on each piece. Recommended Websites http://standards.nctm.org/document/eexamples/chap4/4.4/part2.htm Interactive Tangram Puzzles http://mathforum.org/trscavo/tangrams/construct.html Construct a Tangram Set

GAINS: Tips for Manipulatives

Connecting Cubes

What are Connecting Cubes? Interlocking cubes are available in different sizes and different colours. In a 1-cm cube set, each cube has a mass of 1 g, which is useful for mass explorations. How do Connecting Cubes help students? 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.

How many are recommended? Students usually work in pairs or small groups when using connecting cubes. A class set of about 700 to 1000 pieces allows students enough cubes to do a variety of activities. When interlocking cubes are introduced to the class, allow students time to explore. Note: When students finish an activity, consider instructing students to connect the cubes in “trains” of 10 linked cubes to facilitate collection after an activity, storage between activities, and preparation for the next activity. Sample Activities 1. Build a cube train. What colour is the 200th cube in the train? 2. Represent 347 in expanded form using single cubes, trains of 10 cubes, and large “squares” of 100 cubes. 3. Create a model of one quilting square (or floor tile) using nine cubes of different colours. If four squares are

used to form one larger square, how many different patterns can be formed? 4. How many different ways can you illustrate 3

4 (or a decimal, or a percent)?

5. Use two different colours of cubes to model integer questions. (Note: the number of cubes represents size and the colour of cubes represents sign.)

6. Design a sequence of patterns. Analyse the pattern and determine an attribute of the 100th term in the sequence (connect to algebraic modelling).

7. Explore relationships between perimeter and area, and between surface area and volume. 8. Build a structure then draw the top, front, and side views (or draw the structure on isometric paper). 9. Put different coloured cubes into a paper bag. Determine the probability of choosing a yellow cube. 10. Let b represent the mass of one cube. Determine representations for the masses of different structures. 11. Build an object out of interlocking cubes. Write instructions for making the object. Trade instructions with

another group. 12. Create different “staircase” models to investigate slope. 13. Use connecting cubes to create histograms or frequency charts. 14. Determine how many different structures can be built with four cubes. 15. Use cube links to model different ways to decompose shapes to find area and

perimeter. 16. Use cubes to model algebraic rules. Example: n2 – 1 = (n – 1) (n + 1)

Recommended Websites http://illuminations.nctm.org/Activities.aspx?grade=all&srchstr=isometric Interactive Isometric Drawing Tool http://nlvm.usu.edu/en/nav/search.html?qt=blocks Space Blocks, Algebra. 6-8 (interactive 3-D blocks) http://www.edu.gov.on.ca/eng/curriculum/elementary/math8ex/pattern.pdf Grade 8 Exemplar, Patterning and Algebra activity www.otrnet.com.au/IntegratedMathsModules/B06/B06ApplicationF.pdf Box Design Activity

Examplefor n = 3

(n - 1)(n + 1) n2 - 1

Fraction Strips and Fraction Towers

How do Fraction Towers help students? Fraction towers help students to develop number sense for fractions and mixed numbers, order fractions, and explore number operations with fractions using concrete materials. The towers provide students with a visual and graphical representation because they can stack upwards to show the relationships. The predetermined colours help make these relationships more explicit for students. Fraction towers allow students to explore equivalency as well as the relationship between fractions, decimals and percents. They also help students to see the relationships between the parts and the whole. How many are recommended? Using fraction towers in pairs or small groups is recommended. One set per group of three or four students is ideal but a few sets for a room can be used effectively in conjunction with other manipulatives for representing fractions, decimals and percents. If fewer sets are available, they can be rotated throughout the class so that all students have a chance to explore with the towers and look for relationships; alternatively they can be used in a learning centre. A transparent overhead set or a document camera is effective for whole class demonstrations. Sample Activities 1. Using the towers, students explore, look for relationships between the pieces, and

share their discoveries with their peers. 2. Students explore fractions that are equal to ¼, showing how they know. 3. Working in partners, students put fraction pieces together in various lengths to

determine fraction towers that are close to 0, 1/4, 1/2, 3/4, 1, justifying their answers.

What are Fraction Towers? Fraction towers consist of 51 interlocking blocks that are proportional to each other. There are 9 colours that represent fractions 1 whole, 1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/10, 1/12. These colours help to draw the connections for students among the different fractional parts during their exploration. The colours represented for these fractions match the colours for equivalent decimals and percents if the towers are purchased.

Another option is to purchase fraction tower equivalency cubes which have one side of the cube labeled in fractional notation, the second side in decimal notation, the third in percent notation, and the last side is blank. It was noted above that some resources refer to fraction strips as fraction towers because they are prearranged and stacked beside each other to show relationships in a 2 dimensional form.

Fraction Strips and Fraction Towers

4. Students work in pairs to determine which fraction is larger: 1/2 or 5/8, using multiple ways to show how they know.

5. Students determine the sum 2/8 + 4/8, and show other fractions that also represent this answer.

6. Students compare and order 1/2, 3/8, 2/3, 6/10, explaining their reasoning. 7. Students find other fraction amounts equal to 4/12, showing how they know they

are equal and finding other fractions pairs that are equal. 8. Students solve: James, Gursh and Allan all get the same amount of money for their

monthly allowance. If James spent 20%, Gursh spent 1/3 and Allan spent 0.4 of his allowance who has the most money left over at the end of the month?

9. Students determine how many tenths are in 3/5 and explain how it’s the same as 3/5 ÷ 1/10

10. Students explore: If you added two fractions and your answer is 8/10 what could the two fractions added be, showing as many solutions as they can.

11. Students demonstrate 3 x 1/8 by joining fractional pieces, and then repeat for 3 x 2/8 and describe what they notice?

Recommended Websites http://oame.on.ca/CLIPS/tools/ (electronic fraction strips) http://oame.on.ca/clips/index.html?ePractice=T (interactive fractions activities) http://www.eworkshop.on.ca/edu/pdf/Mod22_fraction_strips.pdf (printable fraction strips) http://www.gradeamathhelp.com/free-fraction-strips.html (downloadable fraction strips and activities) http://nlvm.usu.edu/en/nav/frames_asid_203_g_1_t_1.html?from=topic_t_1.html (electronic fraction strips where the whole can be manipulated) http://nlvm.usu.edu/en/nav/category_g_2_t_1.html (virtual manipulatives) http://www.fractionmonkeys.co.uk/ (fraction game) http://www.maths-games.org/fraction-games.html (interactive fraction games) http://www.edugains.ca/resources/LearningMaterials/GapClosing/Junior/Intermediate/GCU2_SB_ComparingFractions.pdf (fraction towers activity) http://www.learningresources.com/text/pdf/2564book.pdf (shows how you can use fraction towers to change a mixed number into an equivalent fraction before subtraction) Reference: Teaching Student-Centered Mathematics Grades 5-8, John Van de Walle, Pearson, Boston, 2006

Fraction Strips and Fraction Towers

How do Fraction Strips help students? Fraction strips help students to visualize and explore fraction relationships. They allow students to develop a concrete understanding of fractions and mixed numbers, investigate equivalency, compare and order fractions and explore number operations with fractions. Fraction strips help students by allowing them to manipulate parts of the same whole. By their very nature they keep the whole consistent for students. How many are recommended? One set of fraction strips per student is recommended. Teachers can print the coloured versions from online templates or the black and white versions work as well; students can colour the strips themselves. Teachers might consider having students colour the pieces in a consistent manner for ease when referring to pieces during debriefing of the activities. Sample Activities 1. Using non-labeled fraction strips, students cut out and arrange the pieces in relation

to the whole. Students label each of the fraction amounts using proper fractional notation. Engage the class in a vocabulary discussion about numerators and denominators.

2. Students identify 1/2, 1/3, 1/4, 1/8 and explain their reasoning. 3. Students compare various fraction strips and record their observations, and share

their discoveries with the class. Encourage students to begin by counting the pieces of the whole to visualize how fractional notation relates the parts to the whole. Students explore fractions that are part of other fractions. (e.g., 1/10 is 1/2 of 1/5)

4. Students show all the different ways they can represent ½, and describe what they notice about the denominators they have chosen.

5. Students predict which fraction s larger: 2/3 or 2/6? Students use the strips to test their predictions and discuss what they notice?

6. Students order various fraction amounts from greatest to least, explaining what they notice about the denominators.

7. Each student in the class randomly receives a fraction strip ; they find partners so that the total of their strips is one whole.

What are Fraction Strips? Fraction strips are rectangular pieces (electronic or copied on paper strips) to represent different parts of the same whole. They can be cut apart and manipulated to see how various parts can be added together to make the whole or compare different fractional amounts for equivalency. They can be various colours, and sizes. Some resources may call fraction strips fraction towers if they remain intact and pre-arranged.

Fraction Strips and Fraction Towers

8. Students solve and explain their thinking: Three siblings are helping to rake the leaves in the yard of their house. If Hannah rakes 1/4 of the yard and Mya rakes 1/3, how much of the yard is left for Fyona to rake? Who has raked the most leaves, and who has raked the least?

9. Students use the activity in GAP Closing Junior/Intermediate Module 2, page 14 http://www.edugains.ca/resources/LearningMaterials/GapClosing/Junior/Intermediate/GCU2_SB_ComparingFractions.pdf For this activity, the fraction strips are intact. Students use a ruler to measure 1/3 and 2/6 to verify that they both are the same length. Students test other fraction measures to look for equivalent fractions.

10. Students measure from one end of the whole fraction strip to the other end; then measure 1/2 the fraction strip from one end, then ¼, and describe what they notice. If the strips are created with “friendly” numbers from end to end of the whole, (e.g., 40 cm long) it is easier for students to make connections. (e.g., 1/2 is equal to 50% of the whole number measured, and 1/4 is 25% of the whole number measured).

11. Model for students how fraction strips can show multiplication of fractions. (e.g., 1/3 x 3/5 is really 1/3 of 3/5

Start by showing 3/5 Then consider 1/3 of what is coloured This shows the answer is 1/5) Students practice with other fraction multiplication questions. 12. Model for students how fraction strips can show division of fractions (e.g., 2/3 ÷ ¾ Use 12ths to show the whole 2/3 would be ¾ would be

Students write the answer as a fraction showing that 8 pieces out of 9 represents 2/3 ÷ ¾ Therefore 2/3 ÷ ¾ = 8/9)

Students practice with other fraction division questions. Note: Students can visualize that by first choosing the common denominator they can divide the numerators to determine their answer; this eliminates the need for the rule “invert the divisor and multiply”.

31

Fraction Circles

GAINS: Tips for Manipulatives 1

What are Fraction Circles? Fraction circles are a set of nine circles of various colours. Each circle is broken into equal fractional parts and uses the same-sized whole. The circles included are one whole as well as circles divided into halves, thirds, quarters, fifths, sixths, eighths, tenths, and twelfths. Depending on the manufacturer, fraction circles can be transparent for use on an overhead projector for whole class activities or opaque for use at students’ desks or with a document camera. Fraction rings are clear plastic rings that are open in the middle and can hold the fraction circles in the center.

How do Fraction Circles help students? Fraction circles can be used to help students see relationships between fractional parts of the same whole. Students can compare and order fractions, see equivalent fractions, explore common denominators, as well as explore basic operations with fractions. Fraction rings can be used in conjunction with fraction circles to make connections to time, decimals, and percent. They can also be used to make circle graphs. How many are recommended? Students usually use fraction circles in pairs or small groups. Each pair of students will need a set of fraction circles. When students are first exposed to fraction circles they should be given some time for exploration of the circles and the relationships among their parts. Sample Activities 1. Have students orally count the ‘purple’ pieces to create a whole. Have someone else count the ‘blue’ pieces

after that. This allows students to practise saying the names of the parts that make up the whole.

2. Pairs discuss what it means to compare a circle divided into more parts with one divided into a lesser number of parts.

3. Which is larger 48

or 34

? How do you know?

4. Compare and order 1 2 1 3, , , .2 3 8 5

5. How many different ways can you make a half? a quarter? a third? one and one sixth?

6. Is there another fractional part(s) that can cover 912

?

7. Add 1 12 3+ . Are there other equal fractional parts that can fit on top of what was created?

8. If you add 24

and 15

, how much of the circle is not completed?

9. Show 25%, 33%, 50%, 75%, 70% with the fraction circles.

10. Which fractional parts can be combined to equal 12

?

11. Solve this problem, using fraction circles: Sumeet’s team won 13

of their soccer games this season. Wayne’s

team won 25% of their soccer games. Which team won more games?

12. How many tenths are in 35

? Or what is 3 15 10÷ ?

13. Show 138

× by joining fractional pieces. Try 238

× . What do you notice?

Fraction Circles

GAINS: Tips for Manipulatives 2

14. Show 1 32 4× (Prompt students to first find an equivalent of 3

4so that it can be divided in half.). Ask what

students notice about the question and the answer.

15. Solve this problem, using fraction circles: A survey was taken in the class as to the top five sports the class

enjoys. Cricket accounted for 25% of class’ preference, soccer accounted for 38

. The remaining sports of

basketball, swimming and snowboarding were all equal in amount. What fraction or percentages of the class enjoys the last three sports? If you arrange the sports into a circle graph, what are the degree measures of each fractional part in the measure of the central angle?

16. Solve this problem (from eWorkshop): At a camp, the campers stayed in 4 cabins. In the Grizzly Bear

cabin, there were 4 campers, in the Snowy Owl cabin 5 campers, in the Caribou cabin 8 campers, and in the Salmon cabin 6 campers. One day, the campers were treated to pizza. All the pizzas were the same size and could be cut into any number of equal pieces. The pizzas were given out in the following way:

- Grizzly Bear cabin – 3 pizzas - Snowy Owl cabin – 4 pizzas - Caribou cabin – 7 pizzas - Salmon cabin – 5 pizzas

Discuss how the number of pizzas given to each cabin was always one less than the number of campers. Did some campers get more pizza than others, or did all the campers receive the same amount of pizza?

Recommended Websites http://nlvm.usu.edu/en/nav/frames_asid_105_g_4_t_1.html?from=topic_t_1.html equivalent fractions http://www.youtube.com/watch?v=vWAWrf0jPAQ using fraction circles http://www.youtube.com/watch?v=KkEE3hvG-V8&feature=related using fraction circles http://www.visualfractions.com/ – practising with fraction circles http://www.eworkshop.on.ca/edu/pdf/Mod22_lesson_summary.pdf camper problem http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=252 improper fractions and mixed numbers http://www.edu.gov.on.ca/eng/studentsuccess/lms/files/tips4rm/gr7Unit7.pdf fractions and decimals http://www.edu.gov.on.ca/eng/studentsuccess/lms/files/tips4rm/gr8Unit5.pdf fractions and percents

GAINS: Tips for Manipulatives

Geoboards

What are Geoboards? Geoboards are grids of pegs that hold rubber bands in position. Geoboards are available in a variety of sizes, styles, and colours. The transparent geoboard can be placed on an overhead projector to facilitate sharing of student observations and conclusions. Many geoboard activities are available for both the 5-pin × 5-pin and 11-pin × 11-pin geoboard sizes. How do Geoboards help students? Geoboards are useful in developing conceptual understanding of area and perimeter. However, they can be used to explore mathematics from any of the mathematics curriculum strands. Fractions, the Pythagorean theorem, tessellations, transformations, and patterning support the use of this manipulative.

How many are recommended? Geoboard activities are often done by pairs of students, so one geoboard per pair of students is sufficient. When geoboards are introduced to the class, give students some time to experiment and explore. Blackline masters of geoboards can be used to record solutions. Sample Activities 1. Construct a design for a quilt square or a stained glass window. Analyse the design. Enlarge or reduce the

design. Compare your design with another student’s design. (What’s the same? What’s different?) 2. Construct two three-sided (or four-sided) figures that are congruent (or similar). 3. Determine how many different sizes of squares (or equilateral triangles) can be constructed on a

5-pin × 5-pin geoboard. How many can be constructed on a 11-pin × 110-pin geoboard? 4. Construct two pentagons that have the same areas but different perimeters (or vice-versa). 5. Construct a symmetrical design. 6. Construct a diagonal segment. Construct a line segment that is parallel to the first segment. Determine the

lengths and slopes of both segments. 7. Construct a line segment whose length has a measure between 3 and 4 units (or a given slope). 8. Choose any two pegs on the geoboard and determine different paths from the first peg to the second. Record

solutions and determine which path is the longest and which is the shortest. 9. Create a 3-sided (or 4-sided) shape. Compare that shape with a partner’s shape. What is the same? What is

different? 10. Create a shape. Determine its area in more than one way. 11. Design a shape on one half of the geoboard. Construct its reflection. 12. Make a figure that has an area of 4 and a perimeter of 10. 13. Determine the maximum area that can be enclosed on an 11-pin × 11-pin geoboard if the perimeter is 25 units. 14. Determine different ways to divide the geoboard into 4 equal areas. 15. Divide the geoboard into different areas. Express each area as a fraction (decimal, percent) of the whole area. Recommended Websites http://standards.nctm.org/document/eexamples/chap4/4.2/ Interactive Geoboard Activities http://nlvm.usu.edu/en/nav/search.html?qt=frames Virtual Manipulatives Geoboards http://mathforum.org/trscavo/geoboards/contents.html Geoboards in the Classroom (includes dot paper)

Rubber bands

Other Recommended Math Resources

Polydrons

Dice (multi-sided dice)

Calculators

T1-15 calculators (grades 6-8)

T1-10 calculators (grades 3-5)

T1-08 calculators (grades K-2)

Literature

Marilyn Burns Math and Literature Collections (available from K-8)

Big Ideas from Dr. Small (available for K-12)

Good Questions: Great Ways to Differentiate Math Instruction (Marian Small)

Gap Closing (Grade 6 and 9) can also be used for students working below level

TIPS4RM (Grade 7-9)

The Super Source Collection (K-8)

The Guides to Effective Instruction in Mathematics (K-3 and 4-6)

Reflex Math

Literacy and Numeracy Secretariat Resources (LNS)

monographs

webcasts

Useful Math Websites National Library of Virtual Manipulatives (NLVM) - Grades 6-8 http://nlvm.usu.edu/en/nav/category_g_3_t_2.html Explore Learning Gizmos - Grades 7-12 (Math and Science) http://www.explorelearning.com/ Username: contact Password: curriculum Mathwire.com http://www.mathwire.com/ Link to Learning http://www.linktolearning.com/math.htm Math Frog http://cemc2.math.uwaterloo.ca/mathfrog/ BBC www.bbc.co.uk/schools/ Illuminations http://illuminations.nctm.org/ Edugains http://www.edugains.ca/newsite/math2/index.html IXL http://ca.ixl.com/ Ontario Educational Resource Bank (OERB) (Grade 5-12) https://resources.elearningontario.ca/ Username: hscdsbteacher or hscdsbstudent Password: oerbt oerbs Manga High Math (K-12) http://www.mangahigh.com