USING MATH MANIPULATIVES TO BUILD UNDERSTANDING

RekenreksWhat are Rekenreks?Rekenreks are arithmetic racks, developed by Adrian Treffers,a mathematics curriculum researcher at the FreudenthalInstitute in Holland. There are two rods of 10 beads. Each rodhas 5 beads of one colour followed by 5 of another colour. Theorder and colours on the top rod are repeated on the lower rod. The colours most often used arered and white. The starting position should show all the beads pushed to the far right. Thestudent enters a number by sliding the beads to the left in a one-push motion. It is important thateveryone 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 orfinding near doubles as well as thinking in terms of 5s and 10s, instead of counting from oneeach time or counting on in addition and subtraction. Students improve their ability to regroupnumbers 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 theteacher have a large demonstration Rekenrek which is visible tothe whole class for shared activities.Sample Activities1. To introduce the Rekenreks, push all ten beads from the top row to the left and cover thebottom row(s). Students do the same on their individual Rekenreks. Ask the students whatthey notice.2. Push various numbers to the left and ask the students to quickly tell how many beads theysee. Start with 1, then 5, 7, 9, 12, 16, etc. Ask students how they know how many they seeand listen for answers that involve visualizing 5 and 10, or seeing doubles, as opposed tocounting individual beads.3. Reinforce the idea of showing a number on the Rekenrek in “one push.” Ask students toexplain how they knew they were pushing the right number. Notice reasoning that involvesvisualization of 5s and 10s, as well as doubles.4. Model mathematical situations such as: 8 birds were eating crumbs in the park and 4 morecame 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 itas 5 and 5 with one more. Show 5 red on the top rod and 5 red on the bottom rod with oneadditional 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 thebottom set of beads to make the chosen number, e.g., To make 9, push 5 red beads to the leftfrom the top row. Students push 4 on the bottom rod. Look and listen for strategies. Onceshared a few times, students could be encouraged to play the game with a partner.GAINS: Tips for Manipulatives51

Rekenreks8.9.10.11.12.13.14.15.Make numeral cards from 1-20. Hold up a card at random and ask students to show thatnumber on their individual Rekenreks. Allow one push for numbers up to 10 and two pushesfor any number larger. Debrief various solutions with and how students arrived at theposition of beads. To further this activity in pairs, students have a barrier between them. Onestudent draws a card from the pile, saying the number out loud and making that number ontheir Rekenrek in any manner they choose. The first player provides a clue as to how manybeads they pushed on the top rod but the partner must figure out how many are pushed onthe bottom rod to replicate the partner’s solution.Determine all the ways to make 10, using beads from each row.Use the Rekenrek to prove that 3 2 1 4Ask: Is 7 8 8 7? How do you know?Once familiar with the Rekenreks and the number of beads on each row, show some beadsto the class and ask the class to figure out the number of hidden beads to make a certainnumber.There were 12 students playing on the play structure in the playground. Four were on thetop level. How many were on the bottom level?Six students were on the stage in the gym practising for the school play, while four were onthe floor setting up chairs. Three more students came to help. How many students were inthe gym?Out of the 20 cupcakes brought into class for Owen’s birthday only 3 were left. How manywere eaten?Recommended Websiteshttp://therekenrek.com/about the rekenrek.html – about the Rekenrekhttp://therekenrek.com/sample lesson plan.pdf – sample lesson with the kenrek 0308.pdf – using sion01/Rekenrek/REKENREK/index.html – .com/Rekenrek.html – Rekenrek activitieshttp://www.mefeedia.com/watch/26880976 – video of students using oads/Rekenrek%20Activities Directions.pdf –activities and instruction on how to make a RekenrekGAINS: Tips for Manipulatives62

Base 10 BlocksWhat 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 by1 cm by 1 cm. These 10-unit pieces are called rods. The flat square blocks are10 cm by 10 cm by 1 cm and are called flats. The largest blocks are 10 cm by10 cm by 10 cm and are called cubes. These terms are used to signify theinterchangeableness 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 valueconcepts 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 canalso 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-variableactivities. The unit would represent the number. The rod would represent the single variable such as x. The flatwould represent the square of the variable such as x².(x 1) (2x 2) 2x² 4x 2How many are recommended?Students can use the base 10 blocks individually, in pairs, or small groups depending on the activity as well as theconcept being explored. A class set of 1000 unit cubes, 200 rods, 120 flats and 10 cubes will allow students toperform 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 Activities1. Use the base 10 blocks to represent the following numbers: 1342, 211.1, 13.28, 2.5242. 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 apolygon 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 longestperimeter?4. The object of the game is to get closest to one whole after 10 rounds. For this game, a flat is equal to onewhole. Students take turns rolling two numbered cubes in each round and choose how to arrange the digits tomake a number less than one. Students then decide whether they add or subtract that number from 1 and cantrade blocks for their flat, if necessary. After 10 rounds see which player is closest to one whole. Discussstrategies.5. Solve this problem, using base 10 blocks: A video game company wants to pack their games to send out tostores. The game is the same size as a flat. They have decided to fit 12 games in a box. What are their box sizeoptions? 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? Howdo you know?7. How do you know that 16 hundreds are more than 1000? If you know a number is 36 hundreds, how do youknow how many tens it is?GAINS: Tips for Manipulatives71

Base 10 Blocks8. Show how to model 4 22 with the base 10 blocks. Is there another way to arrange the blocks for the sameanswer? What do you notice about the two numbers you multiply in your new arrangement? Try it again for16 23.9. Solve this problem, using base 10 blocks: It cost 120 for 6 people to enter an amusement park. Model howyou 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 same11. Put the amounts in order from least to greatest: 14.2, 1.42, 0.14, 12.4, 1.2412. 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 go4shopping: Josie spends of her money, Christina spends 75% of her money, Audrey spends 0.68 of her517of his money. Who has the largest percentage of money left?money and Manny spends20Recommended Websiteshttp://nlvm.usu.edu/en/nav/frames asid 152 g 1 t 1.html?from topic t 1.html using base 10 blocks torepresent numbershttp://nlvm.usu.edu/en/nav/frames asid 154 g 1 t 1.html?from topic t 1.html additionhttp://nlvm.usu.edu/en/nav/frames asid 155 g 1 t 1.html?from topic t 1.html subtractionhttp://nlvm.usu.edu/en/nav/frames asid 264 g 1 t 1.html?from topic t 1.html adding and its/104455 adding and subtracting e10ideas.html base 10 description.html#algebra base 10 activitiesGAINS: Tips for Manipulatives82

Colour TilesWhat 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 fourcolours of tiles.How do Colour Tiles help students?Colour tiles can be used for explorations, investigations, or games in any of themathematics curriculum strands. The variety of colours allows the tiles to beused for probability, as well as proportional reasoning. Students can use colourtiles to create, identify, and extend patterns. The patterns can be used to developalgebraic 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 piecesallows students to do a variety of activities. Students can make paper versions. Many colour-tile activities can bedone with connecting cubes. When students first use colour tiles, allow for exploration time. A transparent set isuseful for overhead demonstrations by students and/or teachers.Sample Activities1. Build a tile train. What colour is the 200th cube in the train?2. How many different ways can you use tiles to represent34(or a decimal, or a percent)?3. Use two different colours of tiles to model integer questions. (Note: The number of tiles represents size andthe colour of tiles represents sign.)4. Design a sequence of patterns. Analyse the pattern and determine an attribute of the 100th term in thesequence (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, therewill 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 identicalrow and discuss similarities and differences. Continue adding rows until there are 100 tiles in total. Howdoes this illustrate that45is 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 differentcollections and determine an expression for the total value of the new combined collection. Assign a dollarvalue 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 differentalgebraic models can be simplified to showequivalence.light tiles: ndark tiles: n(n-1)light tiles: (n - 2)2dark tiles: 2n 2(n-2)light tiles: 2n - 1dark tiles: (n - 1)2Total number of tiles (number of light tiles) (number of dark tiles), so:n n(n - 1) is equivalent to (n - 2)2 2n 2(n-2) and to (2n - 1) (n - 1)2Recommended Websiteshttp://matti.usu.edu/nlvm/nav/grade g 3.html Interactive Tiles (go to Algebra – Gr. 6-8, ase/RR.09.98/loewen2.4.html Selection of Activitieshttp://math.rice.edu/ lanius//Lessons/Patterns/rect.html Pattern Challenge9GAINS: Tips for Manipulatives

Pattern BlocksWhat are Pattern Blocks?One set of pattern blocks has six colour-coded geometric solids. The top andbottom surfaces of these solids are geometric shapes: hexagon, trapezoid, square,triangle, parallelogram (2). Except for the trapezoid, the lengths of all sides ofthe shapes are the same. This allows students to form a variety of patterns withthese solids.How do Pattern Blocks help students?As their name suggests, pattern blocks are used to create, identify, and extendpatterns. Students use the many relationships among the pieces to explorefractions, 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 1000pieces allows students enough pieces to do a variety of activities. Sometimes a single set of six pieces per pairis sufficient but larger amounts are often required. Allow time for students to explore the blocks and tobecome familiar with their attributes. Discuss the variety of names that can be used for each piece, e.g., thetwo parallelogram faces are also rhombi. The triangle face can also be called an equilateral triangle, an acutetriangle, 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 actuallya 3-D geometric solid. For example, instead of properly naming the yellow block as a hexagonal prism, it isusually called a hexagon.Sample Activities1. 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 theinterior 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 onevertex?8. Determine the size of each different face as a fraction of the size of the hexagon.9. If the hexagon represents56, 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 thesequence.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 inthe shape. Create an expression for the total area. Make several copies of the shape. Create an algebraicexpression for the total area of all of the shapes.Recommended Websiteshttp://nlvm.usu.edu/en/nav/frames asid 169 g 1 t 2.html Virtual Pattern Blockshttp://matti.usu.edu/nlvm/nav/category g 4 t 3.html Interactive Manipulatives and activitieshttp://math.rice.edu/ lanius/Patterns/ Exploring pattern.pdf Explorations with Pattern ml Investigating Tessellations using Pattern l Polygon Playground (interactive)GAINS: Tips for Manipulatives10

TangramsWhat 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 tochallenge individuals to create different

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 c