Multiplication and Division - the CPA Way! Charline Brown

Four Ways to Think About Multiplication Area Model  

Teaching Multiplicative Reasoning with Cuisenaire Rods Of the four ways to show multiplication, the area model will allow students to transition into operations with fractions and algebraic expressions. Take and place the unit cube in front of you on your table. This unit cube demonstrates a rectangle with an area of 1. Is this the only rectangle you can make with an area of 1? Remember, we are working only with whole numbers. Take and place the red rectangle in front of you on your table. This red rectangle demonstrates an area of 2. Is this the only rectangle you can make with an area of 2?

Complete the table using your Cuisenaire Rods   Number Draw the Rectangles Made Factors Notes 1 Only 1 Rectangle can be made. There is also only one factor. 2 3 4 5 6 1, 2 , 3, 6 Two rectangles can be made. There are four factors.

Teaching Multiplicative Reasoning with Cuisenaire Rods What did you notice about the areas (numbers) that could be made with only 1 rectangle? The areas with only one rectangle are prime numbers, with the exception of 1. Other than 1, all other rectangles have exactly 2 dimensions (factors). What did you notice about areas (numbers) that could be made with more that 1 rectangle? The areas with more than one rectangle are composite numbers. These rectangles have more than 2 different dimensions (factors).

Teaching Multiplicative Reasoning with Cuisenaire Rods What did you notice about the areas (numbers) that have an odd number of dimensions that are used to make rectangles? The areas with an odd number of dimensions (factors) are square numbers. One of these rectangles is a square and has the same dimension (factor) on both sides. Notice that 1 can be included here. Which other areas are square numbers? For each area with more than one rectangle, which of your rectangles has the largest perimeter? The rectangle with the farthest distance between the dimensions (factors) has the greatest perimeter. For example, there are three rectangles for the area of 12. Their dimensions are 1 by 12, 2 by 6, and 3 by 4. 1 and 12 are the dimensions farthest apart. This rectangle has the largest perimeter of the three. Can you guess which has the smallest perimeter? Can you determine a rule?

Teaching Long Multiplication with Base Ten Blocks Think back to creating rectangles with the Cuisenaire Rods. You’ll remember that the rectangles allowed us to conceptualize multiplication through the area model. Look at the rectangle below. What is the area (product) represented? What are the dimensions? But, what about conceptualization of multiplication of larger numbers? A more efficient model involves Base Ten Blocks.  

Teaching Long Multiplication with Base Ten Blocks What Base Ten Block would I use to demonstrate an area model for 100? 10 10 x 10 = 100  

Teaching Long Multiplication with Base Ten Blocks How would I demonstrate a rectangle that has dimensions of 10 by 11? 11 10 x 10 = 100 10 x 1 = 10 10    

Teaching Long Multiplication with Base Ten Blocks How would I demonstrate a rectangle that has dimensions of 15 by 12? 12 10 x 10 = 100 5 x 10 = 50 2 x 10 = 20 2 x 5 = 10 15        

Teaching Long Multiplication with Base Ten Blocks It’s Your Turn to Represent it Pictorially and Abstractly.   1 5 X 1 2 2 x 5 = 2 x 10 = 10 x 5 = 10 x 10 =

Teaching Division with Cuisenaire Rods How many groups of 5 are there in 15?         Let’s reorganize our rectangle…       We knew that our area was 15. How many groups of 5 are there in 15? Notice the dimensions of the rectangle are 5 by 3.

Teaching Division with Cuisenaire Rods How many groups of 5 are there in 18?           We don’t have enough room to place another group of 5. What color rod would fit in the space left over?         Let’s reorganize our rectangle. The largest rectangle that we can make is 5 by 3. There are 3 groups of 5 in 18. The remainder is 3, and it is represented by the light green rod.

Teaching Long Division with Base Ten Blocks Think back to decomposing rectangles with the Cuisenaire Rods. You’ll remember that the rectangles allowed us to conceptualize division through the area model. Look at the rectangle below. How many groups of 4 are there in 12? There are 3 groups of 4 in 12. But, what about conceptualization of division of larger numbers? A more efficient model involves Base Ten Blocks.  

Teaching Long Division with Base Ten Blocks How would I demonstrate division of 140 by 10? I know that the area of my rectangle must be 140. I also know that one dimension of the rectangle is 10 because I have groups of 10. 4 10 I know that the answer is at least 10, because 10 groups of 10 is 100. I still have 40 to go! The area of the rectangle is 140. One dimension is 10 and the other is 14. Therefore, there are 14 groups of 10 in 140.          

Teaching Long Division with Base Ten Blocks How would I demonstrate division of 172 by 12? I know that the area of my rectangle must be 172. I also know that one dimension of the rectangle is 12 because I have groups of 12. The rectangle has an area of 168. This is as close as I can get to an area of 172 with one dimension of 12. I would need an area of 180 to have 15 groups of 12. Since I have 172, I will have 4 left over. 168 + 4 = 172 Therefore 172 ÷ 12 = 14 r 4                                        

Teaching Long Division with Base Ten Blocks It’s Your Turn to Represent it Pictorially and Abstractly.   Solve 175 ÷ 13. I start with the largest block to begin making the desired area (dividend). I know I will use 1 hundreds flat. I establish the number of rows (it will match the value of the divisor). I continue to build the rectangle using tens rods until I have an overall rectangle with an area closest to 175. My rectangle has an area of 169. I do not have enough for another group of 13. I have 6 units remaining and I do not have enough for another group of 13. The answer to 175 ÷ 13 is 13 r 6

Games and Online Resources www.kidsnumbers.com/long-division.php www.mangahigh.com/en/maths_games/number/written_methods/long_multiplication National Library of Virtual Manipulatives www.mathplayground.com www.mrnussbaum.com Math Maven Mysteries