Dr Brian Beaudrie pg. 1

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1 Multiplication of Decimals Name: Multiplication of a decimal by a whole number can be represented by the repeated addition model. For example, means add 0.14 three times, regroup, and simplify, as shown below: = roughly There are three separate cases of multiplication, each with its own representation. The first case has already been described: multiplication of a whole number by a decimal. Try these two multiplication problems, modeling what is shown in the example above: = = Dr Brian Beaudrie pg. 1

2 How about this one? = + Multiplication of a whole number by a decimal is an important step in helping a child conceptualize multiplication of two decimals. The second case is the opposite of the first case: multiplication of a decimal by a whole number. For example, suppose you going to find the solution to Of course, you could use the law, which allows you to rewrite the equation as ; but since most 3 rd graders don t understand that law, let s try a different approach: Since we are multiplying a decimal by 2 (in this case) begin with two decimal squares, and shade each with Then, combine them into one decimal square (with regrouping, if necessary) = 0.56 joining regroup Go ahead and try it: (you can combine the joining and regrouping steps): = join and regroup The third case involves multiplying a decimal by a decimal. For this activity, we will only do tenths multiplied by tenths. The method we employ might look a bit familiar we did something similar to it when we did fractions. Dr Brian Beaudrie pg. 2

3 Suppose we wanted to multiply 0.3 by 0.4. Looking at our decimal square we would do the following: We would divide the tenths along the horizontal axis into tenths as well, as many tenths as we needed. From there, we would shade in up to 0.3 and over to 0.4. From that, we would then count the shaded boxes to obtain our answer. So, =.12 Try out this idea on a few examples: = = Questions: 1) How could you represent thousandths using decimal squares? 2) Why do we only do examples that show tenths multiplied by tenths? Why would showing an example having tenths multiplied by hundredths be difficult? 3) What are some of the limitations of using the decimal squares? Dr Brian Beaudrie pg. 3

4 Division of Decimals With division of decimals, there are also three separate cases to consider. The first case, division of a decimal by a whole number, is done using the partitive model of division. Example: The problem is essentially asking to separate 0.54 into three sets of equal size. If we were to estimate a range for our answer, we know that our answer will be more than 0.1 but less than 0.2 because we can split the columns (tenths) into three equal groups once, but not twice. Therefore, to use this method, we will divide up the columns (tenths) first (one for each group), then divide the remaining squares (hundredths) into 3 equal groups, as shown to the right. We can then see that each color uses exactly eighteen squares, so we know: =.18 Use the method (estimating first) described above to do the following examples. a) = b) = For the second case, dividing a whole number by a decimal, the repeated subtraction (measurement) method of division is the model to employ. In this method, you count the number of sets equivalent to the divisor (the second number) that are in the dividend (the first number). For example, in the problem: =, you are being asked how many sets of 0.6 can you take away from three whole units? So, starting with three whole units, you will make groups of size 0.6, as shown below: So, we know that = 5, since we ended up with five different colors. Dr Brian Beaudrie pg. 4

5 Use the repeated subtraction model to find the correct answer to the following problem: = What happens, though, when you don t have a whole number solution? Well, consider The decimal square to the right shows what happens you end up with six groups, each size 0.04, with three extra groups left over. If I look at 0.03/0.04, I can see that it is exactly 0.75 of another group of size Therefore, I have: = 6.75 The third case concerns dividing one decimal by another decimal. For this, you will also use the repeated subtraction model of division. For example, with , you need to find out how many sets of 0.19 exist in So, you would pull out groups of 19 little squares at a time, as shown on the left. So, you will have: = 4. Use your decimal squares and the method described above to find the following quotients. Estimate first where appropriate. a) = b) = Dr Brian Beaudrie pg. 5