Multiplying Fractions Using Arrays

Often, when students are taught to multiply fractions, they learn simple procedures without understanding what they are doing. One of the odd effects of multiplying fractions that are less than one is that the answer is smaller than the largest of the fractions being multiplied (e.g., 2/3 x 3/4 = 6/12 and 6/12 is smaller than 3/4). This is contrary to what happens in whole number multiplication, where the answer is almost always larger than each of the two numbers multiplied. Why is this and what does this say about multiplying fractions?

This lesson uses arrays as a way of demonstrating why "numbers get smaller" when fractions are involved. Once students have been shown the concept of multiplication through arrays, a simple word problem is used to anchor the concept. After students have learned the differences between multiplying whole numbers and fractions they can practice using fraction calculators.

Classroom Discussion and Activities

Computer Lab Activities

Classroom Discussion and Activities

This lesson begins with multiplying fractions. Depending upon the students' ability, you can do this on pencil and paper, on the board, or use fraction calculators.

• Multiply some fractions

 

Sample problems:

 

1/2 x 3/4 1/3 x 6/4

 

3/5 X 2/3 4/3 X 3/8

 

• Discuss the "size" of the answer in relation to the two factors

 

Discuss with students the discrepancy between multiplying fractions and multiplying whole numbers. The following questions can be used to guide the discussion. The exception to the pattern described below is when the two fractions are greater than one. Consequently, stress the fact that we are talking about cases where at least one fraction is less than one.

 

• "Is your answer larger or smaller than the first fraction?" Use a picture to support your answer.

 

• "Is your answer larger or smaller than the second fraction?" Use a picture to support your answer.

 

• "Does the fact that the answer (product) is smaller than the largest fraction make sense to you? How is that different from multiplying whole numbers?"

 

• "Unlike whole numbers, when you multiply two fractions the answer is smaller than either of them. We're going to draw a picture to help explain this."

• Build an array to represent the fraction problem

An array must reflect the denominators of both numbers to be multiplied. For example, 1/2 x 1/3 would indicate a grid 2 rows by 3 columns. The fractions 3/7 x 2/4 would be represented by a grid of 7 rows by 4 columns.

 

• Color the array to represent both fractions.

Use diagonal lines or colored pens to denote the first fraction. Then use diagonal lines that go in the opposite directions or different colors for the second fraction. Make sure that it is colored over the top of the first fraction (see multfrac.xls).

 

Your answer is the number of "double-colored" cells (numerator) over the total number of cells in the array (denominator).

 

Do not simplify fractions yet.

 

• Supply a word problem to explain the process of multiplying fractions

Here is an example word problem:

 

There are twelve students in the third period math class who are going on a field trip. They order sandwiches from the lunch room for their trip. Of the students, 2/3 want chicken sandwiches and the other 1/3 want ham sandwiches.

 

Of the 2/3 who want chicken, 1/4 of them do not want mayonnaise on their sandwiches. How many students do not want mayonnaise on their sandwiches?

 

The problem is 1/4 X 2/3, or 1/4 of the 2/3 who wanted chicken do no want mayonnaise.

 

The array will be a 4 x 3

 

• Reducing fractions (optional)

 

This part of the lesson is optional, and it depends on what students have learned about ratios. You can use it to demonstrate how fractions are reduced to equivalent fractions or ratios using arrays.

 

Construct a fraction like 6/8 using arrays. Use two columns, one for the numerator and one for the denominator.

 

 

Use the equivalence formula to show how fractions are reduced. In this case, you will multiply a fraction by another fraction equal to one. The fraction that is equal to one is made up of factors from the first fraction. This is shown in the example below:

 

Finally, construct columns that show the fraction for 3/4 and discuss how this is an equivalent ratio to 6/8. Your final set of arrays should show how 6 to 8 is equivalent to the ratio of 3 to 4.

 

Computer Lab Activities

Open the spreadsheet file multfrac.xls as an example of using arrays. This file contains two worksheets: one for multiplying fractions and the other for reducing fractions. Use the first worksheet as a model for representing fraction multiplication. The second sheet is optional. It shows how simplified fractions are equivalent ratios. That is, the fraction 6/12 (or 6 to 12) is equivalent to 1/2 (or 1 to 2). This can be seen graphically with the arrays.

If you are unfamiliar with using arrays in spreadsheets, directions can be found in array.doc.