When Small Gets Bigger

Students’ experience with whole numbers leads them to expect that multiplying results in an answer that is “larger” and when dividing that the answer is “smaller” than the dividend. These are not always true for decimal fractions. For example 23.9 ÷ 0.0891 is larger than 23.9 not smaller.

Using Materials

Problem: “Maurice works at the supermarket. He has to cut up 4 kilograms of cheese into 0.5-kilogram packs for sale. How many packs will he make?” “What is 4 ÷ 0.5?”

Write 4 ÷ 0.5 on the board. Get the students to make a strip like this representing the 4 kilograms of cheese:

Then get them to cut the strip into “1-kilogram” pieces. Discuss how to create 0.5- kilogram packs. (Answer: Cut each piece in half.)

Do the cutting. How many pieces are there? (Answer: 8.)

Discuss why 4 ÷ 0.5 = 8. “Is it a surprise that the answer is more than 4?”

Examples. Use paper and cutting for word stories: 3 ÷ 0.5,  2 ÷ 0.25,  3.5 ÷ 0.5,  1 ÷ 0.25,  2.5 ÷ 0.5 ...

Using Imaging

Problem: “Morrie has 5 kilograms of chocolate. He puts the chocolate into separate packets of 0.25 kilograms. How many packets will Morrie make?” “What is 5 ÷ 0.25?”

Write 5 ÷ 0.25 on the board. Get the students to imagine what Morrie would do to 1 kilogram of chocolate. (Answer: He would make 4 packets.)

“How many bags can he make?” (Answer: 5 x 4 = 20.)

Write 5 ÷ 0.25 = 20 on the board.

Examples: Imagine how to solve these problems. Word stories and problems for: 
3.5 ÷ 0.5, 1.25 ÷ 0.25, 8 ÷ 0.5, 2.5 ÷ 0.25 ...

Using Number Properties

Examples: Worksheet (Material Master 8–16).

Understanding Number Properties: How will you decide whether the answers to problems like 345.67 ÷ 1.008 or 2345.09 ÷ 0.012 get bigger or smaller?

Understanding Number Properties:

z ÷ y is always smaller than z when y is ...
z ÷ y is always bigger than z when y is ...