Lesson 11Dividing Numbers that Result in Decimals
Learning Goal
Let’s find quotients that are not whole numbers.
Learning Targets
I can use long division to find the quotient of two whole numbers when the quotient is not a whole number.
Lesson Terms
- long division
Warm Up: Number Talk: Evaluating Quotients
Problem 1
Find the quotients mentally.
Activity 1: Keep Dividing
Problem 1
Here is how Mai used base-ten diagrams to calculate
She started by representing 62.
She then made 5 groups, each with 1 ten. There was 1 ten left. She unbundled it into 10 ones and distributed the ones across the 5 groups.
Here is her diagram for
Discuss these questions with a partner and write down your answers:
Mai should have a total of 12 ones, but her diagram shows only 10. Why?
She did not originally have tenths, but in her diagram each group has 4 tenths. Why?
What value has Mai found for
? Explain your reasoning.
Problem 2
Find the quotient of
Problem 3
Four students share a $271 prize from a science competition. How much does each student get if the prize is shared equally? Show your reasoning.
Activity 2: Using Long Division to Calculate Quotients
Problem 1
Here is how Lin calculated
Discuss with your partner:
Lin put a 0 after the remainder of 2. Why? Why does this 0 not change the value of the quotient?
Lin subtracted 5 groups of 4 from 20. What value does the 4 in the quotient represent?
What value did Lin find for
?
Problem 2
Use long division to find the value of each expression. Then pause so your teacher can review your work.
Problem 3
Use long division to show that:
, or , is 1.25. , or , is 0.8. , or , is 0.125. , or , is 0.04.
Problem 4
Noah said we cannot use long division to calculate
What do you think Noah meant by “there will always be a remainder”?
Do you agree with his statement? Why or why not?
Lesson Summary
Dividing a whole number by another whole number does not always produce a whole-number quotient. Let’s look at
We can see in the base-ten diagram that there are 4 groups of 21 in 86 with 2 ones left over. To find the quotient, we need to distribute the 2 ones into the 4 groups. To do this, we can unbundle or decompose the 2 ones into 20 tenths, which enables us to put 5 tenths in each group.
Once the 20 tenths are distributed, each group will have 2 tens, 1 one, and 5 tenths, so
We can also calculate
The calculation shows that, after removing 4 groups of 21, there are 2 ones remaining. We can continue dividing by writing a 0 to the right of the 2 and thinking of that remainder as 20 tenths, which can then be divided into 4 groups.
To show that the quotient we are working with now is in the tenth place, we put a decimal point to the right of the 1 (which is in the ones place) at the top. It may also be helpful to draw a vertical line to separate the ones and the tenths.
There are 4 groups of 5 tenths in 20 tenths, so we write 5 in the tenths place at the top. The calculation likewise shows