Dividing Decimals

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SWBAT develop an algorithm for dividing decimals.

Big Idea

How can a decimals, fractions, and whole number problem all lead to the same answer?

Do Now

10 minutes

Over the last several lessons we have discussed algorithms for adding, subtracting, and multiplying decimals. Today's Do Now problems target these operations:

Do Now:

1)  13 - 8.647 = 

2)  13.2 x 0.005 =

After 5 minutes, I will randomly call students to the board to complete the problems.  A common mistake for problem #1 is that students do not rewrite 13 as 13.000.  Instead of borrowing they will arrive at an answer of 5.647. If the student at the board makes this mistake, it is important to discuss it as a class.  Chances are several students made the same mistake.  

In addition to having students present at the board, I will ask students to complete a Think Aloud explaining how and why they solved the problem (see Using Student Think Alouds reflection). Depending on how each Think Aloud goes I may have more than one student Think Aloud on each Do Now Problem.


Developing an Algorithm

15 minutes

Today, we will begin our work on an algorithm for dividing decimals. Since students have made contributions to the development of our addition, subtraction, and multiplication algorithms, I will give them opportunities to do so again today. I will, however, lead the discussion through guided questioning. Here's the prompt for our opening conversation:

What do you observe about the problems below?

a)  0.84  ÷  0.42

b)  84/100  ÷ 42/100

c)  84  ÷  42

Possible Student Answers:

  • They're all division problems.
  • They all have the numbers 84 and 42.
  • (b) is the fraction equivalent of (a)

Next, I will ask, "What is the answer for (c)?  What is the answer for (b)?" I will give students time to calculate each expression using what they have previously learned about dividing whole numbers and fractions. Once we establish that b and c have the same answer, I will say, "I did the hard one for you. The answer to a is also 2.  I wonder, how can all three problems have the same answer?" My plan is to give students a few minutes to discuss the three equations answer with their group.  

0.84  ÷  0.42 = 2  84/100  ÷ 42/100 = 2 84  ÷  42 = 2

I expect my students will make some of the following observations:

  • Since (a) is the decimal form of (b), then it makes sense they would have the same answer (this was the case with the other operations as well).
  • If you move the decimal point twice in 0.84 and 0.42, the resulting expression is like the same as (c).

It is important to have these two ideas in the minds of students before moving on to the division algorithm. Once these seeds are sown, I will complete an example with the class, breaking the problem into steps to demonstrate the division of fractions using the long division algorithm.  

Example:   125.01 ÷ 5.4

Step 1 - Make the divisor a whole number.  How can we do that?

Students should suggest that we move the decimal point to the right one place.

Step 2 - Move the decimal point in the dividend the same number of places.

Step 3 - Move the decimal point "up" in the representation of the long division problem.  Do we need to worry about the decimal point anymore?

I will let students discuss this question for a few minutes.

Step 4 - Perform the long division.

Independent Practice

15 minutes

Today's Independent Practice is an opportunity for students to practice the algorithm for dividing decimals on their own and to see the types of word problems that would entail dividing decimals.  Each student will receive a copy of the Dividing Decimals Practice worksheet. Although I would like students to complete the work independently, I will encourage discussion among students if they have questions or need help.  As students work, I will circulate to assist students who have questions.

Exit Ticket

5 minutes

To bring today's lesson to closure I will discuss some of my observations from the Independent Practice session and quickly review the algorithm for dividing decimals with the class. Before we leave for the day, I would also like students to begin thinking about what to do with a remainder when dividing decimals.  This will be covered in the next lesson, Dividing Decimals with Remainders. So, I will give them the following prompts as an Exit Ticket: 

  • When dividing decimals, why is it important to move the decimal point in the divisor?
  • What would you do if you had a remainder?