45Lesson 8: Decimal DivisionCopyright © 2006 by Thomson Nelson

GoalUse decimal division to solve ratio and rate problems.

Prerequisite Skills/Concepts• Divide whole numbers with or

without a calculator, asappropriate.

• Recognize when division is neededto solve a problem.

• Multiply a decimal by 10 or 100.

Decimal Division STUDENT BOOK PAGES 68–71

Direct Instruction

Expectations• [demonstrate an understanding of addition and subtraction of fractions and

integers, and] apply a variety of computational strategies to solve problemsinvolving whole numbers and decimal numbers

• demonstrate an understanding of proportional relationships, using [percent,] ratio, and rate

• solve multi-step problems arising from real-life contexts and involving wholenumbers and decimals, using a variety of tools and strategies

• use estimation when solving problems involving operations with whole numbers,decimals, [and percents,] to help judge the reasonableness of a solution

• determine, through investigation, the relationships among fractions, decimals,[percents,] and ratios

• demonstrate an understanding of rate as a comparison, or ratio, of twomeasurements with different units

• solve problems involving the calculation of unit rates

2.8

Assessment for Feedback What You Will See Students Doing…Students will

• solve problems involving decimals using theappropriate calculation methods

When students understand

• Students will accurately divide a decimal by adecimal using an appropriate calculation method.

• Students will solve rate and ratio problemsinvolving division of two decimal numbers.

If students misunderstand

• Students may have difficulty finding appropriatedecimal equivalents to make the division easier.Have students write the division in fraction form andthen write sequenced equivalent fractions, such as

�30..5264

� �325..46

� �32546

�

• Some students will have difficulty solving wordproblems with decimals because they cannot visualizethe situation. Begin with simpler whole numbers inthe same problem situation and have studentsdescribe what they would do. Then replace the wholenumbers with the decimals from the actual problem.

Preparation and Planning Meeting Individual NeedsExtra Challenge• Students could make a poster to display in the classroom explaining how

to divide a decimal by a decimal.

Extra Support• If students can find appropriate equivalent rations in fraction form, but

make frequent erors in division, allow them to use a calculator to find the final quotient.

Pacing 10 min Introduction(allow 5 min 20 min Teaching and Learningfor previous 10 min Consolidationhomework)

Materials •coloured pencils•10-by-10 Grids, Masters Booklet

p. 44

Workbook p. 21

Recommended 8 (c), 9 (d, f), 10, 11 (Knowledge Practice and Understanding), 13* (Application

of Learning), 16, 19 (Problem Solving/Thinking)

Additional 8 (a, b), 9 (a, b, c, e), 12 Practice (Communications), 14, 15, 17, 18

Extending: 20, 21, 22, 23

Learning Skills Co-operation, Problem Solving

Mathematical Selecting Tools and ComputationalProcesses Strategies, Connecting

Math BackgroundIt can be difficult for students to visualize how many 0.4s there are in 2.56.Thus it is important to model the division using grid paper, and for moreconcrete learners, base ten blocks. Once students can estimate a quotient, we want them to understand that they can change the computation into one that they already know how to do by multiplying by 10, 100, or 1000.

* Key Assessment of Learning Question (See chart on p. 48.)

02b-NEM7-ON-TR-CH02 7/26/05 4:37 PM Page 45

46 Chapter 2: Ratio, Rate, and Percent Copyright © 2006 by Thomson Nelson

Work with the Math

1. Introduction(Whole Class) ➧ about 10 min

Present students with a few warm-up problems that will helpthem to recall division involving decimals, using money as amodel. Begin with division by a whole number, and then askstudents to estimate a quotient when the divisor is a decimal.For example, tell students that you have $45.50 to purchase5 gifts of equal value and want to know how much can bespent on each gift. Have students model how they wouldsolve this and explain why this is a division situation. Thentell the students that you have $3.50 and want to purchasecandy bars that are on sale for 50 cents each. Ask them howthis problem could be solved using mental math.

Sample Discourse“How do you know that the first problem could be solvedusing division?”• The money was going to be shared equally to buy 5 gifts.

This means division.“How did you estimate the number of items that could bebought at 50 cents each?”• I know that there are 2 fifty-cents in a dollar, so there are 6 in 3

dollars, so there would be 7 in $3.50.• I divided 35 by 5 and got 7, so I know that 350 cents ÷ 50 cents

would be 7.“How would $3.50 ÷ 50 cents be written using decimals?”• It would be 3.50 ÷ 0.50.Tell students that in this lesson they will be finding ways todivide decimals, other than using mental math.

2. Teaching and Learning(Small Groups/Whole Class) ➧ about 20 min

Place students into groups of three and distribute six 10-by-10grids or base ten blocks (a set of hundreds, tens, and units) to each group. Read the information and central question onpage 64 together. Assign one example to each member of agroup to study then explain and model for the others in thegroup. Bring the groups together to discuss the examples andanswer any question they might have. Ask students to try toexplain to how their method is similar to a multiplicationstrategy they encountered in Lesson 7.

ReflectingDiscuss the Reflecting questions orally with the class,encouraging a variety of responses.

Answers to Reflecting1. For example, you would get a whole number quotient

without a remainder when you are dividing a number by one of its factors, and 10 is a factor of 100.

2. For example, sometimes a fraction line means “divide by.”So sometimes �

23

� means 2 divided by 3.

3. Consolidation ➧ about 10 min

Solved Examples (Whole Class)

Have students read through the example individually andthen discuss the strategies used. Reinforce the idea that thetwo numbers must be in the same units.

Dealing with Homework (Small Groups) ➧ about 5 min

Have small groups or homework groups meet to comparesolutions for Question 18 (or one other problem that wasassigned for homework) from Lesson 2.7. Ask each groupto make one copy of the solution they all agree on andthen submit it for assessment. All names should be on the solution paper.

Learn about the Math

(Lesson 8 Reflecting Answers continued on p. 79)

02b-NEM7-ON-TR-CH02 7/26/05 4:37 PM Page 46

47Lesson 8: Decimal DivisionCopyright © 2006 by Thomson Nelson

A Checking (Whole Class)

For Question 5, place 10-by-10 grids on the chalkboard or theoverhead. Ask student volunteers to use coloured chalk ormarkers to show each division. When discussing the estimationsin Question 6, encourage students to describe their strategiesand compare them for efficiency. When completing Question 7,students should see that there are several effective ways tocomplete each one. Ask students to explain why mental mathis an efficient strategy for Question 7 a).

Answers to Checking5. a) 2.7 ÷ 0.9 = 3

b) 3.6 ÷ 0.18 = 20

c) 1.24 ÷ 0.5 = 2.48

6. a) For example, 3.6 is close to 4 and 0.9 is close to 1, so 3.6 ÷ 0.9 is close to 4 ÷ 1 = 4.

b) For example, 7.8 is close to 8 and 1.3 is close to 1, so 7.8 ÷ 1.3 is close to 8 ÷ 1 = 8.

7. For example,a) Write in fraction form then multiply numerator and

denominator by 100.

2.7 ÷ 0.3 = �02..073

�

�02..073

××

110000

� = �273

0�

270 ÷ 3 = 90So 2.7 ÷ 0.03 = 90

b) Write in fraction form. c) Write in fraction form.Multiply numerator Multiply numerator and denominator and denominator by 10. by 100.

4.59 ÷ 0.9 = �40.5.9

9� 0.25 ÷ 0.04 = �0

0..20

54

�

�40.5.9

9××

1100

� = �45

9.9� �

00

.

.20

54

××

11

00

00

� = �245�

45.9 ÷ 9 = 5.1 25 ÷ 4 = 6.25So, 4.59 ÷ 0.9 = 5.1 So, 0.25 ÷ 0.04 = 6.25

B Practising (Pairs/Individual)

Modelling Question 10 with play coins will help many studentsvisualize each answer and strengthen their understanding ofdivision by decimals. If students do not notice unreasonableanswers, conference with these students, asking them toestimate an answer and then explain why their solution may beunreasonable. Question 19 relates to Question 20 in Lesson 2.7.Ask students to explain why this is a division situation insteadof multiplication and have them compare the proportions theyhave written for the two questions.

Answer to Key Assessment Question13. (Application of Learning)

For example,a) 80 cm = 0.8 m

11.4 m ÷ 0.8 m = �101..84

�

�101..84

××

1100

� = �11

84

�

114 ÷ 8 = 14.25Nathan will have 14 pieces with some rope left over.(Lesson 8 Key Assessment Answers continued on p. 79)

02b-NEM7-ON-TR-CH02 7/26/05 4:37 PM Page 47

48 Chapter 2: Ratio, Rate, and Percent Copyright © 2006 by Thomson Nelson

C Extending (Individual)

When completing Question 21, students could talk aboutwhat percentage increase occurs when water is frozen. ForQuestion 22, some students may find it easier to visualize0.013 m by looking at a metre stick. Have them find 0.01 m (�

1100� of a metre) before refining the measure to 0.013 m.

Closing (Individual)

Have students complete the following prompt in their journal:“My knowledge of whole number division helps me to dividedecimals because…”

Assessment of Learning—What to Look for in Student Work…Assessment Strategy: Written AnswerApplication of Learning

Key Assessment Question 13• Nathan has 11.4 m of rope. He wants to divide it into equal pieces. How many pieces will there be if the pieces are these lengths?

a) 80 cm long b) 1.4 m long c) 0.7 m long d) half a metre

1 2 3 4

• Demonstrates limited ability toapply mathematical knowledge andskills in familiar contexts (e.g., hasdifficulty using decimal division tosolve familiar or routine problems)

• Demonstrates some ability to apply mathematical knowledge and skills in familiar contexts (e.g.,demonstrates some ability to usedecimal division to solve familiar or routine problems)

• Demonstrates considerable abilityto apply mathematical knowledgeand skills in familiar contexts (e.g.,uses decimal division to solvefamiliar or routine problems)

• Demonstrates sophisticated abilityto apply mathematical knowledgeand skills in familiar contexts (e.g.,demonstrates sophisticated abilityto use decimal division to solvefamiliar or routine problems)

Key Assessment of Learning Question(See chart below.)

Follow-Up and Preparation for Next Class• Have students look through their work in this chapter

and make a list of a) questions they may still have aboutsolving proportions or rate, ratio, and percent problems,and b) questions from their assignments they may stillhave difficulty solving.

02b-NEM7-ON-TR-CH02 7/26/05 4:37 PM Page 48