Multiplication and Division of Fractions and

Decimals

Session 3

January 15, 2013Huntersville Elementary

Solve……….

A number with 3 digits after the decimal point is rounded to 4.8 when rounded to the nearest tenth.

What is the smallest this number could be?

What is the largest this number could be?

Fraction Representations Pizzas, pies, and candy bars

Number lines/Sentence strips

Fraction bars/Centimeter grid paper

Clocks

Percent strips- percent and decimal equivalents

Arrays (4 x 6 and 5 x 12)

Benchmarks (containers)

Decomposing fractions 7/8 = 3/6 + 1/4 + 1/8 (Fraction Track Game)

Pattern blocks (unit fractions)

Investigations’ Bar diagrams

Paper folding and Open arrays

Equations

To multiply unit fractions, multiply the denominator times the denominator and

To multiply any two fractions, multiply the numerator times the numerator, and the denominator times the denominator

When you multiply a whole number times a fraction less than 1, the answer is smaller than the whole number

When you multiply a whole number times a fraction greater than 1, the answer is greater

than the whole number

Fraction Conjectures

Fraction Conjectures continued

When you multiply two fractions that are both less than 1, the product is a fraction smaller than either of the factors

When you divide a whole number by a fraction less than 1, the answer is larger than the whole number

When you divide a fraction by a whole number, the answer is smaller than the whole number and the fraction

10 ÷ ½ = 10 x 2

10 x ½ = 10 ÷ 2

The cafeteria made lunches for the fifth graders going on a field trip. They were in four different groups so the number of sandwiches differed. The sandwiches were all the same size.

Group One had 4 students sharing 3 subsGroup Two had 5 students sharing 4 subsGroup Three had 8 students sharing 7 subsGroup Four had 5 students sharing 3 subsDid each student get a “fair share?”If not, which group ate the least?

Most? How do you know?

Sharing Submarine Sandwiches

Next trip we want to guarantee that each student will receive 2/3 of a sub

Using large paper, create a chart for the cafeteria to help them know how many subs to make for up to 15 students

What patterns do you notice? What strategy could cafeteria workers use

for any number of students? If you knew there were 8 subs made, how

could you figure out how many students could each get 2/3 sub?

Model this situation using numbers and symbols.

Help the Cafeteria Staff

Dividing a Whole Number by a Fraction

What does 6 ÷ ½ mean?

How does this problem relate to multiplication?

How is multiplication and division of whole numbers connected to multiplication and division of fractions?

What is meant by “there are two wholes when dividing fractions?” Give an example..

Using the number line below, label the point

1.68

Rounding Decimals

1.6 1.7

How did you know where to place the number?

Rounding Decimals 1.68

Procedure for Rounding Underline the rounding place Circle the digit to the right If the circled digit is 5 or greater, increase

the underlined digit If the circled digit is less than 5, leave the

underlined digit as it is Drop the digits to the right of the underlined

digit 4.923

Round to the nearest hundredthWhat about 2.97 rounded to the nearest

tenth?

Write the following number using expanded notation: 689,738

Use exponents when possible

What number is 5,000 less than this?What number is 200 more?

Expanded Notation

Note the suggestions in the snap-ins!

Mercedes had 1.86 grams of gold. She used 0.73 grams of it in a piece of jewelry. How much gold does she have left?

Use of Hundredths Grids Subtract in parts Add up from 0.73 to 1, and then from 1 to 1 and 86

hundredths

Decimal Subtraction Problems

Subtract: 0.6 – 0.48

How can equivalent decimals help you to subtract these two numbers?

Multiplying Powers of 10 Use 4 Hundredths grids to show the

following 4 x 0.01

4 x 0.1How can we use what we know about multiplying

fractions to help us solve the two problems above?

4 x 1 = 4 x 0.1 =4 x 10 = 4 x 0.01 =4 x 100 = 4 x 0.001 = ?

What relationships do you notice?

25 x 0.01 = 0.25 25 x 0.1 = 2.5 25 x 1 = 25 25 x 10 = 250 25 x 100 = 2,500

Multiplying by “Small” Numbers

2 x 7 = 2 x 0.7 =

Use a number line from 0 to 2 to show the answer to the second problem

Use the same number line to show the answer to the following: 2 x 0.07

(notice the use of running context p CC110-111)

32 x 0.8 = 2.56 25.6 256

Reasoning with Decimals

Problem: 185 x 0.4 =

If the answer to 185 x 4 is 740, how can we use reasoning to determine the answer to the above problem?

Writing a Rule for Multiplication of Decimals

Multiply the numbers like they are whole numbers and then think about the size of the factors and place the decimal point so the product is the right size.

Multiply a whole number by a decimal, and the answer has the same number of decimal places as the decimal number being multiplied. Or, if each of the numbers has one decimal place, then the answer has two decimal places.

Therefore solve: 42 x 36 = 1,512 4.2 x 3.6 = ?????

Multiplying Tenths….

0.2 x 0.4 =

Think about one of the conjectures we made about multiplying 2 fractions less than 1

How can this conjecture and what you know about the relationship between decimals and fractions

help you solve this problem?

Dividing Powers of 10

2 ÷ 1 = ?2 ÷ 0.1 = ?

2 ÷ 0.01 = ?

What does 2 ÷ 1 mean?

Will the answer to each of these be greater or less than 2? How do you know?

How can we use 2 hundreds grids to represent these situations?

Comparing Multiplication and Division

25 x 100 = 2,500 25 ÷ 100 = ____25 x 10 = 250 25 ÷ 10 = ____25 x 1 = 25 25 ÷ 1 = 2525 x 0.1 = 2.5 25 ÷ 0.1 = 25025 x 0.01 = 0.25 25 ÷ 0.01 = 2,500

What patterns do you notice?How can you use your understanding of fractions to

figure out the answers to the two unsolved problems?

Bonus: use the pattern in the division problems to figure out the answer to 25 ÷ 0.001

Closest Estimate6.8 x 2.3 ≈ 1.4 14 14074 x 8.1 ≈ 5.6 56 560166 x 0.08 ≈ 1.66 16.6 166

What is the closest estimate for each?

Is the closest estimate greater or less than the actual answer?

How do you know?

Dividing Decimals

18 ÷ 6 = 18 ÷ 0.6 =How could you write these problems as

missing factors problems?

Draw a number line from 0-20Solve each of these using the number line.

What are some rules for dividing decimals?

Problems can be solved by thinking about both numbers as whole numbers and then reasoning about where the decimal belongs

Problems can be solved by thinking about the division problem as a missing factor problem to solve or to check the reasonableness of the answer

Will the rule of counting decimal places work for division like it does for multiplication?

97.5÷ 6.5 =

21.52 ÷ 0.8 =

Scooping Trail Mix

Math Reasoning Inventory Questions

www.illustrativemathematics.org

DPI Fraction UnitTest-item bankCMS WikiUnpacking Document

Resources

Why don’t we just teach them the rule?

There are easy rules for multiplying and dividing fractions and decimals. So…..

Reflection……

Thinking about multiplying and dividing fractions, why do you think it has been said that “fractions are the pathway from arithmetic to algebra?

Is multiplying or dividing fractions more difficult? Explain your reasoning.

Is multiplying or dividing decimals more difficult? Explain your reasoning