CMP2:Grade Six Operations with

Fractions

Glenda LappanMilwaukeeApril 2005

Fraction Units

• Bits and Pieces I• Bits and Pieces II• Bits and Pieces III

Bits II (fractions)

Bits II

Bits III (Decimals)

Fractions

• The whole or unit• Partitioning/Re-partitioning • Naming parts• Equivalence— scaling/ ratio/ relative

frequency• Interpretations

– Measures– Indicated division– Operator (stretcher or shrinker —scale factor)– Number (location on a number line)– Ratio

Comparing fraction strips; What is equivalent?

Moving from fraction strips to number lines.

a. On the number line below, carefully label marks

that show where 1/3 and 2/3 are located.

b. What is the distance from the 1/3 mark toThe 1/2 mark on the number line above?

a. What is the distance between the marks for 3/5 and 7/10 on the number line below?

b. Locate marks for 1/10, 2/10, 3/10, 4/10, 5/10, 6/10, 7/10, 8/10, 9/10, and 10/10.

Which of the marks can also be labeled in fifths?

c. Find all fractions with denominators smallerthan 50 that are equivalent to 10/15

Big Ideas

• Equivalence• Operations• Algorithms• Solving problems

Equivalence

• Generating equivalent fractions— ratios/relative frequencies/scaling

• Fraction to equivalent decimal• Decimals to equivalent fractions• Expressions• Mathematical sentences

Operations

• Meaning/ What question(s) does the operation answer? What do the computed answer and remainder tell you?

• Estimating results– Addition– Subtraction– Multiplication– Division

Playing Getting Close

Play Getting Close several timesand keep a record of theestimation strategies you f induseful.

1. Describe or illustrate oneestimation strategy that youfound useful in the game.

2. For which pairs was it easyand for which pairs was ithard to estimate the sum?Why?

Suppose you played Getti ng Close withonly the game cards below:

1.What is the largest sum possible withany two of the game cards shown?

2. What is the smallest sum possible withany two of the game cards shown?

Billie’s grandmother makes thelace edging that is used todecorate the curtains in themodel house. Her grandmotherneeds 5 yards of lace f or thecurtains. She has these lengthsof lace on hand:

1 13 yds

2 56 yds

78 of a yd

512 of a yd

Does Billie’s grandmother hasenough lace.

Stop and think about the size of the answer to a

problem before you do an actual computation. You can

use your knowledge of benchmarking with fractions to

know that 3/ 7 + 9/ 20

is greater than a half, but less than one. This is because

both 3/ 7 and 9/ 20 are less than, but close to 1/ 2 .

.

Estimating Sums and Differences

Benchmark fractions:

0 1/4 1/2 3/4 1 11/4 11/2 13/4 2

Which benchmark is 5/8 nearest?

Here is one way to reason:

Five-eighths is larger than 1/2 , because it is larger than 4/8

Five-eighths is smaller than 3/4, because it is smaller than 6/8

In fact, 5/8 is exactly halfway between 1/2 and 3/4 .

Which benchmark is 0.58 nearest?

Since 0.50 is equal to 1/2 , 0.58 is larger than 1/2 .

We also know that 0.58 is less than 0.75 or 3/4 .

So we can say that 0.58 is between 1/2 and 3/4, but closer to 1/2 .

Solving Problems

• Deciding which operation(s) to use and why

• Computing• Interpreting computed answers

back in original problem

How are operations related?

• Inverse operations– + and -– X and ÷

• Fact families• Finding missing addends and factors• Relationship between + and X• Relationship between - and ÷

Relating computation to what students already

know

Fact Families

2 + 3 = 5 has these two related subtraction

sentences:

5 – 2 = 3 and 5 – 3 = 2

1 2

2

3

3

4 5

This set of sentences is called a fact family.

You can create fact families with fractions also. For example:

3/4 + 1/8 = 7/8 has these two related subtraction sentences:

7/8 – 3/4 = 1/8 and 7/8 – 1/8 = 3/4.

You can see that these facts are related by looking at them on a number line.

7/8

3/4

5/8 3/8 1/4 1/2 3/4

1/81/1/4 0 1 1/8

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Models for Multiplication

A pan of brownies costs $24 dollars. You can buy any fractional part of a pan of brownies. You pay that fraction of $24. For example, half a pan costs 1/2 of $24.

A. Mr. Sims asked to buy half a pan that was 2/3 full. What fraction of a whole pan did Mr. Sims buy and what did he pay?

B. Aunt Serena bought 3/4 of another pan that was half full. What fraction of a whole pan did she buy and how much did she pay?

1. Solve the following problem, but,estimate the product f irst!

2 12

1 16

Draw a picture to prove that yourcalculations make sense.

2. J ohn answers by doing the following:

(2

116) + (

12

1 16)

a. Explain J ohn’s thinking.

b. Try his strategy on

3 45

14 .

c. Does it work? Explain why or why not.

What Motivates Students to Engage with

Mathematical Problems?

Interesting challenges!