Lappan-MTL April 2005

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CMP2:Grade Six Operations with

Fractions

Glenda Lappan

MilwaukeeApril 2005


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Fraction Units

Bits and Pieces I

Bits and Pieces II Bits and Pieces III


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Bits II (fractions)

Bits II

Bits III (Decimals)


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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


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1

2

1

4

2

4

3

4

1

6

2

6

3

6

4

6

5

6

Comparing fraction strips; What is equivalent?


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halves

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Moving from fraction strips to number lines.


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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?


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a. What is the distance between the marks for3/5 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?

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7

10

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


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Big Ideas Equivalence

Operations Algorithms

Solving problems


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Equivalence Generating equivalent fractions

ratios/relative frequencies/scaling

Fraction to equivalent decimal Decimals to equivalent fractions

Expressions

Mathematical sentences


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Operations Meaning/ What question(s) does the

operation answer? What do the computed

answer and remainder tell you? Estimating results Addition Subtraction

Multiplication Division


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0 1 23

Playing Getting Close


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Suppose you played Getting Close withonly the game cards below:

1.What is the largest sum possible with

any two of the game cards shown?

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


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Billies grandmother makes the

lace edging that is used to

decorate the curtains in themodel house. Her grandmother

needs 5 yards of lace for the

curtains. She has these lengths

of lace on hand:

11

3 yds 2 56 yds 78 of a yd5

12 of a yd

Does Billies grandmother has

enough lace.


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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 toknow that

3/ 7 + 9/ 20

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

both3/ 7

and9/ 20

are less than, but close to1/ 2

..

Estimating Sums and

Differences


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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 .


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Solving Problems Deciding which operation(s) to use

and why

Computing

Interpreting computed answers backin original problem


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How are operations related? Inverse operations

+ and -

X and Fact families Finding missing addends and factors

Relationship between + andX

Relationship between - and


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Relating computation to what

students already know


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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.


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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 and7/8 1/8 = 3/4.

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

7/8

3/4

5/83/81/4 1/2 3/4

1/8

1/1/

40 11/8


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Models for Multiplication


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A pan of brownies costs $24 dollars. You can buy anyfractional part of a pan of brownies. You pay thatfraction of $24. For example, half a pan costs 1/2 of$24.

A. Mr. Sims asked to buy half a pan thatwas 2/3 full. What fraction of a whole

pan did Mr. Sims buy and what did hepay?

B. Aunt Serena bought 3/4 of another panthat was half full. What fraction of a

whole pan did she buy and how much didshe pay?


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1. Solve the following problem, but,estimate the product first!

2 12v116Draw a picture to prove that your

calculations make sense.

2. John answers by doing the following:

(2v116) + (1

2 v11

6)

a. Explain Johns thinking.

b. Try his strategy on 3 45 v 14 .c. Does it work? Explain why or why not.


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What Motivates Students to

Engage with MathematicalProblems?


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Interesting challenges!

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