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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
0 112
Moving from fraction strips to number lines.
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0 112
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?
0 135
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!