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Is this you (or someone you know)?
GROUP NORMS AND HOUSEKEEPING
Logistics:
Phone Calls
Rest Rooms
BreaksLunch
PunctualitySharing
Group Norms:Participate
Listen with an open mind
Ask questions
Work toward solutionsLimit side bars
Fifth Grade Big Idea 2Day 1
Develop an understanding of and fluency with
addition and subtraction of fractions and
decimals.
Big Idea 2 Benchmarks
Share 3 donuts between 5 people.
When equally shared, each person gets ⅗ of a donut.
Fractions
The idea of breaking a whole into parts, sharing the parts, and providing names for those parts is the fundamental concept in the development of fraction knowledge.
There are four ways that fractions are used to represent application situations: part of a whole, part of a set, indicates division, and ratio.
Extensive research and observational data demonstrate that few students understand fractions. Therefore major changes must be made in the approach to teaching fractions.
1st meaning: Part/whole:
You take the “whole” and split it into equal parts.
Example 1: A baseball game has nine innings. Seven have been played. What fraction of the game has been played?
Example 2: This class has 19 students. Eighteen are females. What fraction of the class is female?
Three Meanings of a Three Meanings of a FractionFraction
Quotient Implies “division”
Example 1: Pizza for a group of friends: $12 ÷ 3 people (or $ 12/3 each)
Example 2: 3 doughnuts, 5 kids. How much of a doughnut does each kid get?
How could they do the above? (Different from part-whole splitting)
22ndnd Meaning of a Fraction: Meaning of a Fraction:
Ratio: Conceptually different and doesn’t imply dividing a whole into parts or division.
Example.: 1 week: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday.
Weekend days to school days is 2:5 or 2/5.Weekend to whole week is 2:7 or 2/7.
33rdrd Meaning of a Fraction: Meaning of a Fraction:
Fractions—For these problems, circle the greater Fractions—For these problems, circle the greater number of each pair and tell number of each pair and tell the strategiy you the strategiy you
used.used.
€
5
8€
4
5
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4
9
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7
8
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5
4€
11
7
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6
7€
23
10€
4
6
€
2
6
Then: Change 5⅔ to an improper fraction, and Change to a mixed number.
1.
2.
3.
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8
9
1.
2.
3.
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4
10
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13
6
When the whole numbers are different, you only have to compare the whole numbers.
When the whole numbers are different, you only have to compare the whole numbers.
When the numerator is the same, look at the size of the pieces in the denominator.
When the numerator is the same, look at the size of the pieces in the denominator.
Strategies to Compare/Order Fraction
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21
4
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11
2>
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3
5 >
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3
8
•Use benchmark numbers•Use benchmark numbers •Compare missing pieces•Compare missing pieces
Strategies to Compare/Order Fraction
>
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3
8 >
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6
10Think: 3 is less than half of the denominator so the fraction is less than ½.
Think: 6 is more than half of the denominator so the fraction is more than ½.
€
7
8
€
4
5Think: 1/8 is missing to make a whole.
Think: 1/5 is missing. Since 1/5 is a larger missing piece than 1/8 then ….
Revisit these fractions. Compare the strategy Revisit these fractions. Compare the strategy you used previously with one you used this time.you used previously with one you used this time.
€
5
8€
4
5
€
4
9
€
7
8
€
5
4€
11
7
€
6
7€
23
10€
4
6
€
2
6
Then: Change 5⅔ to an improper fraction, and Change to a mixed number.
1.
2.
3.
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8
9
1.
2.
3.
€
4
10
€
13
6
Comparing Fractions
An article by Go Math co-author Juli Dixon, PhD., “An Example of Depth...”.
MA.5.A.2.1Represent addition and subtraction of
decimals and fractions with like and unlike denominators using models, place value or
properties.
MA.5.A.2.2Add and subtract fractions and decimals fluently and verify the
reasonableness of results, including in problem situations.
Models to Add and Subtract Fractions
Manipulatives for Like and Unlike Denominators
Kathy had 2 yards of
ribbon. She gave yard
of her ribbon to Matt. How
much ribbon did Kathy
have left?
The oval track at the horse
race is mile around, but
the horses run for 1
miles during the race. What
length of the track will the
horses run twice?
Fractions Stories…Fractions Stories…
€
1
2
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6
8
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7
8
€
1
16
Real-Life Application of Fractions
You need exactly 1-cup of water for the dessert you are making. You can only find the cup, cup and
cup measuring cups.
How many different ways can you measure out 1-cup of water?
€
1
8
€
1
4
€
1
2
Write stories to support the following:
3/4 + 5/8
4/5 - 1/2
1 1/6 + 2/3
Now it’s your turn to tell Now it’s your turn to tell the story…the story…
MA.5.A.2.1FCAT 2.0 Test Spec.Item
Why Show Fractions in Simplest Forms?
Less pieces and a clearer visualization of the part whole relationship
There were 80 swimming pools at a local store. If of the pools were sold during one hot summer day, how many pools were left for sale after that day?
Jose spent of his money on concert tickets. If he had $200.00 to begin with, how much does he have left?
Singapore Model Drawing Examples
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Answer: B
MA.5.A.2.1FCAT 2.0 Sample Test Questions
MA.5.A.2.3Make reasonable estimates of fraction and
decimal sums and differences, and use techniques for rounding.
When asked this question, only 24% of 13-When asked this question, only 24% of 13-year olds and only 37% of 17-year olds year olds and only 37% of 17-year olds could estimate correctly.could estimate correctly.
Consider this Consider this concerning data…concerning data…
Estimate + .Estimate + .
a)a) 11 b) 2b) 2
c) 19c) 19 d) 21d) 21€
12
13
€
7
8
Consider the highly technical paper Consider the highly technical paper plate…plate…
•What else can you What else can you show me?show me?
•What should I show What should I show you?you?
•Can we use this for Can we use this for decimals?decimals?
1/2 + 2/51/2 + 2/5
2/6 + 3/112/6 + 3/11
2 1/13 + 6/72 1/13 + 6/7
3 4/5 + 1 1/33 4/5 + 1 1/3
1 7/8 - 1/21 7/8 - 1/2
Estimate the Estimate the following:following:
MA.5.A.2.4Determine the prime
factorization of numbers.
Divisibility Rules
How Do We Know They Are Prime?
Composite numbers can be placed into varying types of rectangles
Prime numbers cannot
Let’s look at that…
Composite Numbers
6 2415
Prime Numbers
7 17 29
Prime NumbersEratosthenes’(ehr-uh-TAHS-thuh-
neez) Sieve
276 BC - 194 BC
•Eratosthenes was a Greek mathematician, astronomer, geographer, and librarian atAlexandria, Egypt in 200 B.C. •He invented a method for finding prime numbers that is still used today.•This method is called Eratosthenes’ Sieve.
Eratosthenes’ Sieve
A sieve has holes in it and is used to filter out the juice.
Eratosthenes’s sieve filters out numbers to find the prime numbers.
Definition
Factor – a number that is multiplied by another to give a product.
7 x 8 = 56
Factors
Definition
Prime Number – a number that has exactly two factors.
77 is prime because the only numbersthat will divide into it evenly are 1 and 7.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
Let’s use a number grid from 1 to 100 to see how prime numbers were discovered.
2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
Remove the number 1. It is special number because 1 is its only factor.
2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
Leave the number 2 and remove all its multiples.
2 3 5 7 9
11 13 15 17 19
21 23 25 27 29
31 33 35 37 39
41 43 45 47 49
51 53 55 57 59
61 63 65 67 69
71 73 75 77 79
81 83 85 87 89
91 93 95 97 99
Leave the number 3 and remove all its multiples.
2 3 5 7
11 13 17 19
23 25 29
31 35 37
41 43 47 49
53 55 59
61 65 67
71 73 77 79
83 85 89
91 95 97
Leave the number 5 and remove all its multiples.
2 3 5 7
11 13 17 19
23 29
31 37
41 43 47 49
53 59
61 67
71 73 77 79
83 89
91 97
Leave the number 7 and remove all its multiples.
2 3 5 7
11 13 17 19
23 29
31 37
41 43 47
53 59
61 67
71 73 79
83 89
97
The PRIME Numbers!
GROWING A
FACTOR TREE
Can we grow a tree of
the factors of 180?
180Can you think of one
FACTOR PAIR for 180 ?
This should be two numbers that multiply together to give the
Product 180.
You might see that 180 is an EVEN NUMBER and that
means that 2 is a
factor…
2 x = 180� ?
OrYou might
notice that 180 has a ZERO in
its ONES PLACE which means it is a
multiple of 10.
SO…10 x = 180�
OrYou might
notice that 180 has a ZERO in
its ONES PLACE which means it is a
multiple of 10.
SO…10 x = 180�
10 x 18 = 180
10 18
180
10 18
We “grow” this “tree” downwards since that is how
we write in English (and we are not
sure how big it will be. We could run out of paper if we grew upwards).
NOW
You have to find
FACTOR PAIRS
for 10
and18
10
52
180
18
6 3
2 x 5 = 10 6 x 3 = 18
Find factors for 10 & 18
ARE
WE
DONE
???
2 3 32
5
10
2 6
180
18
3
5
Since 2 and 3
and 5 are PRIME
NUMBERS they do not grow “new branches”. They just
grow down alone.
Since 6 is NOT a prime number - it is
a COMPOSITE NUMBER - it still
has factors. Since it is an EVEN
NUMBER we see that:
6 = 2 x
Answer: C
FCAT 2.0 Sample Test Question
FCAT 2.0 Test Spec.Item
MA.5.A.6.1Identify and relate prime and
composite numbers, factors and multiples within the context of
fractions.