Transcript Grade 5 Big Idea 2 - ElementaryMathematics

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Fifth Grade Big Idea 2
Day 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.
Three Meanings of a Fraction
 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?
nd
2
Meaning of a Fraction:
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)
3rd 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.
Fractions—For these problems, circle the greater
number of each pair and tell the strategiy you used.
1.
4
5
4
9
2.
4
10
5
8
7
8
5
4

3.


1.
4
6
2
6
2.
3
2
10
1
1
7
8
9
6
7

3.


Then: Change 5⅔ to an improper fraction, and Change
mixed number.




13
6
to a
Strategies to Compare/Order Fraction
When the whole numbers are
different, you only have to
compare the whole numbers.
1
2
4

>
1
1
2
When the numerator is the same,
look at the size of the pieces in the
denominator.
3
5


>
3
8
Strategies to Compare/Order Fraction
•Use benchmark numbers •Compare missing pieces
>
3
8
Think: 3 is less
than half of the
denominator
so the fraction
is less than ½.

6
10
7
8
Think: 6 is more
than half of the
denominator so
the fraction is
more than ½.

>
Think: 1/8 is
missing to
make a whole.

4
5
Think: 1/5 is
missing. Since 1/5
is a larger missing
piece than 1/8 then
….
Revisit these fractions. Compare the strategy you
used previously with one you used this time.
1.
4
5
4
9
2.
4
10
5
8
7
8
5
4

3.


1.
4
6
2
6
2.
3
2
10
1
1
7
8
9
6
7

3.


Then: Change 5⅔ to an improper fraction, and Change
mixed number.




13
6
to a
Comparing
Fractions
An article by Go
Math co-author
Juli Dixon, PhD.,
“An Example of
Depth...”.
MA.5.A.2.1
Represent addition and subtraction of decimals and
fractions with like and unlike denominators using
models, place value or properties.
MA.5.A.2.2
Add 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
Fractions Stories…
1
yards of
2
6
ribbon. She gave
yard
8
 Kathy had 2
of her ribbon to Matt. How

much ribbon did Kathy

have left?
 The oval track at the horse
7
mile around, but
8
1
the horses run for 1
16
race is
miles during the race. What

length of the track will the

horses run twice?
Real-Life Application of Fractions
 You need exactly 1-cup of water for the dessert you
are making. You can only find the
1
cup measuring cups.
1
cup, 1 cup
8
4
and
2

How many different ways can
you 
measure out 1cup of water?
Now it’s your turn to tell the
story…
Write stories to support the following:
3/4 + 5/8
4/5 - 1/2
1 1/6 + 2/3
MA.5.A.2.1
FCAT 2.0
Test Spec.
Item
Why Show Fractions in Simplest Forms?
 Less pieces and a clearer visualization of the part
whole relationship
Singapore Model Drawing Examples
 There were 80
swimming pools at a
3
local store. If 5 of the
pools were sold during
one hot summer
day,

how many pools were
left for sale after that
day?
 Jose spent
4
8
of his
money on concert
tickets. If he had

$200.00
to begin with,
how much does he
have left?
MA.5.A.2.1
FCAT 2.0
Sample Test
Questions
Answer: B
MA.5.A.2.3
Make reasonable estimates of fraction and
decimal sums and differences, and use
techniques for rounding.
Consider this
concerning data…
Estimate 12 + 7 .
13
a) 1
 c) 19

8
b) 2
d) 21
When asked this question, only 24% of 13year olds and only 37% of 17-year olds
could estimate correctly.
Consider the highly technical paper
plate…
•What else can you
show me?
•What should I show
you?
•Can we use this for
decimals?
Estimate the
following:
1/2 + 2/5
2/6 + 3/11
2 1/13 + 6/7
3 4/5 + 1 1/3
1 7/8 - 1/2
MA.5.A.2.4
Determine 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
15
24
Prime Numbers
7
17
29
Prime Numbers
Eratosthenes’(ehr-uh-TAHS-thuh-neez)
Sieve
•Eratosthenes was a Greek
mathematician, astronomer,
geographer, and librarian at
Alexandria, Egypt in 200 B.C.
•He invented a method for finding
prime numbers that is still used
today.
•This method is called Eratosthenes’
Sieve.
276 BC - 194 BC
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.
7
7 is prime because the only numbers
that will divide into it evenly are 1 and 7.
Let’s use a number grid from 1 to 100 to see
how prime numbers were discovered.
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
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
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 3 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 5 and remove all its
multiples.
2
11
3
13
23
31
41
37
55
73
49
59
67
77
85
95
19
29
47
65
83
91
25
43
61
7
17
35
53
71
5
79
89
97
Leave the number 7 and remove all its
multiples.
2
11
3
13
5
7
17
23
31
41
29
37
43
47
53
61
71
49
59
67
73
77
83
91
19
79
89
97
The PRIME Numbers!
2
11
3
13
5
7
17
23
31
41
29
37
43
47
53
61
71
19
59
67
73
79
83
89
97
GROWING A
FACTOR TREE
Or
Or
Can
think
You
might
Youyou
might
You
might
of
one
notice
that
notice that 180
180
see
180
FACTOR
Can
we
has
a
ZERO
in
has
athat
ZERO
in
PAIR
for
180
isits
anONES
EVEN
its
ONES
grow
a ?
PLACE
which
NUMBER
PLACE
which
tree
of
means
itit is
means
is a
a
and that
the
This
should
be
multiple
of
10.
multiple
of
10.
means that
two
numbers
factors
SO…
is a
SO…
that2
multiply
of
180?
10
x
•
10
x
•=
= 180
180
factor…
together
to
give the
2 x •= 180 ?
Product
10
x 18 =180.
180
180
10
18
180
NOW
You have
to find
FACTOR
PAIRS
for
10
and
18
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).
180
Find factors
for 10 & 18
18
10
2 x 5 = 10
2
5
6
3
6x3=
18
Since
2ARE
and 3
and 5 are
PRIME
WE
NUMBERS
they do not
DONE
grow “new
branches”.
???
They
just
grow down
alone.
180
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:
18
10
6=2x•
2
2
5
6
3
5 2 3 3
FCAT 2.0
Sample
Test
Question
Answer: C
FCAT 2.0
Test Spec.
Item
MA.5.A.6.1
Identify and relate prime and composite
numbers, factors and multiples within the
context of fractions.