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Fractions and Ratios
Big Idea: Real world situations can be represented by fractions or ratios to better understand them and to apply them to new
situations.
Teacher Essential Question: Can students reason quantitatively and comparatively with fractions, decimals, and percents to
solve problems?
Student Essential Question: How can I problem solve (in context) with fractions, decimal fractions, and percents?
Suggested Instructional Time:7 weeks
Embedded
Objective
Throughout
Topic:
S1C3PO 2. Make estimates appropriate to a given situation
or computation with whole numbers and fractions.
MP. 1 Make sense of problems and persevere in solving them
Page 20 #21-22
Standards
Mathematical Practices
G4S1C1PO 1. Express whole numbers, fractions, decimals, and percents using
and connecting multiple representations.
MP.2. Reason abstractly and
quantitatively.
MP.7. Look for and make use of
structure.
5.MP.1. Make sense of problems and
persevere in solving them.
G6 S1C1PO 3. Demonstrate an understanding of fractions as rates, division of
whole numbers, parts of a whole, parts of a set, and locations on a real number
line.
5.NF.3. Interpret a fraction as division of the numerator by the denominator (
a
b
= ab).
Solve word problems involving division of whole numbers leading to answers in the
form of fractions or mixed numbers, e.g., by using visual fraction models or equations
to represent the problem. For example, interpret
3
4
as the result of dividing 3 by 4,
3
multiplied by 4 equals 3, and that when 3 wholes are shared equally
4
3
among 4 people each person has a share of size . If 9 people want to share a 504
noting that
5.MP.2. Reason abstractly and
quantitatively.
Formatives
Uncovering Student Thinking in
Mathematics Grades 6-12 Page 39
Probe 2
Uncovering Student Thinking in
Mathematics Grades 6-12 Page 44
Probe 2a
5.MP.3. Construct viable arguments
and critique the reasoning of others.
5.MP.4. Model with mathematics.
5.MP.5. Use appropriate tools
strategically.
5.MP.7. Look for and make use of
structure.
pound sack of rice equally by weight, how many pounds of rice should each person
get? Between what two whole numbers does your answer lie?
Connection: 5.SL.1
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G5 S1C1PO 1. Determine
equivalence by converting
between benchmark
fractions, decimals, and
percents.
G4S1C3PO 1. Use benchmarks as meaningful
points of comparison for whole numbers,
decimals, and fractions.
MP.7. Look for and make use of
structure.
G4S1C1PO 5. Use simple ratios to describe problems in context.
G5 S1C1PO 5. Use ratios and unit rates to model, describe and extend problems
in context.
G4/5 S1C2PO 1. Add and subtract decimals through hundredths including
money to $1000.00 and fractions with like and unlike denominators, expressing
answers in simplest form.
Connections
SS04-S5C1-01
4.NF.3. Understand a fraction
a.
b.
2
d.
a
b
with a> 1 as a sum of fractions
1
.
b
MP.1. Make sense of problems and
persevere in solving them.
MP.3. Construct viable arguments
and critique the reasoning of others.
MP.1. Make sense of problems and
persevere in solving them.
MP.3. Construct viable arguments
and critique the reasoning of others.
Page 7 #5
Page 11 #11
Page 17 #18
Uncovering Student Thinking in
Mathematics Grades 6-12 Page 45
Probe 3
Uncovering Student Thinking in
Mathematics Grades 6-12 Page 50
Probe 3a
Page 138 # 25-26
Page 37 #45
Uncovering Student Thinking in
Mathematics Grades 6-12 Page 82
Probe 9a
Uncovering Student Thinking in
Mathematics Grades 6-12 Page 90
Probe10a
Understand addition and subtraction of fractions as joining and separating parts
referring to the same whole.
Decompose a fraction into a sum of fractions with the same denominator in more
than one way, recording each decomposition by an equation. Justify
decompositions, e.g., by using a visual fraction model.
3 1 1 1
= + +
8 8 8 8
1 8 8 1
= + + .
8 8 8 8
Examples:
c.
MP.2. Reason abstractly and
quantitatively.
;
3 1 2 1
= + ;2
8 8 8 8
=1 + 1+
1
;
8
Add and subtract mixed numbers with like denominators, e.g., by replacing each
mixed number with an equivalent fraction, and/or by using properties of
operations and the relationship between addition and subtraction.
Solve word problems involving addition and subtraction of fractions referring to
the same whole and having like denominators, e.g., by using visual fraction
models and equations to represent the problem.
Connections: 4.RI.7; 4.W.2b; ET04-S1C2-02; ET04-S1C4-01
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Uncovering Student Thinking in
Mathematics Grades 6-12 Page 51
Probe 4
Uncovering Student Thinking in
Mathematics Grades 6-12 Page 83
Probe 9b
Uncovering Student Thinking in
Mathematics Grades 6-12 Page 91
Probe 10b
5.NF.1. Add and subtract fractions with unlike denominators (including mixed
numbers) by replacing given fractions with equivalent fractions in such a way as to
produce an equivalent sum or difference of fractions with like denominators. For
example,
2
3
+
5
4
=
8
12
+
15
12
=
23
a
. (In general,
12
b
+
c
d
=
(ad  bc)
bd
5.NF.2. Solve word problems involving addition and subtraction of fractions referring to
the same whole, including cases of unlike denominators, e.g., by using visual fraction
models or equations to represent the problem.Use benchmark fractions and number
sense of fractions to estimate mentally and assess the reasonableness of answers.
For example,recognize an incorrect result
2
5
+
1
2
=
3
3 1
, by observing that < .
7
7 2
Connections: 5.NF.1; 5.RI.7; 5.W.2c; 5.SL.2; 5.SL.3; ET05-S1C2-02
G6 S1C2PO4. Multiply fractions.
4.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction by a
whole number.
a.
b.
c.
a
b
1
Understand a fraction
as a multiple of
. For example, use a visual fraction model to
b
5
1
5
1
represent
as the product 5( ), recording the conclusion by the equation
= 5( ).
4
4
4
4
a
1
Understand a multiple of
as a multiple of
, and use this understanding to multiply a
b
b
2
fraction by a whole number. For example, use a visual fraction model to express 3( ) as
5
1
6
6( ), recognizing this product as
. (In general, n(a/b)=((nxa)/b.)
5
5
4.MP.1. Make sense of problems and
persevere in solving them.
Page 35 #43
4.MP.2. Reason abstractly and
quantitatively.
4.MP.4. Model with mathematics.
4.MP.5. Use appropriate tools
strategically.
4.MP.6. Attend to precision.
Solve word problems involving multiplication of a fraction by a whole number, e.g., by using
visual fraction models and equations to represent the problem. For example, if each person
at a party will eat
3
8
of a pound of roast beef, and there will be 5 people at the party, how
many pounds of roast beef will be needed? Between what two whole numbers does your
answer lie?
Connections: 4.RI.5; 4.W.2e; ET04-S1C2-02
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5.NF.5. Interpret multiplication as scaling (resizing), by:
a.
Comparing the size of a product to the size of one factor on the basis of the size
of the other factor, without performing the indicated multiplication.
b.
Explaining why multiplying a given number by a fraction greater than 1 results in
a product greater than the given number (recognizing multiplication by whole
numbers greater than 1 as a familiar case); explaining why multiplying a given
number by a fraction less than 1 results in a product smaller than the given
number; and relating the principle of fraction equivalence
effect of multiplying
a (na)
=
/(nb)to the
b
5.MP.4. Model with mathematics.
5.MP.6. Attend to precision.
5.MP.7. Look for and make use of
structure.
a
by 1.
b
Connections: 5.RI.3; 5.RI.5; 5.W.2a; 5.W.2b; 5.W.2c; 5.W.2d; 5.W.2e; 5.SL.2; 5.SL.3
5.NF.6. Solve real world problems involving multiplication of fractions and mixed
numbers, e.g., by using visual fraction models or equations to represent the problem.
Connections: 5.RI.7; 5.W.2e; ET05-S1C1-01; ET05-S1C2-02
G4S2C2PO 1. Describe elements of theoretical probability by listing or drawing
all possible outcomes of a given event and predicting the outcome using word
and number benchmarks.
MP. 5 Use appropriate tools strategically
MP. 8 Look for and express regularity in repeated reasoning
Page 178 #20
Page 185 #25
Page 190 #30-31
Page 195 #33
5.MP.2. Reason abstractly and
quantitatively.
5.MP.1. Make sense of problems and
persevere in solving them.
5.MP.3. Construct viable arguments
and critique the reasoning of others.
5.MP.5. Use appropriate tools
strategically.
G4S2C3PO 1. Construct tree
diagrams to solve problems in
context by

representing all possibilities
for a variety of counting
problems,

explaining how its
properties relate to the
problem,

representing the same
counting problem in multiple
ways, and

drawing conclusions.

G6 S2C3PO1 Build and explore
tree diagrams where items repeat.
Page 182 #24 add pact C: If the
flavors are increased by two (mint
and sherbet) and the options also
include a waffle cone and a cup, how
would that change the number of
options?
G4S2C3PO 2. Justify that all
possibilities have been
enumerated without duplication.
MP. 1 Make sense of problems and
persevere in solving them
MP. 3 Construct viable arguments
and critique the reason of others
MP. 5 Use appropriate tools
strategically
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