5th Grade Revised April 2014 (pg. 1)
SMUSD Common Core Math Scope and Sequence
Fifth Grade
Trimester One
Unit 1
Unit 2
Unit 2
Trimester Two
Unit 3
Unit 4
Trimester Three
Unit 5
Unit 6
2 weeks
6 weeks
continued
7 weeks
Place Value
and Whole
Number
Operations
OA.1
OA.2
NBT.1
NBT.2
NBT.5
NBT.6
4 weeks
4 weeks
The Four Operations with
Fractions
NF.1
NF.2
NF.3
NF.4
NF.7
Mid-Unit fraction
assessment for
trimester one report
card
NF.1
NF.2
NF.3
NF.4
NF.5
NF.6
NF.7
7 weeks
Decimals
and
Decimal
Operations
NBT.1
NBT.2
NBT.3
NBT.4
NBT.7
Converting
Measurements
Geometry
and Finding
Volumes
G.3
G.4
End of unit posttest and
performance
task
All units to contain the Standards for Mathematical Practice.
MD.1
MD.3
MD.4
MD.5
NBT.5
4 weeks
Graphing
G.1
G.2
MD.2
OA.3
5th Grade Revised April 2014 (pg. 2)
Standards for Mathematical Practice
Explanations and Examples for Grade Five
MP.1 Make sense of problems and persevere in solving them.
In grade five, students solve problems by applying their understanding of operations with whole numbers, decimals, and fractions including mixed
numbers. They solve problems related to volume and measurement conversions. Students seek the meaning of a problem and look for efficient ways to
represent and solve it. For example, Sonia had 123 sticks of gum. She promised her brother that she would give him ½ of a stick of gum. How much will
she have left after she gives her brother the amount she promised?
Teachers can encourage students to check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make
sense?”, and “Can I solve the problem in a different way?”
MP.2 Reason abstractly and quantitatively.
Students recognize that a number represents a specific quantity. They connect quantities to written symbols and create logical representations of
problems, consider appropriate units and the meaning of quantities. They extend this understanding from whole numbers to their work with fractions
and decimals. Teachers can support student reasoning by asking questions such as, “What do the numbers in the problem represent?” or “What is the
relationship of the quantities?” Students write simple expressions that record calculations with numbers and represent or round numbers using place
value concepts. For example, students use abstract and quantitative thinking to recognize that 0.5 × (300 ÷ 15) is 12 of (300 ÷ 15) without calculating the
quotient.
MP.3 Construct viable arguments and critique the reasoning of others.
In fifth grade students may construct arguments using visual models, such as objects and drawings. They explain calculations based upon models,
properties of operations and rules that generate patterns. They demonstrate and explain the relationship between volume and multiplication. They refine
their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” and “Why is
that true?” They explain their thinking to others and respond to others’ thinking.
Students use various strategies to solve problems and they defend and justify their work with others. For example, two afterschool clubs are having pizza
parties. The teacher will order 3 pizzas for every 5 students in the math club; and 5 pizzas for every 8 students in the student council. If a student is in
both groups, decide which party he/she should to attend. How much pizza will each student get at each party? If a student wants attend the party with
the most pizza (if divided equally between the students at the party), which party should he/she attend?
5th Grade Revised April 2014 (pg. 3)
MP.4 Model with mathematics
In grade five, students experiment with representing problem situations in multiple ways such as using numbers, mathematical language, drawings,
pictures, objects, charts, lists, graphs and equations. Teachers might ask, “How would it help to create a diagram, chart or table?” or “What are some ways
to represent the quantities?” Students need opportunities to represent problems in various ways and explain the connections. Fifth graders evaluate their
results in the context of the situation and they explain whether results to problems make sense. They evaluate the utility of models they see and draw and
can determine which models can be the most useful and efficient to solve problems.
MP.5 Use appropriate tools strategically.
Students consider available tools, including estimation, and decide which tools might help them solve mathematical problems. For instance, students may
use unit cubes to fill a rectangular prism and then use a ruler to measure the dimensions to find a pattern for volume using length of the sides. They use
graph paper to accurately create graphs and solve problems or make predictions from real-world data.
MP.6 Attend to precision.
Students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own
reasoning. Teachers might ask, “How do you know your solution is reasonable?” Students use appropriate terminology when they refer to expressions,
fractions, geometric figures, and coordinate grids. Teachers might ask, “What symbols or mathematical notations are important in this problem?”
Students are careful to specify units of measure and state the meaning of the symbols they choose. For instance, to determine the volume of a rectangular
prism, students record their answers in cubic units.
MP.7 Look for and make use of structure.
Students look closely to discover a pattern or structure. For instance, students use properties of operations as strategies to add, subtract, multiply and
divide with whole numbers, fractions, and decimals. They examine numerical patterns and relate them to a rule or a graphical representation. Teachers
might ask, “How do you know if something is a pattern?” or “What do you notice when…?”
MP.8 Look for and express regularity in repeated reasoning.
Fifth graders use repeated reasoning to understand algorithms and make generalizations about patterns. Students connect place value and their prior
work with operations to understand and use algorithms to extend multi-digit division from one-digit to two-digit divisors and to fluently multiply multidigit whole numbers. They use various strategies to perform all operations with decimals to hundredths and they explore operations with fractions with
visual models and begin to formulate generalizations. Teachers might ask, “Can you explain how this strategy works in other situations?” or “Is this
always true, sometimes true or never true?”
(Adapted from Arizona Department of Education [Arizona] 2012 and North Carolina 97 Department of Public Instruction [N. Carolina] 2013)
5th Grade Revised April 2014 (pg. 4)
Number Talks to support mental computation and flexibility with numbers
*All of the following information and problems have been adapted from Number Talks: Helping Children Build Mental Math and
Computation Strategies by Sherry Parrish. For more detailed information refer to your personal or grade level copy of the book.
During number talks, students should be accurate, efficient and flexible in their thinking. Many other areas can and should be addressed:
number sense, place value, fluency properties, and connecting mathematical ideas. Unpacking each of these components and keeping them
in the forefront during classroom strategy discussions will better prepare your students to be mathematically powerful and proficient.
Number Sense: Every time an answer to problem in a number talk is elicited from a student, and you ask students to share whether the
proposed solutions are reasonable, you are helping students build number sense. When students are asked to give an estimate before they
begin thinking about a specific strategy, number sense is being fostered.
Place Value: Students understand place value if they can apply their understanding in computation. Number talks provide the opportunity
to confront and use place-value understanding on a continual basis.
Fluency: Fluency is much more than fact recall. Fluency is knowing how a number can be composed and decomposed and using that
information to be flexible and efficient with problem solving.
Properties: As students invent their own strategies, they create opportunities to link their thinking to mathematical properties and
understand how they work.
Connecting Mathematical Ideas: Each time you compare student strategies, discuss how addition can be used to solve subtraction
problems, and make links between arrays with multiplication and division you are grounding your students in the idea that mathematical
concepts are related and make sense. Helping students connect mathematical ideas is a critical component of number talks.
The following number talks are crafted to elicit specific strategies; however, you may find that students also share other efficient methods.
Each strategy is labeled to help you better understand its foundation, however strategies are often named for the students who
invented them. Keep in mind the overall purpose is to help students build a toolbox of efficient strategies based on numerical reasoning.
Remember, the ultimate goal of number talks is for students to compute accurately, efficiently, and flexibly.
Once number talks become a regular component of your math instruction, you may consider stopping showing how to solve computation
problems. Instead, provide story problems that incorporate new operations and encourage students to make sense of the problems and
numbers. Using real life context as a spring board for mathematical thinking engages students in mathematics that is relevant, attaches
meaning to the numbers, and helps students access the mathematics.(See pp. 163-167 of Number Talks for further support.)
As you begin to implement number talks in your classroom, start with small numbers. Using small numbers
serves two purposes:
5th Grade Revised April 2014 (pg. 5)
1) Students can focus on the nuances of the strategy instead of the magnitude of the numbers
2) Students are able to build confidence in their mathematical abilities.
As students’ understanding of different strategies develops, you can gradually increase the size of the numbers. When
the numbers become too large, students will rely on less efficient strategies, such as counting all, or resort to paper and
pencil, thereby losing the focus on developing mental strategies.
Trimester One
Multiplication Number Talks:
Instructions: The following number talks consist of three or more sequential problems. The sequence within a given
number talk allows students to apply strategies from previous problems to subsequent problems.
Strategy One: Making Landmark or Friendly Numbers
“A common error students make when changing one of the factors to a landmark number is to forget to adjust the number of groups. The
problem 9 x 25 can help us consider the common errors children make when making this adjustment. If 9 had been changed to 10, then the
product of 250 would need to be adjusted not just by 1 but by one group of 25.” – Number Talks pg. 267
Non- Example:
25
X
Example:
25
9 + 1 x 10
250 – 1 = 249
25
X
25
9 + 1 x 10
250 - 25 = 225
5th Grade Revised April 2014 (pg. 6)
Category 1
5x5
5 x 10
5 x 30
5 x 29
4x5
4 x 10
4 x 50
4 x 49
3 x 10
3 x 50
3 x 100
3 x 149
Category 2
6 x 50
6 x 300
6 x 349
4 x 60
5 x 300
4 x 359
5 x 100
5 x 300
5 x 60
5 x 359
Category 3
6 x 20
30 x 20
36 x 20
36 x 19
3 x 50
50 x 50
53 x 50
53 x 48
10 x 10
10 x 30
2 x 30
12 x 29
Strategy Two: Doubling and Halving
“Halving and doubling is an excellent strategy to restructure a problem with multiple digits and make it easier to solve. Helping students to notice the
relationship between the two factors and the dimensions of the accompanying array is important to understanding the strategy. An equally important
idea in this strategy is that the factors can adjust while the area of the array stays the same.” Number Talks pg. 276
1 x 16
2x8
4x4
5th Grade Revised April 2014 (pg. 7)
Category 1
Category 2
Category 3
1 x 16
2x8
4x4
8x2
1 x 16
8 x 125
4 x 250
2 x 500
3 x 60
6 x 30
12 x 15
1 x 24
2 x 12
4x6
8x3
84 x 5
42 x 10
21 x 20
9 x 56
18 x 28
36 x 14
1 x 48
2 x 24
4 x 12
8x6
16 x 3
35 x 8
70 x 4
140 x 2
2 x 280
4 x 140
8 x 70
16 x 35
1 x 56
2 x 28
4 x 14
8x7
345 x 8
690 x 4
1380 x 2
104 x 3
52 x 6
26 x 12
Strategy Three: Breaking Apart Factors
Breaking apart factors gives students the opportunity to apply the associative property of multiplication.
Example:
8 x 25 = 2 x 4 x 25
or
Category 1
4x3x4
2 x 2 x 12
8x3x2
2x2x3x4
12 x 4
8 x 25 = 8 x 5 x 5
or
8 x 25 = 2 x 4 x 5 x 5
Category 2
Category 3
3x5x4
2 x 15 x 2
15 x 4
3 x 4 x 25
5 x 12 x 5
5 x 2 x 25
12 x 25
5th Grade Revised April 2014 (pg. 8)
Breaking Factors Apart (cont.)
5x2x6
5x4x3
2x2x3x5
5 x 12
5x5x8
2 x 4 x 25
2 x 25 x 4
25 x 8
5x2x4
4x5x2
2x2x5x2
8x5
2 x 4 x 35
8x5x7
8 x 35
2 x 15 x 6
5 x 12 x 3
4x5x3x3
4x5x9
12 x 15
4 x 4 x 25
8 x 2 x 25
1x5x5
16 x 25
Strategy Four: Partial Products
“This strategy is based on breaking one or both factors into addends through using expanded notation and the distributive property. While
both factors can be represented with expanded notation, keeping one number whole is often more efficient.” Number Talks pg. 272
Example:
8 x 25
(4 + 4) x 25 = ( 4 x 25 ) + ( 4 x 25 )
(2 + 2 + 4) x 25 = (2 x 25) + (2x25) + (4 x 25)
8 x (20 + 5) = (8 x 20) + (8 x 5)
8 x (10 + 10 + 5) = (8 x 10) + (8 x 10) + (8 x 5)
Category 1
2x7
4x7
4x8
3x8
8x7
3x8
2x6
6x8
Category 2
Category 3
2 x 125
4 x 25
6 x 100
6x4
6 x 124
3 x 15
10 x 15
13 x 10
13 x 5
13 x 15
2 x 150
5 x 100
5 x 10
5 x 50
5 x 150
15 x 10
15 x 1
10 x 11
5 x 11
15 x 11
5th Grade Revised April 2014 (pg. 9)
Partial Products (cont.)
2x7
4x7
3x7
7x7
4 x 250
4x6
8 x 200
8 x 50
8x6
8 x 256
4 x 22
6 x 11
3 x 22
6 x 22
10 x 22
16 x 22
Trimester Two
Division Number Talks:
Instructions: The following number talks consist of three or more sequential problems. The sequence within a given
number talk allows students to apply strategies from previous problems to subsequent problems.
Strategy One: Partial Quotients
“The Partial Quotients strategy maintains the integrity of place value and allows the students to approach the problem by building on
multiplication problems with friendly multipliers such as 2, 5, 10, powers of 10, and so on. This strategy allows the student to navigate
through the problem by building on what they know, understand, and can implement with ease.” Number Talks pg. 288
Category 1
Category 2
Category 3
300 ÷ 3
40 ÷ 4
100 ÷ 25
120 ÷ 3
16 ÷ 4
250 ÷ 25
420 ÷ 3
56 ÷ 4
500 ÷ 25
30 ÷ 3
24 ÷ 3
54 ÷ 3
50 ÷ 5
30 ÷ 5
80 ÷ 5
40 ÷ 4
24 ÷ 4
67 ÷ 4
30 ÷ 6
18 ÷ 6
300 ÷ 6
348 ÷ 6
100 ÷ 4
200 ÷ 4
40 ÷ 40
16 ÷ 4
256 ÷ 4
400 ÷ 4
80 ÷ 4
16 ÷ 4
120 ÷ 12
240 ÷ 12
368 ÷ 12
70 ÷ 35
105 ÷ 35
350 ÷ 35
525 ÷ 35
150 ÷ 15
300 ÷ 15
600 ÷ 15
5th Grade Revised April 2014 (pg. 10)
Strategy Two: Multiplying Up
“Following the same principle as Adding Up to Subtract, Multiplying Up is an accessible division strategy that capitalizes on the relationship
between multiplication and division. Similar to Partial Quotients, this strategy provides an opportunity for students to gradually build on
multiplication problems they know until they reach the dividend.” Number Talks pg. 292
Category 1
Category 2
Category 2
3x 10
4 x 25
3 x 100
3 x 20
4 x 50
3 x 50
3x3
4 x 100
3x1
3x2
500 ÷4
453 ÷ 3
68 ÷ 3
5x5
8 x 100
4 x 25
5 x 10
8 x 50
4 x 50
5x2
8 x 10
4x3
85 ÷ 5
792 ÷ 8
215 ÷ 4
6 x 10
4 x 25
6 x 100
6x5
4 x 100
6 x 50
6x6
4 x 10
6 x 60
6x2
4 x 20
6x5
99 ÷ 6
484 ÷ 4
536 ÷ 6
4 x 10
7 x 100
8 x 100
4x5
7 x 10
8 x 50
4x8
7x5
8 x 10
4x4
7x2
792 ÷ 8
72 ÷ 4
836 ÷ 7
Strategy Three: Proportional Reasoning
“When division is considered from a fractional perspective, this provides opportunities for students to explore the relationship of the part to
the whole through proportional reasoning using equivalent fractions. Knowing that the divisor and dividend in share common factors,
students can simplify the quantity to any of the following equivalent fractions: , , or .” Number Talks pg. 299
5th Grade Revised April 2014 (pg. 11)
100 ÷ 4
200 ÷ 8
400 ÷ 16
720 ÷ 36
360 ÷ 18
60 ÷ 3
800 ÷ 40
80 ÷ 4
40 ÷2
100 ÷ 4
200 ÷ 8
400 ÷ 16
250 ÷2
500 ÷ 4
1000 ÷8
384 ÷ 16
96 ÷ 4
48 ÷2
172 ÷ 3
144 ÷ 6
288 ÷ 12
46 ÷ 2
92 ÷ 4
184 ÷8
308 ÷ 7
308 ÷ 14
308 ÷ 28
Trimester 3
Addition, Subtraction, Multiplication and Division Number Talks:
Throughout trimester 3 students are encouraged to solve problems using a variety of operations with and without context. Throughout the
trimester students select strategies to solve each problem and then discuss the efficiency of different strategies. The following sample
problems are taken from different sections throughout the book Number Talks. For more samples, or if you would like to continue practice
with only multiplication and division strategies see your grade level copy of the book Number Talks by Sherry Parrish.
28 + 27
342 + 64
18 x 5
35 x 10
64 x 35
116 + 29
35 x 2
16 x 140
39 + 127
35 x 20
4 x 560
114 + 118
35 x 24
1 x 2240
46 + 118
26 x 12
65 + 57
100 – 44
56 ÷ 4
48 ÷ 3
96 ÷ 4
25 x 8
18 x 35
18 x 35
50 – 25
123 – 105
15 x 50
52 – 25
100 – 34
15 x 49
42 x 10
8 x 32
2 x 15 x 6
21 x 20
2 x 128
5 x 12 x 3
496 ÷4
235 ÷ 5
77 ÷ 5
56 – 10
70 – 61
44 + 27
56 – 12
70 – 34
48 + 34
56 – 30
74 – 49
55 + 16
56 – 35
74 – 36
58 + 25
1000 – 674
496 ÷ 8
16 x 35
5th Grade Revised April 2014 (pg. 12)
Standards of Focus to be taught and practiced through problem solving:
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.
NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual models or
equations to represent the problem.
Teaching Mathematics through Problem Solving
(Adapted from Teaching Student-Centered Mathematics)
Teaching mathematics through problem solving is a method of teaching mathematics that helps children develop
relational understanding. With this approach, problem solving is completely interwoven with learning. As children do
mathematics-make sense of cognitively demanding tasks, provide evidence or justification for strategies and solutions,
find examples and connections, and receive and provide feedback about ideas-they are simultaneously engaged in the
activities of problem solving and learning. Teaching mathematics through problem solving requires you to think about
the types of tasks you pose to children, how you facilitate discourse in your classroom, and how you support children’s
variety of representations as tools for problem solving, reasoning, and communication.
Teaching mathematics through problem solving promises to be an effective approach if our ultimate goal is deep
(relational) understanding because it accomplishes these goals:
1) Focuses children’s attention on ideas and sense making. When solving problems, children are reflecting on the
concepts inherent in the problems. Emerging concepts are more likely to be integrated with existing ones, thereby
improving understanding.
2) Emphasizes mathematical processes and practices. Children who are solving problems will engage in all five of
the processes of doing mathematics-problem solving, reasoning, communication, connections, and representation, as
well as the eight-mathematical practices outlined in the Common Core State Standards, resulting in mathematics that is
more accessible, more interesting, and more meaningful.
5th Grade Revised April 2014 (pg. 13)
3) Develops children’s confidence and identities. Every time teachers pose a problem-based task and expect a
solution, they implicitly say to children, “I believe you can do this.” When children are engaged in problem solving and
discourse in which the correctness of the solution lies in justification of the processes, they begin to see themselves as
capable of doing mathematics and that mathematics make sense.
4) Provides a context to help children build meaning for the concept. Using a context facilitates mathematical
understanding, especially when the context is grounded in an experience familiar to children and when the context uses
purposeful constraints that potentially highlight the significant mathematical ideas
5) Allows entry and exit points for a wide range of children. Good problem-solving based tasks have multiple paths
to the solution, so each child can make sense of and solve the task by using his or her own ideas. Furthermore, children
expand their ideas and grow in their understanding as they hear, critique, and reflect on the solution strategies of others.
6) Allows for extensions and elaborations. Extensions and “what if” questions can motivate advanced learners or
quick finishers, resulting in increased learning and enthusiasm for doing mathematics.
7) Engages students so that there are fewer discipline problems. Many discipline issues in a classroom are the
result of children becoming bored, not understanding the teacher directions, or simply finding little relevance in the
task. Most children like to be challenged and enjoy being permitted to solve problems in ways that make sense to them,
giving them less reason to act out or cause trouble.
8) Provides formative assessment data. As children discuss ideas, draw diagrams, or use manipulatives, defend their
solutions and evaluate those of others, and write reports or explanations, they provide the teacher with a steady stream
of valuable information that can be used to inform subsequent instruction.
9) It’s a lot of fun! Children enjoy the creative process of problem solving and sharing how they figured something out.
After seeing the surprising and inventive ways that children think and how engaged children become in mathematics,
very few teachers stop using a teaching-through-problem solving approaching.
5th Grade Revised April 2014 (pg. 14)
A Three-Phase Lesson Format
Before: In the before phase of the lesson you are preparing children to work on the problem. As you plan for the before
part of the lesson, analyze the problem you will give to children in order to anticipate children’s approaches and possible
misinterpretations or misconceptions. This can inform questions you ask in the before phase of the lesson to clarify
children’s understanding of the problem (i.e., knowing what it means rather than how they will solve it.)
During: In the during phase of the lesson children explore the problem (alone, with partners, or in small groups). This is
one of two opportunities you will get in the lesson to find out what the children know, how they think, and how they are
approaching the task you have given them (the other is in the discussion period of the after phase). You want to convey a
genuine interest in what the children are thinking and doing. This is not the time to evaluate or to tell children how to
solve the problem.
When asking whether a result or method is correct, ask children, “How can you decide?” or “Why do you think
that might be right?” or “How can we tell if that makes sense?” Use this time in the during phase to identify different
representations and strategies children used, interesting solutions, and any misconceptions that arise that you will
highlight and address during the after phase of the lesson.
After: In the after phase of the lesson your children will work as a community of learners, discussing, justifying, and
challenging various solutions to the problem that they have just worked on. It is critical to plan for and save ample time
for this part of the lesson. Twenty minutes is not at all unreasonable for a good class discussion and sharing of ideas. It is
not necessary to wait for every child to finish. Here is where much of the learning will occur as children reflect
individually and collectively on ideas they have explored. This is the time to reinforce precise terminology, definitions,
or symbols. After children have shared their ideas, formalize the main ideas of the lesson, highlighting connections
between strategies or different mathematical ideas.
5th Grade Revised April 2014 (pg. 15)
Sample Problem: (NF.2) Jenny was making two different types of cookies. One recipe needed 3/4 cup of
sugar and the other needed 2/3 cup of sugar. How much sugar did he need to make both recipes?
Solutions:
Mental Estimation (MP.2): A student may say that Jenny needs more than 1 cup of sugar but less than 2
cups, since both fractions are larger than ½, while at the same time both fractions are less than 1.
Area Model to Show equivalence (MP.5): A student may choose to represent each partial cup of sugar
using an area model, find equivalent fractions, and then add:
I see that 3/4 of a cup of sugar is equivalent to 9/12 of a cup,
while 2/3 of a cup is equivalent to 8/12 of a cup.
Altogether, I have 17/12 of a cup.
This is more than one cup since
= + =1
Sample
Problem:
(5.NF.6)
5th Grade Revised April 2014 (pg. 16)
Essential Learning for the Next Grade
Adapted from: The Mathematics Framework was adopted by the California State Board of Education on November 6, 2013
Fifth Grade
To be prepared for grade six mathematics students should be able to demonstrate they have acquired certain mathematical concepts and
procedural skills by the end of grade four and have met the fluency expectations for the grade. For fifth graders, the expected fluency is to
multiply multi-digit whole numbers with up to four-digits using the standard algorithm (5.NBT.5). These fluencies and the conceptual
understandings that support them are foundational for work in later grades.
Of particular importance at grade five are concepts, skills, and understandings needed to understand the place value system (5.NBT.1-4▲);
perform operations with multi-digit whole numbers and with decimals to hundredths (5.NBT.5-7▲); use equivalent fractions as a strategy
to add and subtract fractions (5.NF.1-2▲); apply and extend previous understandings of multiplication and division to multiply and divide
fractions. (5.NF.3-7▲); geometric measurement: understand concepts of volume and relate volume to multiplication and to addition
(5.MD.3- 5▲). In addition graphing points on the coordinate plane to solve real-world and mathematical problems (5.G.1-2) is an important
part of a students’ progress to algebra.
Fractions
Student proficiency with fractions is essential to success in algebra at later grades. By the end of grade five students should be able to add,
subtract and multiply any two fractions and understand how to divide fractions in limited cases (unit fractions divided by whole numbers
and whole numbers divided by unit fractions).
Students should understand fraction equivalence and use their skills to generate equivalent fractions as a strategy to add and
subtract fractions, with unlike denominators including mixed fractions. Students should use these skills to solve related word
problems. This understanding brings together the threads of fraction equivalence (grades three through five) and addition and subtraction
(kindergarten through grade four) to fully extend addition and subtraction to fractions. By the end of grade five students know how to
multiply a fraction or whole number by a fraction. Based on their understanding of the relationship between fractions and
division, students divide any whole number by any nonzero whole number and express the answer in the form of a fraction or
mixed number. Work with multiplying fractions extends from students’ understanding of the operation of multiplication. For example, to
multiply a/b x q (where q is a whole number or a fraction), students can interpret x q as meaning a parts of a partition of q into b equal parts.
This interpretation of the product leads to a product that is less than, equal to or greater than q, depending on whether a/b<1, = 1 or >1,
respectively. For a/b<1, this result of multiplying contradicts earlier student experience with whole numbers, so this result needs
to be discussed, explained, and emphasized.
Grade five students divide a unit fraction by a whole number or a whole number by a unit fraction. By the end of grade five students should
know how to multiply fractions to be prepared for division of a fraction by a fraction in grade six.
5th Grade Revised April 2014 (pg. 17)
Decimals
In grade five students will integrate decimal fractions more fully into the place value system as they learn to read, write, compare, and round
decimals. By thinking about decimals as sums of multiples of base-ten units, students extend algorithms for multi-digit operations to
decimals. By the end of grade five students understand operations with decimals to hundredths. Students should understand how to add,
subtract, multiply, and divide decimals to hundredths using models, drawings, and various methods including methods that extend
from whole numbers and are explained by place value meanings. The extension of the place value system from whole numbers to
decimals is a major accomplishment for a student that involves both understanding and skill with base-ten units and fractions. Skill and
understanding with adding, subtracting, multiplying, and dividing multi-digit decimals will culminate in fluency with the standard
algorithm in grade six.
Fluency with Whole Number Operations
In grade five the fluency expectation is to multiply multi-digit whole numbers (one-digit numbers times a number with up to four-digits and
two-digit numbers times two-digit numbers) using the standard algorithm. Students also extend their grade four work in finding wholenumber quotients and remainders to the case of two-digit divisors. Skill and understanding of division with multi-digit whole numbers
will culminate in fluency with the standard algorithm in grade six.
Volume
Students in grade five work with volume as an attribute of a solid figure and as a measurement quantity. Students also relate volume to
multiplication and addition. Students’ understanding and skill with this work supports a learning progression leading to valuable skills in
geometric measurement in middle school.