Mathematics
Grade 5 – Year in Detail (SAMPLE)
Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8
Unit 9
Unit 10
Whole number
operations
Place value
with decimals
Add and Subtract
Decimals
Add and
Subtract
Fractions
Multiply and Divide
Decimals
Multiplying
Fractions
Dividing
Fractions
Volume
Classifying 2-D
Figures
Coordinate
Plane
25 days
20 days
10 days
25 days
20 days
20 days
10 days
10 days
10 days
10 days
5.NBT.A.1
5.NBT.A.1
5.NBT.B.7
5.NF.A.1
5.NBT.A.2
5.NF.B.4
5.NF.B.3
5.MD.C.3
5.G.B.3
5.G.A.1
5.NBT.A.2
5.NBT.A.3
5.MD.A.1
5.NF.A.2
5.NBT.B.7
5.NF.B.5
5.NF.B.7
5.MD.C.4
5.G.B.4
5.G.A.2
5.NBT.B.5
5.NBT.A.4
5.OA.A.1
5.MD.A.1
5.OA.A.1
5.NF.B.6
MP.1
5.MD.C.5
MP.3
5.OA.B.3
5.NBT.B.6
5.MD.A.1
MP.2
5.MD.B.2
MP.2
MP.1
MP.2
MP.4
MP.7
MP.4
5.MD.A.1
MP.7
MP.3
5.OA.A.1
MP.3
MP.2
MP.4
MP.6
4.MD.C.5
MP.6
5.OA.A.1
MP.8
MP.6
MP.1
MP.6
MP.4
MP.6
MP.7
4.MD.C.6
4.OA.C.5
5.OA.A.2
4.NBT.A.2
MP.7
MP.3
MP.7
MP.6
MP.8
4.MD.C.7
MP.2
4.NBT.A.3
4.MD.A.2
MP.4
MP.8
4.NF.B.4
4.MD.A.2
4.G.A.1
MP.6
MP.6
MP.7
4.NF.A.1
4.OA.A.1
4.NF.C.5
4.OA.A.2
4.MD.A.2
4.MD.A.2
4.G.A.2
4.G.A.3
4.OA.A.3
4.NBT.B.4
4.NBT.B.5
4.NBT.B.6
4.MD.A.2
Major Clusters
NBT – Number and Operations in Base Ten (1, 2, 3, 4, 5, 6, 7)
NF – Number and Operations – Fractions (1, 2, 3, 4, 5, 6, 7)
MD – Measurement and Data (3, 4, 5)
Supporting Clusters
MD – Measurement and Data (1, 2)
Additional Clusters
OA – Operations and Algebraic
Thinking (1, 2, 3)
G – Geometry (1, 2, 3, 4)
Other
MP – Standards for Mathematical Practice
Potential Gaps in Student Pre-Requisite Knowledge
4.OA – 1, 2, 3, 5
4.NBT – 2, 3, 4, 5, 6 4.NF– 1, 4, 5
4.MD – 5, 6, 7
4.G - 1, 2, 3
Page 1
Mathematics
Grade 5 – Year in Detail (SAMPLE)
This plan is meant to support districts creating their own curriculum or pacing guides. The scope and sequence of curricular resources such as Eureka Math
and others will likely not match this sample plan exactly. The standards do not require only one order to achieve mastery. Thus, many curricular tools will
suggest different scope and sequences for the standards. Districts should follow the guidance they feel is most appropriate for their students.
Summary of Year for Grade 5 Mathematics
The critical areas of this grade should be focused on three areas:
(1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of
fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions);
(2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with
decimals to hundredths, and developing fluency with whole number and decimal operations; and
(3) developing understanding of volume.
Mathematical Practices Recommendations for Grade 5
Throughout Grade 5, students should continue to develop proficiency with the Common Core’s eight Standards for Mathematical Practice:
1.
2.
3.
4.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
5.
6.
7.
8.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
These practices should become the natural way in which students come to understand and do mathematics. While, depending on the content to be understood
or on the problem to be solved, any practice might be brought to bear, some practices may prove more useful than others. Opportunities for highlighting certain
practices are indicated in different units in this document, but this highlighting should not be interpreted to mean that other practices should be neglected in
those units.
Fluency Requirements for Grade 5
5.NBT.B.5
Fluently multiply multi-digit whole numbers using the standard algorithm.
Page 2
Mathematics
Grade 5 – Year in Detail (SAMPLE)
Possible time frame:
25 days
Students finalize fluency with multi-digit addition, subtraction, multiplication and division and apply whole number operations to measurement conversion (e.g. convert 12 feet
to __yards). Students need experiences with numerical expressions that use grouping symbols to develop understanding of how to use parentheses, brackets, and braces with
whole numbers. Students compare like expressions that are grouped differently as well as place grouping symbols in an equation to make it true. Students write simple
expressions and interpret the meaning of the numerical expression. In prior grades, students used various strategies to multiply. In Grade 5, students must also understand and
be able to use the standard algorithm. In applying the standard algorithm, students recognize the importance of place value. (5.NBT.A.1) In fourth grade, students’ experiences
with division were limited to dividing by one-digit divisors. In Grade 5, students extend their prior experiences to include two-digit divisors. They will demonstrate their ability
with whole number division using strategies, illustrations, and explanations.
Major Cluster Standards
Standards Clarification
Understand the place value systems.
5.NBT.A.1 and 5.NBT.A.2 Work
5.NBT.A.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right
should be limited to whole
and 1/10 of what it represents in the place to its left.
numbers as this standard will be
5.NBT.A.2 Explain patterns in the number of zeros of the product when multiplying a whole number by powers of 10, and explain patterns
revisited in Unit 2 and Unit 5.
5.NBT.B.5 Fluency should be
in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote
attained by the end of the year.
powers of 10.
Following this unit, fluency
Perform operations with multi-digit whole numbers and with decimals to hundredths.
practice with the standard
4.NBT.B.4 Click to see wording of a potential gap whose content is a pre-requisite for Grade 5.
4.OA.A.1 4.OA.A.2 4.OA.A.3 Click to see wording of potential gaps whose content is a pre-requisite for Grade 5.
multiplication algorithm should be
4.NBT.B.5 Click to see wording of a potential gap whose content can be integrated with 5.NBT.B.5.
ongoing.
5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm.
4.NBT.B.6 Click to see wording of a potential gap whose content can be integrated with 5.NBT.B.6
5.NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on
place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation
by using equations, rectangular arrays, and/or area models.
Supporting Cluster Standards
Standards Clarification
Convert like measurement units within a given measurement system
5.MD.A.1 Conversions should be
4.MD.A.2 Click to see wording of a potential gap whose content is a pre-requisite for 5.MD.A.1.
limited to whole numbers.
5.MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g. convert 5 cm to 0.05 m),
4.MD.A.2 Focus on word problems
and use these conversions in solving multi-step, real world problems.
with whole number answers being
sure to address distances, liquid
volumes, masses, and conversions.
Additional Cluster Standards
Standards Clarification
Write and interpret numerical expressions.
Work with these standards should
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
be limited to whole numbers.
5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.
For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large
Unit 1: Whole Number Operations
Page 3
Mathematics
Grade 5 – Year in Detail (SAMPLE)
as 18932 + 921, without having to calculate the indicated sum or product.
Focus Standards for Mathematical Practice
MP.2 Reason abstractly and quantitatively.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
As students solve problems involving measurement conversions, they will need to be able to reason about whether
converting among units will result in a larger or smaller number than they began with and attend to precision as they
work to find the answer (MP.2 and MP.6). Students will also make use of structure as they evaluate and write
expressions involving grouping symbols (MP.7).
Page 4
Mathematics
Grade 5 – Year in Detail (SAMPLE)
Unit 2: Place Value with Decimals
Possible time frame:
15 days
Students extend their understanding of the base-ten system to decimals to the thousandths place, building on their grade 4 work with tenths and hundredths. Students use
base-ten blocks and pictures of base-ten blocks to investigate the relationship between adjacent places, how numbers compare, and how numbers round to thousandths. They
use their understanding of unit fractions to compare decimal places and fractional language to describe those comparisons. Students express their understanding that in multidigit whole numbers, a digit in one place represents 10 times what it represents in the place to its right and 1/10 of what it represents in the place to its left.
Major Cluster Standards
Standards Clarification
Understand the place value system.
5.NBT.A.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and
1/10 of what it represents in the place to its left.
5.NBT.A.1 Work should include
decimals and whole numbers.
4.NBT.A.2 Click to see wording of a potential gap whose content is a pre-requisite for 5.NBT.A.3.
5.NBT.A.3 Read, write, and compare decimals to thousandths.
a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10
+ 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).
b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of
comparisons.
4.NBT.A.3 Click to see wording of a potential gap whose content is a pre-requisite for 5.NBT.A.4.
5.NBT.A.4 Use place value understanding to round decimals to any place.
Supporting Cluster Standards
Standards Clarification
Convert like measurement units within a given measurement system.
4.MD.A.2 Click to see wording of a potential gap whose content is a pre-requisite for 5.MD.A.1.
5.MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g. convert 5 cm to 0.05 m), and
use these conversions in solving multi-step, real world problems.
5.MD.A. 1 Students should use
the centimeter and/or meter
stick to convert within the
same system as it relates to
decimals.
4.MD.A.1 include application
word problems involving
distances, volumes, masses,
and money with decimal values
Focus Standards for Mathematical Practice
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in
repeated reasoning.
Students use their understanding of structure of whole numbers to generalize the understanding to decimals (MP.7). As
students continue to work with measurement conversions, they begin to generalize the procedures used (MP.8).
Page 5
Mathematics
Grade 5 – Year in Detail (SAMPLE)
Unit 3: Add and Subtract Decimals
Possible time frame:
10 days
In this unit students will be adding and subtracting decimals to hundredths. Because of the structure of the base-ten system, students use the same place value understanding
for adding and subtracting decimals that they used for adding and subtracting whole numbers. Like base-ten units must be added and subtracted, so students need to attend to
aligning the corresponding places correctly (this also aligns the decimal points). Students should begin to estimate the addition and subtraction of decimals based on their
understanding of operation and the value of the numbers. Students will use grouping symbols to evaluate expressions containing whole numbers and decimals.
Major Cluster Standards
Standards Clarification
Perform operations with multi-digit whole numbers and with decimals to hundredths.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value,
properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the
reasoning used.
Students should apply addition
and subtraction of decimals as
it relates to metric measures
and money.
Convert like measurement units within a given measurement system.
4.MD.A.2 Click to see wording of a potential gap whose content is a pre-requisite for 5.MD.A.1.
5. MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g. convert 5 cm to 0.05 m), and
use these conversions in solving multi-step, real world problems.
4.MD.A.2 include application
word problems involving
distances, volumes, masses,
and money with decimal values
Write and interpret numerical expressions.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
Work with grouping symbols
should include whole numbers
and decimals.
Supporting Cluster Standards
Additional Cluster Standards
Standards Clarification
Standards Clarification
Focus Standards for Mathematical Practice
MP.2 Reason abstractly and
quantitatively.
MP.3 Construct viable arguments and
critique the reasoning of others.
MP.6 Attend to precision.
MP.7 Look for and make use of
structure.
Instead of just computing answers, students reason about both the relationship between whole number computation and decimal
computation and can explain these processes (MP.2 and MP.3). Encourage students to speak about decimal calculations using precise
language (MP.6). Students will also be able to apply the structure of whole number base-ten calculations to decimal calculations
(MP.7).
Page 6
Mathematics
Grade 5 – Year in Detail (SAMPLE)
Unit 4: Add and Subtract Fractions
Possible time frame:
25 days
Students apply their understanding of fractions and fraction models from previous grades to represent the addition and subtraction of fractions with unlike denominators.
Students will reason about size of fractions to make sense of their answers—e.g. they understand that the sum of 1/2 and 1/3 will be greater than 1. Students will also use
grouping symbols to evaluate expressions containing whole numbers, decimals and/or fractions. It is important to note that in some cases it may not be necessary to find a least
common denominator to add or subtract fractions with unlike denominators. Students should be encouraged to use their conceptual understanding of fractions rather than just
1
using the algorithm for adding and subtracting fractions. In addition, there is no mathematical reason for students to write fractions in simplest form.
Major Cluster Standards
Standards Clarification
Use equivalent fractions as a strategy to add and subtract fractions.
4.NF.A.1 Click to see wording of a potential gap whose content is a pre-requisite for 5.NF.A.1.
4.NF.C.5 Click to see wording of a potential gap whose content can be integrated with 5.NF.A.1.
5.NF.A.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 = 23/12. (in general, a/b + c/d = (ab + bc)/bd.)
5.NF.A.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 = 3/7, by
observing that 3/7 < 1/2.
Supporting Cluster Standards
Convert like measurement units within a given measurement system.
4.MD.A.2 Click to see wording of a potential gap whose content is a pre-requisite for 5.MD.A.1.
5. MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g. convert 5 cm to 0.05
m), and use these conversions in solving multi-step, real world problems.
Represent and interpret data.
5.MD.B.2 Make a line plot to display a data set of measurements in fractions of a unit (½ , ¼, 1/8). Use operations on fractions for this
grade to solve problems involving information presented in line plots. For example, given different measurements or liquid in identical
beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.
Additional Cluster Standards
Write and interpret numerical expressions.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
1
For more information about fractions in Grade 5, see pages 10-13 in the Fractions progression document.
Standards Clarification
5.MD.A.1 Conversion problems
should include fractions of a unit.
5.MD.B.2 This standard is not limited
to measurements between 0 and 1
unit. To support 5.NF.A.1,
measurements should include mixed
numbers.
4.MD.A.2 Emphasize word problems
using addition and subtraction of
fractions.
Standards Clarification
Work in this standard includes whole
numbers, decimals, and fractions.
Page 7
Mathematics
Grade 5 – Year in Detail (SAMPLE)
Focus Standards for Mathematical Practice
MP.1 Make sense of problems and persevere in solving them.
MP.3 Construct viable arguments and critique the reasoning of others.
MP.4 Model with mathematics.
MP.6 Attend to precision.
Students make sense of the problems by using visual models and equations to solve problems
involving the addition and subtraction of fractions (MP.1 and MP.4). Students will attend to precision
as they communicate their reasoning in the problem-solving process (MP.3 and MP.6).
Page 8
Mathematics
Grade 5 – Year in Detail (SAMPLE)
Unit 5: Multiply and Divide Decimals
Possible time frame:
20 days
Students will study the effects of multiplying and dividing by powers of 10 and be able to explain why the product is 10 or 100 times the multiplicand or the quotient is 0.1 or
0.01of the dividend. Students will compute products and quotients of decimals to hundredths efficiently and accurately.
Major Cluster Standards
Understand the place value system.
5.NBT.A.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the
placement of the decimal point when a decimal is multiplies or divided by a power of 10. Use whole-number exponents to denote powers of
10.
Perform operations with multi-digit whole numbers and with decimals to hundredths.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value,
properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the
reasoning used.
Additional Cluster Standards
Standards Clarification
5.NBT.B.7 Divisors could be
less than one.
Standards Clarification
Write and interpret numerical expressions.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
Focus Standards for Mathematical Practice
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique the
reasoning of others.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in repeated
reasoning.
As students work with the operations with decimals, they will reason about the relationship between operations with
whole numbers and operations with decimals (MP.2). Students will explain their reasoning to the method used when
performing operations with decimals using precise language (MP.3 and MP.6). Students will use structure as they evaluate
expressions with grouping symbols (MP.7) and generalize the process for multiplying by powers of 10 (MP.8).
Page 9
Mathematics
Grade 5 – Year in Detail (SAMPLE)
Unit 6: Multiplying Fractions
Possible time frame:
20 days
In fourth grade, students began studying the multiplication of fractions by focusing on multiplying a whole number by a fraction. In Grade 5, students use the meaning of
fractions and of multiplication to understand and explain why the procedure for multiplying fractions makes sense. Students can reason why it works using visual models,
fraction strips, and number line diagrams. For more complicated problems an area model is useful, in which students work with a rectangle that has fractional side lengths,
dividing it into rectangles whose side lengths are the corresponding unit fractions. They will apply the concept of multiplying fractions to find the area of a rectangle with
fractional side lengths. Students learn to see products such as ½ x 3 as an expression that can be interpreted in terms of a quantity, 3, and a scaling factor, ½. They see ½ x 3 as
half the size of 3 without evaluating the product.
Major Cluster Standards
Standards Clarification
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
4.NF.B.4 Click to see wording of a potential gap whose content is a pre-requisite for 5.NF.B.4.
5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷
b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5)
= 8/15. (In general, (a/b) × (c/d) = ac/bd.)
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and how
that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and
represent fraction products as rectangular areas.
5.NF.B.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 a/b = (n×a)/(n×b) to the effect of
multiplying a/b by 1.
4.MD.A.2 Click to see wording of a potential gap whose content is a pre-requisite for 5.NF.B.6.
5.NF.B.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.
Focus Standards for Mathematical Practice
MP.1 Make sense of problems and persevere in solving them.
MP.2 Reason abstractly and quantitatively.
MP.4 Model with mathematics.
Representing multiplication of fractions with visual and concrete models is fundamental to this unit in order
for student to make sense of multiplying fractions by fractions (MP.1 and MP.4). Students reason abstractly
and practice communicating their thinking in real-world situations (MP.2 and MP.6).
MP.6 Attend to precision.
Page 10
Mathematics
Grade 5 – Year in Detail (SAMPLE)
Unit 7: Dividing Fractions
Possible time frame:
10 days
Students use the relationship between division and multiplication to start working with simple fraction division problems. Having seen that division of a whole number by a
whole number, e.g. 5 ÷ 3, is the same as multiplying the number by a unit fraction, 1/3 x 5, they now extend the same reasoning to division of a unit fraction by a whole number.
Seeing for example 1/6 ÷ 3 they reason that 1/6 must be divided into 3 equal parts. Since there are 6 portions of 1/6 in 1 whole and each part must be divided into 3 equal
parts, there would be a total of 18 parts in the whole; therefore 1/6 ÷ 3 = 1/18. In 3 ÷ 1/6, they reason that since there are 6 portions of 1/6 in 1 whole, there must be 3 x 6 in 3
wholes; therefore 3 ÷ 1/6 = 18. Linear and area models will be useful for student to gain a conceptual understanding of division of fractions before mastering the algorithm.
Major Cluster Standards
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
5.NF.B.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, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared
equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-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?
Standards Clarification
5.NF.B.7 Division of a
fraction by a fraction is not
a requirement at this grade.
5. NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3)
÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4
= 1/12 because (1/12) × 4 = 1/3.
b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and
use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20
because 20 × (1/5) = 4.
c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions,
e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3
people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
Focus Standards for Mathematical Practice
MP.1 Make sense of problems and persevere in
solving them.
MP.2 Reason abstractly and quantitatively.
MP.4 Model with mathematics.
In this unit it is critical for students to use concrete objects or pictures to help conceptualize, create, and solve problems
(MP.1 and MP.2). Students will model the relationships in real-world problems and use precise language when explaining
their reasoning (MP.4 and MP.6).
MP.6 Attend to precision.
Page 11
Mathematics
Grade 5 – Year in Detail (SAMPLE)
Unit 8: Volume
Possible time frame:
10 days
Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding the total number of same-size cube units
required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. They select appropriate
units, strategies, and tools for solving problems that involve estimating and measuring volume. They decompose three-dimensional shapes and find volumes of right
rectangular prisms by viewing them as decomposed into layers of arrays of cubes. They measure necessary attributes of shapes in order to determine volumes to solve real
world and mathematical problems.
Major Cluster Standards
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
5.MD.C.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure
volume.
b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
5.MD.C.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
4.MD.A.2 Click to see wording of a potential gap whose content is a pre-requisite for 5.MD.C.5b.
5.MD.C.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving
volume.
a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that
the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the
area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of
multiplication.
b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with wholenumber edge lengths in the context of solving real world and mathematical problems.
c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by
adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
Standards Clarification
Students will find the volume of right
rectangular prisms with edges whose
lengths are whole numbers.
Students learn to determine the volumes
of several right rectangular prisms, using
cubic centimeters, cubic inches, and cubic
feet. With guidance, they learn to
increasingly apply multiplicative reasoning
to determine volumes, looking for and
making use of structure. They understand
that multiplying the length times the
width of a right rectangular prism can be
viewed as determining how many cubes
would be in each layer if the prism were
packed with or built up from unit cubes.
Focus Standards for Mathematical Practice
MP.4 Model with mathematics.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in repeated
reasoning.
Students decompose and recompose geometric figures to make sense of the spatial structure of volume (MP.7). Students
will use cubes to model volume of solid figures and connect the structure to multiplicative reasoning (MP.4 and MP.7).
Students will attend to precision as they state the volume of solid figures using the appropriate units (MP.6). They solve
problems by applying generalized formulas (MP.8).
Page 12
Mathematics
Grade 5 – Year in Detail (SAMPLE)
Unit 9: Classifying 2-D Figures
Possible time frame:
10 days
Students learn to analyze and relate categories of two-dimensional figures explicitly based on their properties. They classify two-dimensional figures in hierarchies. For
example, they conclude that all rectangles are parallelograms, because they are all quadrilaterals with two pairs of opposite, parallel, equal-length sides. In this way, they relate
certain categories of shapes as subclasses of other categories. This leads to students understanding that squares possess all properties of rhombuses and of rectangles.
Additional Cluster Standards
Standards Clarification
4.MD.C.5 4.MD.C.6 4.MD.C.7 4.G.A.3 Click to see wording of potential gaps whose content is a pre-requisite for this unit.
Classify two-dimensional figures into categories based on their properties.
4.G.A.1 4.GA.A.2 Click to see wording of potential gaps whose content is a pre-requisite for 5.G.B.3 and 5.G.B.4.
5.G.B.3 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For
example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
5.G.B.4 Classify two-dimensional figures in a hierarchy based on properties.
Focus Standards for Mathematical Practice
MP.3 Construct viable arguments and
critique the reasoning of others.
MP.7 Look for and make use of
structure.
Students make use of structure to build a logical progression of statements and explore hierarchical relationships among 2dimensional shapes (MP.3 and MP.7).
Page 13
Mathematics
Grade 5 – Year in Detail (SAMPLE)
Unit 10: Coordinate Plane
Possible time frame:
10 days
Students connect ordered pairs of whole-number coordinates to points on the grid, so that these coordinate pairs constitute numerical objects and ultimately can be operated
upon as single mathematical entities. Students solve mathematical and real-world problems using coordinates. Students extend their Grade 4 pattern work by working briefly
with two numerical patterns that can be related and examining these relationships within sequences of ordered pairs and in the graphs in the first quadrant of the coordinate
plane.
Additional Cluster Standards
Graph points on the coordinate plane to solve real-world and mathematical problems.
5.G.A.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin)
arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates.
Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how
far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., xaxis and x-coordinate, y-axis and y-coordinate).
Standards Clarification
Students will graph points in
the first quadrant only.
5.G.A.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret
coordinate values of points in the context of the situation.
Analyze patterns and relationships.
4.OA.C.5 Click to see wording of a potential gap whose content is a pre-requisite for 5.OA.B.3.
5.OA.B.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered
pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the
rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences,
and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
Focus Standards for Mathematical Practice
MP.4 Model with mathematics.
MP.6 Attend to precision.
Students precisely describe the coordinates of points and the relationship of the coordinate plane to the number line (MP.6). Students
both generate and identify relationships in numerical patterns, using the coordinate plan as a way of representing these relationships
and patterns (MP.4).
Page 14
Grade 5 Potential Gaps in Student Pre-Requisite Knowledge
This document indicates pre-requisite knowledge gaps that may exist for Grade 5 students based on what the Grade 4 common core math standards expect.
Column four indicates the Grade 5 common core standard(s) or units which could be affected if the Grade4 gap exists. Other gaps may exist for other reasons;
therefore, it important that teachers diagnose their students’ needs as part of the planning process.
Grade 5
Domain
Grade 4 CCSS
Wording of Grade 4 CCSS Potential Gap
CCSS
Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that
4.OA.A.1
35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of
multiplicative comparisons as multiplication equations.
Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using
4.OA.A.2
drawings and equations with a symbol for the unknown number to represent the problem,
All work in
Operations and
distinguishing multiplicative comparison from additive comparison.
Grade 5
Algebraic Thinking
Solve multistep word problems posed with whole numbers and having whole-number
(OA)
answers using the four operations, including problems in which remainders must be
interpreted. Represent these problems using equations with a letter standing for the
4.OA.A.3
Go to Unit 1
unknown quantity. Assess the reasonableness of answers using mental computation and
Go to Unit 10
estimation strategies including rounding.
Generate a number or shape pattern that follows a given rule. Identify apparent features of the
pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the
starting number 1, generate terms in the resulting sequence and observe that the terms appear
4.OA.C.5
5.OA.B.3
to alternate between odd and even numbers. Explain informally why the numbers will continue to
alternate in this way.
Read and write multi-digit whole numbers using base-ten numerals, number names, and
4.NBT.A.2
expanded form. Compare two multi-digit numbers based on meanings of the digits in each
5.NBT.A.3
place, using >, =, and < symbols to record the results of comparisons.
Use place value understanding to round multi-digit whole numbers to any place.
4.NBT.A.3
5.NBT.A.4
Number and
Fluently
add
and
subtract
multi-digit
whole
numbers
using
the
standard
algorithm.
Whole
Operations in Base
4.NBT.B.4
Number Unit
Ten (NBT)
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two
two-digit numbers, using strategies based on place value and the properties of operations.
Go to Unit 1
4.NBT.B.5
5.NBT.B.5
Illustrate and explain the calculation by using equations, rectangular arrays, and/or area
Go to Unit 2
models.
Find whole-number quotients and remainders with up to four-digit dividends and one-digit
divisors, using strategies based on place value, the properties of operations, and/or the
4.NBT.B.6
5.NBT.B.6
relationship between multiplication and division. Illustrate and explain the calculation by using
equations, rectangular arrays, and/or area models.
Page 15
Grade 5 Potential Gaps in Student Pre-Requisite Knowledge
4.NF.A.1
Number and
Operations—
Fractions (NF)
Go to Unit 4
4.NF.B.4
Go to Unit 6
4.NF.C.5
Measurement and
Data (MD)
Go Unit 1
4.MD.A.2
Go to Unit 2
Go to Unit 3
Go to Unit 4
4.MD.C.5
Go to Unit 6
Go to Unit 8
Go to Unit 9
4.MD.C.6
Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction
models, with attention to how the number and size of the parts differ even though the two
fractions themselves are the same size. Use this principle to recognize and generate equivalent
fractions.
Apply and extend previous understandings of multiplication to multiply a fraction by a whole
number.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to
represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
b. b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a
fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 ×
(1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
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?
Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and
use this technique to add two fractions with respective denominators 10 and 100. For example,
express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
Use the four operations to solve word problems involving distances, intervals of time, liquid
volumes, masses of objects, and money, including problems involving simple fractions or
decimals, and problems that require expressing measurements given in a larger unit in terms
of a smaller unit. Represent measurement quantities using diagrams such as number line
diagrams that feature a measurement scale.
Recognize angles as geometric shapes that are formed wherever two rays share a common
endpoint, and understand concepts of angle measurement:
a. An angle is measured with reference to a circle with its center at the common endpoint of the
rays, by considering the fraction of the circular arc between the points where the two rays
intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,”
and can be used to measure angles.
b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
Measure angles in whole-number degrees using a protractor. Sketch angles of specified
measure.
5.NF.A.1
5.NF.B.4a
5.NF.A.1
5.NF.A.2
5.NF.B.6
5.MD.A.1
5.MD.C.5b
Geometry
Unit
Page 16
Grade 5 Potential Gaps in Student Pre-Requisite Knowledge
Measurement and
Data (MD)
4.MD.C.7
Go to Unit 9
4.G.A.1
Geometry (G)
Go to Unit 9
4.G.A.2
4.G.A.3
Recognize angle measure as additive. When an angle is decomposed into non-overlapping
parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve
addition and subtraction problems to find unknown angles on a diagram in real world and
mathematical problems, e.g., by using an equation with a symbol for the unknown angle
measure.
Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and
parallel lines. Identify these in two-dimensional figures.
Classify two-dimensional figures based on the presence or absence of parallel or perpendicular
lines, or the presence or absence of angles of a specified size. Recognize right triangles as a
category, and identify right triangles.
Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that
the figure can be folded along the line into matching parts. Identify line-symmetric figures and
draw lines of symmetry.
Geometry
Unit
5.G.B.3
5.G.B.3
5.G.B.4
Geometry
Unit
Page 17