---
title: "Math 5"
url: "https://books.hrgrvs.net/2/standards/53/math-5"
---

# Standards for Mathematical Content Grade 5 

## Operations and Algebraic Thinking 

### Write and interpret numerical expressions. 

-   5.OA.1. Use parentheses to construct numerical expressions, and
    evaluate numerical expressions with these symbols.

-   5.OA.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 x (8 + 7). Recognizing that 3 x (18932 + 921) is
    three times as large as 18932 + 921, without having to calculate the
    indicated sum or product.*

### Analyze patterns and relationships. 

-   5.OA.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.*

## Numbers and Operations in Base Ten 

### Understand the place value system. 

-   5.NBT.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.2. Explain and extend the patterns in the number of zeros of
    the product when multiplying a number by powers of 10, and explain
    and extend the patterns 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 powers of 10.

-   5.NBT.3. Read, write, and compare decimals to thousandths.

    - a)  Read and write decimals to thousandths using base-ten numerals,
    number names, and expanded form \[e.g., 347.392 = 3 x 100 + 4 x 10 +
    7 x 1 + 3 (1/10) + 9 (1/100) + 2 (1/1000)\].

    - b)  b\. Compare two decimals to thousandths place based on meanings of
    the digits in each place, using \>, =, and \< symbols to record the
    results of comparisons.

-   5.NBT.4. Use place values understanding to round decimals to any
    place.

### Perform operations with multi-digit whole numbers and with decimals to hundredths. 

-   5.NBT.5. Fluently multiply multi-digit whole numbers using a
    standard algorithm.

-   5.NBT.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,
    number lines, real life situations, and/or area models.

-   5NBT.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 the
    operations. Related the strategy to a written method and explain
    their reasoning in getting their answers.

## Number and Operations - Fractions 

### Use equivalent fractions as a strategy to add and subtract fractions. 

-   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 = 23/12. (In general, a/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 check the reasonableness of
    answers. *For example, recognize an incorrect result 2/5 + 1/2 =
    3/7, by observing that 3/7 \< 1/2.*

### Apply previous understandings of multiplication and division to multiply and divide fractions. 

-   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, 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?*

-   5.NF.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 show 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.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. (Division of a fraction by a
    fraction is not a requirement at this grade.)

-   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).

-   5.NF.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?*

## Measurement and Data 

### Convert like measurement units within a given measurement system and solve problems involving time. 

-   5\. MD.1. Identify, estimate measure, and convert equivalent
    measures within systems English length (inches, feet, yards, miles)
    weight (ounces, pounds, tons) volume (fluid ounces, cups, pints,
    quarts, gallons) temperature (Fahrenheit) Metric length
    (millimeters, centimeters, meters, kilometers) volume (milliliters,
    liters), temperature (Celsius), (e.g., convert 5 cm to 0.05 m), and
    use these conversions in solving multi-step, real world problems
    using appropriate tools.

-   5\. MD.2. Solve problems involving elapsed time between world time
    zones. (L)

### Represent and interpret data. 

-   5.MD.3. Make a line plot to display a data set of measurements in
    fractions of a unit (1/2, 1/4, 1/8). Solve problems involving
    information presented in line plots. *For example, given different
    measurements of liquid in identical beakers, find the amount of
    liquid each beaker would contain if the total amount in all the
    beakers were redistributed equally.*

-   5.MD.4. Explain the classification of data from real-world problems
    shown in graphical representations including the use of terms mean
    and median with a given set of data. (L)

### Geometric measurement: understand concepts of volume and relate to multiplication and addition. 

-   5.MD.5. 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.6. Estimate and measure volumes by counting unit cubes, using
    cubic cm, cubic in, cubic ft, and non-standard units.

-   5.MD7. Relate volume to the operations of multiplication and
    addition and solve real world and mathematical problems involving
    volume.

    - a)  Estimate and 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. Demonstrate the associative property of multiplication by
    using the product of three whole-numbers to find volumes (length x
    width x height).

    - b)  Apply the formulas *V* = *l* × *w* × *h* and *V* = *b* × *h* for
    rectangular prisms to find volumes of right rectangular prisms with
    whole number 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.

## Geometry

### Graph points on the coordinate plane to solve real-world and mathematical problems. 

-   5.G.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., *x*-axis and *x*-coordinate,
    *y*-axis and *y*-coordinate).

-   5.G.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.

### Classify two-dimensional (plane) figures into categories based on their properties. 

-   5.G.3. Understand that attributes belonging to a category of two
    dimensional (plane) 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.4. Classify two-dimensional (plane) figures in a hierarchy based
    on attributes and properties.

# Standards for Mathematical Practice

Instruction around the Standards of Mathematical Practices is delivered
across all grades K-12. These eight standards define experiences that
build understanding of mathematics and ways of thinking through which
students develop, apply, and assess their knowledge.

## Make sense of problems and persevere in solving them.

-   explain correspondences between equations, verbal descriptions,
    tables, and graphs

-   draw diagrams of important features and relationships, graph data,
    and search for regularity or trends

-   use concrete objects or pictures to help conceptualize and solve a
    problem

-   understand the approaches of others to solving complex problems

-   identify correspondences between different approaches

-   check if the solution makes sense

## Reason abstractly and quantitatively.

-   represent a situation symbolically

-   create a coherent representation of the problem

-   have the ability to show how problem has a realistic meaning

-   reflect during the manipulation process in order to probe into the
    meanings for the symbols involved

-   use units consistently

## Construct viable arguments and critique the reasoning of others.

-   construct arguments using concrete referents such as objects,
    drawings, diagrams, and actions

-   justify conclusions, communicate conclusions, listen and respond to
    arguments, decide whether the argument makes sense, and ask
    questions to clarify the argument

-   reason inductively about data, making plausible arguments that take
    into account the context from which the data arose

## Model with Mathematics.

-   apply mathematics to solve problems arising in everyday life

-   identify important quantities in a practical situation and model the
    situation using such tools as manipulatives, diagrams, two-way
    tables, graphs or pictures

-   interpret mathematical results in the context of the situation and
    reflect on whether the results make sense

-   apply mathematical knowledge, make assumptions and approximations to
    simplify a complicated situation

## Use appropriate tools strategically.

-   select the available tools (such as pencil and paper, manipulatives,
    rulers, calculators, a spreadsheet, and available technology) when
    solving a mathematical problem

-   be familiar with tools appropriate for their grade level to make
    sound decisions about when each of these tools might be helpful

-   identify relevant external mathematical resources and use them to
    pose or solve problems

-   use technological tools to explore and deepen their understanding of
    concepts

-   detect possible errors by strategically using estimation and other
    mathematical knowledge

-   know that technology can enable them to visualize the results of
    varying assumptions, explore consequences, and compare predictions
    with data

## Attend to precision.

-   give carefully formulated explanations to each other

-   use clear definitions and reasoning in discussion with others

-   state the meaning of symbols, including using the equal sign
    consistently and appropriately

-   specify units of measure, and label axes to clarify the
    correspondence with quantities in a problem

-   calculate accurately and efficiently

-   express numerical answers with a degree of precision appropriate for
    the problem context

## Look for and make use of structure.

-   discern a pattern or structure

-   understand complex structures as single objects or as being composed
    of several objects

-   check if the answer is reasonable

## Look for and express regularity in repeated reasoning.

-   identify if calculations or processes are repeated

-   use alternative and traditional methods to solve problems

-   evaluate the reasonableness of their intermediate results, while
    attending to the details