Grade 5

Operations and Algebraic Thinking

Number and Operations in Base Ten

Number and Operations – Fractions

Measurement and Data

Geometry

Write and interpret numerical expressions. 

Analyze patterns and relationships.

Understand the place value system. 

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

Use equivalent fractions to add and subtract fractions.

Use previous understandings of multiplication and division to multiply and divide fractions.

Convert like measurement units within a given measurement system.

Represent and interpret data.

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

Graph points on the coordinate plane to solve mathematical problems as well as problems in realworld context.

Classify two-dimensional figures into categories based on their properties.

Use parentheses and brackets in numerical expressions, and evaluate expressions with these symbols (Order of Operations).

Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them (e.g., express the calculation "add 8 and 7, then multiply by 2" as 2 x (8 + 7). Recognize that 3 x (18,932 + 921) is three times as large as 18,932 + 921, without having to calculate the indicated sum or product).

Generate two numerical patterns using two given rules (e.g., generate terms in the resulting sequences). Identify and explain the 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 (e.g., 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).

Understand primes have only two factors and decompose numbers into prime factors.

Apply concepts of place value, multiplication, and division to understand 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.

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 multiplied or divided by a power of 10.

Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form. b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

Use place value understanding to round decimals to any place.

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

Apply and extend understanding of division to find whole-number quotients of whole numbers with up to fourdigit dividends and two-digit divisors.

Add, subtract, multiply, and divide decimals to hundredths, connecting objects or drawings to strategies based on place value, properties of operations, and/or the relationship between operations. Relate the strategy to a written form. 

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 (e.g., 2/3 + 5/4 = 8/12 + 15/12 = 23/12).

Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators by using a variety of representations, equations, and visual models to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers (e.g. recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2).

Interpret a fraction as the number that results from dividing the whole number numerator by the whole number denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers. 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?

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number and a fraction by a fraction. a. Interpret the product (a/b) x q as a parts of a partition of q into b equal parts. For example, use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation. b. Interpret the product of a fraction multiplied by a fraction (a/b) x (c/d). Use a visual fraction model and create a story context for this equation. For example, use a visual fraction model to show (2/3) x (4/5) = 8/15, and create a story context for this equation. In general, (a/b) x (c/d) = ac/bd. c. 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.

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; 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 𝑎𝑎 𝑏𝑏 = 𝑛𝑛 𝑥𝑥 𝑎𝑎 𝑛𝑛 𝑥𝑥 𝑏𝑏 to the effect of multiplying 𝑎𝑎 𝑏𝑏 by 1. 

Solve problems in real-world contexts involving multiplication of fractions, including mixed numbers, by using a variety of representations including equations and models.

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. Use the relationship between multiplication and division to justify conclusions. 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 justify conclusions (e.g., 4 ÷ (1/5) = 20 because 20 x (1/5) = 4). c. Solve problems in real-world context involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, using a variety of representations. 

Convert among different-sized standard measurement units within a given measurement system, and use these conversions in solving multi-step, real-world problems.

Make a line plot to display a data set of measurements in fractions of a unit (1/8, 1/2, 3/4). Use operations on fractions for this grade to 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.

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.

Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

Relate volume to the operations of multiplication and addition and solve mathematical problems and problems in real-world contexts 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. Understand and use the formulas V = l x w x h and V = B x h, where in this case B is the area of the base (B = l x w), for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths to solve mathematical problems and problems in real-world contexts. c. Understand volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms, applying this technique to solve mathematical problems and problems in real-world contexts.

Understand and describe a coordinate system as perpendicular number lines, called axes, that intersect at the origin (0 , 0). Identify a given point in the first quadrant of the coordinate plane using an ordered pair of numbers, called coordinates. Understand that the first number (x) indicates the distance traveled on the horizontal axis, and the second number (y) indicates the distance traveled on the vertical axis.

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. 

Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.

Classify two-dimensional figures in a hierarchy based on properties.

Given a real world problem, write an equation using 1 set of parentheses. 

No CCCs developed for this standard.

Given 2 patterns involving the same context (e.g., collecting marbles) determine the first 5 terms and compare the values.

When given a line graph representing two arithmetic patterns, identify the relationship between the two

Generate or select a comparison between two graphs from a similar situation.

Use up to two rules to verify provided responses or select correct answers (e.g., rules: +3, +2 and table lists pairs, 4:5, 7:7, and 10:9).

No CCCs developed for this standard.

No CCCs developed for this standard.

Identify what an exponent represents (e.g., 8³= 8 x 8 x 8).

Read, write, or select a decimal to the hundredths place.

Read, write or select a decimal to the thousandths place.

Compare two decimals to the thousandths place with a value of less than 1. 

Round decimals to the next whole number.

Round decimals to the tenths place.

Round decimals to the hundredths place.

No CCCs developed for this standard. 

Find whole number quotients up to two dividends and two divisors.

Find whole number quotients up to four dividends and two divisors. 

Solve word problems that require multiplication or division. 

Solve 1 step problems using decimals. 

Add and subtract fractions with unlike denominators by replacing fractions with equivalent fractions (identical denominators). 

Add or subtract fractions with unlike denominators.

Solve word problems involving the addition, subtraction, multiplication or division of fractions.

No CCCs developed for this standard.

Multiply or divide fractions.

Determine whether the product will increase or decrease based on the multiplier.

Determine whether or not the quotient will increase or decrease based on the divisor.

No CCCs developed for this standard.

Solve word problems involving the addition, subtraction, multiplication or division of fractions.

Convert measurements of time.

Convert standard measurements of length. 

Convert standard measurements of mass.

Solve problems involving conversions of standard measurement units when finding area, volume, time lapse, or mass.

Collect and graph data: bar graph, line plots, picture graph (e.g., average height among 3 classrooms, # of boys and girls).

No CCCs developed for this standard.

Use filling and multiplication to determine volume.

No CCCs developed for this standard.

No CCCs developed for this standard.

Apply formula to solve one step problems involving volume.

Locate the x and y axis on a graph. 

Locate points on a graph.

Use order pairs to graph given points.

Find coordinate values of points in the context of a situation.

Recognize properties of simple plane figures.

Distinguish plane figures by their properties.