Grade 5 Math Colorado standards Standards

Geometry

Measurement & Data

Operations & Algebraic Thinking

Number & Operations—Fractions

Number & Operations in Base Ten

Standards for Mathematical Practice

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

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

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).

Use a pair of perpendicular 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 $-$-coordinate, $y$-axis and $y$-coordinate).

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.

Interpret real-world and mathematical problems using given single-digit, first-quadrant coordinate values of points in the context of a situation.

For example, a map aligned with a coordinate plane might indicate a point one block east and two blocks north of the origin can be labeled (1, 2).

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

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

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

Demonstrate an understanding that the attributes of some shapes allow the shape to belong to two categories of shapes.

For example, a square is both a square and a rectangle, while a triangle can be both right and isosceles.

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

Identify and sort two-dimensional figures using the presence or absence of angles of a specified size.

For example, sort right triangles from a group of all triangles.

Convert like measurement units within a given measurement system.

Measurement & Data: Convert like measurement units within a given measurement system.

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.

Convert among different-sized standard measurement units within a given measurement system where the quantities yield whole units (e.g., convert 24 inches to 2 feet).

Represent and interpret data.

Measurement & Data: Represent and interpret data.

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.

Make a line plot to display a data set of measurements in whole and half units.

Use operations to solve problems involving information presented in line plots.

Understand concepts of volume and relate volume to multiplication and to addition.

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

Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

Recognize volume as an attribute of solid figures and understand concepts of volume measurement by filling rectangular prisms with unit cubes.

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.

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.

Measure volumes by counting unit cubes.

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

Solve real-world problems involving volume.

Model 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.

For example, find out how many smaller boxes will fit inside a larger box.

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.

Use the additive nature of volume to 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.

Understand the place value system.

Number & Operations in Base Ten: Understand the place value system.

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.

Recognize that in a multi-digit number with tenths or hundredths, a digit in one place represents 10 times what it represents in the place to its right.

For example, in an amount of money written as $19.99, the nine in the tenths place (90 cents) is ten times as much as the 9 in the hundredths place (9 cents).

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. Use whole-number exponents to denote powers of 10.

Use patterns in the number of zeroes of the product when multiplying oneor two-digit numbers by 10 or one-digit numbers by 100.

Understand that multiplying by 10 twice is the same as multiplying by 100 once because 10 x 10 = 100.

Read, write, and compare decimals to thousandths.

Compare two three-digit whole numbers based on the meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

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+3×110+9×1100+2×11000.

Read and write decimals to hundredths using base-ten numerals, number names, and expanded form, e.g., 4.57 = 4 + 5 x ( 1 ) + 7 X ( 1 ).

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

Compare two decimals to hundredths 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.

Use place value understanding to round whole numbers 1-1000 to any place.

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

Number & Operations in Base Ten: Perform operations with multi-digit whole numbers and with decimals to hundredths.

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

Multiply one-digit whole numbers using models and illustrations using equations, rectangular arrays, and/or area models.

Products should include values up to at least 50.

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.

Find whole-number quotients with dividends up to at least 50 and one-digit divisors, using strategies based on the concept of division using fair and equal shares.

Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

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.

Add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

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

Number & Operations—Fractions: Use equivalent fractions as a strategy to add and subtract fractions.

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.

Add and subtract fractions with single-digit numerators and like denominators up to at least 12.

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.

Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of like denominators up to at least 12, 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: a pizza is cut evenly into 8 slices. If you eat 2 of the pizza and I eat 3 of the 8 8 pizza, what fraction did we eat all together?

Apply and extend previous understandings of multiplication and division.

Number & Operations—Fractions: Apply and extend previous understandings of multiplication and division.

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.

Interpret a fraction as division of the numerator by the denominator (a/b = a divided by b).

Solve word problems involving division of whole numbers up to at least 12 leading to answers in the form of fractions, e.g., by using visual fraction models or equations to represent the problem. For example, interpret the problem of dividing two sandwiches equally amongst 3 people as the fraction 2 , meaning that each person should get 2 of a 3 3 sandwich.

Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Apply and extend previous understandings of multiplication to multiply the following fractions: 1 x 1 , 1 x 1 , 1 x 1 , and 2 x 1 .

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.

Compose and decompose visual fraction models to illustrate these relationships.

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.

Solve word problems involving multiplication of these fractions by using visual fraction models to represent the problem. For example, what is 1 2 of 1 of a pizza?

Interpret multiplication as scaling (resizing), by:

Interpret multiplication as scaling (resizing) by understanding how 8 x 1 2 makes sense when described as “scale 8 so it is half as big,” but does not make sense when described as repeated addition, such as “add 8 to itself half a time.”

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.

Recognize that multiplying by a fraction <1 makes a factor smaller.

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.

Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

Solve real-world problems involving multiplication of fractions, e.g., by using visual fraction models or equations to represent the problem.

Fractions should include single-digit numerators and denominators up to at least 12.

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.)

Apply and extend previous understandings of division to divide unit fractions (1 , 1 , and 1 ) by whole numbers (2, 3, 4) and whole numbers by unit 2 3 4 fractions, e.g., by using visual fraction models or equations to represent the problem.

Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.

For example, students should understand that dividing 1 of a pie into 3 2 equal pieces yields pieces that are 1 of the whole pie. Similarly, students 6 should understand that dividing 2 pies into pieces the size of 1 of each 4 pie yields a total of 8 pieces.

Interpret division of a whole number by a unit fraction, and compute such quotients.

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.

Write and interpret numerical expressions.

Operations & Algebraic Thinking: Write and interpret numerical expressions.

Use grouping symbols (parentheses, brackets, or braces) in numerical expressions, and evaluate expressions with these symbols.

Use one set of parentheses in numerical expressions, and evaluate expressions with these symbols.

For example, in the expression (2+4) , 2 should be added to 4 before 3 dividing by 3.

Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

Identify 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).

Analyze patterns and relationships.

Operations & Algebraic Thinking: Analyze patterns and relationships.

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.

Generate a number pattern that follows a given rule.

Identify a relationship between the sequence number and the number in the pattern. For example, given the rule “add 3,” write the number pattern 3, 6, 9, 12,…and describe 3 as the first number, 9 as the third number, etc.

Algebra and Functions

Data, Statistics, and Probability

Geometry

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.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Number and Quantity