Core Content Connectors by Common Core State Standards: Mathematics 5th Grade

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Grade 5 Overview

Operations and Algebraic Thinking

• Write and interpret numerical expressions.
• Analyze patterns and relationships.

Number and Operations in Base Ten

• Understand the place value system.
• Perform operations with multi-digit whole numbers and with decimals to hundredths.

Numbers and Operations—Fractions

• Use equivalent fractions as a strategy to add and subtract fractions.
• Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Measurement and Data

• 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.

Geometry

• Graph points on the coordinate plane to solve real-world and mathematical problems.
• Classify two-dimensional figures into categories based on their properties.

Operations and Algebraic Thinking	5.OA
Write and interpret numerical expressions.
1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
CCCs linked to 5.OA.A.1	5.SE.1a1 Given a real world problem, write an equation using 1 set of parentheses.
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 as 18932 + 921, without having to calculate the indicated sum or product.
CCCs linked to 5.OA.A.2	5.SE.1a1 Given a real world problem, write an expression using 1 set of parentheses.
Analyze patterns and relationships.
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.
CCCs linked to 5.0A.B.3	5.PRF.1b1 Given 2 patterns involving the same context (e.g., collecting marbles) determine the 1st 5 terms and compare the values.
	5.PRF.1b2 When given a line graph representing two arithmetic patterns, identify the relationship between the two
	5.PRF.1b2 Generate or select a comparison between two graphs from a similar situation.
	6.PRF.2b2 Using provided table with numerical patterns, form ordered pairs.

Numbers and Operations in Base Ten	5.NBT
Understand the place value system.
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.
CCCs linked to 5.NBT.A.1	5.NO.1a1 Compare the value of a number when it is represented in different place values of two 3 digit numbers.
	5.SE.1a1 Given a real world problem, write an expression using 1 set of parentheses.
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 multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
CCCs linked to 5.NBT.A.2	6.NO.1i1 Identify what an exponent represents (e.g., 8³ = 8 x 8 x 8).
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.
CCCs linked to 5.NBT.A.3	5.NO.1b1 Read, write, or select a decimal to the hundredths place.
	5.NO.1b2 Read, write or select a decimal to the thousandths place.
	5.NO.1b3 Compare two decimals to the thousandths place with a value of less than 1.
4. Use place value understanding to round decimals to any place. Perform operations with multi-digit whole numbers and with decimals to hundredths.
CCCs linked to 5.NBT.A.4	5.NO.1b4 Round decimals to the next whole number.
	5.NO.1b5 Round decimals to the tenths place.
	5.NO.1b6 Round decimals to the hundredths place.
Perform operations with multi-digit whole numbers and with decimals to hundredths.
5. Fluently multiply multi-digit whole numbers using the standard algorithm.
CCCs linked to 5.NBT.B.5	5.NO.2a1 Solve problems or word problems using up to three digit numbers and addition or subtraction or multiplication.
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.
CCCs linked to 5.NBT.B.6	5.NO.2a3 Find whole number quotients up to two dividends and two divisors.
	5.NO.2a4 Find whole number quotients up to four dividends and two divisors.
	5.NO.2a5 Solve word problems that require multiplication or division.
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.
CCCs linked to 5.NBT.B.7	5.NO.2c1 Solve 1 step problems using decimals.

Numbers and Operations--Fractions	5.NF
Use equivalent fractions as a strategy to add and subtract fractions.
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.)
CCCs linked to 5.NF.A.1	5.NO.2b1 Add and subtract fractions with unlike denominators by replacing fractions with equivalent fractions (identical denominators).
	5.NO.2b2 Add or subtract fractions with unlike denominators.
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. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
CCCs linked to 5.NF.A.2	5.NO.2c2 Solve word problems involving the addition, subtraction, multiplication or division of fractions.
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
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?
CCCs linked to 5.NF.B.3	5.NO.2b4 Divide unit fractions by whole numbers and whole numbers by unit fractions.
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.
CCCs linked to 5.NF.B.4	5.NO.2b3 Multiply a fraction by a whole or mixed number.
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.
CCCs linked to 5.NF.B.5	5.PRF.1a1 Determine whether the product will increase or decrease based on the multiplier.
	6.PRF.1a2 Determine whether or not the quotient will increase or decrease based on the divisor.
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.
CCCs linked to 5.NF.B.6	5.NO.2b3 Multiply a fraction by a whole or mixed number.
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?
CCCs linked to 5.NF.B.7	5.NO.2b4 Divide unit fractions by whole numbers and whole numbers by unit fractions.
	6.NO.2c4 Solve word problems involving the addition, subtraction, multiplication or division of fractions.

Measurement and Data	5.MD
Convert like measurement units within a given measurement system.
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.
CCCs linked to 5.MD.A.1	5.ME.1b1 Convert measurements of time.
	5.ME.1b2 Convert standard measurements of length.
	5.ME.1b3 Convert standard measurements of mass.
	5.ME.2a1 Solve problems involving conversions of standard measurement units when finding area, volume, time lapse, or mass.
Represent and interpret data.
2. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). 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. Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
CCCs linked to 5.MD.B.2	5.DPS.1c1 Collect and graph data: bar graph, line plots, picture graph (e.g., average height among 3 classrooms, # of boys and girls).
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
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.
CCCs linked to 5.MD.C.3	5.ME.2b1 Use filling and multiplication to determine volume.
Represent and interpret data.
4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
CCCs linked to 5.MD.C.4	5.ME.2b1 Use filling and multiplication to determine volume.
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 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.
CCCs linked to 5.MD.C.5	6.ME.2b3 Decompose complex 3-D shapes into simple 3-D shapes to measure volume.
	5.ME.2b2 Apply formula to solve one step problems involving volume.

Geometry	5.G
Graph points on the coordinate plane to solve real-world and mathematical problems.
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).
CCCs linked to 5.G.A.1	5.GM.1c1 Locate the x and y axis on a graph.
	5.GM.1c2 Locate points on a graph.
	5.GM.1c3 Use order pairs to graph given points.
	6.GM.1c4 Locate points on a graph.
	6.GM.1c5 Use order pairs to graph given points.
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 figures into categories based on their properties.
CCCs linked to 5.G.A.2	6.GM.1c6 Find coordinate values of points in the context of a situation.
Classify two-dimensional figures into categories based on their properties.
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.
CCCs linked to 5.G.B.3	5.GM.1a1 Recognize properties of simple plane figures.
4. Classify two-dimensional figures in a hierarchy based on properties.
CCCs linked to 5.G.B.4	5.GM.1b1 Distinguish plane figures by their properties.