Massachusetts State Standards for Mathematics: Grade 5

MA.5.N. Number Sense and Operations: Students engage in problem solving, communicating, reasoning, connecting, and representing.

5.N.1. Demonstrate an understanding of (positive integer) powers of ten.

5.N.2. Demonstrate an understanding of place value through millions and thousandths.

5.N.3. Represent and compare large (millions) and small (thousandths) positive numbers in various forms, such as expanded notation without exponents, e.g., 9724 = 9 x 1000 + 7 x 100 + 2 x 10 + 4.

5.N.4. Demonstrate an understanding of fractions as a ratio of whole numbers, as parts of unit wholes, as parts of a collection, and as locations on the number line. This standard is intentionally the same as standard 6.N.4.

5.N.5. Identify and determine common equivalent fractions and mixed numbers with denominators 2, 4, 5, 10 only, decimals, and percents (through one hundred percent), e.g., 3/4 = 0.75 = 75%.

5.N.6. Find and position whole numbers, and positive fractions, positive mixed numbers, and positive decimals on a number line.

5.N.7. Compare and order whole numbers, positive fractions, positive mixed numbers, positive decimals, and percents.

5.N.8. Apply the number theory concepts of common factor, common multiple, and divisibility rules for 2, 3, 5, and 10 to the solution of problems. Demonstrate an understanding of the concepts of prime and composite numbers.

5.N.9. Solve problems involving multiplication and division of whole numbers, and multiplication of a positive fraction with a whole number.

5.N.10. Demonstrate an understanding of how parentheses affect expressions involving addition, subtraction, and multiplication, and use to solve problems, e.g., 3 x (4 + 2) = 3 x 6.

5.N.11. Demonstrate an understanding of the inverse relationship of addition and subtraction, and use that understanding to simplify computation and solve problems. This standard is intentionally the same as standard 6.N.12.

5.N.12. Accurately and efficiently add and subtract whole numbers and positive decimals; multiply and divide (with double-digit divisors) whole numbers; multiply positive decimals with whole numbers.

5.N.13. Accurately and efficiently add and subtract positive fractions and mixed numbers with like denominators and with unlike denominators (2, 4, 5, 10 only); multiply positive fractions with whole numbers. Simplify fractions in cases when both the numerator and the denominator have 2, 3, 4, 5, or 10 as a common factor.

5.N.14. Estimate sums and differences of whole numbers, and positive fractions and decimals. Estimate products of whole numbers, and positive decimals with whole numbers. Use a variety of strategies and judge the reasonableness of the answer.

MA.5.P. Patterns, Relations, and Algebra: Students engage in problem solving, communicating, reasoning, connecting, and representing.

5.P.1. Analyze and determine the rules for extending symbolic, arithmetic, and geometric patterns and progressions, e.g., ABBCCC; 1, 5, 9, 13?; 3, 9, 27... This standard is intentionally the same as standard 6.P.1.

5.P.2. Replace variables with given values and evaluate/simplify. This standard is intentionally the same as standard 6.P.2.

5.P.3. Use the properties of equality to solve problems with whole numbers, e.g., if __ + 7 = 13, then __ = 13 - 7, therefore __ = 6 ; if 3 x __ = 15, then __ = 15 / 3, therefore __ = 5.

5.P.4. Represent real situations and mathematical relationships with concrete models, tables, graphs, and rules in words and with symbols, e.g., input-output tables. This standard is intentionally the same as standard 6.P.4.

5.P.5. Solve problems involving proportional relationships using concrete models, tables, graphs, and paper-pencil methods.

5.P.6. Interpret graphs that represent the relationship between two variables in everyday situations.

MA.5.G. Geometry: Students engage in problem solving, communicating, reasoning, connecting, and representing.

5.G.1. Identify, describe, and compare special types of triangles (isosceles, equilateral, right) and quadrilaterals (square, rectangle, parallelogram, rhombus, trapezoid), e.g., Recognize that all equilateral triangles are isosceles, but not all isosceles triangles are equilateral.

5.G.2. Identify, describe, and compare special types of three-dimensional shapes (cubes, prisms, spheres, pyramids) based on their properties, such as edges and faces.

5.G.3. Identify relationships among points and lines, e.g., intersecting, parallel, perpendicular.

5.G.4. Using ordered pairs of whole numbers (including zero), graph, locate, and identify points, and describe paths on the Cartesian coordinate plane.

5.G.5. Describe and perform transformations on two-dimensional shapes, e.g., translations, rotations, and reflections.

5.G.6. Identify and describe line symmetry in two-dimensional shapes, including shapes that have multiple lines of symmetry.

5.G.7. Determine if two triangles or two quadrilaterals are congruent by measuring sides or a combination of sides and angles, as necessary; or by motions or series of motions, e.g., translations, rotations, and reflections.

MA.5.M. Measurement: Students engage in problem solving, communicating, reasoning, connecting, and representing.

5.M.1. Apply the concepts of perimeter and area to the solution of problems involving triangles and rectangles. Apply formulas where appropriate.

5.M.2. Identify, measure, describe, classify, and draw various angles. Draw triangles given two sides and the angle between them, or two angles and the side between them, e.g., draw a triangle with one right angle and two sides congruent.

5.M.3. Solve problems involving simple unit conversions within a system of measurement.

5.M.4. Find volumes and surface areas of rectangular prisms. This standard is intentionally the same as standard 6.M.6.

5.M.5. Find the sum of the measures of the interior angles in triangles with and without measuring the angles.

MA.5.D. Data Analysis, Statistics, and Probability: Students engage in problem solving, communicating, reasoning, connecting, and representing.

5.D.1. Given a set of data, find the median, mean, mode, maximum, minimum, and range, and apply to solutions of problems.

5.D.2. Construct and interpret line plots, line graphs, and bar graphs. Interpret and label circle graphs.

5.D.3. Predict the probability of outcomes of simple experiments (e.g., tossing a coin, rolling a die) and test the predictions.

MA.CC.5.OA. Operations and Algebraic Thinking

5.OA.1. Use parentheses, brackets, or braces in numerical expressions, and evaluate 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 * (8 + 7). Recognize that 3 * (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

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.

MA.CC.5.NBT. Number and Operations in Base Ten

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 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.

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

5.NBT.3.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).

5.NBT.3.b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

5.NBT.4. Use place value understanding to round decimals to any place.

5.NBT.5. Fluently multiply multi-digit whole numbers using the 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, and/or area models.

5.NBT.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.

MA.CC.5.NF. Number and Operations-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 assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

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.

5.NF.4.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.)

5.NF.4.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:

5.NF.5.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.

5.NF.5.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.

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.

5.NF.7.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.

5.NF.7.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.

5.NF.7.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?

MA.CC.5.NS. The Number System

5.NS.MA.1. Use positive and negative integers to describe quantities such as temperature above/below zero, elevation above/below sea level, or credit/debit.

MA.CC.5.MD. Measurement and Data

5.MD.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.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.

5.MD.3. Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

5.MD.3.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.

5.MD.3.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.4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

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

5.MD.5.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.

5.MD.5.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.

5.MD.5.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.

MA.CC.5.G. Geometry