Massachusetts State Standards for Mathematics: Grade 4

MA.4.N. Number Sense and Operations: Students engage in problem solving, communicating, reasoning, connecting, and representing.

4.N.1. Exhibit an understanding of the base ten number system by reading, modeling, writing, and interpreting whole numbers to at least 100,000; demonstrating an understanding of the values of the digits; and comparing and ordering the numbers.

4.N.2. Represent, order, and compare large numbers (to at least 100,000) using various forms, including expanded notation, e.g., 853 = 8 X 100 + 5 X 10 + 3.

4.N.3. Demonstrate an understanding of fractions as parts of unit wholes, as parts of a collection, and as locations on the number line.

4.N.4. Select, use, and explain models to relate common fractions and mixed numbers (1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/10, 1/12, and 1 1/2), find equivalent fractions, mixed numbers, and decimals, and order fractions.

4.N.5. Identify and generate equivalent forms of common decimals and fractions less than one whole (halves, quarters, fifths, and tenths).

4.N.6. Exhibit an understanding of the base ten number system by reading, naming, and writing decimals between 0 and 1 up to the hundredths.

4.N.7. Recognize classes (in particular, odds, evens; factors or multiples of a given number; and squares) to which a number may belong, and identify the numbers in those classes. Use these in the solution of problems.

4.N.8. Select, use, and explain various meanings and models of multiplication and division of whole numbers. Understand and use the inverse relationship between the two operations.

4.N.9. Select, use, and explain the commutative, associative, and identity properties of operations on whole numbers in problem situations, e.g., 37 X 46 = 46 X 37, (5 X 7) X 2 = 5 X (7 X 2).

4.N.10. Select and use appropriate operations (addition, subtraction, multiplication, and division) to solve problems, including those involving money.

4.N.11. Know multiplication facts through 12 X 12 and related division facts. Use these facts to solve related multiplication problems and compute related problems, e.g., 3 X 5 is related to 30 X 50, 300 X 5, and 30 X 500.

4.N.12. Add and subtract (up to five-digit numbers) and multiply (up to three digits by two digits) accurately and efficiently.

4.N.13. Divide up to a three-digit whole number with a single-digit divisor (with or without remainders) accurately and efficiently. Interpret any remainders.

4.N.14. Demonstrate in the classroom an understanding of and the ability to use the conventional algorithms for addition and subtraction (up to five-digit numbers), and multiplication (up to three digits by two digits).

4.N.15. Demonstrate in the classroom an understanding of and the ability to use the conventional algorithm for division of up to a three-digit whole number with a single-digit divisor (with or without remainders).

4.N.16. Round whole numbers through 100,000 to the nearest 10, 100, 1000, 10,000, and 100,000.

4.N.17. Select and use a variety of strategies (e.g., front-end, rounding, and regrouping) to estimate quantities, measures, and the results of whole-number computations up to three-digit whole numbers and amounts of money to $1000, and to judge the reasonableness of the answer.

4.N.18. Use concrete objects and visual models to add and subtract common fractions.

MA.4.P. Patterns, Relations, and Algebra: Students engage in problem solving, communicating, reasoning, connecting, and representing.

4.P.1. Create, describe, extend, and explain symbolic (geometric) and numeric patterns, including multiplication patterns like 3, 30, 300, 3000).

4.P.2. Use symbol and letter variables (e.g., s, x) to represent unknowns or quantities that vary in expressions and in equations or inequalities (mathematical sentences that use =, <, >).

4.P.3. Determine values of variables in simple equations.

4.P.4. Use pictures, models, tables, charts, graphs, words, number sentences, and mathematical notations to interpret mathematical relationships.

4.P.5. Solve problems involving proportional relationships, including unit pricing (e.g., four apples cost 80 cents, so one apple costs 20 cents) and map interpretation (e.g., one inch represents five miles, so two inches represent ten miles).

4.P.6. Determine how change in one variable relates to a change in a second variable, e.g., input-output tables.

MA.4.G. Geometry: Students engage in problem solving, communicating, reasoning, connecting, and representing.

4.G.1. Compare and analyze attributes and other features (e.g., number of sides, faces, corners, right angles, diagonals, and symmetry) of two- and three-dimensional geometric shapes.

4.G.2. Describe, model, draw, compare, and classify two- and three-dimensional shapes, e.g., circles, polygons - especially triangles and quadrilaterals - cubes, spheres, and pyramids.

4.G.3. Recognize similar figures.

4.G.4. Identify angles as acute, right, or obtuse.

4.G.5. Describe and draw intersecting, parallel, and perpendicular lines.

4.G.6. Using ordered pairs of numbers and/or letters, graph, locate, identify points, and describe paths (first quadrant).

4.G.7. Describe and apply techniques such as reflections (flips), rotations (turns), and translations (slides) for determining if two shapes are congruent.

4.G.8. Identify and describe line symmetry in two-dimensional shapes.

4.G.9. Predict and validate the results of partitioning, folding, and combining two- and three-dimensional shapes.

MA.4.M. Measurement: Students engage in problem solving, communicating, reasoning, connecting, and representing.

4.M.1. Demonstrate an understanding of such attributes as length, area, weight, and volume, and select the appropriate type of unit for measuring each attribute.

4.M.2. Carry out simple unit conversions within a system of measurement, e.g., hours to minutes, cents to dollars, yards to feet or inches, etc.

4.M.3. Identify time to the minute on analog and digital clocks using a.m. and p.m. Compute elapsed time using a clock (e.g., hours and minutes since) and using a calendar (e.g., days since).

4.M.4. Estimate and find area and perimeter of a rectangle, triangle, or irregular shape using diagrams, models, and grids or by measuring.

4.M.5. Identify and use appropriate metric and English units and tools (e.g., ruler, angle ruler, graduated cylinder, thermometer) to estimate, measure, and solve problems involving length, area, volume, weight, time, angle size, and temperature.

MA.4.D. Data Analysis, Statistics, and Probability: Students engage in problem solving, communicating, reasoning, connecting, and representing.

4.D.1. Collect and organize data using observations, measurements, surveys, or experiments, and identify appropriate ways to display the data.

4.D.2. Match representations of a data set such as lists, tables, or graphs (including circle graphs) with the actual set of data.

4.D.3. Construct, draw conclusions, and make predictions from various representations of data sets, including tables, bar graphs, pictographs, line graphs, line plots, and tallies.

4.D.4. Represent the possible outcomes for a simple probability situation, e.g., the probability of drawing a red marble from a bag containing three red marbles and four green marbles.

4.D.5. List and count the number of possible combinations of objects from three sets, e.g., how many different outfits can one make from a set of three shirts, a set of two skirts, and a set of two hats?

4.D.6. Classify outcomes as certain, likely, unlikely, or impossible by designing and conducting experiments using concrete objects such as counters, number cubes, spinners, or coins.

MA.CC.4.OA. Operations and Algebraic Thinking

4.OA.1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 * 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

4.OA.2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

4.OA.3. Solve multi-step word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

4.OA.4. Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.

4.OA.5. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule ''Add 3'' and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

MA.CC.4.NBT. Number and Operations in Base Ten

4.NBT.1. Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 / 70 = 10 by applying concepts of place value and division

4.NBT.2. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

4.NBT.3. Use place value understanding to round multi-digit whole numbers to any place.

4.NBT.4. Fluently add and subtract multi-digit whole numbers using the standard algorithm.

4.NBT.5. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

4.NBT.MA.5a. Know multiplication facts and related division facts through 12 x 12.

4.NBT.6. Find whole-number quotients and remainders with up to four-digit dividends and one-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.

MA.CC.4.NF. Number and Operations-Fractions

4.NF.1. Explain why a fraction a/b is equivalent to a fraction (n * a)/(n * b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

4.NF.2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

4.NF.3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

4.NF.3.a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

4.NF.3.b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

4.NF.3.c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

4.NF.3.d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

4.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

4.NF.4.a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 * (1/4), recording the conclusion by the equation 5/4 = 5 * (1/4).

4.NF.4.b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 * (2/5) as 6 * (1/5), recognizing this product as 6/5. (In general, n * (a/b) = (n * a)/b.)

4.NF.4.c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

4.NF.5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.

4.NF.6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

4.NF.7. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.

MA.CC.4.MD. Measurement and Data

4.MD.1. Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...

4.MD.2. Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

4.MD.3. Apply the area and perimeter formulas for rectangles in real-world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

4.MD.4. 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 addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.

4.MD.5. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:

4.MD.5.a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a ''one-degree angle,'' and can be used to measure angles.

4.MD.5.b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

4.MD.6. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

4.MD.7. Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

MA.CC.4.G. Geometry