# California Standard Study Island Topic Common Core Standard

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1 State: CA Subject: Math Grade Level: 4 California Standard Study Island Topic Standard NUMBER SENSE 1.0: Students understand the place value of whole numbers and decimals to two decimal places and how whole numbers and decimals relate to simple fractions. Students use the concepts of negative numbers: Number Sense Read and write whole numbers in the millions. Reading & Writing Numbers Covered by Second Grade Number Sense Order and compare whole numbers and decimals to two decimal places. Number Sense Round whole numbers through the millions to the nearest ten, hundred, thousand, ten thousand, or hundred thousand. Ordering Numbers Rounding Numbers 4.NBT.2. Read and write multidigit whole numbers using baseten 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.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 the number line or another visual model. 4.NBT.3. Use place value understanding to round multidigit whole numbers to any place. Number Sense Decide when a rounded solution is called for and explain why such a solution Not Tested

2 may be appropriate. Number Sense Explain different interpretations of fractions, for example, parts of a whole, parts of a set, and division of whole numbers by whole numbers; explain equivalents of fractions (see Standard 4.0). Number Sense Write tenths and hundredths in decimal and fraction notations and know the fraction and decimal equivalents for halves and fourths (e.g., 1/2 = 0.5 or.50; 7/4 = 1 3/4 = 1.75). Fractional Parts of Objects Fractions & Decimals 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.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/ 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. Number Sense Write the fraction represented by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line. Fractional Parts of Objects Number Lines 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.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.

3 Number Sense Use concepts of negative numbers (e.g., on a number line, in counting, in temperature, in "owing"). Negative Numbers Covered by Sixth Grade Number Sense Identify on a number line the relative position of positive fractions, positive mixed numbers, and positive decimals to two decimal places. Number Lines 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. NUMBER SENSE 2.0: Students extend their use and understanding of whole numbers to the addition and subtraction of simple decimals: Number Sense Estimate and compute the sum or difference of whole numbers and positive decimals to two places. Number Sense Round two-place decimals to one decimal or the nearest whole number and judge the reasonableness of the rounded answer. Estimate Sums & Differences Rounding Numbers 4.NBT.4. Fluently add and subtract multi-digit whole numbers using the standard algorithm. 4.NBT.3. Use place value understanding to round multidigit whole numbers to any place. NUMBER SENSE 3.0: Students solve problems involving addition, subtraction, multiplication, and division of whole numbers and understand the relationships among the operations: Number Sense Demonstrate an Addition & Subtraction

4 understanding of, and the ability to use, standard algorithms for the addition and subtraction of multidigit numbers. Number Sense Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a twodigit number and for dividing a multidigit number by a one-digit number; use relationships between them to simplify computations and to check results. Multiplication & Division 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.4. Find all factor pairs for a whole number in the range Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range is a multiple of a given one-digit number. Determine whether a given whole number in

5 Number Sense Solve problems involving multiplication of multidigit numbers by two-digit numbers. Multiplication & Division Real World Problems the range is prime or composite. 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.5.1. Solve problems involving multiplication of multidigit numbers by two-digit numbers. 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. 4.OA.4. Find all factor pairs for a whole number in the range Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range is a multiple of a given one-digit number. Determine whether a given whole number in the range is prime or composite.

6 Number Sense Solve problems involving division of multidigit numbers by one-digit numbers. Multiplication & Division Real World Problems 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.5.1. Solve problems involving multiplication of multidigit numbers by two-digit numbers. 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.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.

7 NUMBER SENSE 4.0: Students know how to factor small whole numbers: Number Sense Understand that many whole numbers break down in different ways (e.g., 12 = 4 x 3 = 2 x 6 = 2 x 2 x 3). Number Sense Know that numbers such as 2, 3, 5, 7, and 11 do not have any factors except 1 and themselves and that such numbers are called prime numbers. Factorization Factorization 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.4. Find all factor pairs for a whole number in the range Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range is a multiple of a given one-digit number. Determine whether a given whole number in the range is prime or composite. ALGEBRA AND FUNCTIONS 1.0: Students use and interpret variables, mathematical symbols, and properties to write and simplify expressions and sentences: Algebra and Functions Use letters, boxes, or other symbols to stand for any number in simple expressions or equations (e.g., demonstrate an understanding and the use of the concept of a variable). Algebra and Functions Interpret and evaluate mathematical expressions Number Sentences Variables & Expressions Simplify Expressions 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. Covered by Fifth Grade

8 that now use parentheses. Algebra and Functions Use parentheses to indicate which operation to perform first when writing expressions containing more than two terms and different operations. Simplify Expressions Covered by Fifth Grade Algebra and Functions Use and interpret formulas (e.g., area = length x width or A = lw) to answer questions about quantities and their relationships. Algebra and Functions Understand that an equation such as y = 3x + 5 is a prescription for determining a second number when a first number is given. Formulas Algebraic Techniques Formulas Number Sentences 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. ALGEBRA AND FUNCTIONS 2.0: Students know how to manipulate equations:

9 Algebra and Functions Know and understand that equals added to equals are equal. Algebra and Functions Know and understand that equals multiplied by equals are equal. Algebraic Techniques Number Sentences Algebraic Techniques Number Sentences 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. MEASUREMENT AND GEOMETRY 1.0: Students understand perimeter and area: Geometry Measure the area of rectangular shapes by using appropriate units, such as square centimeter (cm 2 ), square meter (m 2 ), square kilometer (km 2 ), square inch (in 2 ), square yard (yd 2 ), or square mile (mi 2 ). Perimeter & Area 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. Geometry Recognize that rectangles that have the same area can have different perimeters. Geometry Understand that rectangles that have the same perimeter can have different areas. Perimeter & Area Perimeter & Area 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.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

10 Geometry Understand and use formulas to solve problems involving perimeters and areas of rectangles and squares. Use those formulas to find the areas of more complex figures by dividing the figures into basic shapes. Perimeter & Area the length, by viewing the area formula as a multiplication equation with an unknown factor. 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. MEASUREMENT AND GEOMETRY 2.0: Students use two-dimensional coordinate grids to represent points and graph lines and simple figures: Geometry Draw the points corresponding to linear relationships on graph paper (e.g., draw 10 points on the graph of the equation y = 3x and connect them by using a straight line). Graph Lines Covered by Sixth Grade Geometry Understand that the length of a horizontal line segment equals the difference of the x-coordinates. Graph Lines Covered by Sixth Grade Geometry Understand that the length of a vertical line segment equals the difference of the y- coordinates. Graph Lines Covered by Sixth Grade MEASUREMENT AND GEOMETRY 3.0: Students demonstrate an understanding of plane and solid geometric objects and use this knowledge to show relationships and solve problems: Geometry Identify lines that are parallel and Angles & Lines 4.G.1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines.

11 perpendicular. Geometry Identify the radius and diameter of a circle. Circles Identify these in two-dimensional figures. Covered by Seventh Grade Geometry Identify congruent figures. Symmetry & Congruence Covered by Eighth Grade Geometry Identify figures that have bilateral and rotational symmetry. Geometry Know the definitions of a right angle, an acute angle, and an obtuse angle. Understand that 90, 180, 270, and 360 are associated, respectively, with 1/4, 1/2, 3/4, and full turns. Symmetry & Congruence Angles & Lines 4.G.3. Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify linesymmetric figures and draw lines of symmetry. 4.G.1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. Geometry Visualize, describe, and make models of geometric solids (e.g., prisms, pyramids) in terms of the number and shape of faces, edges, and vertices; interpret two-dimensional representations of threedimensional objects; and draw patterns (of faces) for a solid that, when cut and folded, will make a model of the solid. 3-Dimensional Solids Folding & Combining Shapes Covered by Sixth Grade Geometry Know the definitions of different triangles (e.g., equilateral, isosceles, scalene) and identify their attributes. Triangles & Quadrilaterals 4.G.2. Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and

12 Geometry Know the definition of different quadrilaterals (e.g., rhombus, square, rectangle, parallelogram, trapezoid). Triangles & Quadrilaterals identify right triangles. (Two dimensional shapes should include special triangles, e.g., equilateral, isosceles, scalene, and special quadrilaterals, e.g., rhombus, square, rectangle, parallelogram, trapezoid.) 4.G.2. Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles. (Two dimensional shapes should include special triangles, e.g., equilateral, isosceles, scalene, and special quadrilaterals, e.g., rhombus, square, rectangle, parallelogram, trapezoid.) STATISTICS, DATA ANALYSIS, AND PROBABILITY 1.0: Students organize, represent, and interpret numerical and categorical data and clearly communicate their findings: Statistics, Data Analysis, and Probability Formulate survey questions; systematically collect and represent data on a number line; and coordinate graphs, tables, and charts. Statistics, Data Analysis, and Probability Identify the mode(s) for sets of categorical data and the mode(s), median, and any apparent outliers for numerical data sets. Graphs Median & Mode 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. Covered by Sixth Grade Statistics, Data Analysis, and Probability Interpret one-and twovariable data graphs to answer questions about a situation. Graphs Covered by Seventh Grade

13 STATISTICS, DATA ANALYSIS, AND PROBABILITY 2.0: Students make predictions for simple probability situations: Statistics, Data Analysis, and Probability Represent all possible outcomes for a simple probability situation in an organized way (e.g., tables, grids, tree diagrams). Possible Outcomes Covered by Seventh Grade Statistics, Data Analysis, and Probability Express outcomes of experimental probability situations verbally and numerically (e.g., 3 out of 4; 3 /4). Probability Covered by Seventh Grade MATHEMATICAL REASONING 1.0: Students make decisions about how to approach problems: Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns Determine when and how to break a problem into simpler parts. MATHEMATICAL REASONING 2.0: Students use strategies, skills, and concepts in finding solutions: Use estimation to verify the reasonableness of calculated results Apply strategies and

14 results from simpler problems to more complex problems Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy Make precise calculations and check the validity of the results from the context of the problem. MATHEMATICAL REASONING 3.0: Students move beyond a particular problem by generalizing to other situations: Evaluate the reasonableness of the solution in the context of the original situation Note the method of deriving the solution and

15 demonstrate a conceptual understanding of the derivation by solving similar problems Develop generalizations of the results obtained and apply them in other circumstances. Fourth Grade Standards not Covered in California Fourth Grade 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. 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 = 10 by applying concepts of place value and division. 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. a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 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 = /8 = 8/8 + 8/8 + 1/8. 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. 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. a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to

16 represent 5/4 as the product 5 (1/4), recording the conclusion by the equation 5/4 = 5 (1/4). 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.) 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.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.5. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: 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. 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.

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