Notes on Reading the Washington State Mathematics Standards Transition Documents This document serves as a guide to translate between the 2008 Washington State K-8 Learning Standards for Mathematics and the Common Core State Standards (CCSS) for Mathematics. It begins with the Standards for Mathematical Practice which are the backbone of the CCSS for Mathematics. These practices highlight the change in focus, through instructional practices, of developing these habits of mind in our students. One or more of these Standards for Mathematical Practice should be intentionally incorporated in the development of any concept or procedure taught. The Standards for Mathematical Practice are followed by the key critical areas of focus for a grade-level. These critical areas are the overarching concepts and procedures that must be learned by students to be successful at the next grade level and beyond. As units are planned, one should always reflect back on these critical areas to ensure that the concept or fluency developed in the unit is tied directly to one of these. The CCSS were developed around these critical areas in order for instruction to be deep and focused on a few key topics each year. By narrowing the focus and deepening the understanding, increases in student achievement will be realized. The body of this document includes a two-column table which indicates the alignment of the 2008 Washington State K-8 Learning Standards for Mathematics to the CCSS for Mathematics at a grade level. It is meant to be read from left to right across the columns. The right column contains all of the CCSS for Mathematics for that grade. The left column indicates the grade-level Washington standard that most aligns to it. Bolded words are used to describe the degree of alignment between these sets of standards. If the words bolded are Continue to, this indicates that the CCSS standard and the Washington standard are closely aligned. The teacher should read the wording carefully on the CCSS standard because often there is a more in-depth development of the aligned Washington standard and often there are more than one standard that address a particular Washington standard. If the word extend is bolded that indicates that the Washington and CCSS standards are similar but the CCSS takes the concept further than the Washington standard. Lastly, if the words Move students to are bolded, then the CCSS standards take the Washington standard to a deeper or further understanding of this particular cluster concept. If the left-hand side is blank, the CCSS are new material that does not match the Washington standards at this grade level. Sometimes there can be a page or more of these unaligned standards. One is reminded that while this is new material for this grade level, other standards currently taught at this grade level in the 2008 Washington standards will have moved to other grades. The movement of these unaligned standards is laid out on the last pages of this document. Collaboratively developed by OSPI and ESD Regional Mathematics Coordinators Page 1
Washington State Grade 4 Mathematics Standards Transition Document The Standards for Mathematical Practice describes varieties of expertise that mathematics educators at all levels should seek to develop in their students. These standards should be integrated throughout the teaching and learning of the content standards of the Common Core State Standards. 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Critical Areas of Focus for Grade 4 With full implementation of the Common Core State Standards for mathematics in Grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; (3) understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry. Collaboratively developed by OSPI and ESD Regional Mathematics Coordinators Page 2
Operations and Algebraic Thinking 2008 WA Grade 4 Learning Standards Grade 4 CCSS Students currently: Students need to: Use the four operations with whole numbers to solve problems. Move students to 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.1) 4.1.I Solve single- and multi-step word problems involving multi-digit multiplication and verify the solutions. 4.1.J. Solve single- and multi-step word problems involving division and verify the solutions. 4.4.A Represent an unknown quantity in simple expressions, equations, and inequalities using letters, boxes, and other symbols. 4.1.H Estimate products to approximate solutions to problems and determine reasonableness of answers. Continue to 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. 1 (4.OA.2) Solve multistep 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.3) Gain familiarity with factors and multiples. 4.1.B Identify factors and multiples of a number. Continue to 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.4) Collaboratively developed by OSPI and ESD Regional Mathematics Coordinators Page 3
Generate and analyze patterns. Move students to 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.OA.5) 1 See Glossary, Table 2. Number and Operations in Base Ten² 2008 WA Grade 4 Learning Standards Grade 4 CCSS Students currently: 4.1.D Multiply by 10, 100, and 1,000. 4.1.E Compare the values represented by digits in whole numbers using place value. Students need to: Generalize place value understanding for multi-digit whole numbers. Continue to 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.1) 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.2) Move students to use place value understanding to round multidigit whole numbers to any place. (4.NBT.3) Collaboratively developed by OSPI and ESD Regional Mathematics Coordinators Page 4
Use place value understanding and properties of operations to perform multi-digit arithmetic. Move students to fluently add and subtract multi-digit whole numbers using the standard algorithm. (4.NBT.4) 4.1.C Represent multiplication of a two-digit number by a two-digit number with place value models. 4.1.F Fluently and accurately multiply up to a three-digit number by one- and two digit numbers. Continue to multiply a whole number of up to four digits by a onedigit 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) Move students to 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.NBT.6) ² Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000. Number and Operations - Fractions³ 2008 WA Grade 4 Learning Standards Grade 4 CCSS Students currently: 4.2.F Write a fraction equivalent to a given fraction. 4.2.G Simplify fraction using common factors. Students need to: Extend understanding of fraction equivalence and ordering. Continue to 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.1) Collaboratively developed by OSPI and ESD Regional Mathematics Coordinators Page 5
4.2.E Compare and order fractions (including mixed numbers) on the number line, list, and the symbols <, >, or =. 4.2.C Convert a mixed number to a fraction and vice versa, and visually represent the number. 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.2) Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. There are no WA Grade 4 learning standards to match these CCSS on fractions. Move students to 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. 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. 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. Collaboratively developed by OSPI and ESD Regional Mathematics Coordinators Page 6
Move students to apply and extend previous understandings of multiplication to multiply a fraction by a whole number. (4.NBT.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). 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? Understand decimal notation for fractions, and compare decimal fractions. 4.2.D Convert a decimal to a fraction and vice versa, and visually represent the number. 4.2.B Read, write, compare, and order decimals through hundredths. Continue to 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. 4 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. (4.NBT.5) 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.NBT.6) Collaboratively developed by OSPI and ESD Regional Mathematics Coordinators Page 7
4.2.A Represent decimals through hundredths with place value models, fraction equivalents, and the number line. 4.2.E Compare and order decimals on the number line, lists, and the symbols <, >, =. 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. (4.NBT.7) 3 Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, 100. 4 Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade. Measurement and Data 2008 WA Grade 4 Learning Standards Grade 4 CCSS Students currently: 4.4.B Solve single- and multi-step problems involving familiar unit conversions, including time, within either the U.S. customary or metric system. 4.2.I Solve single- and multi-step word problems involving comparison of decimals and fractions (including mixed numbers), and verify the solutions. Students need to: Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. Continue to 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.1) Continue to 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 Collaboratively developed by OSPI and ESD Regional Mathematics Coordinators Page 8
4.4.C Estimate and determine elapsed time using a calendar, a digital clock, and an analog clock. 4.3.C Determine the perimeter and area of a rectangle using formulas, and explain why the formulas work. 4.3.F Solve single- and multi-step word problems involving perimeters and areas of rectangles and verify the solutions. 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.2) Continue to 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) Represent and interpret data. Move students to 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.4) Geometric measurement: understand concepts of angle and measure angles. There are no WA Grade 4 learning standards that match these CCSS angle standards. Move students to 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. Collaboratively developed by OSPI and ESD Regional Mathematics Coordinators Page 9
b) An angle that turns through n one-degree angles is said to have an angle measure of n degrees. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. (4.MD.6) 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. (4.MD.7) Geometry 2008 WA Grade 4 Learning Standards Grade 4 CCSS There are no WA Grade 4 learning standards that match with these CCSS geometry standards. Students need to: Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Move students to draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. (4.G.1) 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. (4.G.2) 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 line-symmetric figures and draw lines of symmetry. (4.G.3) Collaboratively developed by OSPI and ESD Regional Mathematics Coordinators Page 10
With full implementation of the CCSS, Grade 4 teachers will no longer be responsible for teaching students the standards listed below. The grade level where these standards will be emphasized is in parentheses. 4.1.A Quickly recall multiplication facts through 10 x 10 and the related division facts. (Grade 3) 4.3.B Determine the approximate area of a figure using square units. (Grade 3) 4.3.D Determine the areas of figures that can be broken down into rectangles. (Grade 3) 4.3.E Demonstrate that rectangles with the same area can have different perimeters, and that rectangles with the same perimeter can have different areas. (Grade 3) 4.4.D Graph and identify points in the first quadrant of the coordinate plane using ordered pairs. (Grade 5) 4.4.E Determine the median, mode, and range of a set of data and describe what each measure indicates about the data. (Grade 6) 4.4.F Describe and compare the likelihood of events. (Grade 7) 4.4.H Display the results of probability experiments and interpret the results. (Grade 7) With full implementation of the CCSS, only a portion of this WA Grade 4 learning standards is taught. The portion not taught will be emphasized at the grade level indicated in parentheses. 4.1.F Fluently and accurately multiply up to a three-digit number by one- and two-digit numbers using the standard multiplication algorithm. (Fluent and accurate multiplication is developed in Grade 4, in Grade 5 students use the algorithm) While not Grade 4 CCSS, attention to these WA Grade 4 learning standards will be a natural part of developing understanding of concepts. 4.1.G Mentally multiply two-digit numbers by numbers through 10 and by multiples of 10. (Grade 3) 4.2.H Round fractions and decimals to the nearest whole number. (Grade 5) 4.3.A Determine congruence of two-dimensional figures. This standard will no longer be emphasized at any grade level. 4.4.G Determine a simple probability from a context that includes a picture. Collaboratively developed by OSPI and ESD Regional Mathematics Coordinators Page 11