GRADE 4 MATH COMPETENCY STATEMENTS & PERFORMANCE INDICATORS. Major Clusters Supporting Clusters Additional Clusters
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1 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. 1. Students generalize their understanding of place value to 1,000,000, understanding the relative sizes of numbers in each place. They apply their understanding of models for multiplication (equal-sized groups, arrays, area models), place value, and properties of operations, in particular the distributive property, as they develop, discuss, and use efficient, accurate, and generalizable methods to compute products of multi-digit whole numbers. Depending on the numbers and the context, they select and accurately apply appropriate methods to estimate or mentally calculate products. They develop fluency with efficient procedures for multiplying whole numbers; understand and explain why the procedures work based on place value and properties of operations; and use them to solve problems. Students apply their understanding of models for division, place value, properties of operations, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multi-digit dividends. They select and accurately apply appropriate methods to estimate and mentally calculate quotients, and interpret remainders based upon the context. 2. Students develop understanding of fraction equivalence and operations with fractions. They recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent fractions. Students extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number. 3. Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and analyzing two-dimensional shapes, students deepen their understanding of properties of two-dimensional objects and the use of them to solve problems involving symmetry. Items in bold are items directly from the CCSS. Major Clusters Supporting Clusters Additional Clusters Revised September 2017 Page 1
2 Competency 1: Operations and Algebraic Thinking - Students will demonstrate the ability to compute accurately, make reasonable estimates, understand meanings of operations and use algebraic notation to represent and analyze patterns and relationships. C1PI1a. Students will use the four operations with whole numbers to solve problems. 4.OA.1 C1PI1b. 4.OA.2 C1PI1c. 4.OA.3 C1PI2. Students will generate and analyze patterns. 4.OA.5 Given a multistep equation, I can create a logical word problem that accurately reflects the mathematical relationships between the known(s) and unknown(s). Evidence for this A-level performance level descriptor can be found in indicator C1PI1a. Evidence for this A-level performance level descriptor can be found in indicator C1PI1a. I can identify a number pattern that follows a multiple step rule and extend the pattern. I can accurately identify and explain in writing which estimation strategy is most appropriate to use given the particular word problem. I can verify my solutions by determining the reasonableness of my answer. Evidence for this B-level performance indicator C1PI1a. Evidence for this B-level performance indicator C1PI1a. I can accurately use a t-chart, table, and/or diagram to identify and describe features of a number or shape pattern by identifying the rule and other features explicit in the rule. For example: There are four beans in the jar. Each day 3 beans are added. How many beans are in the jar for each of the first five days? I can interpret a multiplicative equation as a comparison. I can represent verbal statements of multiplicative comparisons as multiplication equations. For example: interpret 35 = 5 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. I can accurately multiply or divide to solve word problems involving multiplication or division with an unknown. For example: By using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. I can accurately solve multistep word problems posed with whole numbers using the four operations, including problems in which remainders must be interpreted. I can accurately represent these problems using equations with a letter standing for the unknown quantity and assess the reasonableness of my answers using mental computation and estimation strategies including rounding. I can accurately generate a number or shape pattern that follows a given rule and 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. NYC - Inaccurate and/or inconsistent demonstration of the C-level expectation Revised September 2017 Page 2
3 Competency 2: Number and Numeration in Base Ten/Fractions - Students will demonstrate the ability to understand the meanings, uses, and representations of numbers as well as equivalent names of numbers. Number & Operations in Base Ten: G4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000. C2PI1a. Students will be able to generalize place value understanding for multi-digit whole numbers. 4.NBT.1 4.NBT.3 C2PI1b. 4.NBT.2 C2PI2a. Students will be able to use place value understanding and properties of operations to perform multi-digit arithmetic. 4.NBT.4 I accurately convert between place values through millions. For example: 90,000 =? ones;? tens,? hundreds,? thousands,? ten thousands. Evidence for this A-level performance indicator C2PI1a. Given a solution to an addition or subtraction problem, I can analyze the performed steps to determine errors and explain the computational misconceptions. I can accurately read and write multi-digit numbers in a variety of different forms. For example: 285 = 28 tens + 5 ones; 1 hundred + 18 tens + 5 ones; ; two hundred eighty-five; 2 hundred eighty five. Evidence for this B-level performance level descriptor can be found in indicator C2PI1a. I can accurately check multi-digit whole number addition and subtraction problems by using the appropriate inverse operation. I accurately 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. I can accurately use place value understanding to round multi-digit whole numbers to any place. I can accurately read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form and compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. I can accurately and fluently add and subtract multi-digit whole numbers using the standard algorithm. NYC - Inaccurate and/or inconsistent demonstration of the C-level expectation Revised September 2017 Page 3
4 C2PI2b. 4.NBT.5 I can accurately check whole number division problems without remainders with up to four digit dividends and two digit divisors by finding the related multiplication equation - OR -Given a solution to a multiplication problem, I can analyze the performed steps to determine errors and explain the computational misconceptions. I can accurately multiply a whole number of up to four digits by a two-digit whole number using a variety of strategies. I can accurately 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. I can illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. C2PI2c. 4.NBT.6 4.OA.4 Evidence for this A-level performance indicator C2PI2b. Evidence for this B- level performance level descriptor can be found in indicator C2PI2b. I can accurately 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. I can illustrate and explain the calculation by using equations, rectangular arrays, and/or models. I can accurately: find all factor pairs for a whole number in the range I can recognize that a whole number is a multiple of each of its factors. I can determine whether a given whole number in the range is a multiple of a given one-digit number. I can determine whether a given whole number in the range is prime or composite. Revised September 2017 Page 4
5 Number & Operations-Fractions: G4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100. 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 G4. C2PI3. Students will extend an understanding of fraction equivalence and ordering. 4.NF.1 4.NF.2 I can explain how I ordered the three fractions. I can accurately compare three or more fractions with unlike numerators and denominators. I can accurately explain why a fraction a/b is equivalent to another fraction (nxa)/(nxb) 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. I can use this principle to recognize and generate equivalent fractions. I can accurately 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 ½. I can recognize that comparisons are valid only when the two fractions refer to the same size whole. I can record results with symbols <, =, and >. For example: ½ or 1/8 of a medium pizza is different from ½ or 1/8 of a large pizza. I can record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. C2PI4a. Students will be able to build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF.3, 4.NF.3a, 4.NF.3b, and 4.NF.3c When solving word problems with mixed numbers or fractions with unlike denominators, I can use mathematical vocabulary to explain my process for determining common denominators and/or simplest form. I can accurately add and subtract fractions including mixed numbers, with unlike benchmark denominators and express my answer in simplest form OR I can solve word problems involving addition and subtraction of fractions and mixed numbers with unlike benchmark denominators. (2, 3, 4, 5, 6, 8, 10, 12, and 100). For example: ¼ + ½ + 2/8 = 1 I can accurately add and subtract fractions including 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. I can accurately 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. C2PI4b. 4.NF.3d Evidence for this A- level performance level descriptor can be found in indicator C2PI4a. Evidence for this B-level performance indicator C2PI4a. I can solve word problems involving addition and subtraction of fractions and mixed numbers referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. For example: A cake recipe calls for ¾ cup of milk, ¼ cup oil, and 2/4 cup of water. How much liquid was needed to make the cake. Revised September 2017 Page 5
6 C2PI5a. Students will be able to apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 4.NF.4, 4a, and 4.NF.4b I can express my answer in simplest form. I can accurately multiply a whole number by a fraction without using a visual model. I can accurately demonstrate an understanding that a/b is 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.) C2PI5b. 4.NF.4c I can accurately solve and explain word problems involving multiplication of a fraction by a whole number. I can accurately solve word problems involving multiplication of a fraction by a whole number without visual models. I can 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? C2PI6a. Students will understand decimal notation for fractions, and compare decimal fractions. 4.NF.5 4.NF.6 I can use decimal notation for fractions with denominators 10 or 100 in a word problem. I can accurately add and subtract using equivalent fractions with denominators of 10 and 100. For example: 6/10 5/100 = 60/100 5/100 = 55/100 - OR - I can accurately explain multiple ways to decompose (break apart) a decimal using addition. For example:.56 = ; or.56 = ; or.56 = ; or I can accurately express a fraction with a denominator of 10 as an equivalent fraction with a denominator of 100, and use this technique to add two fractions with respective denominators of 10 and 100. For example: express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100 I can 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. C2PI6b. 4.NF.7 I can accurately compare a set of more than three decimals to hundredths and explain my reasoning. I can accurately compare two decimals using at least two different models and/or explain how I know my comparisons are valid. I can accurately compare two decimals to hundredths by reasoning about their size in relation to the same size whole. I recognize that comparisons are valid only when the two decimals refer to the same whole and I can record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. Revised September 2017 Page 6
7 Competency 3: Measurement - Students will demonstrate the ability to understand the systems and processes of measurement; use appropriate techniques, tools, units and formulas in making measurements. C3PI1a. Students will be able to solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. 4.MD.1 4.MD.2 C3PI1b. 4.MD.3 4.OA.4 C3PI2. Students will understand concepts of angle and measure angles. 4.MD.5 4.MD.5a 4.MD.5b 4.MD.6 4.MD.7 I can accurately convert given measurements to various unit measures i.e. 5 cups = x pints and y cups. I can accurately solve word problems resulting in fractional units. I can accurately apply the area and perimeter formulas for rectilinear shapes with missing side lengths. I can explain how to use given angle measures to determine missing angle measures. I can accurately represent and explain how measurement diagrams help me solve word problems. Given the area of a rectangular shape, I can identify all of its possible side lengths. I can accurately estimate angle measurements and support my estimate in writing with relevant math vocabulary. OR - I can estimate the measure of an angle in writing before checking its measure with a protractor. I can accurately demonstrate understanding of relative sizes of measurement units within one system and convert larger units to smaller units by recording them in a t-chart or graphic organizer. Units include: km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. I can accurately 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. I can accurately represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. I can accurately apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example: Find the width of a rectangular room given its area and length. I can accurately: find all factor pairs for a whole number in the range I can recognize that a whole number is a multiple of each of its factors. I can determine whether a given whole number in the range is a multiple of a given one-digit number. I can determine whether a given whole number in the range is prime or composite. I can accurately demonstrate understanding of the relationship between angle measurement and a circle. I can define angles as geometric shapes that are formed wherever two rays share a common endpoint. For example: An angle that turns through 1/360 of a circle is called a one-degree angle, and can be used to measure angles. I can accurately demonstrate understanding that an angle that turns through n one-degree angles is said to have an angle measure of n degrees. I can accurately measure angles in whole-number degrees using a protractor within 3 degrees and sketch angles of specified measure. I can accurately recognize angle measure as additive by using addition and subtraction to determine the sums and differences of two adjacent angle measures. NYC - Inaccurate and/or inconsistent demonstration of the C-level expectation Revised September 2017 Page 7
8 Competency 4: Data - Students will demonstrate the ability to represent and analyze data. C4PI1. Students will be able to represent and interpret data. 4.MD.4 Given a set of data, I can make accurate predictions about a more broad population. I can accurately interpret a line plot and communicate generalizations or create questions from it. I can accurately make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8) and solve problems involving addition and subtraction of fractions by using information presented in line plots. NYC - Inaccurate Competency 5: Geometry - Students will demonstrate the ability to investigate characteristics and properties of two and/or three- dimensional geometric shapes and apply transformations and symmetry in geometric situations. C5PI1a. Students will be able to draw and identify lines and angles, and classify shapes by properties of their lines and angles. 4.G.1 4.G.2 I can accurately classify quadrilaterals and triangles based on their properties and explain my classifications. I can accurately use a set of attributes to describe shapes OR to construct them. I can accurately draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. I can identify these in 2D figures. I can accurately classify two-dimensional figures based on the properties of parallel or non-parallel lines, perpendicular or non-perpendicular lines, and angle size. I can recognize right triangles as a category and identify right triangles. NYC - Inaccurate and/or C5PI1b. 4.G.3 I can accurately classify polygons with more than two lines of symmetry and explain patterns of symmetry in regular polygons. I can accurately classify polygons that have more than one line of symmetry. I can accurately identify line-symmetric figures and draw lines of symmetry for two-dimensional figures. Competency 6: Fact Fluency - Students will demonstrate the ability to quickly and accurately verbalize and compute fact fluency. C6PI1. Students will be able to demonstrate memorization of products and related quotients up to 12 x 12. I can demonstrate memorization of products and related quotients up to 12 x 12. Not Yet Inaccurate and/or inconsistent demonstration of the C-level expectation. Revised September 2017 Page 8
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