GRADE 4. M : Solve division problems without remainders. M : Recall basic addition, subtraction, and multiplication facts.


 Hector Gardner
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1 GRADE 4 Students will: Operations and Algebraic Thinking Use the four operations with whole numbers to solve problems. 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. [4OA1] M : Use arrays to show equal groups in multiplication. M : Recall basic multiplication facts. M : Interpret the products of whole numbers. M : Demonstrate computational fluency, including quick recall of addition and subtraction facts. M : Recognize multiplication as repeated addition. 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. (See Appendix A, Table 2.) [4OA2] M : Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. M : Recognize key terms to solve word problems. Examples: in all, how much, how many, in each M : Apply properties of operations as strategies to add. M : Recall basic multiplication facts. M : Demonstrate computational fluency, including quick recall of addition and subtraction facts. 3. Solve multistep word problems posed with whole numbers and having wholenumber 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. [4OA3] M : Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. M : Solve singlestep word problems. M : Recognize key terms to solve word problems. Examples: in all, how much, how many, in each M : Solve division problems without remainders. M : Recall basic addition, subtraction, and multiplication facts. Curriculum Guide to the Alabama Course of Study: Mathematics 44
2 Gain familiarity with factors and multiples. 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 onedigit number. Determine whether a given whole number in the range is prime or composite. [4OA4] M : Define factors, prime number, and composite number. M : Apply properties of operations as strategies to multiply and divide. M : Identify all factor pairs for a whole number in the range M : Name the first ten multiples of each onedigit natural number. M : Recall basic multiplication facts. M : Count within 1000; skipcount by 5s, 10s, and 100s. Generate and analyze patterns. 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. [4OA5] 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. M : Identify arithmetic patterns, including patterns in the addition table or multiplication table; and explain them using properties of operations M : Recognize arithmetic patterns (including geometric patterns or patterns in the addition table or multiplication table). Examples: Continue a geometric pattern Ο Ο by drawing the next three shapes. Sample Answer: Ο Complete the numerical pattern for the following chart when given the rule, Input + 5 = Output. Sample Answer: Input 5, Output 10; Input 9, Output 14. Input Output ? ? M : Construct repeating and growing patterns with a variety of representations. M : Continue an existing pattern. M : Identify arithmetic patterns. M : Demonstrate computational fluency, including quick recall, of addition multiplication facts. Curriculum Guide to the Alabama Course of Study: Mathematics 45
3 Number and Operations in Base Ten (Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.) Generalize place value understanding for multidigit whole numbers. 6. Recognize that in a multidigit whole number, a digit in one place represents ten times what it represents in the place to its right. [4NBT1] Example: Recognize that = 10 by applying concepts of place value and division. M : Use place value understanding to round whole numbers to the nearest 10 or 100. M : Add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. M : Multiply onedigit whole numbers by multiples of 10 in the range (e.g., 9 80, 5 60) using strategies based on place value and properties of operations. M : Recall basic multiplication facts. M : Recall that the three digits of a threedigit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. M : Recognize that the numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones) M : Recognize that 100 can be thought of as a bundle of ten tens, called a hundred. 7. Read and write multidigit whole numbers using baseten numerals, number names, and expanded form. Compare two multidigit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. [4NBT2] M : Compare two threedigit numbers based on meanings of the hundreds, tens, and ones digits using >, =, and < symbols to record the results of comparisons. M : Read and write numbers to 1000 using baseten numerals, number names, and expanded form. M : Convert a number written in expanded notation to standard form. 8. Use place value understanding to round multidigit whole numbers to any place. [4NBT3] M : Use place value understanding to round whole numbers to the nearest 10 or 100. M : Model rounding whole numbers to the nearest 100. M : Round whole numbers from 100 to 999 using whole numbers from 10 to 99. M : Model rounding whole numbers to the nearest 10. M : Round whole numbers from 10 to 99 using whole numbers from 1 to 9. M : Round whole numbers from 1 to 9 and model to show proficiency. Curriculum Guide to the Alabama Course of Study: Mathematics 46
4 Use place value understanding and properties of operations to perform multidigit arithmetic. 9. Fluently add and subtract multidigit whole numbers using the standard algorithm. [4NBT4] M : Add and subtract within M : Apply signs +, , and = to actions of joining and separating sets. M : Add and subtract singledigit numbers. M : Recall basic addition and subtraction facts. 10. Multiply a whole number of up to four digits by a onedigit whole number, and multiply two twodigit 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. [4NBT5] M : Multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 5 = 40, one knows 40 5 = 8) or properties of operations. M : Multiply singledigit numbers. M : Recall basic multiplication facts. M : Apply concepts of multiplication through the use of manipulatives, number stories, skipcounting arrays, area of a rectangle, or repeated addition. Examples: Array 8 3 Repeated addition = Find wholenumber quotients and remainders with up to fourdigit dividends and onedigit 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. [4NBT6] M : Divide within 100, using strategies such as the relationship between multiplication and division (e.g. knowing that 8 x 5 = 40, one knows 40 5 = 8). M : Divide within 100, using strategies such as properties of operations. M : Multiply within 100, using strategies such as properties of operations. M : Multiply within 100, using strategies such as the relationship between multiplication and division (e.g. knowing that 8 x 5 = 4=, one knows 40 5 = 8). M : Recall products of two onedigit numbers. M : Name the first 10 multiples of each onedigit natural number. Example: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70 M : Recall basic addition, subtraction, and multiplication facts. Curriculum Guide to the Alabama Course of Study: Mathematics 47
5 Number and Operations Fractions (Grade 4 expectations in this domain are limited to fractions with denominations 2, 3, 4, 5, 6, 8, 10, 12, and 100.) Extend understanding of fraction equivalence and ordering. 12. Explain why a fraction a ( nxa) is equivalent to a fraction by using visual fraction models, with b ( nxb) 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. [4NF1] M : Define fraction, numerator and denominator. M : Recognize fraction 1 b equal parts. Example: as the quantity formed by 1 part when a whole is partitioned into b 2 6 M : Identify the parts of a fraction a b as the quantity formed by a parts and size 1 b. In the example above, a = 2 parts of the fraction numerator b = the whole part of the fraction (6 parts) denominator M : Recognize fractions as numerals that may represent division problems. M : Label numerator, denominator, and fraction bar. M : Identify parts of a whole with two, three, or four equal parts. M : Recognize that equal shares of identical wholes need not have the same shape. M : Distinguish between equal and nonequal parts. 13. 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. Recognize 2 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. [4NF2] M : Identify fraction 1 b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a b as the quantity formed by a parts and size 1. b M : Identify a fraction as a number on the number line; represent fractions on a number line diagram. M : Recognize a fraction as a number on the number line. M : Represent fractions on a number line diagram. M : Recognize fractions as numerals that may represent division problems. Curriculum Guide to the Alabama Course of Study: Mathematics 48
6 M : Label numerator, denominator, and fraction bar. M : Identify parts of a whole with two, three, or four equal parts. M : Distinguish between equal and nonequal parts. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 14. Understand a fraction a b with a > 1 as a sum of fractions 1 b. [4NF3] M : Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. M : Identify two fractions as equivalent (equal) if they are the same size or the same point on a number line. M : Recognize and generate simple equivalent fractions, e.g., 1 = 2, 4 = 2. Explain why the fractions are equivalent, e.g., by using a visual fraction model. M : Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3 ; recognize that = 6; locate 4 4 and 1 at the same point of a number line diagram. M : Label a fraction with multiple representations. M : Recognize pictorial representations of equivalent fractions. M : Recognize different interpretations of fractions, including parts of a set or a collection, points on a number line, numbers that lie between two consecutive whole numbers, and lengths of segments on a ruler. M : Recognize that a whole can be partitioned into differing equal parts (halves, fourths, eighths, etc.). a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. [4NF3a] M. 4.14a.1: Identify numerator and denominator. M. 4.14a.2: Recall basic addition and subtraction facts. 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. [4NF3b] Examples: 3 8 = ; 3 8 = ; = = M. 4.14b.1: Demonstrate an understanding of fractional parts. M. 4.14b.2: Recall basic addition and subtraction facts. Curriculum Guide to the Alabama Course of Study: Mathematics 49
7 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. [4NF3c] M. 4.14c.1: Define mixed numbers. M. 4.14c.2: Recall basic addition and subtraction facts. 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. [4NF3d] M. 4.14d.1: Demonstrate an understanding of fractional parts. M. 4.14d.2: Solve basic word problems using whole numbers. M. 4.14d.3: Express parts of a whole as a fraction. M. 4.14d.4: Write number sentences for word problems. M. 4.14d.5: Identify key terms in word problems. M. 4.14d.6: Recall basic addition and subtraction facts. 15. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. [4NF4] M : Recognize fractions in their simplest forms. M : Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. M : Demonstrate an understanding of fractional parts. M : Apply properties of operations as strategies to multiply and divide. M : Recall basic multiplication facts. a. Understand a fraction a b as a multiple of 1 b. [4NF4a] Example: Use a visual fraction model to represent 5 as the product 5 ( 1 ), recording the 4 4 conclusion by the equation 5 4 = 5 ( 1 4 ). M. 4.15a.1: Define multiple. M. 4.15a.2: Compare two fractions with the same numerator or the same denominator by reasoning about their size. M. 4.15a.3: Recognize that comparisons are valid only when the two fractions refer to the same whole. M. 4.15a.4: Record results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. M. 4.15a.5: Name the first ten multiples of each onedigit natural number. M. 4.15a.6: Recall basic multiplication facts. Curriculum Guide to the Alabama Course of Study: Mathematics 50
8 b. Understand a multiple of a as a multiple of 1, and use this understanding to multiply a fraction b b by a whole number. [4NF4b] Example: Use a visual fraction model to express 3 ( 2 ) as 6 ( 1 ), recognizing this product as 5 5 nxa 6. (In general, n ( a ) = ( ).) 5 b b M. 4.15b.1: Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3 ; recognize that = 6; locate 4 4 and 1 at the same point of a number line diagram. M. 4.15b.2: Solve simple fractions using multiplication strategies. M. 4.15b.3: Recognize equivalent forms of fractions. 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. [4NF4c] Example: If each person at a party will eat 3 of a pound of roast beef, and there will be 5 8 people at the party, how many pounds of roast beef will be needed? Between which two whole numbers does your answer lie? M. 4.15c.1: Multiply proper fractions with common denominators M. 4.15c.2: Solve word problems using whole numbers. M. 4.15c.3: Write number sentences for word problems. M. 4.15c.4: Identify key terms in word problems. M. 4.15c.5: Multiply and divide within 100. Curriculum Guide to the Alabama Course of Study: Mathematics 51
9 Understand decimal notation for fractions, and compare decimal fractions. 16. 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. (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.) [4NF5] Example: Express as, and add + 4 = M : Recognize equivalent forms of fractions and decimals. M : Demonstrate equivalent fractions using concrete objects or pictorial representation. M : Recognize pictorial representations of equivalent fractions and decimals in tenths and hundredths. Example: 0.60 = 0.6 M : Identify place value of decimals to the tenths and hundredths. M : Use place value understanding to round whole numbers to the nearest 10 or Use decimal notation for fractions with denominators 10 or 100. [4NF6] Example: Rewrite 0.62 as 62 ; describe a length as 0.62 meters; locate 0.62 on a number line diagram. 100 M : Define tenths, hundredths, decimal notation. M : Recognize equivalent forms of fractions and decimals. M : Recognize that endpoints locate a b on a number line. M : Identify place value of decimals to the tenths and hundredths. M : Label fraction parts. Examples: numerator, denominator, fraction bar M : Use place value understanding to round whole numbers to the nearest 10 or 100. Curriculum Guide to the Alabama Course of Study: Mathematics 52
10 18. 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. [4NF7] M : Compare two fractions with the same numerator or the same denominator by reasoning about their size. M : Recognize that comparisons are valid only when the two fractions refer to the same whole. M : Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. M : Convert fractions to decimals. M : Compare two decimals to tenths. M : Compare whole numbers. M : Identify comparison symbols. Examples: >, <, and = Measurement and Data Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. 19. Know relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, oz; l, ml; and 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 twocolumn table. [4MD1] Examples: 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) M : Define conversion. M : Define length, kilometers, meters and centimeters. M : Define weight, kilograms, grams, pounds, ounces, liters and milliliters. M : Define hour, minute, second. M : Measure and estimate liquid volumes and masses of objects using standard units of grams, kilograms, and liters. M : Identify standard units of measurement equivalents. Examples: 60 minutes equals 1 hour, 16 ounces equals 1 pound M : Match measurement units to abbreviations. Examples: kilometers (km), meters (m), centimeters (cm), kilograms (kg), grams (g), pounds (lb), ounces (oz), liters (l), milliliters (ml) Curriculum Guide to the Alabama Course of Study: Mathematics 53
11 20. 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. [4MD2] M : Define distance, time, elapsed time, volume, mass. M : Determine elapsed time to the day with calendars and to the hour with a clock. M : Express liquid volumes and masses of objects using standard units of grams, kilograms, and liters. M : Use addition, subtraction, multiplication and division to solve one and twostep word problems. M : Recognize key terms to solve word problems. M : Recall basic facts for addition, subtraction, multiplication, and division. M : Identify monetary equivalents. Examples: four quarters equal one dollar, five onedollar bills equals five dollars 21. Apply the area and perimeter formulas for rectangles in realworld and mathematical problems. [4MD3] 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. M : Recall the formula for area (L X W). M : Recognize that unit squares are equal. M : Recall the formula for perimeter (P= L+L+W+W or P=2L + 2W). M : Recall basic addition and multiplication facts. Represent and interpret data. 22. Make a line plot to display a data set of measurements in fractions of a unit ( 1, 1, 1 ). Solve problems involving addition and subtraction of fractions by using information presented in line plots. [4MD4] Example: From a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection. M : Display data by making a line plot where the horizontal scale is marked off in appropriate units whole numbers, halves, or quarters. M : Interpret data using graphs including bar, line, and circle graphs, and Venn diagrams. M : Identify the parts of a line plot. M : Recognize a line plot. M : Draw a scaled picture graph and a scaled bar graph to represent a data set. Curriculum Guide to the Alabama Course of Study: Mathematics 54
12 Geometric measurement: understand concepts of angle and measure angles. 23. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement. [4MD5] M : Define degree, angle, ray, and vertices. M : Recognize and draw shapes having specified attributes such as a given number of angles or a given number of equal faces. M : Estimate angle measures using 45, 90, 180, 270, or 360. M : Identify angle, ray, and vertices. M : Draw shapes to possess defining attributes. 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 measure angles. [4MD5a] of a circle is called a onedegree angle and can be used to M. 4.23a.1: Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. Example: Partition a shape into 4 parts with equal area, and describe the area of each part of the area of the shape. as 1 4 M. 4.23a.2: Partition circles and rectangles into two, three, or four equal shares; describe the shares using the words halves, thirds, half of, a third of, etc.; and describe the whole as two halves, three thirds, or four fourths. M. 4.23a.3: Recognize that equal shares of identical wholes need not have the same shape. M. 4.23a.4: Demonstrate equivalent fractions using concrete objects or pictorial representations. Examples: 2 6 = 1 3 Curriculum Guide to the Alabama Course of Study: Mathematics 55
13 b. An angle that turns through n onedegree angles is said to have an angle measure of n degrees. [4MD5b] M. 4.23b.1: Define center, radius, and diameter of a circle. M. 4.23b.2: Identify realworld examples of radius and diameter. Examples: bicycle wheel, pizza, pie M. 4.23b.3: Identify intervals of 1 between 0 and 5 on a protractor. M. 4.23b.4: Skip count by fives and tens on a protractor. 24. Measure angles in wholenumber degrees using a protractor. Sketch angles of specified measure. [4MD6] M : Define symmetry. M : Model using a protractor to draw angles. M : Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen. M : Measure the length of an object by selecting and using appropriate tools such as a ruler. M : Measure length using standard and nonstandard units of measurement. M : Plot points on grids, graphs, and maps using coordinates. M : Draw points, lines, line segments, and parallel and perpendicular lines, angles, and rays. M : Identify lines of symmetry on onedimensional figures. 25. Recognize angle measure as additive. When an angle is decomposed into nonoverlapping 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 realworld or mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. [4MD7] M : Identify straight angles. M : Recognize angle measures such as 45, 90, 180, 270, 300. M : Recall basic addition and subtraction facts. M : Skip count by fives and tens. Geometry Draw and identify lines and angles, and classify shapes by properties of their lines and angles. 26. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in twodimensional figures. [4G1] M : Define points, lines, line segments, rays, right angle, acute angle, obtuse angle, perpendicular lines, and parallel lines. M : Define twodimensional figure. M : Recognize onedimensional points, lines, and line segments. M : Model shapes in the world by building shapes from components. Curriculum Guide to the Alabama Course of Study: Mathematics 56
14 27. Classify twodimensional 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. [4G2] M : Define right angle. M : Recognize that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). M : Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. M : Recognize and draw shapes having specified attributes such as a given number of angles or a given number of equal faces. M : Identify triangles. 28. Recognize a line of symmetry for a twodimensional 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. [4G3] M : Identify line symmetric figures. M : Draw lines of symmetry on a onedimensional figure. M : Recognize lines of symmetry on a onedimensional figure. Curriculum Guide to the Alabama Course of Study: Mathematics 57
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