Scholastic Fraction Nation

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2 Scholastic Correlated to the Grade 4 Number Operations and the Problems They Solve Problem solving with the four operations 3. Solve problems posed with both whole numbers and fractions. Understand that while quantities in a problem might be described with whole numbers, fractions, or decimals, the operations used to solve the problem depend on the relationships between the quantities regardless of which number representations are involved. 4. Assess the reasonableness of answers using mental computation and estimation strategies including rounding to the nearest 10 or 100. Number Fractions Operations on fractions 1. Understand addition of fractions: a. Adding or subtracting fractions with the same denominator means adding or subtracting copies of unit fractions. For example, 2/3 + 4/3 is 2 copies of 1/3 plus 4 copies of 1/3, or 6 copies of 1/3 in all, that is 6/3. b. Sums of related fractions can be computed by replacing one with an Lesson 53: Use a number line to add and subtract fractions. Demonstrate addition and subtraction of fractions on a number line (same denominator). Lesson 54: Add and subtract fractions with the same denominator. Add and subtract fractions and locate the sum and difference on a number line (same denominator). Lesson 55: Add mixed numbers with the same denominator. Add mixed numbers and locate the sum on a number line (same denominator). Lesson 56: Subtract mixed numbers with the same denominator. line (same denominator). Lesson 59: Find common denominators. Name the least common denominator of a pair of fractions. Lesson 60: Add fractions with different denominators. Add fractions and locate the sum on a number line (unlike Lesson 61: Subtract fractions with different denominators. Subtract fractions and locate the difference on a number line (unlike Lesson 62: Add mixed numbers with different denominators. Add mixed numbers and locate the sum on a number line (unlike Lesson 63: Subtract mixed numbers with different denominators. line (unlike Lesson 58: Estimate sums and differences of fractions with different denominators. Estimate sums and differences of fractions by locating the estimate on a number line (unlike Lesson 53: Use a number line to add and subtract fractions. Demonstrate addition and subtraction of fractions on a number line (same denominator). Lesson 54: Add and subtract fractions with the same denominator. Add and subtract fractions and locate the sum and difference on a number line (same denominator). Lesson 59: Find common denominators. Name the least common denominator of a pair of fractions. Lesson 60: Add fractions with different denominators. and /Page 1

3 Scholastic Correlated to the Grade 4 equivalent fraction that has the same denominator as the other. For example, the sum of the related fractions 2/3 and 1/6 can be computed by rewriting 2/3 as 4/6 and computing 4/6 + 1/6 = 5/6. 2. Compute sums and differences of fractions with like denominators, add and subtract related fractions within 1 (e.g., 1/2 + 1/4, 3/10 + 4/100, 7/8 1/4), and solve word problems involving these operations. 3. Understand that the meaning of multiplying a fraction by a whole number comes from interpreting multiplication by a whole number as repeated addition. For example, 3 x 2/5 = 6/5 because 3 x 2/5 = 2/5 + 2/5 + 2/5 = 6/5. Decimal concepts 7. Understand that a two-digit decimal is a sum of fractions with denominators 10 and 100. For example, 0.34 is 3/10 + 4/ Use decimals to hundredths to describe parts of wholes; compare and order decimals to hundredths based on meanings of the digits; and write fractions of the form a/10 or a/100 in decimal notation. Use > and < symbols to record the results of comparisons. Add fractions and locate the sum on a number line (unlike Lesson 53: Use a number line to add and subtract fractions. Demonstrate addition and subtraction of fractions on a number line (same denominator). Lesson 54: Add and subtract fractions with the same denominator. Add and subtract fractions and locate the sum and difference on a number line (same denominator). Lesson 40: Name equivalent fractions by multiplication. Identify equivalent fractions by using the multiplicative identity. Lesson 21: Introduction to decimal numbers. Express a fraction less than one as a decimal by using a whole Lesson 22: Identify decimal tenths. Express a fraction greater than one as a decimal by using a whole Lesson 24: Identify decimal hundredths. Express a fraction as a decimal by using a whole divided into equal parts (hundredths). Lesson 26: Estimate decimal numbers. Estimate decimal magnitude using whole-number benchmarks (up to 10). Lesson 47: Introduction to equivalent decimals. Express equivalent decimals to the thousandths place value. Lesson 48: Use place value to name equivalent decimals. Express equivalent decimals using expanded forms of place value. Lesson 51: Express fractions as equivalent decimals. Express fractions as equivalent decimals by using denominators of 10 and 100. Lesson 52: Express fractions as equivalent decimals through division. Express fractions as equivalent decimals by representing fractions as a division expression. Lesson 21: Introduction to decimal numbers. Express a fraction less than one as a decimal by using a whole Lesson 22: Identify decimal tenths. Express a fraction greater than one as a decimal by using a whole Lesson 23: Compare unit fractions to decimals. Compare a pair of numbers, one of which is a unit fraction and one of which is a decimal, and identify the greater fraction. and /Page 2

4 Scholastic Correlated to the Grade 4 Measurement and Data The number line and units of measure 1. Understand that the unit length on a number line (interval from 0 to 1) can be divided into parts of equal fractional length. Draw number line representations of problem situations involving length, height, and distance including fractional or decimal units. For example, show distances along a race course to tenths of a mile on a number line, by dividing the unit length into 10 equal parts to get parts of length 1/10; the endpoint of the segment of 1/10 length from 0 represents 1/10 of a mile from the starting point of the race. In Grade 4, all numbers lines begin with zero. Lesson 24: Identify decimal hundredths. Express a fraction as a decimal by using a whole divided into equal parts (hundredths). Lesson 25: Identify decimals as part of an area. Identify a decimal less than or greater than 1, using the area of a square (tenths, hundredths). Lesson 27: Compare decimal numbers. Compare decimal tenths and decimal hundredths and identify the greater fraction. Lesson 49: Name a decimal between two numbers. Use equivalence to identify a decimal between two decimals. Lesson 50: Compare decimals by using equivalence. Compare decimals by using place value. Lesson 51: Express fractions as equivalent decimals. Express fractions as equivalent decimals by using denominators of 10 and 100. Lesson 52: Express fractions as equivalent decimals through division. Express fractions as equivalent decimals by representing fractions as a division expression. Lesson 16: Introduction to number lines. Locate the position of whole numbers on number lines with various features (start and end points, tick marks, labels). Lesson 17: Use a number line with fractions. Locate the position of proper and improper fractions on number lines with various features (start and end points, tick marks, labels). and /Page 3

5 Scholastic to the Grade 5 Number Base Ten Decimal concepts 5. Read, write, and compare numbers expressed as decimals. Understand that a digit in one place represents ten times what it represents in the place to its right. For example, 7 in the hundredths place represents 10 times as many as 7 in the thousandths place. 6. Round decimals (to hundredths) to the nearest whole number. 7. Write fractions in decimal notation for fractions with denominators 2, 4, 5, 8, 10, and 100. Operations on decimals 8. Understand that in adding or subtracting finite decimals, one adds or subtracts like units (tenths and tenths, hundredths and hundredths, etc.) and sometimes it is necessary to compose or decompose a higher value unit. 9. Fluently find 0.1 more than a number and less than a number; 0.01 more than a number and less than a number; and more than a number and less than a number, for numbers expressed as finite decimals. 11. Compute sums, differences, products, and quotients of finite decimals using strategies based on Lesson 21: Introduction to decimal numbers. Express a fraction less than one as a decimal by using a whole Lesson 22: Identify decimal tenths. Express a fraction greater than one as a decimal by using a whole Lesson 24: Identify decimal hundredths. Express a fraction as a decimal by using a whole divided into equal parts (hundredths). Lesson 27: Compare decimal numbers. Compare decimal tenths and decimal hundredths and identify the greater fraction. Lesson 47: Introduction to equivalent decimals. Express equivalent decimals to the thousandths place value. Lesson 48: Use place value to name equivalent decimals. Express equivalent decimals using expanded forms of place value. Lesson 49: Name a decimal between two numbers. Use equivalence to identify a decimal between two decimals. Lesson 50: Compare decimals by using equivalence. Compare decimals by using place value. Lesson 26: Estimate decimal numbers. Estimate decimal magnitude using whole-number benchmarks (up to 10). Lesson 23: Compare unit fractions to decimals. Compare a pair of numbers, one of which is a unit fraction and one of which is a decimal, and identify the greater fraction. Lesson 51: Express fractions as equivalent decimals. Express fractions as equivalent decimals by using denominators of 10 and 100. Lesson 52: Express fractions as equivalent decimals through division. Express fractions as equivalent decimals by representing fractions as a division expression. and /Page 1

6 Scholastic to the Grade 5 place value, the properties of operations, and/or the inverse relationships between addition and subtraction and between multiplication and division; explain the reasoning used. For example, transform into 15 3 = Explain why strategies and algorithms for computations with finite decimals work. Include explanations supported by drawings, equations, or both. A range of reasonably efficient algorithms may be covered, not only the standard algorithm. 13. Use the standard algorithm for each of the four operations on decimals (to hundredths). Number Fractions Fraction equivalence 1. Understand fraction equivalence: a. Multiplying the numerator and denominator of a fraction by the same nonzero whole number produces an equivalent fraction. For example, 2/3 = (2 x 4)/(3 x 4) = 8/12. (1/3 is 4 copies of 1/12, so 2/3 is 8 copies of 1/12.) b. Equivalent fractions correspond to the same point on a number line. In Grade 5, all numbers lines begin with zero. c. When the numerators of equivalent fractions are their denominators, the resulting quotients are the same. Lesson 28: Express whole numbers as equivalent fractions. Express whole numbers (up to 10) as equivalent fractions. Lesson 29: Express fractions as equivalent whole numbers. Express fractions as equivalent whole numbers (up to 10). Lesson 32: Express mixed numbers as equivalent fractions. Express mixed numbers (up to 10) as equivalent improper fractions. Lesson 33: Express fractions as equivalent mixed numbers. Express improper fractions as equivalent mixed numbers (up to 10). Lesson 34: Name equivalent fractions for ½. Name fractions that are equivalent to ½. Lesson 35: Name equivalent fractions for unit fractions. Name fractions equivalent to unit fractions with denominators up to 24. Lesson 36: Name more equivalent fractions. Name equivalent fractions with denominators up to 24. Lesson 37: Identify equivalent fractions of an area. Identify equivalent fractions for fractions represented by the area of a circle and a rectangle. Lesson 38: Identify equivalent fractions of a set. Identify equivalent fractions for unit fractions represented by a set divided into equal parts. Lesson 39: Identify more equivalent fractions of a set. Identify equivalent fractions for proper fractions represented by a set divided into equal parts. Lesson 40: Name equivalent fractions by multiplication. Identify equivalent fractions by using the multiplicative identity. Lesson 44: Determine if fractions are equivalent. Express fractions with common denominators to determine equivalence. and /Page 2

7 Scholastic to the Grade 5 2. Identify pairs of equivalent fractions; given two fractions with unlike denominators, find two fractions with the same denominator and equivalent to each. 3. Compare and order fractions with like or unlike denominators, e.g., by finding equivalent fractions with the same denominator, and describe the sizes of fractional quantities from a context with reference to the context. Compare using the fractions themselves, tape diagrams or number line representations, and area models. Operations on fractions 4. Understand that sums and differences of fractions with unlike denominators can be computed by replacing each with an equivalent fraction so that the resulting fractions have the same denominator. For example, 2/3 + 5/4 = 8/ /12 = 23/ Compute sums and differences of fractions with like or unlike denominators, and solve word problems involving addition and subtraction of fractions. Estimate fraction sums and differences to assess the reasonableness of results. Lesson 31: Compare fractions by using equivalence. Compare improper fractions by using equivalence to identify whole-number benchmarks. Lesson 40: Name equivalent fractions by multiplication. Identify equivalent fractions by using the multiplicative identity. Lesson 44: Determine if fractions are equivalent. Express fractions with common denominators to determine equivalence. Lesson 45: Compare fractions by using equivalence. Compare fractions by using equivalence to express fractions with a common denominator. Lesson 4: Compare unit fractions. Compare unit fractions and identify the greater fraction. Lesson 10: Compare fractions with the same denominator. Compare fractions with the same denominator and identify the greater fraction. Lesson 11: Compare fractions with the same numerator. Compare fractions with the same numerator and identify the greater fraction. Lesson 31: Compare fractions by using equivalence. Compare improper fractions by using equivalence to identify whole-number benchmarks. Lesson 45: Compare fractions by using equivalence. Compare fractions by using equivalence to express fractions with a common denominator. Lesson 46: Name a fraction between two other fractions. Identify a fraction between two fractions with like denominators by using equivalent forms of the two fractions. Lesson 59: Find common denominators. Name the least common denominator of a pair of fractions. Lesson 60: Add fractions with different denominators. Add fractions and locate the sum on a number line (unlike Lesson 61: Subtract fractions with different denominators. Subtract fractions and locate the difference on a number line (unlike Lesson 62: Add mixed numbers with different denominators. Add mixed numbers and locate the sum on a number line (unlike Lesson 63: Subtract mixed numbers with different denominators. line (unlike Lesson 53: Use a number line to add and subtract fractions. Demonstrate addition and subtraction of fractions on a number line (same denominator). Lesson 54: Add and subtract fractions with the same denominator. Add and subtract fractions and locate the sum and difference on a number line (same denominator). Lesson 55: Add mixed numbers with the same denominator. Add mixed numbers and locate the sum on a number line (same denominator). Lesson 56: Subtract mixed numbers with the same denominator. and /Page 3

8 Scholastic to the Grade 5 6. Understand that multiplying a fraction by a/b means taking a parts of a decomposition of the fraction into b equal parts. For example, to multiply 2/3 x 4/5 = 8/15, one may decompose a whole of size 4/5 into 3 equal parts; each part has size 4/15. Two of these parts then make 8/15, so 2/3 x 4/5 = 8/15. (In general, a/b xp/q = ap/bq.) This standard includes multiplication of a whole number by a fraction, by writing the whole number as fraction with denominator Explain and justify the properties of operations with fractions, e.g., by using equations, number line representations, area models, and story contexts. line (same denominator). Lesson 58: Estimate sums and differences of fractions with different denominators. Estimate sums and differences of fractions by locating the estimate on a number line (unlike Lesson 59: Find common denominators. Name the least common denominator of a pair of fractions. Lesson 60: Add fractions with different denominators. Add fractions and locate the sum on a number line (unlike Lesson 61: Subtract fractions with different denominators. Subtract fractions and locate the difference on a number line (unlike Lesson 62: Add mixed numbers with different denominators. Add mixed numbers and locate the sum on a number line (unlike Lesson 63: Subtract mixed numbers with different denominators. line (unlike Lesson 40: Name equivalent fractions by multiplication. Identify equivalent fractions by using the multiplicative identity. Lesson 53: Use a number line to add and subtract fractions. Demonstrate addition and subtraction of fractions on a number line (same denominator). Lesson 54: Add and subtract fractions with the same denominator. Add and subtract fractions and locate the sum and difference on a number line (same denominator). Lesson 55: Add mixed numbers with the same denominator. Add mixed numbers and locate the sum on a number line (same denominator). Lesson 56: Subtract mixed numbers with the same denominator. line (same denominator). and /Page 4

9 Scholastic to the Grade Understand that a mixed number such as 3 2/5 represents the sum of a whole number and a fraction less than one. Because a whole number can be represented as a fraction (3 = 3/1), and the sum of two fractions is also a fraction, a mixed number also represents a fraction (3 2/5 = 3 + 2/5 = 15/5 + 2/5 = 17/5). Write fractions as equivalent mixed numbers and vice versa. Lesson 58: Estimate sums and differences of fractions with different denominators. Estimate sums and differences of fractions by locating the estimate on a number line (unlike Lesson 59: Find common denominators. Name the least common denominator of a pair of fractions. Lesson 60: Add fractions with different denominators. Add fractions and locate the sum on a number line (unlike Lesson 61: Subtract fractions with different denominators. Subtract fractions and locate the difference on a number line (unlike Lesson 62: Add mixed numbers with different denominators. Add mixed numbers and locate the sum on a number line (unlike Lesson 63: Subtract mixed numbers with different denominators. line (unlike Lesson 30: Estimate improper fractions using whole numbers. Estimate improper-fraction magnitude using whole-number benchmarks (up to 10). Lesson 31: Compare fractions by using equivalence. Compare improper fractions by using equivalence to identify whole-number benchmarks. Lesson 32: Express mixed numbers as equivalent fractions. Express mixed numbers (up to 10) as equivalent improper fractions. Lesson 33: Express fractions as equivalent mixed numbers. Express improper fractions as equivalent mixed numbers (up to 10). and /Page 5

10 Scholastic Correlated to the Grade 6 The Number System The system of rational numbers 5. Understand that a number is a point on the number line. 8. Find and position rational numbers, including integers, on a number line. Lesson 16: Introduction to number lines. Locate the position of whole numbers on number lines with various features (start and end points, tick marks, labels). Lesson 17: Use a number line with fractions. Locate the position of proper and improper fractions on number lines with various features (start and end points, tick marks, labels). Lesson 16: Introduction to number lines. Locate the position of whole numbers on number lines with various features (start and end points, tick marks, labels). Lesson 17: Use a number line with fractions. Locate the position of proper and improper fractions on number lines with various features (start and end points, tick marks, labels). Lesson 21: Introduction to decimal numbers. Express a fraction less than one as a decimal by using a whole Lesson 23: Compare unit fractions to decimals. Compare a pair of numbers, one of which is a unit fraction and one of which is a decimal, and identify the greater fraction. and /Page 1

11 Scholastic Correlated to the Grade 7 The Number System The system of rational numbers 2. Understand and perform addition and subtraction with rational numbers: a. Understand that on a number line, the sum p + q is the number located a distance q from p, to the right of p if q is positive and to the left of p if q is negative. A number and its opposite are additive inverses (i.e., their sum is zero). b. Compute sums of signed numbers using the laws of arithmetic. For example, 7 + ( 3) = 4 because 7 + ( 3) = (4 + 3) + ( 3) = 4 + [3 + ( 3)] = 4 + [0] = 4. c. Understand that subtraction of rational numbers is defined by viewing a difference as the solution of an unknown-addend addition problem. Subtraction of a rational number gives the same answer as adding its additive inverse. d. Explain and justify rules for adding and subtracting rational numbers, using a number line and practical contexts. For example, relate r + ( s) = r s to a bank transaction; explain why p (q + r) = p q r. e. Understand that the additive inverse of a sum is the sum of the additive inverses, that is (p + q) = p + q. For example, (6 + 2) = ( 6) + 2 because [6 + ( 2)] + [( 6) + 2] = [6 + ( 6)] + [ ( 2) + 2] = [0] + [0] = 0. Lesson 53: Use a number line to add and subtract fractions. Demonstrate addition and subtraction of fractions on a number line (same denominator). Lesson 54: Add and subtract fractions with the same denominator. Add and subtract fractions and locate the sum and difference on a number line (same denominator). Lesson 55: Add mixed numbers with the same denominator. Add mixed numbers and locate the sum on a number line (same denominator). Lesson 56: Subtract mixed numbers with the same denominator. line (same denominator). Lesson 58: Estimate sums and differences of fractions with different denominators. Estimate sums and differences of fractions by locating the estimate on a number line (unlike Lesson 59: Find common denominators. Name the least common denominator of a pair of fractions. Lesson 60: Add fractions with different denominators. Add fractions and locate the sum on a number line (unlike Lesson 61: Subtract fractions with different denominators. Subtract fractions and locate the difference on a number line (unlike Lesson 62: Add mixed numbers with different denominators. Add mixed numbers and locate the sum on a number line (unlike Lesson 63: Subtract mixed numbers with different denominators. line (unlike and /Page 1

12 Scholastic Correlated to the Grade 8 The Number System The system of real numbers 1. Understand informally that every number on a number line has a decimal expansion, which can be found for rational numbers using long division. Rational numbers are those with repeating decimal expansions (this includes finite decimals which have an expansion that ends in a sequence of zeros). Lesson 21: Introduction to decimal numbers. Express a fraction less than one as a decimal by using a whole Lesson 22: Identify decimal tenths. Express a fraction greater than one as a decimal by using a whole Lesson 23: Compare unit fractions to decimals. Compare a pair of numbers, one of which is a unit fraction and one of which is a decimal, and identify the greater fraction Lesson 24: Identify decimal hundredths. Express a fraction as a decimal by using a whole divided into equal parts (hundredths). Lesson 47: Introduction to equivalent decimals. Express equivalent decimals to the thousandths place value. Lesson 48: Use place value to name equivalent decimals. Express equivalent decimals using expanded forms of place value. Lesson 49: Name a decimal between two numbers. Use equivalence to identify a decimal between two decimals. Lesson 50: Compare decimals by using equivalence. Compare decimals by using place value. Lesson 51: Express fractions as equivalent decimals. Express fractions as equivalent decimals by using denominators of 10 and 100. Lesson 52: Express fractions as equivalent decimals through division. Express fractions as equivalent decimals by representing fractions as a division expression. and /Page 1