5.NF.1
“Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)”
Questions 1 to 4 rewrite thirds, fourths, fifths and sixths over common denominators of 12, 10 and 15, with regrouping in question 4.
5.NF.2
“Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 <1/2.”
Question 5 uses the 1/2 benchmark: Marcus's 4/10 sits below 1/2, so it cannot be the sum of 3/8 and 1/2.
5.NF.3
“Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?”
Question 10 turns 9 ÷ 4 into 9/4 = 2 1/4 pounds of trail mix and asks which two whole numbers the answer lies between.
5.NF.4.a
“Interpret the product (a/b) x q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) x (c/d) = ac/bd.)”
Question 6 multiplies numerators and denominators directly: 3/4 × 5/6 = 15/24 = 5/8 and 2/5 × 15 = 6.
5.NF.4.b
“Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.”
Question 7 finds the area of a 3/4 yard by 2/3 yard garden bed, which simplifies to 1/2 square yard.
5.NF.5.a
“Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.”
Question 8's table compares three products, including 6 × 3/5 and 4/7 × 5/5, to the first factor without computing.
5.NF.5.b
“Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.”
The Reason column in question 8 asks students to justify each comparison, including why a factor of 5/5 changes nothing.
5.NF.6
“Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.”
The garden bed in question 7 is the real-world multiplication problem on this test.
5.NF.7.a
“Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3)÷4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3)÷4 = 1/12 because (1/12) × 4 = 1/3.”
Questions 9a and 9c compute 1/4 ÷ 3 and 1/6 ÷ 2, and question 12 divides half a brownie pan among 4 classmates to get 1/8.
5.NF.7.b
“Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4÷(1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4÷(1/5) = 20 because 20 × (1/5) = 4.”
Question 9b computes 5 ÷ 1/3 = 15, and question 11 uses the same idea to get 24 batches from 6 cups of flour.
5.NF.7.c
“Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?”
Questions 11 and 12 are the story versions: 1/4-cup flour batches and equal shares of Maya's leftover brownies.
