Trail-Mix Fractions: 5/7, 5/12, and 17/12
Mrs. Adeyemi's Grade 5 class is scaling a trail-mix recipe. One batch uses 3/4 cup of raisins and 2/3 cup of oats. Students answer three questions: how much of each ingredient is needed for 1 1/2 batches, how much of the two ingredients together is in one batch, and how many 1/4-cup servings of raisins are in 3 cups.
Work samples from the quiz:
- Priya wrote 3/4 + 2/3 = 5/7 cup for one batch.
- Marco wrote 3/4 + 2/3 = 5/12 cup.
- Jade wrote 3/4 + 2/3 = 17/12, then renamed it as 1 5/12 cups.
- Tessa wrote 3 ÷ 1/4 = 3/4 and 1/4 ÷ 3 = 1/12.
Mrs. Adeyemi has fraction strips, two number lines (one marked in fourths, one in twelfths), and a set of measuring cups. She can plan one 20-minute intervention before the performance task, and she wants it aimed at the misconception shared by the most students rather than at the single most dramatic wrong answer.
1. Identify the misconception Priya and Marco share, choose the model that best confronts it, and explain how Jade's paper shows the target understanding.
Read the worked answer
Priya computed (3 + 2)/(4 + 3) = 5/7; Marco computed 5/12 by adding numerators and multiplying denominators. Both treat the numerator and denominator as separate whole numbers to be operated on, instead of as counts of equal-size pieces. That is the shared misconception, and it is the one to attack.
The fraction strips or the twelfths number line is the right model. Lay a 3/4 strip and a 2/3 strip end to end against the twelfths strip: 3/4 = 9/12 and 2/3 = 8/12, so the total is 17/12. The strips make two things visible that symbols alone do not: the pieces must be the same size before they can be counted, and the sum of two positive fractions must be larger than either addend.
That last check is something students can own. 5/7 is less than 3/4 and 5/12 is less than 2/3, so both answers are impossible on their face. Jade's paper shows the target: a common unit, equivalent fractions, add the counts, then rename 17/12 as 1 5/12 because the pieces exceed a whole.
2. Compute the amount of each ingredient for 1 1/2 batches and explain, using the size of the factor, why each result compares to the one-batch amount the way it does.
Read the worked answer
1 1/2 batches means 3/2 copies of each one-batch amount.
- Raisins: 3/2 × 3/4 = 9/8 = 1 1/8 cups.
- Oats: 3/2 × 2/3 = 6/6 = 1 cup.
Both products are larger than the single-batch amount, because the factor 3/2 is greater than 1: each ingredient is counted once and then half again. That is the general principle students should state - multiplying by a factor greater than 1 increases, by a factor less than 1 decreases, by exactly 1 leaves the amount unchanged.
Reasonableness bounds for each: 1 1/2 × 3/4 must fall between 3/4 and 1 1/2, and 1 1/8 does. 1 1/2 × 2/3 must fall between 2/3 and 1, and it lands exactly on 1 because two thirds plus half of two thirds is 2/3 + 1/3. A student who writes 9/8 for both, or who tries to add 3/4 + 1/2 by adding across, can be sent back to the bounds rather than to a rule.
3. Contrast 3 ÷ 1/4 with 1/4 ÷ 3: give a story context for each, state both answers, and diagnose what produced Tessa's 3 ÷ 1/4 = 3/4.
Read the worked answer
3 ÷ 1/4 = 12. Context: how many 1/4-cup servings of raisins can be scooped from 3 cups? Each cup holds four quarter-cups, so three cups hold twelve. The quotient is larger than the dividend because the question counts groups, and the groups are small.
1/4 ÷ 3 = 1/12. Context: three bowls share 1/4 cup of raisins equally - how much does each bowl get? Cutting a quarter cup into three equal parts gives twelfths, so each bowl gets 1/12 cup. The quotient is smaller than the dividend because a small amount is being shared.
The two problems look alike but the divisor plays a different role in each, which is exactly why Grade 5 separates 'divide by a unit fraction' from 'divide a unit fraction by a whole number.'
Tessa's 3/4 suggests she multiplied 3 by 1/4 instead of asking how many quarter-cups fit in 3. A number-line count - four 1/4 jumps per cup, twelve jumps to reach 3 - confronts that directly, and pairs naturally with 1/4 ÷ 3 as one jump cut into three pieces.