Problem 1
Find 2/3 + 1/4.
Show worked solution
8/12 + 3/12 = 11/12.
MathematicsGrade 5About 90 minutes
Add and subtract unlike fractions using equivalent units
Ohio learning standards this draft targets
math-2017:5.NF.1math-2017:5.NF.2
Read the standard text
math-2017:5.NF.1 Add and subtract fractions with unlike denominators (including mixed numbers and fractions greater than 1) 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, use visual models and properties of operations to show 2/3 + 5/4 = 8/12 + 15/12 = 23/12. In general, a/b + c/d = (a/b x d/d) + (c/d x b/b) = (ad + bc)/bd.
math-2017:5.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.
Study these connected lessons together: Add and subtract mixed numbers with justified exchanges.
Section 1 · LearnLearn the idea
Learning goal: Add and subtract fractions with unlike denominators by renaming them in equal units. Use a common whole, explain the equivalence and check the answer with benchmark fractions. The denominator names the size of each unit; the numerator counts how many units are present.
Scope: This lesson teaches the common-denominator method and the benchmark checks of 5.NF.1 and 5.NF.2, including a fraction greater than one. Adding and subtracting mixed numbers, which 5.NF.1 also names, is taught in the companion lesson on mixed-number sums and differences.
A recipe uses 2/3 cup of one ingredient and 1/4 cup of another. To find their combined volume in this simplified measuring context, add 2/3 + 1/4 cups. Thirds and fourths are different-sized units. Adding their counts without renaming would be like combining a count of feet with a count of inches and calling the result either feet or inches.
Partition each third into four equal parts. The whole cup is now divided into twelve equal parts, and two thirds contain eight twelfths. Thus 2/3 = 8/12. Partition each fourth into three equal parts, and one fourth becomes three twelfths: 1/4 = 3/12. The quantities have not changed, only the names of the units used to describe them.
Now combine like units: 8/12 + 3/12 = 11/12 cup. The denominator remains twelve because the parts remain twelfths. The model uses equal-length whole strips to preserve the same unit. Each full strip represents one cup, including unselected portions. The selected lengths total eleven of the twelve equal parts in a whole cup.
Multiplying numerator and denominator by the same nonzero whole number produces an equivalent fraction because it partitions each selected and unselected region in the same way. In 2/3 = (2 × 4)/(3 × 4), the number of counted parts grows fourfold while each part becomes one fourth as large. The original selected amount stays constant.
A common denominator need not be the least possible denominator to be valid. Twenty-fourths also work: 2/3 = 16/24 and 1/4 = 6/24, giving 22/24. That is equivalent to 11/12. Smaller common units can make the arithmetic less efficient, but a correct equivalent fraction still names the answer. Explain equivalence rather than treating simplification as a different amount.
For subtraction, compare 3/4 cup and 1/6 cup. Rename them as 9/12 and 2/12, then subtract to obtain 7/12 cup. The larger numerator counts how many equal pieces were available, and subtracting two twelfths leaves seven. The whole cup and the size of a twelfth stay fixed throughout.
Use benchmarks to test the result. Two thirds plus one fourth must exceed two thirds but remain below one whole because one fourth is less than the missing one third. Eleven twelfths meets these conditions. The incorrect answer 3/7 is less than one half, so it cannot equal a positive addition beginning with two thirds. Number sense can expose the error before a detailed calculation.
Section 2 · ModelWork through the examples
Worked addition: 3/5 + 1/2 = 6/10 + 5/10 = 11/10 = 1 1/10. The result exceeds one because six tenths and five tenths contain eleven tenth units. A fraction greater than one is valid; an answer need not be forced below one.
Worked subtraction with a fraction greater than one: 5/4 − 2/3 = 15/12 − 8/12 = 7/12. Five fourths is more than one whole, but it is renamed in twelfths exactly like a proper fraction.
Worked subtraction: 5/6 − 1/4 = 10/12 − 3/12 = 7/12. Check by adding 7/12 + 3/12 = 10/12 = 5/6. The inverse operation preserves the same whole.
Section 3 · DoPractice, demonstrate, and explain
Guided practice: Build equivalent fractions with paper strips of the same whole length. Overlay the partitions to find a common unit, then connect the counted regions to equations. Explain why changing only a denominator changes the amount.
Independent performance: Solve one addition and one subtraction story with unlike denominators and the same measurement unit. Show an equivalent-fraction model, equation and benchmark check. Create a plausible-looking wrong answer and explain the unit mistake. Assess model consistency, equivalence reasoning and arithmetic separately.
Section 4 · Check8-question lesson practice test
An adult may read the prompts aloud. Answer independently with words, objects, drawings or a demonstration before opening the answer guide. Award 1 point for a correct result or claim and 1 point for a matching explanation or demonstration; where a question asks only for a result (a corrected sentence, a spelling, a count), a fully correct result earns both points, and the answer guide shows what a complete response contains. Record prompting separately. The total (two points per question) is local practice feedback, not an Ohio proficiency level.
Problem 2
In 2/3 + 1/4, why can’t you add the numerators and the denominators to obtain 3/7?
Show worked solution
Thirds and fourths are different-sized units, so their counts cannot be combined until both are renamed as the same unit (twelfths). Adding 3 + 4 to get sevenths changes the unit size instead; 3/7 is even less than 2/3, the larger addend.
Problem 3
Find 3/4 − 1/6.
Show worked solution
9/12 − 2/12 = 7/12.
Problem 4
Is 22/24 a valid equivalent answer to 2/3 + 1/4?
Show worked solution
Yes. It names the same amount as 11/12, although twelfths give a more compact calculation.
Problem 5
Find 3/5 + 1/2.
Show worked solution
11/10, or 1 1/10, after renaming the addends as 6/10 and 5/10.
Problem 6
Use a benchmark to reject 2/3 + 1/4 = 3/7.
Show worked solution
3/7 is below one half, but adding a positive amount to two thirds must produce more than two thirds.
Problem 7
A trail mix uses 3/4 cup of oats and 2/3 cup of raisins, each measured separately. What is the sum of the two measured amounts, before mixing? Check with a benchmark.
Show worked solution
9/12 + 8/12 = 17/12 = 1 5/12 cups. (Once mixed, small pieces can settle into gaps, so the mixture may fill a little less than this.) Each amount is more than half a cup, so a total above one cup is reasonable.
Problem 8
A jug contains 5/6 liter of water. After 3/8 liter is poured out, how much water is left?
Show worked solution
20/24 − 9/24 = 11/24 liter. Both amounts are fractions of the same liter. Benchmark check: 5/6 − 1/3 = 1/2, and 3/8 is more than 1/3, so removing 3/8 must leave less than 1/2. 11/24 is just under 12/24, or one half, so it fits.