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.)
Official wording from the Common Core State Standards for Mathematics (Β© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Fractions can only be added or subtracted directly when they count the same size of piece. Thirds and fourths are different units, so 2/3 + 1/4 cannot be found by adding tops and bottoms. Fifth graders rewrite both fractions as equivalent fractions with a shared denominator: 2/3 = 8/12 and 1/4 = 3/12, so the sum is 11/12. The pieces are now all twelfths, and twelfths can be counted.
The same idea works for mixed numbers. To find 3 1/2 - 1 2/3, students can rename the fractions as sixths (3 3/6 - 1 4/6), regroup one whole as 6/6 when the fraction part is too small, and subtract to get 1 5/6. Another valid route converts both mixed numbers to fractions greater than 1. Any common denominator works; the least common denominator often keeps numbers smaller, but finding it is not required. The general pattern a/b + c/d = (ad + bc)/bd shows why multiplying the denominators always produces a common one.
Writing 1/2 + 1/3 = 2/5 treats the fractions like whole numbers. A quick check shows 2/5 is less than 1/2, so the sum cannot be right.
Students who rewrite 2/3 as 2/12 have changed the size of the pieces without changing how many there are. Both numerator and denominator must be multiplied by 4 to get 8/12.
In 4 1/5 - 2 3/5 some students compute 3/5 - 1/5 and get 2 2/5. The top fraction is smaller, so a whole must be regrouped: 3 6/5 - 2 3/5 = 1 3/5.
Any common multiple works. Using 24 instead of 12 for thirds and eighths still gives a correct, equivalent answer that can be simplified.
Find 2/3 + 5/4 and write the answer as a mixed number.
Answer: 2/3 + 5/4 = 23/12 = 1 11/12.
Fraction strips or area models that are cut into a common unit make the need for a common denominator obvious. Once students see that thirds and fourths both fit into twelfths, the symbolic method follows naturally.
Items include straight computations, mixed-number subtraction that requires regrouping, and error analysis such as explaining why a student's 3/8 + 1/4 = 4/12 is wrong.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 5/6
Use sixths: 1/2 = 3/6 and 1/3 = 2/6. 3/6 + 2/6 = 5/6.
Answer: 7/12
Use twelfths: 3/4 = 9/12 and 1/6 = 2/12. 9/12 - 2/12 = 7/12.
Answer: D) 3 9/10
Use tenths: 1/2 = 5/10 and 2/5 = 4/10. Whole numbers 2 + 1 = 3, fractions 5/10 + 4/10 = 9/10. The sum is 3 9/10.
Answer: 2 7/12
Use twelfths: 4 4/12 - 1 9/12. 4/12 is too small, so regroup: 3 16/12 - 1 9/12 = 2 7/12.
Answer: B) 5/8
1/4 = 2/8, so 3/8 + 2/8 = 5/8. Adding the denominators gave twelfths, which are the wrong size of piece.
Answer: 1 5/24
Use 24ths: 5/6 = 20/24 and 3/8 = 9/24. 20/24 + 9/24 = 29/24 = 1 5/24.
A full lesson with slides, activities and an exit ticket on adding and subtracting unlike fractions, pitched to grade 5 and editable in PowerPoint or Google Slides.
Make a lesson βA printable, differentiated worksheet on 5.NF.A.1 with an answer key, ready in about a minute.
Make a worksheet βTurn adding and subtracting unlike fractions into a quiz students answer online that marks itself, with a class summary for you.
Build a test βNo. Any common denominator gives a correct answer. The least common denominator is often more efficient, and answers can be simplified afterward.
Yes. The standard explicitly includes adding and subtracting mixed numbers with unlike denominators.