Compose and Decompose to Add and Subtract

10 min

Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for subtracting fractions and mixed numbers, particularly cases where it is necessary to rewrite a whole number as a fraction, or decompose it into a different whole number and fraction, in order to perform subtraction. These understandings will be helpful later in this lesson when students subtract multi-digit numbers that involve decomposing units.

Launch

  • Display one expression.
  • “Give me a signal when you have an answer and can explain how you got it.”
Teacher Instructions
  • 1 minute: quiet think time
  • Record answers and strategy.
  • Keep expressions and work displayed.
  • Repeat with each expression.

Student Task

Find the value of each expression mentally.

  • 234−1142\frac{3}{4} - 1\frac{1}{4}
  • 114−341\frac{1}{4} - \frac{3}{4}
  • 518−2385\frac{1}{8} - 2\frac{3}{8}
  • 3210−27103\frac{2}{10} - 2\frac{7}{10}

Sample Response

  • 1241\frac{2}{4} or 1121\frac{1}{2}: I subtract the whole numbers and then subtract the fractions.
  • 24\frac{2}{4} or 12\frac{1}{2}: I take 34\frac{3}{4} from 1 or 44\frac{4}{4}, and then add 14+14\frac{1}{4} + \frac{1}{4}
  • 2682\frac{6}{8} or 2342\frac{3}{4}: I subtract the whole numbers to get 318−383\frac{1}{8} - \frac{3}{8}. Then I rewrite 3183\frac{1}{8} as 2+982 + \frac{9}{8}, and then subtract 38\frac{3}{8}
  • 510\frac{5}{10} or 12\frac{1}{2}: I subtract the whole numbers to get 1210−7101\frac{2}{10} - \frac{7}{10}. Then I rewrite it as 1210−710\frac{12}{10} - \frac{7}{10}.

Synthesis

  • Highlight strategies in which students decomposed the first mixed number in each expression.
  • Consider asking:
    • “Who can restate _______ 's reasoning in a different way?”
    • “Did anyone have the same strategy but would explain it differently?”
    • “Did anyone approach the expression in a different way?”
    • “Does anyone want to add on to____’s strategy?”
Standards
Addressing
  • 4.NF.3.c·Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
  • 4.NF.B.3.c·Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

15 min

20 min