Ways to Find Unknown Length (Part 2)

10 min

Narrative

The purpose of this Warm-up is to elicit strategies and understandings students have for adding, subtracting, and multiplying fractions and mixed numbers. The series of equations prompt students to use properties of operations (associative and commutative properties in particular) in their reasoning, which will be helpful when students solve geometric problems involving fractional lengths (MP7).

Launch

  • Display one equation.
  • “Give me a signal when you know whether the equation is true and can explain how you know.”
  • 1 minute: quiet think time
Teacher Instructions
  • Share and record answers and strategy.
  • Repeat with each statement.

Student Task

Decide if each statement is true or false. Be prepared to explain your reasoning.

  • 115+225+335+445=121\frac{1}{5} + 2\frac{2}{5} + 3\frac{3}{5} + 4\frac{4}{5} = 12
  • 10−12−22−32−42=510 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - \frac{4}{2} = 5
  • 116+226+336+446+556=15361\frac{1}{6} + 2\frac{2}{6} + 3\frac{3}{6} + 4\frac{4}{6} + 5\frac{5}{6} = 15\frac{3}{6}
  • 13+23+33=3×23\frac{1}{3} + \frac{2}{3} + \frac{3}{3} = 3 \times \frac{2}{3}

Sample Response

  • True: 1+2+3+4=101 + 2 + 3 + 4 = 10 and 15+25+35+45=105=2\frac{1}{5} + \frac{2}{5} + \frac{3}{5} + \frac{4}{5} = \frac{10}{5}=2, and 10+2=1210 + 2 = 12.
  • True: a total of 102\frac{10}{2} or 5 is being subtracted from 10.
  • False: the whole numbers add up to 15 and the fractions will add up to more than 36\frac{3}{6}.
  • True: the fractions on the left add up to 63\frac{6}{3} and 3 groups of 23\frac{2}{3} is also 63\frac{6}{3}.
Activity Synthesis (Teacher Notes)
  • “What strategies did you find useful for adding or subtracting these numbers with fractions?” (Sample response:
    • Add whole numbers separately than fractions.
    • Notice that 1+2+3+41 + 2 + 3 + 4 is 10 and use that fact to add or subtract fractions.
    • Combine fractions that add up to 1 (such as 15+45\frac{1}{5} + \frac{4}{5} and 25+35\frac{2}{5} + \frac{3}{5}).
    • In the second equation, add up the fractions and subtract the sum from 10, instead of subtracting each fraction individually.)
  • 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.4·Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
  • 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.
  • 4.NF.B.4·Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

20 min

15 min