The Numbers in Subtraction

10 min

Narrative

This Number Talk encourages students to use what they learned about products of a whole number and a fraction, the relationship between each pair of factors, and the structure in the expressions to mentally solve problems.

Students may write all the products as fractions, including products greater than 1. If everyone expresses the last three products only as 1812\frac{18}{12}, 3612\frac{36}{12}, and 36012\frac{360}{12}, then during the Activity Synthesis, ask if students could write whole-number or mixed-number equivalents for these fractions. The reasoning elicited here will be helpful later in the lesson when students decompose whole numbers in order to subtract fractional amounts.

Launch

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

Student Task

Find the value of each expression mentally.

  • 2×3122 \times \frac{3}{12}
  • 6×3126 \times \frac{3}{12}
  • 12×31212 \times \frac{3}{12}
  • 12×301212 \times \frac{30}{12}

Sample Response

Sample response:
  • 612\frac{6}{12}: 2 groups of 3 twelfths makes 6 twelfths.
  • 1812\frac{18}{12}:  6×36\times3 is 18 (or 16121\frac{6}{12}, because two 612\frac{6}{12}s make 1 whole, and one more 612\frac{6}{12} makes 16121\frac{6}{12}).
  • 3 (or 3612\frac{36}{12}): every 4 groups of 312\frac{3}{12}s make 1212\frac{12}{12} or 1 whole, so 8 groups of 312\frac{3}{12}s makes 2 and 12 groups of 312\frac{3}{12}s makes 3.
  • 30 (or 36012\frac{360}{12}): 3012\frac{30}{12} is 10 times 312\frac{3}{12}, so 12×301212 \times \frac{30}{12} is 10 times 12×31212 \times \frac{3}{12} or 10×310 \times 3.
Activity Synthesis (Teacher Notes)
  • “In which expressions might it be helpful to use 1 whole—or 12 twelfths—to find the product? How?” (The first three. We know 4 groups of 312\frac{3}{12} make 1 whole, so:
    • 2 groups of 312\frac{3}{12} make 12\frac{1}{2}
    • 6 groups of 312\frac{3}{12} make  1121\frac{1}{2}
    • 12 groups of 312\frac{3}{12} make 3
  • “Why might it be a little harder to think of the last expression in terms of 12 twelfths?” (The 30 in 3012\frac{30}{12} is not a factor or a multiple of 12.)
  • 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
Building On
  • 4.NF.4.b·Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. <em>For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)</em>
  • 4.NF.B.4.b·Understand a multiple of <span class="math">\(a/b\)</span> as a multiple of <span class="math">\(1/b\)</span>, and use this understanding to multiply a fraction by a whole number. <span>For example, use a visual fraction model to express <span class="math">\(3 \times (2/5)\)</span> as <span class="math">\(6 \times (1/5)\)</span>, recognizing this product as <span class="math">\(6/5\)</span>. (In general, <span class="math">\(n \times (a/b) = (n \times a)/b.\)</span>)</span>

20 min

15 min