Types of Angles

10 min

Narrative

This Number Talk elicits the strategies students have for multiplying a fraction by a whole number, and what they know about the sizes of fractions and equivalent fractions. Students have learned that a fraction ab\frac{a}{b} is aa multiples of 1b\frac{1}{b}. They use these insights and properties of operations to find the products of a whole number and a fraction. The work here helps students develop fluency and will be helpful later in the next lesson, when students find angle measurements formed by the hands of a clock.

The progression of expressions encourages students to look for and make use of structure (MP7) in each expression and across expressions. Noticing the connections between the whole-number and fractional factors can help students find each product efficiently.

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.

  • 12×11212 \times \frac{1}{12}
  • 120×112120 \times \frac{1}{12}
  • 360×112360 \times \frac{1}{12}
  • 360×312360 \times \frac{3}{12}

Sample Response

  • 1: There are 12 twelfths in 1 whole. 
  • 10: I know that 120 is 10 times 12, so 120×112120 \times \frac{1}{12} is 10 times 12×11212 \times \frac{1}{12} or 10 times 1. 
  • 30: There are 3 groups of 120 in 360, so 360×112360 \times \frac{1}{12} is 3 times the previous product or 3×103 \times 10.
  • 90:
    • 312\frac{3}{12} is 3 times 112\frac{1}{12}, so 360×312360 \times \frac{3}{12} is 3 times the previous product or 3×303 \times 30.
    • 312\frac{3}{12} is equivalent to 14\frac{1}{4}, which is half of 12\frac{1}{2}. I know 360×12=180360 \times \frac{1}{2} = 180, so 360×14360 \times \frac{1}{4} is half of 180, which is 90.

Synthesis

  • “What connections did you see between the factors in the four expressions?” (The whole numbers are multiples of 12. The fractions are all twelfths. 360 is a multiple of 120.)
  • “How did those observations help you find the value of the last expression?”
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>

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