How Many Groups? (Part 2)

10 min

Teacher Prep
Setup
2–3 minutes of quiet work time.

Narrative

In this Warm-up, students continue to think of division in terms of equal-size groups, using fraction strips as an additional tool for reasoning.

Monitor for how students transition from concrete questions (the first three) to symbolic ones (the last three). Framing division expressions as “How many of this fraction is in that number?” may not yet be intuitive to students. They will further explore that connection in a later activity. For now, support them using whole-number examples such as by asking “How do you interpret 6÷26 \div 2?”

The divisors used here involve both unit fractions and non-unit fractions. The last question shows a fractional divisor that is not on the fraction strips. This encourages students to transfer the reasoning used with fraction strips to a new problem, or to use an additional strategy, such as by first writing an equivalent fraction.

As students work, identify those who are able to modify their reasoning effectively, even if the approach may not be efficient (such as adding a row of 110\frac{1}{10}s to the fraction strips). Ask them to share later.

Launch

Give students 2–3 minutes of quiet work time. 

Student Task

Write a fraction or whole number as an answer for each question. If you get stuck, use the fraction strips. Be prepared to share your reasoning.

  1. How many 12\frac 12s are in 2?
  2. How many 15\frac 15s are in 3?
  3. How many 18\frac {1}{8}s are in 1141\frac 14?
  4. 1÷26=?1 \div \frac {2}{6} = {?}
  5. 2÷29=?2 \div \frac 29 = {?}
  6. 4÷210=?4 \div \frac {2}{10} = {?}

Fraction strips depicting 2 in 8 different ways.
Fraction strips depicting 2 in 8 different ways, by rows. First row, two 1s. Second row, 4 of the fraction one over two. Third row, 6 of the fraction one over three. Fourth row, 8 of the fraction one over four. Fifth row, 10 of the fraction one over five. Sixth row, 12 of the fraction one over six. Seventh row, 16 of the fraction one over eight. Eight row, 18 of the fraction one over nine.

 

Sample Response

  1. 4
  2. 15
  3. 10
  4. 3
  5. 9
  6. 20
Activity Synthesis (Teacher Notes)

Focus the discussion on how students interpreted division expressions such as 1÷261 \div \frac{2}{6} and found their values. Invite students to share their responses and reasoning. Highlight observations that finding the value of 1÷261 \div \frac{2}{6} is like finding how many 26\frac{2}{6}s are in 1, or finding the unknown factor in ?⋅26=1? \boldcdot \frac{2}{6} = 1.

For the last question, if no students mention using a unit fraction that is equivalent to 210\frac{2}{10}, ask them to discuss this idea.

Anticipated Misconceptions

Because the fraction strips do not show tenths, students might be unsure how to approach the last question. Ask questions such as:

  • “Could any of the given fraction strips help you reason about 210\frac{2}{10}?”
  • “How did you reason about the two earlier questions? Could you use the same reasoning to answer this question?”
Standards
Building On
  • 5.NF.7·Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
  • 5.NF.B.7·Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.<span>Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.</span>
Addressing
  • 6.NS.1·Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. <em>For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?</em>
  • 6.NS.A.1·Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. <span>For example, create a story context for <span class="math">\((2/3) \div (3/4)\)</span> and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that <span class="math">\((2/3) \div (3/4) = 8/9\)</span> because <span class="math">\(3/4\)</span> of <span class="math">\(8/9\)</span> is <span class="math">\(2/3\)</span>. (In general, <span class="math">\((a/b) \div (c/d) = ad/bc\)</span>.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? </span>

25 min