Part-Part-Whole Ratios

5 min

Teacher Prep
Setup
Display one problem at a time. 1 minute of quiet think time, followed by a whole-class discussion.

Narrative

This Warm-up focuses on multiplication of a whole number and a unit fraction. It encourages students to use the meaning of fractions and properties of operations to reason about equations. While students may evaluate each side of the equation to determine if it is true or false, elicit the following ideas:

  • The first equation: Dividing is the same as multiplying by the reciprocal of the divisor.
  • The second equation: Adjusting the factors adjusts the products. If both factors increase, the resulting product will be greater than the original.
  • The third equation: Multiplication is commutative. Changing the order of the factors doesn’t change the product.
  • The fourth equation: Decomposing a dividend into two numbers and dividing each by the divisor is a way to find the quotient of the original dividend.

In explaining their reasoning, students need to be precise in their word choice and use of language (MP6).

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

  • Give students quiet think time and ask them to give a signal when they have an answer and a strategy.
  • Invite students to share their strategies and record and display their responses for all to see.
  • Use the questions in the activity synthesis to involve more students in the conversation before moving to the next problem. 
  • Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Decide mentally whether each statement is true.

  • 15⋅45=455\frac15 \boldcdot 45 = \frac{45}{5}
  • 15⋅20=14⋅24\frac15 \boldcdot 20 = \frac14 \boldcdot 24
  • 42⋅16=16⋅4242 \boldcdot \frac16 = \frac16 \boldcdot 42
  • 486⋅112=48012+612486 \boldcdot \frac{1}{12} = \frac{480}{12}+\frac{6}{12}

Sample Response

  • True. Sample reasoning: 
    • One-fifth of 45 is 9, so is 45÷545 \div 5.
    • 15⋅45\frac{1}{5} \boldcdot 45 is the same as 45⋅1545 \boldcdot \frac{1}{5}, which is 455\frac{45}{5}.
    • Division is the same as multiplying by the reciprocal.
  • False. Sample reasoning: 
    • The expression on the left has a value of 4. The expression on the right has a value of 6.
    • Both factors on the right are greater than those on the left.
  • True. Sample reasoning: The two factors are the same, just switched in order, which doesn’t affect the product (commutative property of multiplication).
  • True. Sample reasoning: 486=480+6486 = 480 + 6, so 48612\frac{486}{12} is 48012+612\frac{480}{12} + \frac{6}{12}.
Activity Synthesis (Teacher Notes)

To involve more students in the conversation, consider asking:

  • “Who can restate ‾\underline{\hspace{.5in}}’s reasoning in a different way?”
  • “Did anyone use the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to ‾\underline{\hspace{.5in}}’s strategy?”
  • “Do you agree or disagree? Why?”
  • “What connections to previous problems do you see?”

After each true equation, ask students if they could rely on the reasoning used on the given problem to think about or solve other problems that are similar in type. After each false equation, ask students how we could make the equation true.

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing
Standards
Building On
  • 3.OA.5·Apply properties of operations as strategies to multiply and divide.
  • 3.OA.B.5·Apply properties of operations as strategies to multiply and divide.<span>Students need not use formal terms for these properties.</span> <span>Examples: If <span class="math">\(6 \times 4 = 24\)</span> is known, then <span class="math">\(4 \times 6 = 24\)</span> is also known. (Commutative property of multiplication.) <span class="math">\(3 \times 5 \times 2\)</span> can be found by <span class="math">\(3 \times 5 = 15\)</span>, then <span class="math">\(15 \times 2 = 30\)</span>, or by <span class="math">\(5 \times 2 = 10\)</span>, then <span class="math">\(3 \times 10 = 30\)</span>. (Associative property of multiplication.) Knowing that <span class="math">\(8 \times 5 = 40\)</span> and <span class="math">\(8 \times 2 = 16\)</span>, one can find <span class="math">\(8 \times 7\)</span> as <span class="math">\(8 \times (5 + 2) = (8 \times 5) + (8 \times 2) = 40 + 16 = 56\)</span>. (Distributive property.)</span>
  • 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>

15 min

15 min