The Distributive Property, Part 1

5 min

Teacher Prep
Setup
Display one problem at a time. Allow 30 seconds of quiet think time per problem, followed by a whole-class discussion.

Narrative

This Math Talk focuses on multiplication of multi-digit numbers. It encourages students to think about decomposition of numbers by place value to rely on what they know about properties of operations to mentally solve problems. The strategies elicited here will be helpful in upcoming activities and lessons as students build on their informal understanding of the distributive property to generate and justify equivalent expressions.

Students must be precise in their word choice and use of language when describing how they may decompose numbers, how they multiply, and the sums or differences they create to find the product (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.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Find the value of each product mentally.

  • 5⋅1025 \boldcdot 102
  • 5⋅985 \boldcdot 98
  • 5⋅9995 \boldcdot 999
  • 5⋅(0.999)5 \boldcdot (0.999)

Sample Response

  • 510. Sample reasoning:
    • 5⋅102=5⋅100+5⋅25 \boldcdot 102 = 5\boldcdot 100 + 5\boldcdot 2, which is 500+10500 + 10.
    • 10⋅10210 \boldcdot 102 is 1,020 and 5 is half of 10, so 5⋅1025 \boldcdot 102 is half of 1,020, which is 510.
  • 490. Sample reasoning:
    • 98 is 2 less than 100, so 5⋅985 \boldcdot 98 is 5⋅25 \boldcdot 2, or 10, less than 500, which is 490. 
    • 5⋅98=5⋅90+5⋅8=450+405 \boldcdot 98 = 5 \boldcdot 90 + 5 \boldcdot 8 = 450 + 40, which is 490.
      98 is 4 less than 102, so 5⋅985 \boldcdot 98 is 5⋅45 \boldcdot 4, or 20, less than 510, which is 490.
  • 5,015. Sample reasoning:
    • 999 is 1 less than 1,000, so the product is 5⋅15 \boldcdot 1 less than 5⋅1,0005 \boldcdot 1,000 or 5,000−55,000-5, which is 4,995.
    • 999=900+90+9999 = 900 + 90 + 9. Multiplying 900, 90, and 9 each by 5 gives 4,500, 450, and 45. Adding them up gives 4,995.
  • 4,995. Sample reasoning:
    • The product is one-thousandth of 4,995, because 0.999 is a thousandth of 999.
    • 0.999=1−0.0010.999 = 1 - 0.001, so 5⋅(0.999)=5⋅1−5⋅(0.001)5 \boldcdot (0.999) = 5 \boldcdot 1 - 5 \boldcdot (0.001), or 5−0.0055 - 0.005, which is 4.995.
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?”

Once students have had a chance to share a few different ways of reasoning about each product, focus on explanations using the distributive property and record the steps of reasoning for all to see. For example, when students find 5⋅985 \boldcdot 98 by thinking of 98 as 100−2100 - 2, record:

5⋅1025⋅(100+2)5⋅100+5⋅2500+105105 \boldcdot 102\\ 5 \boldcdot (100 + 2)\\ 5 \boldcdot 100 + 5 \boldcdot 2\\ 500 + 10\\ 510

Explain to students that the strategies that involve decomposing one factor as a sum or difference of numbers and then multiplying each part by the other factor demonstrate the distributive property of multiplication. In the shown example, we are “distributing” the multiplication of 5 to the 100 and the 2. Applying the distributive property allows us to write an expression that is equivalent to a given expression but is easier to calculate. Let students know that they will spend the next few lessons deepening their understanding of this property.

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I ‾\underline{\hspace{.5in}} because . . . .” or “I noticed ‾\underline{\hspace{.5in}} 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
  • 4.NBT.5·Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
  • 4.NBT.B.5·Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
  • 5.NBT.7·Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
  • 5.NBT.B.7·Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
Building Toward
  • 6.EE.3·Apply the properties of operations to generate equivalent expressions. <em>For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.</em>
  • 6.EE.A.3·Apply the properties of operations to generate equivalent expressions. <span>For example, apply the distributive property to the expression <span class="math">\(3 (2 + x)\)</span> to produce the equivalent expression <span class="math">\(6 + 3x\)</span>; apply the distributive property to the expression <span class="math">\(24x + 18y\)</span> to produce the equivalent expression <span class="math">\(6 (4x + 3y)\)</span>; apply properties of operations to <span class="math">\(y + y + y\)</span> to produce the equivalent expression <span class="math">\(3y\)</span>.</span>
  • 6.NS.4·Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1—100 with a common factor as a multiple of a sum of two whole numbers with no common factor. <em>For example, express 36 + 8 as 4 (9 + 2).</em>
  • 6.NS.B.4·Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.<span> For example, express <span class="math">\(36 + 8\)</span> as <span class="math">\(4 (9 + 2)\)</span>.</span>

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