Subtract Strategically

10 min

Narrative

The purpose of this Number Talk is to elicit the strategies students have for finding products of single-digit factors. These reasoning strategies help students develop fluency and will be helpful later in this unit when students solve two-step word problems.

When students use strategies based on the properties of multiplication to find unknown products, they look for and make use of structure (MP7). Students may reverse the order of the factors to create a multiplication fact they know. Students may think about “one more group” as they move from the first expression to the second expression (or the third to the fourth). Also, students may say that they “just know” the product. All of these responses are acceptable because students will be in different stages as they progress toward fluency.

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.

  • 2×62 \times 6
  • 3×63 \times 6
  • 2×72 \times 7
  • 3×73 \times 7
Solution Steps (4)
  1. 1
    2×6
    →2 groups of 6 = 12
  2. 2
    3×6
    →One more group of 6: 12+6=18
  3. 3
    2×7
    →2 groups of 7: 7+7=14
  4. 4
    3×7
    →One more group of 7: 14+7=21

Sample Response

  • 12: I counted by 2. I just knew it.
  • 18: I knew it would be 1 more group of 6 than the first expression, and 12 plus 6 is 18. I just knew it.
  • 14: I knew it would be 1 more group of 2 than the first problem. It's 2 groups of 7, so I found 7+77+7, which is 14.
  • 21: It would be 1 more group of 7 than the last problem. 14 plus 7 is 21. It's 3 more than 3×63×6, or 3 more than 18, which is 21.
Activity Synthesis (Teacher Notes)
  • “What pattern do you see as you look at the expressions and their values? Why is that happening?” (When the first factor increases by 1, the value increases by the other number because it’s like adding another group. The value of 2×72×7 is 2 more than the value of 2×62×6 because there’s 1 more in each group or 1 more group of 2.)

  • As needed, record student thinking, using equal-groups drawings or arrays to help all students visualize the pattern.

  • Consider asking:    

    • “Did anyone notice a different pattern?”
    • “Did anyone notice the same pattern but would explain it differently?”
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>
Addressing
  • 3.OA.7·Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
  • 3.OA.9·Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. <em>For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.</em>
  • 3.OA.C.7·Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that <span class="math">\(8 \times 5 = 40\)</span>, one knows <span class="math">\(40 \div 5 = 8\)</span>) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
  • 3.OA.D.9·Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. <span>For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.</span>

20 min

15 min