Solve Problems Involving Arrays

10 min

Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have about equal groups in multiplication expressions and to see a pattern as one factor is decreased. These understandings help students develop fluency and will be helpful later in this lesson when students will need to use multiplication to answer questions about array situations.

When students notice that the product decreases by a group of 2 as the number being multiplied by 2 decreases by 1, they look for and express regularity in repeated reasoning (MP8).

Launch

  • Display the first 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.

  • 10×210\times2
  • 9×29\times2
  • 8×28\times2
  • 7×27\times2
Solution Steps (5)
  1. 1
    Find 10×2 mentally
    →10 groups of 2 = 20 (or skip count by 2 ten times)
  2. 2
    Find 9×2 mentally
    →9 groups of 2 = 18 (or one less group of 2 than 20: 20-2=18)
  3. 3
    Find 8×2 mentally
    →8 groups of 2 = 16 (or one less group of 2 than 18: 18-2=16)
  4. 4
    Find 7×2 mentally
    →7 groups of 2 = 14 (or one less group of 2 than 16: 16-2=14)
  5. 5
    Notice the pattern
    →Product decreases by 2 each time because there's one less group of 2

Sample Response

  • 20: 2 groups of 10 is 20.
  • 18: I counted by 2 (2, 4, 6, 8, 10, 12, 14, 16, 18).
  • 16: I counted by 2, or it’s one group of 2 less than the last equation.
  • 14: I counted by 2, or it’s one group of 2 less than the last equation.
Activity Synthesis (Teacher Notes)
  • “What pattern do you see as you look at the expressions and their values? Why is that happening?” (The factor that isn’t 2 goes down by 1 each time. The value of the product decreases by 2 because I have one less 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
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>

25 min

10 min