Ways to Represent Multiplication of Teen Numbers

10 min

Narrative

The purpose of this Warm-up is to elicit the idea that while there are many ways to represent 2 groups of 12, some ways are more useful than others. While students may notice and wonder many things about the images, how 2 of the images show the groups of 12 organized using place value and how this type of decomposition can be helpful in finding the total are the important discussion points.

Launch

  • Groups of 2
  • Display the image.
  • “What do you notice? What do you wonder?”
  • 1 minute: quiet think time
Teacher Instructions
  • “Discuss your thinking with your partner.”
  • 1 minute: partner discussion
  • Share and record responses.

Student Task

What do you notice? What do you wonder?

2 groups of 12 squares.

Base ten blocks. 2 groups of 1 ten and 2 ones.

<p>Base ten blocks. 2 sets of 1 ten and 2 ones.</p>

Solution Steps (2)
  1. 1
    Identify equal groups in each diagram
    →All show 2 groups of 12
  2. 2
    Notice place value decomposition
    →Base-ten diagrams show 2 tens and 4 ones = 24 total

Sample Response

Students may notice:
  • All the diagrams show 2 groups.
  • Each group has 12 squares.
  • Two of the diagrams show tens and ones.
  • It’s easier to count the 12 in the diagrams with groups of ten because I can think 10, 11, 12.
  • Each diagram has the same number of squares.
  • You can think of all the diagrams as 2×122 \times 12.
  • In the two diagrams on the right, I see 20 and 4.
Students may wonder:
  • How could we multiply 2×122 \times 12?
  • Why are the squares scattered in one diagram?
  • Why are there tens and ones in some of the diagrams?
  • Why are the tens and ones turned sideways in the last diagram?
Activity Synthesis (Teacher Notes)
  • “The image on the left is a drawing of equal groups. The other images are base-ten diagrams. What is the same and different about these representations?” (They all show 12. They all show 2 groups of the same size. In the base-ten diagrams, we can see the tens easier. It’s harder to see the tens in the first drawing.)
Standards
Building Toward
  • 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>

20 min

15 min