Lesson 9
Patterns in the Multiplication Table
Warmup: Notice and Wonder: Multiplication Table (10 minutes)
Narrative
The purpose of this warmup is to elicit the idea that the product of two factors on the multiplication table is found where the row and column of each factor intersect. While students may notice and wonder many things about these products, the patterns in the multiplication table and how the table is structured 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
Activity
 “Discuss your thinking with your partner.”
 1 minute: partner discussion
 Share and record responses.
Student Facing
Student Response
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Activity Synthesis
 If not mentioned in students’ responses, explain: “A multiplication table uses rows and columns to show products of two numbers. The numbers in the leftmost column and the top row are factors.”
 “Each number in the (nonshaded part of the) table is the result of multiplying the two factors in the same row and column as that number.”
 “What are some patterns that you see in the multiplication table and why do they work?” (As we move right on the 3s row or down in the 3s column, the products increase by 3, because we are adding groups of 3. The number 15 appears in two places because we can find \(3 \times 5\) or \(5 \times 3\) to get 15. We see 12 in two places in the table because we can get 12 by counting by 3 like 3, 6, 9 12 or counting by 4 like 4, 8, 12.)
 “Find all the places where 20 appears. Which pairs of factors multiply to 20?” (4 and 5)
Activity 1: Products in the Table (20 minutes)
Narrative
The purpose of this activity is for students to apply multiplication strategies based on properties of operations to find products in a multiplication table. While students may use various strategies based on properties of operations, look for opportunities to highlight strategies based on the commutative property. Students consider how known products that are already in the table can help find an unknown product in the multiplication table.
When students use a multiplication fact that they know to determine a multiplication fact that they don’t know, they look for and make use of structure (MP7).
Advances: Conversing, Reading
Launch
 Groups of 2
 “We’ll work with another multiplication table in this activity. How is this table different from the first table we saw?” (It has more products than the first table. It doesn’t have all of the products in it. Some of the boxes have letters in them.)
 1 minute: quiet think time
 Share responses.
Activity
 “Use the numbers in the table to help you find the numbers that should replace the letters A–G. Think about how the numbers that are already in the table might help.”
 “Afterwards, find numbers that should go in three other empty cells in the table. Be prepared to explain your reasoning.”
 5–7 minutes: independent work time
 “Share with your partner how you found the missing numbers in the table.”
 3–5 minutes: partner discussion
 Monitor for students who:
 use \(7\times2\), which is in the table, to find \(2\times7\) or A
 add one more group of 4 to 20 to find C
 use a product from the 9s row to find a product in the 9s column
Student Facing
Here is a partially completed multiplication table.
 Use the products in the table to help you find the numbers that should replace letters A–G. Be prepared to explain your reasoning.

Find the number that should go in three other empty cells in the table. Use:
 7 as a factor
 9 as a factor
 10 as a factor
Student Response
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Activity Synthesis
 Select previously identified students to share how they used the numbers that were in the table to find unknown products. If possible, display and annotate the table to illustrate students’ reasoning.
Activity 2: If I Know, Then I Know: Multiplication (15 minutes)
Narrative
The purpose of this activity is for students to articulate how they use known products to find unknown products, using a structure similar to that used in an earlier lesson. Students may describe strategies that are based on any property of operations. The focus should be on the description of the strategy (such as “multiplying two numbers in any order gives the same product”) rather than remembering the property on which the strategy is based (such as “commutative property”).
Supports accessibility for: Memory, Conceptual Processing
Launch
 Groups of 2
Activity
 “In the right column, work independently to write down at least two multiplication facts you can figure out because you know the given multiplication fact in the left column.”
 3–5 minutes: independent work time
 “Now, share the facts that you found with your partner. Record any facts that your partner found that you didn’t find. Be sure to explain your reasoning.”
 3–5 minutes: partner work time
Student Facing
 In each row, write down at least two multiplication facts you can figure out because you know the given multiplication fact in the left column. Be prepared to share your reasoning.
If I know . . . , then I also know . . . \(2 \times 4\) \(4 \times 2\), \(4 \times 4\), \(2 \times 8\) \(3 \times 5\) \(4 \times 10\) \(7 \times 2\) \(5 \times 8\)  If time permits, complete the rest of the multiplication table. Use the multiplication facts you know to find those you don’t know.
Student Response
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Activity Synthesis
 For each given product, invite 1–2 students to share the products they found and how they were related to the given product.
Lesson Synthesis
Lesson Synthesis
“Today we used products that we knew to find products that we didn’t know.”
“What patterns did you find helpful?” (We can write the factors in any order, the result is still the same, like \(3 \times 6\) has the same value as \(6 \times 3\). If we know \(3 \times 5\) is 15 and 6 is \(2 \times 3\), then \(6 \times 5\) is twice \(3 \times 5\) or \(2 \times (3 \times 5)\), or twice 15, which is 30. We can find the value of \(8 \times 2\) by thinking of 8 as \(3 + 5\) and then finding \(3 \times 2\) and \(5 \times 2\). When 2, 4, 6, 8, and 10 is a factor, the product is even. When 5 is a factor, the product alternates between 5 and 10. When 10 is a factor, the product ends in 0.)
Record the patterns students noticed.
Cooldown: Find the Missing Product (5 minutes)
CoolDown
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