Lesson 8

Multiplying Expressions

These materials, when encountered before Algebra 1, Unit 7, Lesson 8 support success in that lesson.

8.1: Math Talk: Combining the Similar Numbers (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for combining the same number in a variety of ways. These understandings help students develop fluency and will be helpful later in the supported Algebra 1 lesson when students will need to be able to factor quadratic expressions.

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole class discussion.

Student Facing

Evaluate mentally.

\(100 \boldcdot 100\)

\(\text{-}3 \boldcdot 3\)

\(\text{-}300 + 300\)

\(1,\!279 + \text{-}1,\!279\)

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
  • “Did anyone have 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?”

8.2: A Method for Multiplying (15 minutes)

Activity

In this activity, students learn and practice a method for multiplying two values of the form \((10^n - a)(10^n+a)\). Through the discussion, students should begin to recognize that two of the terms from the expanded form are opposites. In the associated algebra lesson, students examine quadratic expressions that can be factored into the form \((x-a)(x+a)\).

Student Facing

Here is a method for multiplying 97 and 103:

97 is \(100 - 3\)

103 is \(100 + 3\)

So \(97 \boldcdot 103 = (100-3)(100+3)\)

  100 -3
100 10,000 -300
3 300 -9
  1. Explain how this diagram is used to compute \(97 \boldcdot 103 = 9,\!991\).
  2. Draw a similar diagram that helps you mentally compute \((30+1)(30-1)\). What is the result? What multiplication problem did you just solve?
  3. Use this method to compute:
    1. \(7 \boldcdot 13\)
    2. \(102 \boldcdot 98\)
    3. \(995 \boldcdot 1,\!005\)
  4. Create a challenge problem for your partner, that could use this method. Create a diagram that shows the answer before giving the problem to your partner.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The purpose of the discussion is to notice the pattern that two of the cells in the table are opposites when the multiplication is done for numbers in this form. Select students to share their solutions and challenge problems. Ensure that students are working with numbers in the form \((10^n - a)(10^n+a)\).

Ask students,

  • “What do you notice about the values inside the table after you multiply the appropriate values?” (The number on the bottom left and top right are opposites.)
  • “Although this method can be used to multiply any numbers, why is this setup easier to work with than something like \(97 \boldcdot 105\)?” (With other values, there will not be 2 parts of the table that are opposites, so the entire table would need to be filled in and summed. When the values are like the ones in the activity, only the top left and bottom right of the table need to be combined.)

8.3: Find the Missing Pieces (20 minutes)

Activity

In this activity, students find missing parts of a multiplication diagram. This strategy will be useful in the associated Algebra 1 lesson when they factor polynomials. Students look for and make use of structure (MP7) when they use tables to understand the relationships between 2 numbers.

Student Facing

Complete each diagram. Use the diagram to write some equivalent expressions that could be solved using the diagram.

  1.   10 5
    10 100  
        45
  2.     7
    10    
    -7 -70  
  3.   \(x\) 8
    \(x\)    
    -8    
  4.   \(a\) -9
        \(\text{-}9a\)
    9    
  5.   \(b\) \(\frac12\)
    \(b\) \(b^2\)  
        \(\text{-}\frac{1}{4}\)
  6.   7  
    \(c\)   \(\text{-}c^2\)
    7 49  

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The purpose of the discussion is to gain fluency with using the diagram to find missing parts and rewriting the diagram as an expression. Select students to share their solutions and methods. Ask students,

  • “When you know a value in the middle of the diagram, how do you use it to find a missing value to the top or side?” (When you know one of the values on the side or top, you can divide the value in the middle of the diagram by the value you know to find the other value on the top or side that is missing.)
  • “Once you have the entire diagram completed, how do you write an expression for the multiplication question that could be answered by the diagram?” (One factor is the sum of the values in the first row and the other factor is the sum of the values in the first column. Multiply the two factors to write the expression.)