Lesson 19

Expanding and Factoring

19.1: Number Talk: Parentheses (10 minutes)

Warm-up

The purpose of this number talk is to remind students that when we evaluate expressions, we multiply before we add or subtract. Parentheses are used to indicate that the order should be different. Remembering how this works will be important for an activity in this lesson.

Launch

Display one problem at a time. Give students 30 seconds of 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.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Find the value of each expression mentally.

\(2 + 3 \boldcdot 4\)

\((2+3)(4)\)

\(2 - 3 \boldcdot 4\)

\(2 - (3 + 4)\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The second question is an opportunity to remind students that “next to” implies “multiply.” The second expression could be rewritten \((2+3) \boldcdot 4\). Point out that we can also know that the second expression is 20 by using the distributive property: \(2 \boldcdot 4 + 3 \boldcdot 4\).

The fourth expression could also be rewritten \(2-3-4\).

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 ___’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 _____’s strategy?”
  • “Do you agree or disagree? Why?”
Speaking: MLR8 Discussion Supports.: Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . . ." Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

19.2: Factoring and Expanding with Negative Numbers (20 minutes)

Activity

This activity is an opportunity for students to practice rewriting expressions using the distributive property. It is a step up from the same type of work in grade 6 because arithmetic with signed numbers is required.

The row with \(6a-2b\) is designed to allow students to figure out how to factor by reasoning based on structure they already understand, instead of learning how to factor based on a procedure that a teacher demonstrates first. The rows with \(k(4 - 17)\) and \(10a-13a\) are designed to prepare students for combining like terms in the next lesson.

Launch

Draw students’ attention to the organizers that appear above the table, and tell them that these correspond to the first three rows in the table. Let them know that they are encouraged to draw more organizers like this for other rows as needed.

Arrange students in groups of 2. Instruct them to take turns writing an equivalent expression for each row. One partner writes the equivalent expression and explains their reasoning while the other listens. If the partner disagrees, they work to resolve the discrepancy before moving to the next row.

Action and Expression: Internalize Executive Functions. Provide students with blank diagrams to organize their work for the factored and expanded expressions. Include example diagrams labeled “expanded” and “factored.”
Supports accessibility for: Language; Organization

Student Facing

In each row, write the equivalent expression. If you get stuck, use a diagram to organize your work. The first row is provided as an example. Diagrams are provided for the first three rows.

Three area diagrams.
factored expanded
\(\text- 3(5 - 2y)\) \(\text- 15 + 6y\)
\(5(a-6)\)
\(6a-2b\)
\(\text- 4(2w-5z)\)
\(\text- (2x-3y)\)
\(20x-10y+15z\)
\(k(4-17)\)
\(10a-13a\)
\(\text- 2x(3y-z)\)
\(ab-bc-3bd\)
\(\text- x(3y-z+4w)\)

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Expand to create an equivalent expression that uses the fewest number of terms: \(\left(\left(\left(\left(x\strut+1\right)\frac12\right)+1\right)\frac12\right)+1\). If we wrote a new expression following the same pattern so that there were 20 sets of parentheses, how could it be expanded into an equivalent expression that uses the fewest number of terms? 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students are unsure how to proceed, remind them of tools and understandings they have seen recently that would be helpful. For example, “Draw an organizer and think about how the organizer represents terms in the expression.” Also, “Rewrite subtraction as adding the opposite.”

Activity Synthesis

Much of the discussion will take place in small groups. Display the correct equivalent expressions and work to resolve any discrepancies. Expanding the term \(\text-(2x-3y)\) may require particular care. One way to interpret it is to rewrite as \(\text-1 \boldcdot (2x-3y).\) If any confusion about handling subtraction arises, encourage students to employ the strategy of rewriting subtraction as adding the opposite.

To wrap up the activity, ask:

  • “Which rows did you and your partner disagree about? How did you resolve the disagreement?”
  • “Which rows are you the most unsure about?” 
  • “Describe a process or procedure for taking a factored expression and writing its corresponding expanded expression.”
  • “Describe a process or procedure for taking an expanded expression and writing its corresponding factored expression.”
Writing: MLR3 Clarify, Critique, Correct. Present an incorrect response for one of the expanded expressions in the table. For example, “An equivalent expression for \(6a − 2b\) is \(4(a−b)\) because \(6−2\) is 4 and \((a−b)\) is left on its own.” Prompt students to clarify any of the language and reasoning in the incorrect response and then to identify the error(s). Invite students to work with a partner to write a correct response. This helps students evaluate and improve on the written mathematical arguments of others.
Design Principle(s): Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

  • To write an equivalent expression by factoring means to use the distributive property to write a sum as a product.
  • To write an equivalent expression by expanding means to use the distributive property to write a product as a sum.

Ask students to give an example of each.

19.3: Cool-down - Equivalent Expressions (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

We can use properties of operations in different ways to rewrite expressions and create equivalent expressions. We have already seen that we can use the distributive property to expand an expression, for example \(3(x+5) = 3x+15\). We can also use the distributive property in the other direction and factor an expression, for example \(8x+12 = 4(2x+3)\).

We can organize the work of using distributive property to rewrite the expression \(12x-8\). In this case we know the product and need to find the factors.

The terms of the product go inside:

Area diagram, 1 row, 2 columns. Inside the diagram, starting wtih the first box, 12 x, negative 8. To the left of the row and above each column, blank boxes indicate missing labels.

We look at the expressions and think about a factor they have in common. \(12x\) and \(\text- 8\) each have a factor of 4. We place the common factor on one side of the large rectangle:

Area diagram, 1 row, 2 columns.  To the left of the row, 4. Inside the boxes, 12 x, negative 8. Above each row a blank box indicates a missing label.

Now we think: "4 times what is 12\(x\)?" "4 times what is -8?" and write the other factors on the other side of the rectangle:

Area diagram, 1 row, 2 columns. To the left of the row, 4. Above each column, 3 x, negative 2. Beginning with the first box, 12 x, negative 8.

So, \(12x-8\) is equivalent to \(4(3x-2)\).