Lesson 7

Rewriting Quadratic Expressions in Factored Form (Part 2)

7.1: Sums and Products (5 minutes)

Warm-up

This warm-up serves two purposes. The first is to recall that if the product of two numbers is negative, then the two numbers must have opposite signs. The second is to review how to add two numbers with opposite signs.

Student Facing

1. The product of the integers 2 and -6 is -12. List all the other pairs of integers whose product is -12.
2. Of the pairs of factors you found, list all pairs that have a positive sum. Explain why they all have a positive sum.
3. Of the pairs of factors you found, list all pairs that have a negative sum. Explain why they all have a negative sum.

Student Response

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

Activity Synthesis

Ask students to share their list of factors. Once all the pairs are listed, highlight that each pair has a positive number and a negative number because the product we are after is a negative number, and the product of a positive and a negative number is negative.

Then, invite students to share their responses for the next two questions. Consider displaying a number line for all to see and using arrows to visualize the additions of factors. Make sure students understand that when adding a positive number and a negative number, the result is the difference of the absolute values of the numbers, and that sum takes the sign of the number that is farther from zero.

7.2: Negative Constant Terms (15 minutes)

Activity

In this activity, students encounter quadratic expressions that are in standard form and that have a negative constant term. They notice that, when such expressions are rewritten in factored form, one of the factors is a sum and the other is a difference. They connect this observation to the fact that the product of a positive number and a negative number is a negative number.

Students also recognize that the sum of the two factors of the constant term may be positive or negative, depending on which factor has a greater absolute value. This means that the sign of the coefficient of the linear term (which is the sum of the two factors) can reveal the signs of the factors.

Students use their observations about the structure of these expressions and of operations to help transform expressions in standard form into factored form (MP7).

Launch

Arrange students in groups of 2. Give students a few minutes of quiet time to attempt the first question. Pause for a class discussion before students complete the second question. Make sure students recall that when transforming an expression in standard form into factored form, they are looking for two numbers whose sum is the coefficient of the linear term and whose product is the constant term.

Next, display a completed table for the first question and the incomplete table for the second question for all to see. Ask students to talk to their partner about at least one thing they notice and one thing they wonder about the expressions in the table.

Students may notice that:

• The constant terms in the right column of the first table are 30 and 18. They are both positive.
• The constant terms in the right column of the second table are all -36.
• The linear terms in the right column of the first table are positive and negative. That is also the case for the linear terms in the second table.
• The pairs of factors in the first table are both sums or both differences, while the factors in the second table are a sum and a difference.

Students may wonder:

• why all the expressions in standard form in the second table have the same squared term and constant term but different linear terms
• whether the missing expression in the first row of the second table will also have the same squared term and constant term as the rest of the expressions in standard form
• whether the factors will be sums, differences, or one of each

Next, ask students to work quietly on the second question before conferring with their partner.

Conversing: MLR8 Discussion Supports. In groups of two, ask students to take turns describing the differences they notice between the expressions in each table. Display the following sentence frames for all to see: “_____ and _____ are different because . . .”, “One thing that is different is . . .” and “I noticed _____, so . . . .” Encourage students to challenge each other when they disagree. This will help students clarify their reasoning about the relationship between quadratic expressions with a negative constant term written in factored and standard form.
Design Principle(s): Support sense-making; Maximize meta-awareness
Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. During the launch, take time to review terms students will need to access for this activity. Invite students to suggest language or diagrams to include that will support their understanding of factored form, standard form, linear term, constant term, coefficient, squared term, expression, and factors.
Supports accessibility for: Conceptual processing; Language

Student Facing

1. These expressions are like the ones we have seen before.

factored form standard form
$$(x+5)(x+6)$$
$$x^2+13x+30$$
$$(x-3)(x-6)$$
$$x^2-11x+18$$

Each row has a pair of equivalent expressions.

Complete the table. If you get stuck, consider drawing a diagram.

2. These expressions are in some ways unlike the ones we have seen before.

factored form standard form
$$(x+12)(x-3)$$
$$x^2-9x-36$$
$$x^2-35x-36$$
$$x^2+35x-36$$

Each row has a pair of equivalent expressions.

Complete the table. If you get stuck, consider drawing a diagram.

3. Name some ways that the expressions in the second table are different from those in the first table (aside from the fact that the expressions use different numbers).

Student Response

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

Activity Synthesis

Display the incomplete second table for all to see. Invite some students to complete the missing expressions and explain their reasoning. Discuss questions such as:

• “How did you know what signs the numbers in the factored expressions would take?” (The two numbers must multiply to -36, which is a negative number, so one of the factors must be positive and the other must be negative.)
• “How do you know which factor should be positive and which one negative?” (If the coefficient of the linear term in standard form is positive, the factor of 36 with the greater absolute value is positive. If the coefficient of the linear term is negative, the factor of 36 with the greater absolute value is negative. This is because the sum of a positive and a negative number takes the sign of the number with the greater absolute value.)

If not mentioned in students’ explanations, point out that all the factored expressions in the second table contain a sum and a difference. This can be attributed to the negative constant term in the equivalent standard form expression.

7.3: Factors of 100 and -100 (15 minutes)

Activity

This activity aims to solidify students’ observations about the structure connecting the standard form and factored form. Students find all pairs of factors of a number that would lead to a positive sum, a negative sum, and a zero sum. They then look for patterns in the numbers and draw some general conclusions about what must be true about the numbers to produce a certain kind of sum. Along the way, students practice looking for regularity through repeated reasoning (MP8).

Launch

Give students time to complete the first two questions and then pause for a class discussion. If time is limited, consider arranging students in groups of 2 and asking one partner to answer the first question and the other to answer the second question. Alternatively, consider offering one pair of factors as an example in each table.

Before students answer the last question, consider displaying the completed tables for all to see and inviting students to observe any patterns or structure in them. Discuss questions such as:

• “How are the pairs of factors of 100 like or unlike those of -100?” (The numbers are the same but the signs are different.)
• “In the first two tables, what do you notice about factor pairs that give positive $$b$$ values?” (They are both positive.) “What about factor pairs that give negative $$b$$ values?” (They are negative.)
• “In the next two tables, what do you notice about factor pairs that give positive $$b$$ values?” (One factor is positive and the other is negative. The positive number has a greater absolute value.)
• “What about factor pairs that give negative $$b$$ values?” (One factor is positive and the other is negative. The positive number has a greater absolute value.)
• “What do you notice about the pair of factors that give a $$b$$ value of 0?” (They are opposites.)

Encourage students to use these insights to answer the last question.

Action and Expression: Provide Access for Physical Action. Support effective and efficient use of tools and assistive technologies. To use graphing technology, some students may benefit from a demonstration or access to step-by-step instructions for a method to find pairs of factors for a given number.
Supports accessibility for: Organization; Memory; Attention

Student Facing

1. Consider the expression $$x^2 + bx +100$$.

Complete the first table with all pairs of factors of 100 that would give positive values of $$b$$, and the second table with factors that would give negative values of $$b$$.

For each pair, state the $$b$$ value they produce. (Use as many rows as needed.)

positive value of $$b$$

factor 1 factor 2 $$b$$ (positive)

negative value of $$b$$

factor 1 factor 2 $$b$$ (negative)

2. Consider the expression $$x^2 + bx -100$$.

Complete the first table with all pairs of factors of -100 that would result in positive values of $$b$$, the second table with factors that would result in negative values of $$b$$, and the third table with factors that would result in a zero value of $$b$$.

For each pair of factors, state the $$b$$ value they produce. (Use as many rows as there are pairs of factors. You may not need all the rows.)

positive value of $$b$$

factor 1 factor 2 $$b$$ (positive)

negative value of $$b$$

factor 1 factor 2 $$b$$ (negative)

zero value of $$b$$

factor 1 factor 2 $$b$$ (zero)

3. Write each expression in factored form:

1. $$x^2 - 25x + 100$$
2. $$x^2 + 15x - 100$$
3. $$x^2 - 15x - 100$$
4. $$x^2 + 99x - 100$$

Student Response

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

Student Facing

Are you ready for more?

How many different integers $$b$$ can you find so that the expression $$x^2+10x+b$$ can be written in factored form?

Student Response

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

Anticipated Misconceptions

When completing the tables to find $$b$$, some students may multiply the factors rather than add them. Remind them that what we are looking for is the coefficient of the linear term.

Consider completing one row of the table and displaying a rectangle diagram to remind students how the value of $$b$$ is obtained when we rewrite an expression such as $$(x+20)(x+5)$$ in standard form. Applying the distributive property gives $$x^2+20x+5x+100$$ or $$x^2+25x+100$$. Point out that in standard form, the product of the factors,100, is the constant term. If the coefficient of the linear term is what we are after, we need to find the sum of the factors.

Activity Synthesis

Ask students to share their responses to the last question. Discuss how the work in the first two questions helped them rewrite the quadratic expressions in factored form.

Highlight that the sign of the constant term can help us anticipate the signs of the numbers in the factors, making it a helpful first step in rewriting quadratic expressions in factored form. If the constant term is positive, the factors will have two negative numbers or two positive numbers. If the constant term is negative, the factors will have one positive number and one negative number. From there, we can determine which two factors give the specified value of $$b$$ in $$x^2+bx+c$$.

Lesson Synthesis

Lesson Synthesis

To help students consolidate the observations and insights from this lesson, consider asking them to describe to a partner or write down their responses to prompts such as:

• “How would you explain to a classmate who is absent today how to rewrite $$x^2 +16x -36$$ in factored form?”
• “How would you explain how to rewrite $$x^2 - 5x - 24$$ in factored form?”
• “Suppose you are rewriting the quadratic expression $$x^2 + bx + c$$ in factored form $$(x+m)(x+n)$$. How will the factors be different when the $$c$$ is positive versus when $$c$$ is negative?”

7.4: Cool-down - The Missing Symbols (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

When we rewrite expressions in factored form, it is helpful to remember that:

• Multiplying two positive numbers or two negative numbers results in a positive product.
• Multiplying a positive number and a negative number results in a negative product.

This means that if we want to find two factors whose product is 10, the factors must be both positive or both negative. If we want to find two factors whose product is -10, one of the factors must be positive and the other negative.

Suppose we wanted to rewrite $$x^2 -8x + 7$$ in factored form. Recall that subtracting a number can be thought of as adding the opposite of that number, so that expression can also be written as $$x^2 + \text-8x + 7$$. We are looking for two numbers that:

• Have a product of 7. The candidates are 7 and 1, and -7 and -1.
• Have a sum of -8. Only -7 and -1 from the list of candidates meet this condition.

The factored form of $$x^2 -8x + 7$$ is therefore $$(x + \text-7)(x + \text-1)$$ or, written another way, $$(x-7)(x-1)$$.

To write $$x^2 + 6x - 7$$ in factored form, we would need two numbers that:

• Multiply to make -7. The candidates are 7 and -1, and -7 and 1.
• Add up to 6. Only 7 and -1 from the list of candidates add up to 6.

The factored form of $$x^2 + 6x - 7$$ is $$(x+7)(x-1)$$.