Lesson 8

Reasoning About Equations

These materials, when encountered before Algebra 1, Unit 2, Lesson 8 support success in that lesson.

8.1: Math Talk: Multiplying (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for multi-digit multiplication and using the distributive property. These understandings help students develop fluency with multiplication and be explicit about the properties of operations.

In this activity, students have an opportunity to notice and make use of structure (MP7) as they compare different representations of the same multiplication problem.

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.

\(3 \boldcdot 10\)

\(3 \boldcdot 13\)

Apply the distributive property.

\(3(13+x)\)

\(3x(8-y)\)

Student Response

Student responses to this activity are available at 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?”

As you record, consider using notation that makes properties of operations, such as the associative and distributive properties, visible. For example, if a student evaluates \(3\boldcdot13\) and describes it as an extension of evaluating \(3\boldcdot10\), record that as \(3\boldcdot13=(3\boldcdot10)+(3\boldcdot3)\) and ask students if your recording matches the student’s thinking. Ask students how they know that \(3\boldcdot13=(3\boldcdot10)+(3\boldcdot3)\). It’s fine if students don’t name the distributive property, as long as they recognize that you can multiply 10 and 3 by 3 separately and add the results, or add first and then multiply. 

8.2: Keeping It Equal (20 minutes)

Activity

The purpose of this activity is to help students focus on not just the what, but the why and how of moves that are made when solving equations. Students who struggle often have a hard time seeing the steps that are taken to solve an equation as purposeful moves and struggle to understand the choices made. Focusing on the why and the justification will help students when they work to make sense of solving systems of equations and solving equations with multiple variables for one of the variables.

Monitor for students who make different choices in solving the equation in the second problem, either distributing first, or multiplying or dividing to get rid of the coefficient on the quantity in parentheses.

Launch

Display Andre’s work for all to see.

Andre’s work: 

\(2x+1 = 5\)

Hanger diagram.

\(2x= 4\)

Hanger diagram.

\(x=2\)

Hanger diagram.

Ask students about the moves Andre did. Focus on:

  1. What did Andre do? How do you see it in the hanger diagram and the equations? (He simplified the hanger diagram until there was only one square and two pentagons left. In the equation, he isolated \(x \) and found that it equals 2. )

  2. Why did Andre do that? How did it help him find the weight of the blue square? How did it help him find the value of \(x \) that made the equation true? (He found the weight of one square because it shows that 1 square weighs the same as 2 pentagons, which sheds light on the value of \(x \).)

  3. How did Andre know his moves would keep the hangers balanced? Why do moves that keep the hangers balanced keep the equation true? How could he justify each move? (He found the value of \(x \) by removing equal weight from each side of the diagram until there was only one square left on one side. He had to remove the same weight from each side of the hanger diagram in order for it to remain balanced.)

Display Clare’s work for all to see:

\(\displaystyle \begin{align} 4(7 - x) &= 18 \\ \frac14 \boldcdot 4(7 - x) &= \frac14 \boldcdot 18 \\ 7 - x &= 4.5 \\ 7 - x + x &= 4.5 + x \\ 7 &= 4.5 + x \\ 7 - 4.5 &= 4.5 + x - 4.5 \\ 2.5 &= x \\ \end{align}\)

Ask students about the moves Clare did. Focus on:

  1. What did Clare do? (She found the value of \(x \) by adding the same number to each side or multiplying each side by the same number.)

  2. Why did Clare do that? How did it help her find the value of \(x \) that made the equation true? (She used acceptable moves until the equation looked like \(x \) = a number)

  3. How did Clare know her moves would keep the equations equivalent? How could she justify each move? (Think about the hanger diagrams. For example,  if you add the same weight to each side, the hanger stays balanced.)

  4. Are there are any similarities in what Clare and Andre did their problems? Clare would have a hard time using a hanger diagram to represent her work. Why? How can she use the same reasoning to show that each side of her equation is still equal or “balanced”? (In each step, Clare and Andre both removed equal amounts from each side which kept the equation and hanger diagram balanced throughout the solving process. Clare would have a hard time representing her work with a hanger diagram because the equation involves a complex quantity with parentheses and a decimal which can be difficult to represent with one shape.)

Student Facing

  1. Here is Diego’s work.

    \(\displaystyle \begin{align*} \frac{(4x + 1)}{5} &= 2x \\ 5 \boldcdot \frac{(4x + 1)}{5} &= 5 \boldcdot 2x \\ 4x + 1 &= 10x \\ 4x + 1 - 4x &= 10x - 4x \\ 1 &= 6x \\ \frac16 \boldcdot 1 &= \frac16 \boldcdot 6x \\ \frac16 &= x \\ \end{align*} \)

    For each step, explain:

    1. What did Diego do?
    2. Why did Diego do that? How did it help him find the value of \(x\) that made the equation true?
    3. How could Diego justify each move?
  2. Here is an equation and the solution. What moves could you make to get from the equation to the solution? Justify each move you make:

    \(\displaystyle \begin{align*} 12(x - 4)& = 2 \\ x &= 4\frac16\ \\ \end{align*} \)

Student Response

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

Activity Synthesis

Invite students to share their explanations for Diego’s moves. Ask previously identified students to share the different moves they used to solve the second problem. If students would benefit from seeing the moves represented with a hanger diagram to help them justify why the equations stay “balanced,” identify different students to represent those moves with a hanger diagram, and talk about the whys and hows of each move. Ask students about what some of the acceptable moves are that keep equations equivalent. (Multiplying or dividing each side by the same (non-zero) amount, adding or subtracting the same amount from each side.)

8.3: Turn Up the Volume (15 minutes)

Activity

In this activity, students get a chance to practice working with formulas and writing them in equivalent forms, or expressing regularity in repeated reasoning as they find the values of unknown quantities in the formulas.

The practice will pay off when they solve formulas and equations to isolate chosen variables in the associated Algebra 1 lesson.

Monitor for students who come up with steps or algorithms or even write rules or formulas that they use for each of the problems. Let students know you will ask them to share their algorithm, rule, or process, and that they should be prepared to explain it to the class.

Launch

Display the formulas for all to see, and make sure students understand what each letter represents.

Perimeter of a Rectangle: \(P = 2l+2w\)

Area of a Rectangle: \(A = lw\)

Area of a Triangle: \(A = \frac{1}{2}bh\)

Volume of a Cube: \(V = s^3\)

Volume of a Sphere: \(V = \frac{4}{3}\pi r^3\)

Volume of a Cylinder: \(V = \pi r^2 h\)

Assign each student or group one of the four sets of problems. Explain to students that as they solve the problems, they should look for any patterns they notice in the process or the answers to help them come up with a rule, procedure, or shortcut for answering similar questions. They might notice that the problems get easier and more routine as they do several. They should be prepared to explain any regularity they notice.

Note that problem 4, regarding cylinders, is the most challenging formula to work with.

After students have worked on their problems, arrange students in groups of 4 (or 3 if you didn’t assign anyone the cylinder problems), with each student in the group having solved a different problem. Have students teach each other a formula, rule, pattern, or steps for:

  1. finding the length of a rectangle given its width and perimeter
  2. finding the length of a rectangle given its area and perimeter
  3. finding the base of a triangle given its area and height
  4. finding the height of a cylinder given its volume and radius (if used)

Student Facing

Here are some geometric formulas. In the given problems, you will get some information and be asked to figure out one of the measurements.

As you work, look for patterns or a set of steps that you could use to quickly figure out one measurement, given the others.

Perimeter of a Rectangle: \(P = 2l + 2w\)

Area of a Rectangle: \(A = lw\)

Area of a Triangle: \(A = \frac12bh\)

Volume of a Cylinder: \(V = \pi r^2 h\)

  1. Find the missing measurement of the rectangle.
    1. A rectangle has a length of 3.5 units and a width of 9 units. Find its perimeter.
    2. A rectangle has a perimeter of 25 units and a width of 9 units. Find its length.
    3. A rectangle has a perimeter 18 units and a width of 4 units. Find its length.
    4. Look at your steps and answers so far. Are there any patterns you could use to help you solve the next two problems easily?
    5. A rectangle has a perimeter of 24 units and a width of 11 units. Find its length.
    6. A rectangle has a perimeter of 15 units and a width of 3 units. Find its length.
    7. How would you teach someone else to find the length of a rectangle using the patterns you noticed?
  2. Find the missing measurement of the rectangle.
    1. A rectangle has a length of 4 units and a width of 9 units. Find its area.
    2. A rectangle has an area of 36 square units and a width of 9 units. Find its length.
    3. A rectangle has an area 50 square units and a width of 10 units. Find its length.
    4. Look at your steps and answers so far. Are there any patterns you could use to help you solve the next two problems easily?
    5. A rectangle has an area of 25 square units and a width of 5 units. Find its length.
    6. A rectangle has an area of 39 square units and a width of 6 units. Find its length.
    7. How would you teach someone else to find the length of a rectangle using the patterns you noticed?
  3. Find the missing measurement of the triangle.
    1. A triangle has a base of 5 units and a height of 4 units. Find its area.
    2. A triangle has an area of 10 square units and a height of 4 units. Find its base.
    3. A triangle has an area of 12 square units and a height of 8 units. Find its base.
    4. Look at your steps and answers so far. Are there any patterns you could use to help you solve the next two problems easily?
    5. A triangle has an area of 6 square units and a height of 3 units. Find its base.
    6. A triangle has an area of 13 square units and a height of 5 units. Find its base.
    7. How would you teach someone else to find the base of a triangle using the patterns you noticed?
  4. Find the missing measurement of the cylinder.
    1. A cylinder has a height of 3 units and a radius of 5 units. Find its volume.
    2. A cylinder has a volume of \(75\pi\) cubic units and a radius of 5 units. Find its height.
    3. A cylinder has a volume of \(90\pi\) cubic units and a radius of 3 units. Find its height.
    4. Look at your steps and answers so far. Are there any patterns you could use to help you solve the next two problems easily?
    5. A cylinder has a volume of \(20\pi\) cubic units and a radius of 2 units. Find its height.
    6. A cylinder has a volume of 100 cubic units and a radius of 5 units. Find its height.
    7. How would you teach someone else to find the height of a cylinder using the patterns you noticed?

Student Response

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

Activity Synthesis

After students have worked on their problems, arrange students in groups of 4 (or 3 if you didn’t assign anyone the cylinder problems), with each student in the group having solved a different problem. Have students teach each other a formula, rule, pattern, or steps for:

  1. finding the length of a rectangle given its width and perimeter

  2. finding the length of a rectangle given its area and perimeter

  3. finding the base of a triangle given its area and height

  4. finding the height of a cylinder given its volume and radius (if used)

Ask students to share any rules, patterns, or steps they came up with for each of the scenarios.

Record students' shortcuts symbolically. For example, if a student says, “I always divided the perimeter by 2 and subtracted the width to find the length,” record that as \(\frac{P}{2} - w = l\) and ask the student to check that what you recorded matches what they did.

Display the formulas for perimeter of a rectangle, area of a rectangle, area of a triangle, and volume of a cylinder next to students’ processes for isolating a variable in question. For example:

\(P = 2l + 2w\)

\(l = \frac{P}{2} - w\) or \(l = \frac{P - 2w}{2}\)

\(A = lw\)

\(l = \frac{A}{w}\)

\(A = \frac12bh\)

\(b = \frac{2A}{h}\)

\(V = \pi r^2 h\)

\(h = \frac{V}{\pi r^2}\)

Ask students what they notice and what they wonder about the pairs or sets of rules. Focus on the noticing and wondering, and give students time to sit with the patterns. They will have opportunities to rewrite equations to isolate variables in their Algebra 1 class. This activity gives students more time to notice and develop intuitions around the relationships among two equivalent formulas. Ask students what they predict this equation might be useful for.