Lesson 16

Representing Contexts with Equations

16.1: Don't Solve It (5 minutes)

Warm-up

For this warm-up students use what they have learned about arithmetic with negative and positive numbers to determine the sign of solutions to equations. Students who focus on the signs of the numbers and the relative magnitudes without actually computing are at an advantage.

Launch

Display one problem at a time. Give students 30 seconds of quiet think time per problem and ask them to give a signal when they have an answer. Follow with a whole-class discussion.

Student Facing

Is the solution positive or negative?

\((\text-8.7)(1.4) = a\)

\(\text- 8.7b = 1.4\)

\(\text-8.7 + c = \text- 1.4\)

\(\text-8.7 - d = \text- 1.4\)

Student Response

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Activity Synthesis

For each question, ask at least one student to explain their reasoning. Make sure there is agreement for each question on whether the solution is positive or negative. Ask if anyone used the third question to help them answer the fourth.

16.2: Warmer or Colder than Before? (10 minutes)

Activity

In this activity, students work with changing temperatures to build understanding of equations that represent situations with negative coefficients, variables, and solutions. Students choose from a bank of equations to find two equations, one that represents the situation using a variable and the other representing the path to solve for the variable. They interpret the meaning of the variable in the context of each situation, solve for the value of the variable that makes the equations true, and explain how the equations and their solutions describe the situation. Students engage in MP2 as they contextualize and decontextualize between the contexts of changing temperatures and the equations that represent them. 

Note that the last question involves some ambiguity. In order to select the anticipated response, students need to use the assumption that the temperature is less at midnight than it is at 9. Since it's the last question, they could also use some process of elimination to help lead them to making the assumption.

Launch

Give students 5 minutes of quiet work time followed by whole-class discussion.

Students using the digital activity have a thermometer applet to help visualize the changes in temperature.

Representation: Internalize Comprehension. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. For example, to get students started, provide a smaller bank of equations and only the first two situations. Once students have successfully completed the three steps for each, introduce the remaining equations and situations.
Supports accessibility for: Conceptual processing; Organization
Representing, Listening: MLR8 Discussion Supports. Demonstrate the first situation to clarify what students need to do by thinking aloud using mathematical language and reasoning. This will help invite more student participation, conversations, and meta-awareness of language representing situations with negative coefficients, defining variables, and determining values that make equations true.
Design Principle(s): Support sense-making; Maximize meta-awareness

Student Facing

For each situation,

  • Find two equations that could represent the situation from the bank of equations. (Some equations will not be used.)

  • Explain what the variable \(v\) represents in the situation.

  • Determine the value of the variable that makes the equation true, and explain your reasoning.

Bank of equations:

\(\text-3v=9\) \(v = \text-16 + 6\) \(v = \frac13 \boldcdot (\text-6)\) \(v+12=4\) \(\text-4\boldcdot 3=v\)
\(v = 4 + (\text-12)\) \(v = \text-16 - (6)\) \(v=9+3\) \(\text-4 \boldcdot \text-3 = v\) \(\text-3v = \text-6\)
\(\text-6 + v = \text-16\) \(\text-4 = \frac13v\) \(v=\text-\frac13 \boldcdot 9\) \(v = \text-\frac13 \boldcdot (\text-6)\) \(v = 4 + 12\)
  1. Between 6 a.m. and noon, the temperature rose 12 degrees Fahrenheit to 4 degrees Fahrenheit.
  2. At midnight the temperature was -6 degrees. By 4 a.m. the temperature had fallen to -16 degrees.
  3. The temperature is 0 degrees at midnight and dropping 3 degrees per hour. The temperature is -6 degrees at a certain time.
  4. The temperature is 0 degrees at midnight and dropping 3 degrees per hour. The temperature is 9 degrees at a certain time.
  5. The temperature at 9 p.m. is one third the temperature at midnight.

You may use this applet if you find it helpful. 

Student Response

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Launch

Give students 5 minutes of quiet work time followed by whole-class discussion.

Students using the digital activity have a thermometer applet to help visualize the changes in temperature.

Representation: Internalize Comprehension. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. For example, to get students started, provide a smaller bank of equations and only the first two situations. Once students have successfully completed the three steps for each, introduce the remaining equations and situations.
Supports accessibility for: Conceptual processing; Organization
Representing, Listening: MLR8 Discussion Supports. Demonstrate the first situation to clarify what students need to do by thinking aloud using mathematical language and reasoning. This will help invite more student participation, conversations, and meta-awareness of language representing situations with negative coefficients, defining variables, and determining values that make equations true.
Design Principle(s): Support sense-making; Maximize meta-awareness

Student Facing

For each situation,

  • Find two equations that could represent the situation from the bank of equations. (Some equations will not be used.)

  • Explain what the variable \(v\) represents in the situation.

  • Determine the value of the variable that makes the equation true, and explain your reasoning.

Bank of equations:

\(\text-3v=9\)

\(\text-4\boldcdot 3=v\)

\(\text-4 \boldcdot \text-3 = v\)

\(v=\text-\frac13 \boldcdot 9\)

\(v = \text-16 + 6\)

\(v = 4 + (\text-12)\)

\(\text-3v = \text-6\)

\(v = \text-\frac13 \boldcdot (\text-6)\)

\(v = \frac13 \boldcdot (\text-6)\)

\(v = \text-16 - (6)\)

\(\text-6 + v = \text-16\)

\(v = 4 + 12\)

\(v+12=4\)

\(v=9+3\)

\(\text-4 = \frac13v\)

\(4 = 3v\)

  1. Between 6 a.m. and noon, the temperature rose 12 degrees Fahrenheit to 4 degrees Fahrenheit.

  2. At midnight the temperature was -6 degrees. By 4 a.m. the temperature had fallen to -16 degrees.

  3. The temperature is 0 degrees at midnight and dropping 3 degrees per hour. The temperature is -6 degrees at a certain time.
  4. The temperature is 0 degrees at midnight and dropping 3 degrees per hour. The temperature is 9 degrees at a certain time.
  5. The temperature at 9 p.m. is one third the temperature at midnight.

Student Response

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Anticipated Misconceptions

For students who struggle to understand how equations represent these situations, suggest that they draw a number line in the form of a thermometer and show the changes along the number line while reasoning about the rising and falling temperatures.

Activity Synthesis

This discussion has two goals: First, for students to connect the equation that represents a situation to the equation that represents the solution strategy, and second, to ensure that students understand how to represent negative quantities in an equation. Ask students to share examples of how they chose:

  • the equation that represents the situation
  • the equation that represents the solution strategy

and in what order they chose them. Also ask for examples of how they knew that a situation required an equation with negative values.  

16.3: Animals Changing Altitudes (15 minutes)

Optional activity

In this optional activity, students use expressions and number line diagrams to represent situations involving the changing height and depth of sea animals. They discuss how there is more than one correct way to write an equation that represents each situation. As students work, identify students who are writing their equations differently.

Launch

Reading, Representing, Conversing: MLR2 Collect and Display. While students work, circulate and listen to students talk about the similarities and differences between the changing height and depth of sea animals. Write down common or important phrases you hear students say about there being more than one correct way to write an equation that represents each representation. Write the students’ words on a visual display of the expressions and number line diagrams. This will help students read and use mathematical language during their paired and whole-group discussions.
Design Principle(s): Support sense-making

Student Facing

  1. Match each situation with a diagram.

    1. A penguin is standing 3 feet above sea level and then dives down 10 feet. What is its depth?
       
    2. A dolphin is swimming 3 feet below sea level and then jumps up 10 feet. What is its height at the top of the jump?
       
    3. A sea turtle is swimming 3 feet below sea level and then dives down 10 feet. What is its depth?
       
    4. An eagle is flying 10 feet above sea level and then dives down to 3 feet above sea level. What was its change in altitude?
       
    5. A pelican is flying 10 feet above sea level and then dives down reaching 3 feet below sea level. What was its change in altitude?
       
    6. A shark is swimming 10 feet below sea level and then swims up reaching 3 feet below sea level. What was its change in depth?
       
  2. Next, write an equation to represent each animal's situation and answer the question. Be prepared to explain your reasoning.

Diagrams


A

A number line with the numbers negative 10 through 10 indicated. An arrow starts at 3, points to the left, and ends at negative 7. A second arrow starts at 0, points to the right, and ends at 3.


B

A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the right, and ends at 10. A solid dot is indicated at 3.


C

A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the left, and ends at negative 10. A solid dot is indicated at negative 3.


D

A number line. 


E

A number line. 


F

A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the right, and ends at 10. A solid dot is indicated at negative 3.

 

Student Response

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Anticipated Misconceptions

Some students may struggle to match the verbal descriptions with the number line diagrams. Prompt them to examine whether each situation is looking for the animal's final altitude or change in altitude. If it is asking for a change in altitude, the number line will have one arrow and one point, instead of two arrows.

Activity Synthesis

Poll the class on which diagram matched which situation. The majority of the discussion should focus on how the students wrote the equations to represent each situation. Have previously identified students share their equations with the class and have the class discuss whether their equations correctly represent the situation. Try to get at least two different equations for every situation.

16.4: Equations Tell a Story (20 minutes)

Activity

Unlike the previous activities, this activity gives students a chance to generate an equation by themselves, in preparation for the work in the upcoming lessons.

Launch

Arrange students in groups of 2–3 and provide tools for making a visual display. Assign one situation to each group. Note that the level of difficulty increases for the situations, so this is an opportunity to differentiate by assigning more or less challenging situations to different groups.

Engagement: Develop Effort and Persistence. Provide prompts, reminders, guides, rubrics, or checklists that focus on increasing the length of on-task orientation in the face of distractions. For example, create an exemplar display including all required components, highlighting the explanation for how inverses were used in problem solving.
Supports accessibility for: Attention; Social-emotional skills
Speaking, Listening, Representing: MLR7 Compare and Connect. As students prepare the visual display of their given situation, look for students with different strategies for writing and solving their equations. As students analyze each other’s work, ask students to share what worked well in a particular approach when representing the situations with equations. Then encourage students to make connections between the use of multiplicative and arithmetic inverses when solving for the unknown quantities. During this discussion, listen for and amplify language students use to make sense of how the operations in the equation represent the relationships in the situation. This will foster students’ meta-awareness and support constructive conversations as they compare strategies for writing and solving variable equations, and make connections between the situations and visual representations of equations and solutions on the visual display.
Design Principle(s): Cultivate conversation; Maximize meta-awareness

Student Facing

Your teacher will assign your group one of these situations. Create a visual display about your situation that includes:

  • An equation that represents your situation

  • What your variable and each term in the equation represent

  • How the operations in the equation represent the relationships in the story

  • How you use inverses to solve for the unknown quantity

  • The solution to your equation

  1. As a \(7\frac14\) inch candle burns down, its height decreases \(\frac34\) inch each hour. How many hours does it take for the candle to burn completely?

  2. On Monday \(\frac19\) of the enrolled students in a school were absent. There were 4,512 students present. How many students are enrolled at the school?

  3. A hiker begins at sea level and descends 25 feet every minute. How long will it take to get to an elevation of -750 feet?

  4. Jada practices the violin for the same amount of time every day. On Tuesday she practices for 35 minutes. How much does Jada practice in a week?

  5. The temperature has been dropping \(2\frac12\) degrees every hour and the current temperature is \(\text-15 ^\circ\text{F}\). How many hours ago was the temperature \(0^\circ\text{F}\)?

  6. The population of a school increased by 12%, and now the population is 476. What was the population before the increase?

  7. During a 5% off sale, Diego pays $74.10 for a new hockey stick. What was the original price?

  8. A store buys sweaters for $8 and sells them for $26. How many sweaters does the store need to sell to make a profit of $990?

Student Response

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Student Facing

Are you ready for more?

Diego and Elena are 2 miles apart and begin walking towards each other. Diego walks at a rate of 3.7 miles per hour and Elena walks 4.3 miles per hour. While they are walking, Elena's dog runs back and forth between the two of them, at a rate of 6 miles per hour. Assuming the dog does not lose any time in turning around, how far has the dog run by the time Diego and Elena reach each other?

Student Response

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Activity Synthesis

Invite groups to present their solutions or to view all the solutions on display. The discussion should focus on

  • how students decided what their variable would represent
  • how to write the equation
  • how to solve the equation
  • how to interpret the solution in terms of the context.

Students should also address any equations with negative quantities and discuss how they represent the situations. 

Lesson Synthesis

Lesson Synthesis

  • When writing an equation to represent a situation, how do you decide what your variable represents?
  • How do you solve the equation?

16.5: Cool-down - Floating Above a Sunken Canoe (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

We can use variables and equations involving signed numbers to represent a story or answer questions about a situation.

For example, if the temperature is \(\text-3^\circ\text{C}\) and then falls to \(\text-17^\circ\text{C}\), we can let \(x\) represent the temperature change and write the equation:

\(\displaystyle \text-3 + x = \text- 17\)

We can solve the equation by adding 3 to each side. Since \(\text-17 + 3 = \text-14\), the change is \(\text-14^\circ\text{C}\).

Here is another example: if a starfish is descending by \(\frac32\) feet every hour then we can solve \(\displaystyle \text-\frac32h=\text-6\) to find out how many hours \(h\) it takes the starfish to go down 6 feet. 

We can solve this equation by multiplying each side by \(\text-\frac23\). Since \(\text-\frac23\boldcdot \text-6 = 4\), we know it will take the starfish 4 hours to descend 6 feet.