Lesson 6

Using Equations to Solve Problems

6.1: Number Talk: Quotients with Decimal Points (5 minutes)

Warm-up

The purpose of this Number Talk is to elicit strategies and understandings students have for determining how the size of a quotient changes when the decimal point in the divisor or dividend moves. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to check the reasonableness of their answers. While four problems are given in the first problem, it may not be possible to share answers for all of them. 

Launch

Arrange students in groups of 2. Display the first question. Give students 2 minutes of quiet think time and ask them to give a signal when they have an answer, and reasoning, to support their answer. Follow with a whole-class discussion. Display the second question and give students 1 minute of quiet think time.

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

Student Facing

Without calculating, order the quotients of these expressions from least to greatest.

\(42.6 \div 0.07\)

\(42.6 \div 70\)

\(42.6 \div 0.7\)

\(426 \div 70\)

Place the decimal point in the appropriate location in the quotient: \(42.6 \div 7 = 608571\)

Use this answer to find the quotient of one of the previous expressions.

Student Response

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

Activity Synthesis

Ask students to share where they placed the decimal point in the second question and their reasoning. After students share, ask the class if they agree or disagree. Ask selected students, who chose different problems to solve, to share quotients for the problems in the first question. 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?”

Emphasize student reasoning based in place value that involve looking at the relationship between the dividend and divisor to determine the size of the quotient.

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)

6.2: Concert Ticket Sales (15 minutes)

Activity

This activity requires students to work with larger numbers, which is intended to encourage students to use an equation and notice the efficiencies of doing so. It also emphasizes the interpretation of the constant of proportionality in the context. In this case, the constant represents the cost of a single ticket, and makes it easy to identify which singer would make more money for similar ticket sales in a concert series. Note that asking students to give the revenues for different ticket sales encourages looking for and expressing regularity in repeated reasoning (MP8). The last set of questions ask students to interpret the constant of proportionality as represented in an equation in terms of the context (MP2).

Monitor for students who solve the problems using the following strategies and invite them to share during the whole-class discussion.

  • writing many calculations, without any organization
  • creating a table to organize their results
  • writing an equation to encapsulate repeated reasoning

Launch

Provide access to calculators. Consider using the names of actual performers to make the task more interesting to students.

Representation: Internalize Comprehension. Activate or supply background knowledge. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing

Student Facing

A performer expects to sell 5,000 tickets for an upcoming concert. They want to make a total of \$311,000 in sales from these tickets.

  1. Assuming that all tickets have the same price, what is the price for one ticket?
  2. How much will they make if they sell 7,000 tickets?
  3. How much will they make if they sell 10,000 tickets? 50,000? 120,000? a million? \(x\) tickets?
  4. If they make \$404,300, how many tickets have they sold?
  5. How many tickets will they have to sell to make \$5,000,000?

Student Response

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

Activity Synthesis

Select student responses to be shared with the whole class in discussion. Sequence their explanations from less efficient and organized to more efficient and organized. Discuss how the solutions are the same and different, and the advantages and disadvantages of each method. An important part of this discussion is correspondences and connections between different approaches.

Speaking, Listening: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each explanation that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they heard to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making

6.3: Recycling (15 minutes)

Activity

This activity is intended to further develop students’ ability to write equations to represent proportional relationships. It involves work with decimals and asks for equations that represent proportional relationships of different pairs of quantities, which increases the challenge of the task.

Students may solve the first two problems in different ways. Monitor for different solution approaches such as: using computations, using tables, finding the constant of proportionality, and writing equations. 

Launch

Arrange students in groups of 2. Provide access to calculators. Give 5 minutes quiet work time followed by sharing work with a partner.

Representation: Internalize Comprehension. Represent the same information through different modalities by using tables. If students are unsure where to begin, suggest that they draw a table to help organize the information provided.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Reading: MLR6 Three Reads. Use this routine to support reading comprehension, without solving, for students. In the first read, students read the problem with the goal of comprehending the situation (e.g., The situation involves weight of cans and the amount of money made from recycling). In the second read, ask students to look for quantities without focusing on specific values. Listen for, and amplify, the quantities that vary in relation to each other in this situation: number of aluminum cans; total weight of aluminum cans, in kilograms; money earned, in dollars. In the third read, ask students to brainstorm possible strategies to calculate the weight of aluminum in one can and the amount of money earned from one can. This helps students connect the language in the word problem and the reasoning needed to solve the problem keeping the intended level of cognitive demand in the task.
Design Principle(s): Support sense-making

Student Facing

Aluminum cans can be recycled instead of being thrown in the garbage. The weight of 10 aluminum cans is 0.16 kilograms. The aluminum in 10 cans that are recycled has a value of \$0.14.

  1. If a family threw away 2.4 kg of aluminum in a month, how many cans did they throw away? Explain or show your reasoning.
  2. What would be the recycled value of those same cans? Explain or show your reasoning.
  3. Write an equation to represent the number of cans \(c\) given their weight \(w\).
  4. Write an equation to represent the recycled value \(r\) of \(c\) cans.
  5. Write an equation to represent the recycled value \(r\) of \(w\) kilograms of aluminum.

Student Response

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

Student Facing

Are you ready for more?

The EPA estimated that in 2013, the average amount of garbage produced in the United States was 4.4 pounds per person per day. At that rate, how long would it take your family to produce a ton of garbage? (A ton is 2,000 pounds.)

Student Response

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

Anticipated Misconceptions

If students have trouble getting started, encourage them to create representations of the relationships, like a diagram or a table. If they are still stuck, suggest that they first find the weight and dollar value of 1 can.

Activity Synthesis

Select students to share their methods: using computations, using tables, finding the constant of proportionality, writing equations. If students did not use equations to solve the first two problems, ask them how they can use the equations they found later in the activity to answer the first two questions.

If time permits, highlight connections between the equations generated, illustrated by the sequence of equations below. \(\displaystyle r = 0.014c\) \(\displaystyle r = 0.014(62.5w)\) \(\displaystyle r = 0.875w\)

Lesson Synthesis

Lesson Synthesis

The activities in this lesson removed some scaffolds used in previous lessons (e.g., presenting a table) and included features (e.g., large numbers) intended to motivate use of equations. Remind students that throughout this lesson, they considered problem situations and created organized ways to get answers. Whether the numbers in the problem are whole numbers, large numbers, or decimals, if there is a proportional relationship between two quantities, their relationship can be represented by an equation of the form \(y = kx\). The situations provided demonstrate the efficiency of equations for certain types of problems. Finding how many tickets should be sold in order to earn \$5 million and finding the relationship between number of cans and weight and recycled value are more elegantly and efficiently handled by equations than by calculations or tables.

  • What were some helpful ways we organized information?
  • What were some equations we found in this lesson?
  • In each equation, what did the letters represent? What did the number mean? \(y=62.2x\), \(c = 62.5w\), \(r = 0.014c\), \(r = 0.875w\)

6.4: Cool-down - Granola (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

Remember that if there is a proportional relationship between two quantities, their relationship can be represented by an equation of the form \(y = k x\). Sometimes writing an equation is the easiest way to solve a problem.

For example, we know that Denali, the highest mountain peak in North America, is 20,300 feet above sea level. How many miles is that? There are 5,280 feet in 1 mile. This relationship can be represented by the equation

\(\displaystyle f=5,\!280 m\)

where \(f\) represents a distance measured in feet and \(m\) represents the same distance measured miles. Since we know Denali is 20,310 feet above sea level, we can write

\(\displaystyle 20,\!310=5,\!280 m\)

So \(m = \frac{20,310}{5,280}\), which is approximately 3.85 miles.