## There Has to Be a System for This Sweet Problem

• Lesson
9-12
1

We are confronted with problems on a regular basis. Some of these are easy to solve, while others leave us puzzled. In this lesson, students use problem-solving skills to find the solution to a multi-variable problem that is solved by manipulating linear equations. The problem has one solution, but there are multiple variations in how to reach that solution.

### Warm-Up

Based on the level of your students, you may want to consider using a warm up activity to ensure students have sufficient background knowledge of multiple linear equations. Have students work together to solve the problem on the Movie Rental Overhead.

Once students have had adequate time to reach a solution, go over the answer below. Make sure students understand that the more complex problem presented later in this lesson uses systems differently from the way this warm-up uses them. The warm-up activity finds a solution to the system, while the candy problem does not.

Solution

With f = flat rate and r = rental rate:

### The Candy Problem

Present to students the challenge on the Candy Problem Activity Sheet.

 Daniel bought 1 pound of jelly beans and 2 pounds of chocolates for $2.00. A week later, he bought 4 pounds of caramels and 1 pound of jelly beans, paying$3.00. The next week, he bought 3 pounds of licorice, 1 pound of jelly beans, and 1 pound of caramels for $1.50. How much would he have to pay on his next trip to the candy store if he bought 1 pound of each of the 4 kinds of candy? When solving the problem, students can work individually or with a partner. If you want to encourage mathematical discourse, have students work in pairs. Ask students to individually take at least 5 minutes to read over and write down their thoughts about the problem before discussing it with a partner. They should use Questions 1–4 to aid in this discussion. Encourage students to use the questions on the handout to scaffold their problem solving approach. If students are strong problem solvers, or if a problem-solving approach has previously been established in the classroom, students may choose to skip to Question 5. Walk around the room and check what students have written down for Questions 1–4. Students who do not read the problem correctly will venture off on false starts, which may lead to growing frustration and declaring the problem to be unsolvable. Aid student understanding by pointing out there are 4 unknowns and only 3 equations. This is a multi-variable problem, but they are not being asked to find the value of the individual variables. This problem asks to find the cost of the next purchase, which requires manipulating the system but not finding the solution of it. As you walk around viewing student work, consider asking students one or more of the following questions: • What symbolic representation did you use for the unknowns and why did you pick these? [Answers may vary, but some students may choose the first letter of the candy, such as j, c, and l with k representing caramels since c was used for chocolates. Remind students to write down what the variables represent, both to remind them as they delve deeper into the problem and as a legend for others to understand their work.] • Share with me what you are looking for in this problem and why. [If students have read the problem correctly, they will share that they are looking for the total cost of the fourth purchase, which consisted of 1 lb of jelly beans + 1 lb of chocolates + 1 lb of caramels + 1 lb of licorice. If students tell you that they are looking for the cost of the individual candy types, ask them to reread the problem to ensure they have properly identified what they need to know.] • Can you share with me your approach to answering the question? [Students should first write equations, as directed in Questions 1–4, so that linear transformations can be used. Their equations may read as follows:  j + 2c + + = 2.00 j + + 4k + = 3.00 j + + k + 3l = 1.50 j + c + k + l = y A table is a good way for students to organize their work. If any students use guess-and-check, consider challenging them to find the same solution using an algebraic approach. See the Candy Problem Answer Key for a more detailed solution to this problem.] Once all students have completed the problem have them document their solution on chart paper so they can share it with the class. Answers to the candy problem and activity sheet questions can be found on the Candy Problem Answer Key. Expect multiple ways of manipulating the problem even though there is only one solution, y =$2. Some students will remove the decimal from the pricing, thus working in pennies — which is fine, provided they convert their final answer from 200 to $2.00. If some students have finished their work ahead of others, consider using them as coaches for student pairs who are still working. ### Suggestions for Differentiation Using Ability Grouping Pairing students with like abilities provides stronger students with the option of working ahead quickly, while allowing you to facilitate learning by coaching weaker pairs. Pairing students with mixed abilities allows the stronger student an opportunity to coach the weaker student. In this problem, the weaker student is often the one who reads the problem correctly, while the stronger student may try to undertake the impossible task of finding the value of each unknown. This type of pairing can be rewarding for both students. ### Reference Jim Wilson, University of Georgia. The Candy Problem Assessment Options 1. Collect completed activity sheets. 2. Have students present their findings and results. 3. Make up another word problem involving more unknowns than equations. To design your own question, you may want to initially assign values to the unknowns, but make sure none of these values are needed in the final problem to be presented. Extensions 1. Consider asking students to write a similar problem, complete with solution, that they will share with another class or group. For more advanced students, you may consider having them write a problem that could be used on a future assessment. 2. For students looking for another challenge, you may consider sharing the 7-11 problem: You go to 7-11 and take 4 items to the checkout. The clerk tells you that the cash register is broken, so she’ll use her calculator to find the total. You notice that she enters the price of each item, but she multiplies the 4 prices instead of adding them. When she finishes, her total is$7.11. You are about to say something, but then you realize that the total would have been the same if she had added the prices, so you just keep your mouth shut. If there is no sales tax on the items, what are the 4 prices?
[$3.16,$1.25, $1.50, and$1.20 have both a sum of 7.11 and a product of 7.11.]

Questions for Students

1. Did any of you have false starts? If so, please share the path you went down at first. What made you recognize that this method was not leading to a solution?

[Most false starts are due to students not reading the problem or over generalizing and trying to solve for the cost of each candy. Questions involving systems of equations generally ask students to find the value of variables. In this problem, this is not possible because there are more unknowns than equations.]

2. What information would you need to solve for the individual price of the 4 kinds of candy?

[The most common suggestion is the value of 1 unknown or a fourth equation (purchase).]

3. Is there a graphical solution to this problem? Why or why not?

[The graphical solution to a system of equations is the point where the lines representing all the equations intersect. When this intersection is a point, it is called a solution to the system. However, since the problem statement does not provide enough information to find the values of the unknowns, the coordinates (which correspond to these values) also cannot be determined. Therefore, there is no graphical solution to this problem.]

4. How is your solution to the candy problem similar to the solution of the movie rental problem? Why is the solution to the candy problem not the same as the solution to the system of equations?

[The problems are similar because they both involve linear transformations. However, the definition of a solution of a system of linear equations is the set of values for all variables in the system. The candy problem does not find the individual values, while the movie rental problem does.]

Teacher Reflection

• Did your students have sufficient background knowledge to complete this activity? If not, what would you change?
• Were students frustrated with the problem? What supports did you or could you provide to reduce frustration?
• Were students able to clearly articulate their understanding?
• Were students’ levels of enthusiasm and involvement high or low? Explain why.
• Did you challenge the high achievers? How?
• Were students who are typically low achievers able to complete the problem? How could their learning have been scaffolded differently?
• Did you set clear expectations so that students knew what was expected of them? If not, how could you make them more clear?
• Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments? Were these adjustments effective?
• What worked with classroom behavior management? What didn’t work? How would you change what didn’t work?
• What did you want your students to accomplish via this task? Was it accomplished?

### Learning Objectives

Students will:

• Solve problems using systems of equations.
• Demonstrate knowledge of systems of equations through clear articulation of their mathematical thinking.

### NCTM Standards and Expectations

• Write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases.
• Use symbolic algebra to represent and explain mathematical relationships.