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 verhead.
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.
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:
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.]
|j ||+||2c ||+|| ||+|| ||=||2.00|
|j ||+|| ||+||4k ||+|| ||=||3.00|
|j ||+|| ||+||k ||+||3l ||=||1.50|
|j ||+||c ||+||k ||+||l ||=||y |
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.