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.
With f = flat rate and r = rental rate:
|8r + f ||=||16.50|
|–(6r + f ||=||14.00)|
| || || |
|8(1.25) + f ||=||16.50|
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
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
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,
othe 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.