Standards-based assessment and Instruction

# Math 6-8

## Which Is the Best Buy?

Eric has opened a new video store in town. He is advertising \$1.50 for each video rented. He claims that his videos are the cheapest. However, Sarah’s Short Stop says their videos are the better buy. For just an \$18 membership fee, videos only cost \$1 for each. Use mathematics to decide which video rental is the best buy.

#### More Accessible Version:

Eric has opened a new video store in town. He is advertising \$1.50 for each video rented. He claims that his videos are the cheapest. However, Sarah’s Short Stop says their videos are the better buy. For just an \$18 membership fee, videos only cost \$1 for each. If a customer wants to rent 2 videos a month for 1 full year, use mathematics to decide which video rental is the best buy.

#### More Challenging Version:

The original version, and …
Research the rates at a video store near you. How does its rates compare to those of Eric’s and Sarah’s?

### NCTM Content Standards and Evidence

#### Algebra Standard for Grades 6-8

Instructional programs from Pre Kindergarten through grade 12 should enable students to ...

• Understand patterns, relations, and functions
• Use mathematical models to represent and understand quantitative relationships
• NCTM Evidence: Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules
• NCTM Evidence:Relate and compare different forms of representation for a relationship
• NCTM Evidence:Model and solve contextualized problems using various representations, such as graphs, tables, and equations
• Exemplars Task Specific Evidence: This task requires students to find the break-even point of two sets of data.

### Time/Context/Qualifiers/Tip(s) From Piloting Teacher

This task takes one to two 45-minute class periods to complete.

This task could link to other “better buy” tasks that require finding the break-even point to solve, such as the “Calculating Calculator Costs” also featured in this issue of Exemplars.

### Common Strategies Used to Solve This Task

Most students will start with a chart, translate the data to a double line graph, and use the intersection of the two graphs to determine the break-even point.

### Possible Solutions

#### Original Version:

 # of videos cost at sarah's Cost at eric's 1 \$19.00 \$1.50 2 \$20.00 \$3.00 3 \$21.00 \$4.50 4 \$22.00 \$6.00 5 \$23.00 \$7.50 6 \$24.00 \$9.00 7 \$25.00 \$10.50 8 \$26.00 \$12.00 9 \$27.00 \$13.50 10 \$28.00 \$15.00 11 \$29.00 \$16.50 12 \$30.00 \$18.00 13 \$31.00 \$19.50 14 \$32.00 \$21.00 15 \$33.00 \$22.50 16 \$34.00 \$24.00 17 \$35.00 \$25.50 18 \$36.00 \$27.00 19 \$37.00 \$28.50 20 \$38.00 \$30.00 21 \$39.00 \$31.50 22 \$40.00 \$33.00 23 \$41.00 \$34.50 24 \$42.00 \$36.00 25 \$43.00 \$37.50 26 \$44.00 \$39.00 27 \$45.00 \$40.50 28 \$46.00 \$42.00 29 \$47.00 \$43.50 30 \$48.00 \$45.00 31 \$49.00 \$46.50 32 \$50.00 \$48.00 33 \$51.00 \$49.50 34 \$52.00 \$51.00 35 \$53.00 \$52.50 36 \$54.00 \$54.00 37 \$55.00 \$55.50 38 \$56.00 \$57.00 39 \$57.00 \$58.50 40 \$58.00 \$60.00 41 \$59.00 \$61.50 42 \$60.00 \$63.00 43 \$61.00 \$64.50 44 \$62.00 \$66.00 45 \$63.00 \$67.50 46 \$64.00 \$69.00 47 \$65.00 \$70.50 48 \$66.00 \$72.00 49 \$67.00 \$73.50

36 videos is the break even point, after that Sarah's is the better buy.

The task can also be approached by solving systems of linear equations.

y = 1.5 x
y = x + 18

Use the substitution method:

1.5 x = x + 18
.5 x = 18
x = 36 videos is the break-even point

#### More Accessible Version:

 # of Videos cost at Sarah's cost at eric's 1 \$19.00 \$1.50 2 \$20.00 \$3.00 3 \$21.00 \$4.50 4 \$22.00 \$6.00 5 \$23.00 \$7.50 6 \$24.00 \$9.00 7 \$25.00 \$10.50 8 \$26.00 \$12.00 9 \$27.00 \$13.50 10 \$28.00 \$15.00 11 \$29.00 \$16.50 12 \$30.00 \$18.00 13 \$31.00 \$19.50 14 \$32.00 \$21.00 15 \$33.00 \$22.50 16 \$34.00 \$24.00 17 \$35.00 \$25.50 18 \$36.00 \$27.00 19 \$37.00 \$28.50 20 \$38.00 \$30.00 21 \$39.00 \$31.50 22 \$40.00 \$33.00 23 \$41.00 \$34.50 24 \$42.00 \$36.00

For 24 videos, Eric has the better buy.

#### More Challenging Version:

Results will vary depending on the data the student collects on an additional video store.

General Notes: This task lends itself well to students using math representations to solve the task and communicate the solution.

Click on a level for student example.
Novice The student has limited understanding of the task. No progress is made toward a solution. No math language or representations are used.
Apprentice Partial understanding is demonstrated. An incorrect conclusion is achieved. Computation
or reasoning errors are present. Some math language and representations are used. Some
awareness of audience is evident.
Practitioner A correct solution is achieved. All work is shown and accurate. Math representations are used as a strategy to solve the task and to communicate the solution.
Expert All work is correct, organized, and accurate. Math representations are used as a strategy to solve the task and to communicate the solution. Awareness of audience is intentional. Algebraic notation is used to communicate the solution.

Our teacher-friendly tasks are designed to support both the Common Core and Citywide instructional expectations. GO Math! alignments are also available.
Set up your FREE 30-day Trial today!

Explore our latest K-5 math material and begin using it in your classroom.
Set up your FREE 30-day Trial today!

## Here's What People Are Saying

The Southwest Vermont Supervisory Union (SVSU) in Bennington was trying to help its local community gain a better understanding of what students need to demonstrate in order to be successful on their math assessments. One of the most difficult sections of the exam is problem solving. Here, students are not only expected to determine the correct strategy, but also to explain their answers in writing, using math language and representation. For four years SVSU has used Exemplars as a curriculum tool in teaching its students how to solve these types of problems...

Steve Cleveland

Information Technology Coordinator

Exemplars | 271 Poker Hill Road | Underhill, Vermont 05489 | ph: 800-450-4050 | fax: 802-899-4825 | infoREMOVETHISBEFORESENDING@exemplars.com