POSTER PROBLEMS
Rating Rate Plans
Sixth Grade Poster Problem
Expressions and Equations
Students learn how to describe cell phone rate plans using
symbolic expressions. Also, students interpret expressions
explaining the rate plans of three fictional cell phone companies
and make recommendations about which is the best for
certain users.
There is also a bit misleading advertising thrown into this
problem for fun!
The way this works: one lesson in six phases
Learning Objectives: • Interpret and translate real-world situations into numerical
expressions using variables.
• Write and evaluate numerical expressions involving wholenumber exponents, variables, and specific values.
• Evaluate expressions with various values of their variables
utilizing the Order of Operations.
Common Core State Standards
for Mathematics:
LAUNCH
Teachers set the stage by leading an introductory discussion that
orients students to the context of the problem.
POSE A PROBLEM
Day 1
Teachers introduce a mathematical way of thinking about the
context and engage students in a preliminary approach that
opens the door to the workshop phase.
Teacher Tune Up:
• Variable, Parameter, and Unknown. What’s the Diff?
• How can focusing on expressions prevent students
from rushing to answer-getting? WORKSHOP
The workshop starts with a more challenging and more openended extension of the problem. In teams, students plan and
produce mathematical posters to communicate their work.
FLEXIBLE
6.EE - 1, 2a, 2c
POST, SHARE, COMMENT
Teams display their posters in the classroom, get to know other
teams’ posters, and attach questions/comments by way of small
adhesive notes (or similar).
STRATEGIC TEACHER-LED DISCUSSION
Day 2
Teachers then compare, contrast and connect several posters. In
the process they highlight a progression from a more basic
approach to a more generalizable one. By doing this, teachers
emphasize standards-aligned mathematics using studentgenerated examples.
FOCUS PROBLEM: SAME CONCEPT IN A NEW CONTEXT
Serving as a check for understanding, this more focused problem
gives teachers evidence of student understanding.
SERP 2014!
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Rating Rate Plans - Sixth Grade Poster Problem !
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1
1. LAUNCH
Directions for teacher:
Engage students in a discussion about cell phones, text usage,
voice usage and rate plans. You will want to lead the discussion
to elicit ideas about:
• Per-text or per-minute charges
• Flat rate charges for a fixed number of text messages or call
minutes per month.
• Overages and other additional charges and fees
Ask the students to analyze a sample bill (Handout #1). Discuss
cellular charges for text and voice calls. Guide students through
the sample bill to understand and identify:
• What is the “billing cycle?”
• How many minutes did this person use for calls? How were
the charges calculated? • How many text messages were sent and received? How
were the charges calculated?
• How would you describe the type of plan this person has?
Unlimited? Pay-as-you-go? Other?
Also, consider a discussion to help students think about how
realistic numbers might be. For example, ask, “Suppose you got
a bill with 1000 texts for a month. Is that realistic? Is it even
possible?” [Yes. 1000÷30 is about 33. And some people easily
send 33 texts a day.]
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Sample Bill - Handout
1
E
L
P
M
L
A
L
I
S B
(512) 555-3333
Alexander Paul Wilksinson
1234 Telephone Lane
Austin, TX 78748-1234
Mobile Services, Inc.
Monthly Charges - Nov 1 thru Nov 30
Family Tree Plan 900
$69.99
Message More Plan 300
$30.00
Total Monthly Charges
$99.99
Other Charges and Credits
Voice Usage Summary - Family Tree Plan 900
Minutes Used
1076
overage $.10/min
176
$17.60
Messaging Summary - Message More Plan 300
Text/Instant Msg Incoming
185
Text/Instant Msg Outgoing
160
overage $.05/msg
45
$2.25
Total Other Charges and Credits
$19.85
Surcharges and Other Fees
Texas State Telecom Tax
$5.93
Austin City Telecom Tax
$0.95
911 Service Fee
$0.50
Federal Universal Service Charge
$2.23
Total Surcharges and Other Fees
$9.61
$129.45
Total for (512) 555-3333
In your opinion, is the bill expensive or not very
expensive? Explain.
SERP 2014!
Student Name: ____________________________________________________
Rating Rate Plans
2014 http://math.serpmedia.org/diagnostic_teaching/
Rating Rate Plans - Sixth Grade Poster Problem !
Sample Bill - Handout #1
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2
2. POSE A PROBLEM
Directions for teacher:
Show Slide #1.
This phase of the lesson gets students accustomed
to using algebraic expressions to compare monthly
rate plans.
Help students determine the meaning of m (number of minutes)
and t (number of texts). Working together, put in sample values
for the variables to better understand the formulas. For example,
you could determine how much Mobile Services, Inc. would
charge for 1,000 minutes and 500 texts. You will want to focus on how the formula is more complex when
the user exceeds the “base” number of minutes or text
messages. This concept should be familiar, however students
are unlikely to have seen it expressed as a formula. Mobile Services, Inc.
Voice calls:
Text messages:
Family Tree Plan 900
Message More Plan
$69.99/month for 900 minutes
$30.00/month for 300 text messages
What if you go over 900 minutes?
What if you go over 300 text messages?
$.10 per minute added to the standard
$.05 per message added to the standard $69.99 charge:
$30.00 charge:
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.1(m – 900) + 69.99
.05(t – 300) + 30
POSTER PROBLEMS - R ATING R ATE PLANS S LIDE #1
Slide #1
Use Handout #2 to give individual or pairs of students practice
at determining monthly costs for four sample customers with
Mobile Services, Inc. rates. Answers to Handout #2:
#1 Karyn
Student Name: ____________________________________________________
Rating Rate Plans
50 minutes voice, $69.99
Sample Customers - Handout
2
Instructions
Here is how Mobile Services, Inc.
calculates what to charge
customers. Please determine how
much Karyn, Nathan, Tara, and
Matt would owe the company when
they get their monthly bills.
100 text messages, $30.00
Total: $99.99
Mobile Services, Inc.
#2 Nathan
Voice calls:
Text messages:
Family Tree Plan 900
Message More Plan
$69.99/month for 900 minutes
$30.00/month for 300 text messages
What if you go over 900 minutes?
What if you go over 300 text messages?
$.10 per minute added to the standard
$.05 per message added to the standard
$69.99 charge:
$30.00 charge:
.1(m – 900) + 69.99
.05(t – 300) + 30
POSTER PROBLEMS - R ATING R ATE PLANS SLIDE #1
150 minutes voice, $69.99
Customer #1: Karyn
Karyn uses 50
minutes on calls
each month but
sends twice as
many text messages.
0 text messages, $30.00
Total: $99.99
Nathan
Customer #2:
3 times
Nathan uses
minutes
the number of
h
Karyn uses eac
r,
month. Howeve
d text
sen
he doesn’t
all.
messages at
Voice Charge
#3 Tara
Customer #3:
Tara
On average, Tara
uses 500 text
messages eve
ry
month and 200
minutes of talk
time
Customer #4: Matt
Matt uses half the
number of texts
compared to Tara
but his voice calls
average 1000
minutes per month.
Text Messages Charge
Total
#1: Karyn
____ minutes voice
200 minutes voice, $69.99
____ text messages
#2: Nathan
500 text messages, $40.00
____ minutes voice
____ text messages
#3: Tara
Total: $109.99
____ minutes voice
____ text messages
#4: Matt
____ minutes voice
#4 Matt
____ text messages
2014 http://math.serpmedia.org/diagnostic_teaching/
1,000 minutes voice, $79.99
Sample Customers - Handout #2
250 text messages, $30.00
Total: $109.99
SERP 2014!
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Rating Rate Plans - Sixth Grade Poster Problem !
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3. W ORKSHOP
Slide #2
Directions for teacher:
Show Slide #2. Ask, “What can we determine about
Transparent Technologies’ rate plan?”
TRANSPARENT TECHNOLOGIES
YOUR PHONE BILL SHOULD
NEVER BE A SURPRISE!
Note: This company uses a flat fee (unlimited use). The
monthly bill is always $99.99.
Show Slide #3. Ask, “What can we determine about
Commendable Communications’ rate plan?”
Note: This company uses a fee structure that always
charges $.10 per minute for voice and begins charging for
individual text messages after 100. They also add a $6.99
fee each month.
99.99 + 0(t) + 0(m)
Slide #3
Commendable
Communications
P
P
-R
R
P
S
#2
OSTER
ROBLEMS
ATING
ATE
LANS
LIDE
Your first 100
texts are on us!
Show Slide #4. Ask, “What can we determine about
Square Deal Media’s rate plan?”
Note: This company indeed offers the first text for the only
half a penny; however, the cost of numerous texts would be
astronomical due to the exponent in the formula. (For fun,
let students discover the effect of the exponent on their
own before you explain it to them).
if t ≤ 100
.10(m) + 6.99
if t > 100
.10(m) + .25(t – 100) + 6.99
Slide #4
POSTER PROBLEMS - R ATING R ATE PLANS S LIDE #3
2
Each company makes a claim that may be true on
the surface. But once the students analyze how each
company determines their monthly fees, these claims
may or may not hold up.
Show Slide #5. Challenge students: “Create a display
that analyzes the rate plans and compares the three
companies. Be sure to include examples of what the
charges would be for a variety of customers. Some might
text more, talk for more minutes, etc. Your poster should
help someone understand which rate plan would be the
best choice for them.”
SQUARE DEAL MEDIA
Half a penny for your first text!
How can you beat that?
0.005t2 + 0(m) + 4.95
Slide #5
TRANSPARENT TECHNOLOGIES
99.99 + 0(t) + 0(m)
POSTER PROBLEMS - R ATING R ATE PLANS S LIDE #4
Commendable Communications
if t ≤ 100
.10(m) + 6.99
if t > 100
.10(m) + .25(t – 100) + 6.99
SQUARE DEAL MEDIA
0.005t2 + 0(m) + 4.95
POSTER PROBLEMS - R ATING R ATE PLANS S LIDE #5
SERP 2014!
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Rating Rate Plans - Sixth Grade Poster Problem !
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4
4. POST, SHARE, COMMENT
Directions for teacher:
Have students post their posters around the classroom.
Encourage students to travel around to view the posters
created by other groups. Students should be
encouraged to pose questions to other groups by
attaching a small adhesive notes to their posters. During this time, the teacher should be review all the
posters and consider which to highlight during the
Phase 5 discussion.
A
Sample Posters:
Poster A displays three sample customers
with different use patterns and lists the sums
each should expect on the monthly bill. This
group also provided a summary statement
about the kind of customer who would
benefit from the specific rate plan.
Poster B shows the work of the calculations
and uses sample users who provide a much
more dramatic difference in costs. They also
made a preliminary attempt to display data a
graph. The graph is unlabeled, but is likely to
be more illustrative than data-based. Notice
that the graph for “Square Deal” is not
curved.
B
C
Poster C focuses on the skyrocketing of the
texting charge with Square Deal’s rate plan.
Poster D compares all three plans, showing
the effect of increasing the number of
minutes for a small fixed number of texts
(left) and increasing the number of texts for a
small fixed number of minutes (right). This
shows ingenuity and understanding, even
though the curve for Square Deal does not
go up as fast as it should!
D
SERP 2014!
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Rating Rate Plans - Sixth Grade Poster Problem !
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5
5. STRATEGIC TEACHER-LED DISCUSSION
Directions for teacher:
Facilitate a discussion about why the “best plan” for a user depends on that particular person's calling and texting
habits. Select a sequence of posters to use during the teacher-led discussion that will help move all students from
their current thinking (often Levels 1–3 below) up to Level 4 or 5.
Level 1: Calculations show final results but don’t show how the student got there. In particular, the work may not
yet connect the algebraic expressions to each company's billing approach. Level 2: Students makes generalizations with limited data. They do not reliably select sample customers to
give a bigger picture.
Level 3: Students coordinate both the voice and messaging charges and intentionally create sample customers to
highlight how changing the voice minutes and text messages change the final bill.
Level 4: Students use multiple representations to illustrate break-even points and how particular patterns of use
affect the bill.
Level 5: Patterns, relationships, and well-articulated caveats are explained with mathematics. Students articulate
the absurdity of using an exponent in a rate plan. They demonstrate visually how dramatic (and Draconian) its
increases are. Additional advances in mathematical thinking to press for during the discussion:
We use variables in different ways. One traditional type of algebra problem presents students with an equation to
solve, such as 2x + 3 = 7. The student learns to “isolate” x to solve the equation, and get the answer, x = 2.
In this activity, however, we don’t need to solve equations. Instead, we present students with expressions that
contain variables. The students have to find and compare the values of these expressions (the total monthly cost)
for different values of the variables (the number of minutes and texts). The students do more plugging in than
solving—but it’s “plugging with a purpose”: the expressions describe something real (the rate plans) using symbolic
mathematics, and the students try different values in order to uncover the patterns behind the mathematical
expressions. The underlying purpose is to find the best plan, where “best” depends on how you use your phone.
Key idea: A variable is a letter that can take on different values within a particular expression, equation,
or situation.
A graph shows us, visually, the relationship between two quantities. Traditionally, the dependent or response
variable goes on the vertical axis, and we often label it “y.” The independent or predictor variable goes on the
horizontal (“x”) axis.
But what do you do when you have two predictors (in this activity, talk minutes and texts)? One strategy is to hold
one variable constant. For example, you could make a graph of how the cost depends on the number of texts,
given that you talk for 500 minutes. (See Poster D at Step 4.)
Key Idea: 2-D graphs show relationships between two variables. With some creativity, however, you can explore
three or more variables, as long as you look at only two at a time.
SERP 2014!
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Rating Rate Plans - Sixth Grade Poster Problem !
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6. FOCUS PROBLEM : SAME CONCEPT IN A N EW CONTEXT
Directions for teacher:
Display Slide #6.
Ask students to talk to a partner to become familiar with the table.
Ask several students to share information that they noticed.
• Food truck and restaurant prices differ.
• All the prices within a category are the same at each location.
At the close of the strategic teacher-led discussion with the
students, distribute Handout #3 to help determine whether
students can use their experience with the rate plans to formulate
an algebraic expression for the total amount a family would pay at
their favorite eatery. Kasa Indian Eatery
Food Truck Prices
Restaurant Prices
$8
$15
Sides!
Pappadum!
Daal!
Samosa
$5
$9
Beverages!
Chai!
Lassi!
Soda
$2
$3
Entrees!
Chicken Tikka!
Lamb Curry!
Gobi Aloo (Vegetarian)
POSTER PROBLEMS - R ATING R ATE PLANS S LIDE #6
Answers to Handout #3:
Slide #6
1.
8e + 5s + 2b is the expression for the food truck
15e + 9s + 3b is the expression for the restaurant
The prices are shown as coefficients in the expressions.
2.
Student Name: ____________________________________________________
Rating Rate Plans
From Rate Plans to Restaurants - Handout
Food Truck Prices
Kasa Indian Eatery
Food truck
Entrees
3
Restaurant Prices
$8.00
$15.00
$5.00
$9.00
$2.00
$3.00
• Chicken Tikka
8(5) + 5(6) + 2(6)
• Lamb Curry
• Gobi Aloo (Vegetarian)
40 + 30 + 12
Sides
• Pappadum
• Daal
82
• Samosa
Beverages
$82.00 (plus tax and tip?)
• Chai
• Lassi
• Soda
Instructions
Restaurant
Trey’s family loves eating at Kasa.
Sometimes they eat at the restaurant and
sometimes they go to Kasa’s food truck
and eat in a park. Either way, they use
algebraic expressions to know what they
will have to pay.
15(5) + 9(6) + 3(6)
75 + 54 + 18
1.
3.
8e + 5s + 2b
15e + 9s + 3b
Is this expression for the
Is this expression for the
food truck or the
147
Variables:
e = number of entrees ordered
s = number of sides ordered
b = number of beverages ordered
food truck or the
restaurant?
restaurant?
How do you know?
How do you know?
$147.00 (plus tax and tip?)
2.
Use the expressions to find out how much Trey’s family would pay if they ordered the meal marked on
the menu above at the food truck. Then do the same for the restaurant.
Encourage the use of an expression that can be used
regardless of the number of items ordered. (prices as
coefficients and numbers of the entrees, sides, and
beverages each as variables)
3.
Finally, change all the prices at Kasa. Write new expressions that could be used for your new prices.
SERP 2014!
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2014 http://math.serpmedia.org/diagnostic_teaching/
From Rate Plans to Restaurants - Handout #3
Rating Rate Plans - Sixth Grade Poster Problem !
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