Topic A: Proportional Relationships

Topic A: Proportional
Relationships
Lesson 2
Proportional Relationships
LEARNING TARGET
Lesson 2: Proportional Relationships – Day 1
Today I can identify the constant of proportionality (k) and write an equation from a table in
the form y = kx.
STANDARDS
7.RP.2b Identify the constant of
proportionality (unit rate) in tables, graphs,
equations, diagrams, and verbal
descriptions of proportional relationships.
7.RP.c Represent proportional relationships
by equations.
KEY VOCABULARY
Proportional
Constant
Constant of Proportionality
AGENDA
• (10 min) Review Key Vocabulary
• (5 min) Example 1: Pay by the Ounce
Frozen Yogurt
• (10 min) Discussion
• (10 min) Write an Equation
• (5 min) Example 2: A Cooking Cheat
Sheet!
• (10 min) Discussion
• (10 min) Example 2 (Continued)…
• (5 min) Exit Ticket
• (20-30 min) Online Practice
Review Key Vocabulary
• Proportional – when two quantities that simplify to
the same ratio.
• Constant – a quantity having a value that does not
change or vary.
• Constant of Proportionality - a constant value of the
ratio of two proportional quantities.
Example 1: Pay by the Ounce Frozen Yogurt
A new self-serve frozen yogurt store opened this
summer that sells its yogurt at a price based upon the
total weight of the yogurt and its toppings in a dish.
Each member of Isabelle’s family weighed their dish and
this is what they found.
Weight (ounces)
12.5
10
5
8
Cost ($)
5
4
2
3.20
Discussion
1. Does everyone pay the same cost per ounce?
How do you know?
2. Isabelle’s brother takes an extra-long time to
create his dish. When he puts it on the scale,
it weighs 15 ounces. If everyone pays the
same rate in this store, how much will his
dish cost? How did you calculate this cost?
3. What happens if you don’t serve yourself any
yogurt or toppings, how much do you pay?
Write an Equation
Weight (ounces)
12.5
10
5
8
Cost ($)
5
4
2
3.20
For any measure x, how do we find y?
Example 2: A Cooking Cheat Sheet!
In the back of a recipe book, a diagram provides
easy conversions to use while cooking.
Discussion
1.What does the diagram tell us?
2.Is the number of ounces proportional to the
number of cups? How do you know?
3.Is there another way to represent this same
information?
A Cooking Cheat Sheet! (Continued…)
Cups (x)
0
½
1
1½
2
4
5
8
Ounces
(y)
0
4
8
12
16
?
?
?
1. For any number of cups x, how do we find the number
of ounces, y?
2. If we want to verify our equation, which 𝑥 and 𝑦
values can we use to see if it is true? How do you
know?
Exit Ticket – Day 1
1. Explain how we found the constant of
proportionality?
2. Explain how we used the constant of
proportionality to find missing values in the
table.
LEARNING TARGET
Lesson 2: Proportional Relationships – Day 2
Today I can determine if one value is proportional to another value by testing equivalent
ratios in a table.
STANDARDS
7.RP.2a Decide whether two quantities are
in a proportional relationship, e.g., by testing
for equivalent ratios in a table or graphing
on a coordinate plane
and observing whether the graph is a
straight line through the origin.
KEY VOCABULARY
Proportional
AGENDA
• (10 min) Review Key Vocabulary
• (5 min) Exercise 1: Calories Burned
• (10 min) Discussion
• (5 min) Example 3: Summer Job
• (10 min) Discussion
• (15 min) Partner Work & Share Out
• (5 min) Exit Ticket
• (20-30 min) Online Practice
Review Key Vocabulary
• Proportional – when two quantities that simplify to
the same ratio.
• Constant – a quantity having a value that does not
change or vary.
• Constant of Proportionality - a constant value of the
ratio of two proportional quantities.
Exercise 1: Calories Burned
During Jose’s physical education class today,
students visited activity stations. Next to each
station was a chart depicting how many Calories
(on average) would be burned by completing
the activity.
Discussion
1. Is the number of calories burned
proportional to time? How do you know?
2. If Jose jumped rope for 6.5 minutes, how
many calories would he expect to burn?
Example 3: Summer Job
Alex spent the summer helping out at his family’s business. He
was hoping to earn enough money to buy a new $220 gaming
system by the end of the summer. Halfway through the
summer, after working for 4 weeks, he had earned $112. Alex
wonders, “If I continue to work and earn money at this rate,
will I have enough money to buy the gaming system by the
end of the summer?”
To check his assumption, he decided to make a table. He
entered his total money earned at the end of week 1 and his
total money earned at the end of Week 4.
Week
Total
Earnings
0
1
$28
2
3
4
$112
5
6
7
8
Discussion
1. How much do you think Alex earned by the end
of 2 weeks?
2. How will a table help us to check Alex’s
prediction?
3. Where did the two given pairs of data come
from?
4. Is this reasonable?
5. What other pair could we complete fairly easily?
6. How will we find out his earnings after 2 weeks?
3 weeks?
Partner Work
1. Work with a partner to answer Alex’s
question.
2. Are Alex’s total earning proportional to the
number of weeks he worked? How do you
know.
Exit Ticket – Day 2
1. How did you determine if Alex’s earning was
proportional to the number of weeks he
worked?
2. What is a situation where earning is not
proportional to the number of weeks
worked?
LEARNING TARGET
Lesson 2: Proportional Relationships – Day 3
Today I can determine if two quantities are proportional from a word problem.
STANDARDS
7.RP.2a Decide whether two quantities are
in a proportional relationship, e.g., by testing
for equivalent ratios in a table or graphing
on a coordinate plane and observing
whether the graph is a straight line through
the origin.
KEY VOCABULARY
Proportional
Constant
Constant of Proportionality
AGENDA
• (10 min) Review Key Vocabulary
• (15 min) Extension & Share Out
• (20 min) Problem Set
• (5 min) Lesson Summary
• (20-25 min) Lesson 2 – Quiz
• (10 min) Grade & Correct
• (5 min) Collect Class Data
Review Key Vocabulary
• Proportional – when two quantities that simplify to
the same ratio.
• Constant – a quantity having a value that does not
change or vary.
• Constant of Proportionality - a constant value of the
ratio of two proportional quantities.
Extension
Ms. Albero decided to make juice to serve along
with the pizza at the Student Government party.
The directions said to mix 2 scoops of powdered
drink mix with a half a gallon of water to make
each pitcher of juice. One of Ms. Albero’s
students said she will mix 8 scoops with 2
gallons of water to get 4 pitchers. How can you
use the concept of proportion to decide
whether the student is correct?
Problem Set
1 Point
(Unsatisfactory)
2 Points
(Partially Proficient)
3 Points
(Proficient)
A correct answer
Missing or incorrect Missing or incorrect with some evidence
answer and little
answer but
of reasoning or an
evidence of
evidence of some
incorrect answer
reasoning
reasoning
with substantial
evidence
4 Points
(Advanced)
A correct answer
supported by
substantial
evidence of solid
reasoning
Lesson Summary
1. How do we know if two quantities are
proportional to each other?
2. How can we recognize a proportional
relationship when looking at a table or a set
of ratios?
Lesson 2 - Quiz
1 Point
(Unsatisfactory)
2 Points
(Partially Proficient)
3 Points
(Proficient)
A correct answer
Missing or incorrect Missing or incorrect with some evidence
answer and little
answer but
of reasoning or an
evidence of
evidence of some
incorrect answer
reasoning
reasoning
with substantial
evidence
4 Points
(Advanced)
A correct answer
supported by
substantial
evidence of solid
reasoning