Transcript Ratio and Proportion - Region 11 Math and Science

Ratio and Proportion
Today’s Goals
1.
2.
3.
4.
5.
Get acquainted;
Review our learning related to the
concept of equality;
Learn strategies to help students
think proportionally and
multiplicatively;
Differentiate proportional from nonproportional situations;
Understand the PLC tasks related to
ratio and proportion strategies.
T or F:
5 x 84 = 10 x 42
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By calculating
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By thinking relationally
10 x 42 = 420
5 x 84
= 5 x (2 x 42)
= (5 x 2) x 42
= 10 x 42
So, 5 x 84 = 10 x 42
…or?
5 x 84 = 420
TASK
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Work as a group
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Solve your equation in two ways:
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
by calculating
by thinking relationally
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Show your thinking
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Be ready to explain your work to the larger
group
Relational Thinking…
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Focuses on relations rather than only on
calculating answers (red)
Uses fundamental properties of arithmetic to
relate or transform expressions (green)
Looks at expressions and equations in their
entirety rather than as procedures to be
carried out step by step (orange)
Flexibly uses operations and relations
(yellow)
from Tom Carpenter’s slide show on September 23rd
Walk-about and Discuss

Take a ‘wall walk’ with your group

Look for examples of these relational thinking
characteristics in these problems.

Use post-its to identify examples when you
see them
So….

What is the connection between
relational thinking and solving
equations?
TASK
1.
Select a PLC task of your choice:
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2.
3.
4.
5.
6.
Baseline Assessment
Teaching a lesson
Interviewing students
Post Assessment
Find 2-3 others who share your interest
Find the related discussion Q’s
Discuss the Q’s with your small group
Discuss the Follow-up Q’s
Be ready to report out
Summary Questions
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What do we want our kids to know
about equality?
What can we do to improve their
understanding of equality as an
equivalence relationship?
How will this work impact students
understanding of algebra?
Quantitative Comparison
TASK:

Work as a group

Examine the student work and be able
to explain the strategy used

Compare the samples of student work

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What strategies are similar?
What strategies are different?
Student Responses
(from Glenda Lappan)
Student Responses
(from Glenda Lappan)
Student Responses
(from Glenda Lappan)
Student Responses
(from Glenda Lappan)
Student Responses
(from Glenda Lappan)
Wes Problem #1
Two baseballs cost $5. How
much will seven baseballs
cost?
Two baseballs cost $5. How much
will seven baseballs cost?
The unit rate approach is characterized when
students find the constant factor across or
between the two measure spaces.
baseballs
2
7
x 2.5
x 2.5
$
5
? 17.50
Two baseballs cost $5. How much
will seven baseballs cost?
The scale factor approach is characterized
when students find the multiplicative
relationship among elements within the
measurement space.
baseballs
X
3.5
2
7
$
5
17.50?
X
3.5
Two baseballs cost $5. How much
will seven baseballs cost?
baseballs
The table strategy
is a strategy that is a
mix of unit rate and
factor of change.
2
4
6
7
8
$
5
10
15
17.50
20
Two baseballs cost $5. How much
will seven baseballs cost?
y = 2.5x
The graphing
strategy has an
interesting equation.
20
baseballs
$
2
5
4
10
6
15
7
17.50
8
20
Dollars ($)
17.50
15
10
5
0
0
2
4
6
baseballs
7
8
Two baseballs cost $5. How much
will seven baseballs cost?
The fraction strategy is devoid of the problem
context.
2
5
x 3.5
x 3.5
7
=
?
17.50
Two baseballs cost $5. How much
will seven baseballs cost?
Cross-multiplication is also a strategy but is
hard for students to explain why it works and
they tend not to use it when solving problems
on their own.
2
7
=
5
x
2 x  35
baseball-dollar
Wes Problem #2
I can buy 6 baseballs for $20.
How much will 18 baseballs
cost?
Wes Problem #3
I can buy 6 baseballs for $18.
How much will 11 baseballs
cost?
Wes Problem #4
I can buy 7 baseballs for $12.
How much will 11 baseballs
cost?
The Marissa Problem
Marissa bought 0.46 of a pound of
wheat flour for which she paid
$0.83. How many pounds could
she buy for one dollar?
The Track Problem
Wendy and Brooke run at the same
speed around a track. Wendy was
running lap 3 when Brooke was running
lap 7. What lap will Brooke be on when
Wendy is running lap 9?
Proportional Situation or
Not?
Taxi cabs in Minneapolis charge $2.50 for
the first one-fifth mile and each
succeeding fifth will cost $.38. How much
will the fare be for a 2 mile trip?
Proportional Situation or
Not?
The action figure of a famous wrestler
has a scale of 1/12. If the doll is 7 inches
tall, how tall is the actual wrestler?
Proportional Situation or
Not?
Dara swims four laps across the pool in the
same time as Michael swims three laps.
They begin swimming at the same time.
How far has Michael swum when Dara
completes lap 10?
Proportional Situation or
Not?
Jake typically exchanges 200 text messages a
day. The plan he signed up for charges a $30
flat fee per month and does not charge for the
first 1000 text messages exchanged. Each
message exchanged after the first 1000 costs
$0.05. What is the amount for Jake’s bill for the
month of October?
Proportional Situation or
Not?
Lance is riding his bicycle in France. He rides
18 km in half an hour. If he rides at the same
rate, how long will it take him to ride 162 km?
Proportional Situation or
Not?
Two U.S. dollars can be exchanged for three
Euros. How many U.S. dollars can be
exchanged for 21 Euros?
Proportional Situation or
Not?
Proportional Situation or
Not?
Proportional Situation or
Not?
Proportional Situation or
Not?
Proportional Situation or
Not?
Proportional Situation or
Not?
What is proportional thinking?
(partial list adapted from Lamon)
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the ability to use proportionality as a mathematical model
to organize information in appropriate contexts;
the ability to distinguish situations in which
proportionality is not an appropriate mathematical model
from situations in which it is useful;
knowing that the graph of a direct proportional situation
is a straight line through the origin;
the ability to explain the difference between functions of
the form y = mx and functions of the form y = mx + b. In
the latter function, y is not proportional to x, but rather
the change in y is proportional to the change in x.
Minnesota State Algebra
Standards (7th grade)
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Understand the concept of proportionality in real-world and
mathematical situations, and distinguish between proportional and
other relationships.
Recognize proportional relationships in real-world and mathematical
situations; represent these and other relationships with tables,
verbal descriptions, symbols and graphs; solve problems involving
proportional relationships and explain results in the original context.
Apply understanding of order of operations and algebraic properties
to generate equivalent numerical and algebraic expressions
containing positive and negative rational numbers and grouping
symbols; evaluate such expressions.
Represent real-world and mathematical situations using equations
with variables. Solve equations symbolically, using the properties of
equality. Also solve equations graphically and numerically. Interpret
solutions in the original context.
Upcoming PLC Tasks
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Administer a Baseline Assessment on
Ratio and Proportion to at least one
class
Teach a Lesson
Interview at least 3 students
Administer a Post Assessment
Meet in your PLCs to discuss your
observations/insights
Be a Student…
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Work the Baseline Assessment
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Predict how your students will respond
to the questions
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Ask Q’s about anything that is unclear
Baseline Assessment #1
2 baseballs cost $5. How
much will 7 baseballs cost?
Answer:
Use drawings, words, or numbers to show
how you got your answer.
Baseline Assessment #2
A sixth-grade class needs 5 leaves each day
to feed its 2 caterpillars. How many leaves
would they need each day for 12 caterpillars?
(NAEP, 1996)
Answer:
Use drawings, words, or numbers to show how you got your
answer.
Baseline Assessment #3
The length of Mr. Short is 4
large buttons.
The length of Mr. Tall is 6
large buttons.
When paper clips are used to
measure Mr. Short and Mr.
Tall:
The length of Mr. Short is 6
paper clips.
What is the length of Mr. Tall
in paper clips?
Baseline Assessment #4
Wendy and Brooke run at the same
speed around a track. Wendy was
running lap 3 when Brooke was
running lap 7. What lap will Brooke
be on when Wendy is running lap
9?
Baseline Assessment #5
Doughman Instant Pancake mix
requires 3 cups of water for every 4
cups of mix. Xiong needs to make a
large number of pancakes. She has
23 cups of mix and wants to use it
all. How much water does Xiong
need to add to make her
pancakes?
TASK:
Prepare to teach a lesson
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Turn to page 7 in the handout.
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Work with a partner to create sample
problems for each of the strategies.
Summary for the Day
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Keep reviewing equality -- students develop their
understanding over time.
Use multiple strategies to model problems involving
proportional thinking.
 Unit rate
 Scale factor
 Tables
 Graphs
 Fractions
Think deeply before you teach cross multiplication.
Teach what proportions ARE and what they ARE NOT