PREPARING FOR SUCCESS IN ALGEBRA
DEMONSTRATION CENTER
A Collaboration among:
 Los Angeles USD
 University of California, San Diego
 San Diego State University
 University of California, Irvine
Why is Proportional Reasoning Important?
It is primary goal of the CCSS in grades 5 - 8
 Scaling a problem allows you to solve real life
problems such as recipe modification, scale
drawings or photo enlargements.
 An ability to reason proportionally indicates a
facility with rational numbers and their
multiplicative concepts

Different views of fractions
Fractions as measures
 Fractions as part/whole representations
 Fractions as operators i.e. scaling
 Fractions as ratios/percents

Different views of fractions
Fractions as measures
 Fractions as part/whole representations
 Fractions as operators i.e. scaling
 Fractions as ratios/percents

Fractions as measures
The number line is the primary tool for this view.
A group of Ms. Guzman’s students was following
a set of directions to move a paper frog along a
number line. Their last direction took them to
1/2. The next direction says: Go 1/3 of the
way to 3/4. What number will the frog land
on?
Different views of fractions
Fractions as measures
 Fractions as part/whole representations
 Fractions as operators i.e. scaling
 Fractions as ratios/percents

Fractions as part/whole representations
For most students, this is the most common and
perhaps their only interpretation.
Jose bought 8 pizzas for the 28 students in his
class. Normally each pizza is cut into 8 pieces
but the pizzeria was willing to cut them into a
different number (the same for each pizza
however). Jose wanted everyone to receive the
same number of pieces. What is the smallest
number he could choose for each pizza?
Different views of fractions
Fractions as measures
 Fractions as part/whole representations
 Fractions as operators i.e. scaling
 Fractions as ratios/percents

Fractions as operators
Scale factors. This is a functional view of
fractions where there is an input and a
resulting output after performing a fixed
operation. Scaling up or down is a natural
interpretation e. g. scaling up or down a recipe
depending on the number of guests and
recognizing when multiple objects are related
by the same scale factor. Producing an
enlargement of a photograph.
Different views of fractions
Fractions as measures
 Fractions as part/whole representations
 Fractions as operators i.e. scaling
 Fractions as ratios/percents

Fractions as ratios and percents
Ratios are ordered pairs (as are fractions) but
care must be used in interpretation. (2:3) is
usually interpreted as 2/3 but may also be
represented as 2/5 or 40%. This latter view
indicates the first entry is 40% of the two
quantities combined. Rates may be interpreted
as ratios e.g. 35 mph could be interpreted as
(35:1) or (70:2) or (87.5:2.5). Thirty five
percent may be represented as (35:100).
Theater Seats
There are 100 seats in a theater, with 30 in the
balcony and 70 on the main floor. 80 tickets
were sold including all the seats on the main
floor. What is the ratio of empty seats to
occupied seats? What is the ratio of empty
seats to occupied seats in the balcony?
What is the meaning of 24x?
The camera has a zoom lens ranging from a wide angle of
25mm to a full telephoto of 600mm. (Both 35mm equivalents)
These two numbers produce a ratio of 600:25 or the equivalent ratio of 24:1
The ratio of 24:1 is represented in cameras as 24x
Estimating populations
One method of estimating the number of
fish in a lake is to catch and tag a collection
of fish and then release them. At a later
time, we return and catch a sample of fish
and determine the proportion of tagged fish.
We can then use that proportion to estimate
the total fish population in the lake.
Estimating Populations (Continued)
For example, suppose that we initially
caught, tagged, and released a total of 50
fish. If we return later and catch a total of 60
fish, 18 of which were tagged, we can then
estimate the number of fish in the lake?
Fish population estimate
Using a Tape Model
?
50
Tag
All fish in the lake
18
60
Tag
sample
Fish population simulation
All Proportional Reasoning Problems
1
2
3
4
There are 4 quantities. If
you know any 3 of them,
you can calculate the 4th.
1/2 = 3/4 or 1/3 = 2/4
Ratings and Shares
A rating is defined as the percentage of all
TVs in the market that are tuned in to a
particular program.
 The share is defined as the percentage of the
TVs tuned in to a particular program among all
TVs that are actually ON.

Consider a Market consisting of 5
televisions.
Set 1
Dodger
Game
Set 2
Dodger
Game
Set 3
Set 4
Set 5
Other
Other
Off
Program Program
What rating and share would the Dodger game receive?
Create a tape diagram to illustrate your solution.
5 Set Market
The Dodger Game in a
5 Set Market
4
TVs on during game
5
TVs on during game
All TVs
In This Market


The Dodger game rating would be 40(%)
because 2 out of 5 are watching the game.
The Dodger game share would be 50(%)
since 2 out of 4 sets which are ON during the
game are actually tuned to the game.
July
nd
22
, 2009
was Manny Ramirez’ bobblehead night at
Dodger stadium. Since he had injured his
hand the previous night, he was not
expected to play and in fact, he came into
the game as a pinch-hitter and faced only
one pitch. He did hit a grand slam with that
one pitch and that was the difference in the
game – just Manny being Manny.
Manny Continued
MLB.tv had a 15 rating (15% of all TVs in
the Los Angeles area were tuned in) and a
25 share (25% of all TVs in the Los Angeles
area that were actually ON were tuned in to
the game).
What fraction of TV's (in the Los Angeles
area) were ON during the game, that is,
tuned into some program during the time slot
of the game?
An informal Solution
Solve the problem by considering a sample
of 100 televisions in the LA area. Note that
percent means the number per one hundred
so 100 is a natural choice to use for a
sample size for computational purposes.
A Dodger game TV ratings
ALL TVS
TVs on During the Game
The Dodger Game
with a Tape Diagram
25 rating :15 share
TVs on during
the game
TVs on during game
All TVs
The Dodger Game
with a Tape Diagram
25 rating :15 share
25%
25
25
25
15
15
15
Dodger
Game`
15%
60% of all TVs on during game
All TVs
TVs on during
the game
The students in Ms. Baca’s art class were mixing yellow and blue paint. She told
them that two mixtures will be the same shade of green if the blue and yellow paint
are in the same ratio.
The table below shows the different mixtures of paint that the students made.
A
B
C
D
E
Yellow
1 part
2 parts
3 parts
4 parts
6 parts
Blue
2 part
3 parts
6 parts
6 parts
9 parts
How many different shades of paint did the students
make?
Some of the shades of paint were bluer than others.
Which mixture(s) were the bluest? Show your work or
explain how you know.
Carefully plot a point for each mixture on a coordinate plane
like the one that is shown in the figure. (Graph paper might help.)
Yellow-blue paint
A
B
C
D
E
Yellow
1 part
2 parts
3 parts
4 parts
6 parts
Blue
2 part
3 parts
6 parts
6 parts
9 parts
The students made two different shades: mixtures A and C are the same, and
mixtures B, D, and E are the same because they represent equivalent ratios.
To make A and C, you add 2 parts blue to 1 part yellow. To make mixtures B, D, and E, you
add 3/2 parts blue to 1 part yellow. Mixtures A and C are the bluest because you add more
blue paint to the same amount of yellow paint.
If two mixtures produce the
same shade, they lie on the
same line through the point (0,0).
And Finally
Even if solving problems algebraically is not
appropriate for your students – true for nearly
all of us, you are laying the foundation for
them to do so in the future. That number to
symbol sense transition is critical and we want
to help you help them make that transition.