Transcript Ratio and Proportion ver 2

Developing Concepts of
Ratio and Proportion
By David Her
∫
54 ÷ cos2θ dθ = 86 ÷ 317
Big Ideas
1. A ratio is a comparison of any two
quantities.
2. Proportions involve multiplicative rather
than additive comparison.
3. Proportional thinking is developed
through activities involving comparing
and determining the equivalence of ratios
and solving proportions without recourse
to rules.
Proportional Reasoning
1. Provide ratio and proportion tasks in a wide of
context. i.e. measurements, prices, geometric
or visual context.
2. Encourage discussion and experimentation in
predicting and comparing ratios.
3. Relate proportional reasoning to existing
process.
4. Recognize that symbolic or mechanical
methods, such as cross product, for solving
proportions does not develop proportional
reasoning.
Question #1)
Two weeks ago, a rose
was measured at 8
inches and a sunflower
at 12 inches. Today the
rose is 11 inches and
the sunflower is 15
inches tall. Did the rose
or the sunflower grow
more?
Informal Activities to Develop
Proportional Reasoning
•
•
•
•
Equivalent Ratio Selections
Comparing Ratios
Scaling with Ratios Tables
Construction and Measuring Activities
Equivalent Ratio Selections
• Select an equivalent ratio from those that
are presented.
• Focus on an intuitive rationale why the
pairs are selected.
Question #2)
On which cards is the ratio of trucks to boxes the same? Also compare trucks
to trucks and boxes to boxes.
Question #3)
Which rectangles are similar?
Question #3)
Which rectangles are similar?
Question #3)
Which rectangles are similar?
Comparing Ratios
• An understanding of proportional
situations includes being able to
distinguish between ratios as well as to
identify those ratios that are equivalent.
Question #4)
Some of the hens in Farmer
Brown’s chicken farm lay brown
eggs and the others lay white
eggs. Farmer Brown noticed that
in the large hen house he
collected about 4 brown eggs for
every 10 white ones. In the
smaller hen house the ratio of
brown to white was 1 to 3. In
which hen house do the hens lay
more brown eggs?
Scaling with Ratios Tables
• Ratio table or charts show how two variable
quantities are related. This is a good way to
organize information.
Question #5)
Fill in the chart then graph it.
Acres
5
10
15
Pine
trees
75
150 225
20
25
Question #5)
Fill in the chart then graph it.
Acres
5
10
15
20
25
30
35
Pine
trees
75
150 225 350 425 550 600
Construction and Measuring
Activities
• In these activities, students make
measurement or visual models of
equivalent ratios in order to provide
tangible example of a proportion as well as
look at numeric relationships.
Question #6)
Create a similar object that has 4 times the
volume as shown below.
Solving Proportions
• Within and Between Ratios
• An informal approach
• Cross Product Algorithm
Within and Between Ratios
•
•
A ratio of two measures in the same setting is a within ratio.
A between ratio is a ratio of two corresponding measures in different
situations.
Between
a
A
B
b
Within
Within
Between
Within
A
B
=
a
b
Between
A
B
=
a
b
An informal approach
• Students find ways to solve proportions
using their own ideas first. If you have
been exploring proportions informally,
students will have a good foundation on
which to build their own approaches.
Question #7)
The price of a box of 2 dozen candy bars is $4.80.
Bridget wants to buy 5 candy bars. What will she have
to pay?
Money
money
There
1So
And
So
$2.40
dozen
each
15
dozen
are
candy
divided
candy
candy
12candy
candy
bars
bars
by
bar12
times
bars
cost
bars
cost
is
money.$0.20
Money
20half
cents
cost
in20aor
ofcents
dozen
$2.40
20
$4.80
is $1.00
cents
Cross Product Algorithm
• Sketch a simple picture that will determine
what parts are related.
Question #8)
Apples are 3 pounds for 89 cents. How much
should you pay for 5 pounds? Hint use within
and between ratio.
5 pounds
3 pounds
89 cents
Within
3
=
89
n cents
5
n
Between
3
5
=
89
n
Reference
John A. Van De Walle. Elementary and
Middle School Mathematics, 5th Edition.
Pearson Education Inc., 2004