Grow Beasts: Growing students’ understanding of ratio, proportions and slope
Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by...
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![Page 1: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/1.jpg)
Problem Solving in Geometry with
Proportions
![Page 2: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/2.jpg)
Ratio
A ratio is a comparison of two quantities by division
The ratio of a and b can be represented three ways: a/b a:b a to b An extended ration compares three values (i.e.
a:b:c)
![Page 3: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/3.jpg)
Ratio Example
![Page 4: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/4.jpg)
Extended Ratio Example
![Page 5: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/5.jpg)
Proportion
A Proportion is an equation that states two proportions are equal
First and last numbers are the Extremes Middle two numbers are the Means
![Page 6: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/6.jpg)
Cross Products Property
![Page 7: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/7.jpg)
Additional Properties of Proportions
ab =
cd
ac =
bd, then
ab =
cd
a + b
b = d, then
IF
IFc + d
![Page 8: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/8.jpg)
Ex. 1: Using Properties of Proportions
p6 =
r10
pr =
35, then
IF
p6 =
r10
pr =
610
Given
a
b=
c
d , thena
c=
b
d
![Page 9: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/9.jpg)
Ex. 1: Using Properties of Proportions
pr =
35
IF
Simplify
The statement is true.
![Page 10: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/10.jpg)
Ex. 1: Using Properties of Proportions
a3 =
c4
Given
a + 3
3 = 4a
b=
c
d , thena + b
b=
c + d
d
c + 4
a + 3
3 ≠ 4c + 4 Because these
conclusions are not equivalent, the statement is false.
![Page 11: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/11.jpg)
Ex. 2: Using Properties of Proportions
In the diagram
AB=
BD
AC
CE
Find the length of
BD.
10
30
x
16
A
DE
B C
Do you get the fact that AB ≈ AC?
![Page 12: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/12.jpg)
Solution
AB = AC
BD CE
16 = 30 – 10
x 10
16 = 20
x 10
20x = 160
x = 8
Given
Substitute
Simplify
Cross Product Property
Divide each side by 20.
10
30
x
16
A
DE
B C
So, the length of BD is 8.
![Page 13: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/13.jpg)
Geometric Mean
The geometric mean of two positive numbers a and b is the positive number x such that
ax =
xb
If you solve this proportion for x, you find that x = √a ∙ b which is a positive number.
![Page 14: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/14.jpg)
Geometric Mean Example
For example, the geometric mean of 8 and 18 is 12, because
and also because x = √8 ∙ 18 = x = √144 = 12
812 = 18
12
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Ex. 3: Using a geometric mean
PAPER SIZES. International standard paper sizes are commonly used all over the world. The various sizes all have the same width-to-length ratios. Two sizes of paper are shown, called A4 and A3. The distance labeled x is the geometric mean of 210 mm and 420 mm. Find the value of x.
420 mmA3
x
x
210 mm
A4
![Page 16: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/16.jpg)
Solution:
420 mmA3
x
x
210 mm
A4
210
x=
x
420
X2 = 210 ∙ 420
X = √210 ∙ 420
X = √210 ∙ 210 ∙ 2
X = 210√2
Write proportion
Cross product property
Simplify
Simplify
Factor
The geometric mean of 210 and 420 is 210√2, or about 297mm.
![Page 17: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/17.jpg)
Using proportions in real life
In general when solving word problems that involve proportions, there is more than one correct way to set up the proportion.
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Ex. 4: Solving a proportion
MODEL BUILDING. A scale model of the Titanic is 107.5 inches long and 11.25 inches wide. The Titanic itself was 882.75 feet long. How wide was it?
Width of Titanic Length of Titanic
Width of model Length of model=
LABELS:
Width of Titanic = x
Width of model ship = 11.25 in
Length of Titanic = 882.75 feet
Length of model ship = 107.5 in.
![Page 19: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/19.jpg)
Reasoning:
Write the proportion.
Substitute.
Multiply each side by 11.25.
Use a calculator.
Width of Titanic Length of Titanic
Width of model Length of model
x feet 882.75 feet
11.25 in. 107.5 in.
11.25(882.75)
107.5 in.
=
=x
x ≈ 92.4 feet
=
So, the Titanic was about 92.4 feet wide.
![Page 20: Problem Solving in Geometry with Proportions. Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649f155503460f94c2aa02/html5/thumbnails/20.jpg)
Note:
Notice that the proportion in Example 4 contains measurements that are not in the same units. When writing a proportion in unlike units, the numerators should have the same units and the denominators should have the same units.
The inches (units) cross out when you cross multiply.