RATIO AND PROPORTION


Graphing Ratios
(Teacher Copy)
Activity 7: Graphing Ratios
Students identify ratios of quantities with like and unlike units,
write ratios in colon and fraction formats, and make xy line graphs of
ratios. .
To understand the concept of ratio using tables, graphs, and a formal
definition.
Introduction:
In the previous activities, we constructed a number of tables.
Once we knew the first numbers in the table, we were often able to predict
what the next numbers would be. Whenever we can predict numbers in
one row of a table by multiplying numbers in another row of a table by
a given number, we call the relationship between the numbers a ratio.
There are ratios in which both items have the same units (they are often
called proper ratios). For example, when we compared the diameter
of a circle to its circumference, both measured in centimeters, we were
using a sameunits ratio. Miles per gallon is a good example of a
differentunits ratio. If we did not specifically state that we were
comparing miles to gallons, there would be no way to know what was being
compared!
When both quantities in a ratio have the same units, it is not necessary
to state the unit. For instance, let's compare the quantity of chocolate
chips used when Mary and Quinn bake cookies. If Mary used 6 ounces
and Quinn used 9 ounces, the ratio of Mary's usage to Quinn's would be
2 to 3 (note that the order of the numbers must correspond to the verbal
order of the items they represent). How do we get this?
One way would be to build a table where the second row was always one and
a half times as much as the first row. This is the method we used
in the first two lessons. Another way is to express the items being
compared as a fraction complete with units:
Notice that both numerator and denominator have the same units and thus
we can "cancel out" the units. Notice also that both numerator and
denominator have values that are divisible by three. When expressing
ratios, we generally treat them like fractions and "reduce" or simplify
them to the smallest numbers possible (fraction and colon forms use two
numbers, as a 3:1 ratio, whereas the decimal fraction form uses a single
number—for example, 3.0—that is implicitly compared to the whole number
1).
Part A
Gear 
Ratio 
Gear 
Ratio 
Gear 
Ratio 
1 
1:1, 1/1 
5 
7:4, 7/4 
9 
38:21, 38/21 
2 
7:6, 7/6 
6 
2:1, 2/1 
10 
19:9, 19/9 
3 
4:3, 4/3 
7 
19:14, 19/14 
11 
19:8, 19/8 
4 
14:9, 14/9 
8 
19:12, 19/12 
12 
19:7, 19/7 
The gear ratios in Part B of Activity 6 were written as decimal fractions.
Ratios expressed as decimal fractions implicitly use the whole number 1
as the number of comparison. These ratios make it easier to compare
the relative size of different ratios.
Part B
Twentyfive parts per hundred (25:100 or 1:4) represents the ratio
of the number of defective parts to the number of parts produced.
Parts produced 
100 
200 
300 
400 
500 
Defective parts 
25 
50 
75 
100 
125 
Numbers in the bottom row are multiplied by 4 to get the corresponding
number in the top row. Numbers in the top row are multiplied by 0.25
(or divided by 4) to get the corresponding numbers in the bottom row.
Part C
Sample Graph: Lightning Electric Company's Defect Rate
Horizontal distance: 38 mm (on 4squaresperinch graph paper)
Vertical distance: 9.5 or 10 mm
The vertical distance must be multiplied by about 4 to get the horizontal distance.
The coordinates for the new point (or any point) on the drawn line will
be in a ratio of 1:4, the same as the ratio in the table for number of
defective parts to number of parts produced.
No. Because the graphed line is straight, the horizontal and vertical
distances from any point on the line to the x and y axes will be proportionate.
Closing Discussion

Can all ratios that are represented by tables be represented by graphs?
Is the graph of a ratio always a straight line? Yes, all ratios represented
by tables can be represented by xy graphs. No, only constant ratios
are straight lines; geometric and inverse ratios are not straight lines.
All ratios shown so far in this unit are constant ratios and thus do form
straight lines.

How could you make a graph from the colon form of a ratio without making
a table first? Write several instances of the ratio (e.g., 1:2; 2:4;
3:6, etc.). Then, plot the ratios as points on the graph by using
the first number of each ratio for the distance along one axis and the
second number of each ratio for the second axis.

How is the slope of a line related to the ratio between the vertical and
horizontal distance between two points on the line? The slope of
the line is the ratio of the yaxis difference between two values to the
difference between the corresponding xaxis values.
Activity 8: Using Ratio to Approximate
Students approximate Ö2 using ratios
and squares, and approximate distance using similar triangles.
To understand the concept of ratio using tables, graphs, and a formal definition.
Introduction:
Who knows how to find the length of the hypotenuse of a right triangle?
Does anyone know the Pythagorean Theorem? With the students, find
the length of the hypotenuse of a right triangle with oneunit sides.
Part A
Using their calculators, students should find that Ö2
» 1.4.
Length of the side (inches) 
3 
4 
6 
8 
Length of the diagonal (inches) 
» 4.2 in. 
» 5.6 in. 
» 8.5 in. 
» 11.3 in. 
Divide the diagonal by side 
» 1.4 
» 1.4 
» 1.4 
» 1.4 
The ratio of the length of the diagonal to one of the sides of a square
will always be Ã2. Students may say that the length of the
diagonal gets shorter/longer proportionally as the length of a side gets
shorter/longer.
Part B
1. » 4 1/4 (4.25) inches
2. » 2 inches
3. » 2.125
4. » 141 feet
Summary Questions

When the speedometer registers 55, the ratio is 55 miles to one hour.
The units are different: miles and hours. Ratios might be stated
as "55 to 1" and written 55:1 or 55/1.

Answers may vary. Students should recognize that one row of the table
would have different speeds in miles per hour and another would show corresponding
speeds in kilometers.
Closing Discussion
In the first lesson in this unit, we referred to the number p.
Does p represent a ratio? Why or why not?
Although the term ratio was never used in the first lesson, p
represents the ratio of circumference to diameter of a circle.
© Copyright
Area 10 Mathematics and Technology Professional Development Center
Permission is granted to duplicate these materials for classroom use.
Last updated on 1/30/1999
Comments: egalindo@indiana.edu
http://www.indiana.edu/~atmat/units/ratio/ratio_t3.htm
