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PADMAPRIYA SHIRALI
RATIO
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INTRODUCTION
How does one introduce a topic like ratio, which is so widely present in daily life and so intimately
connected with human experiences? Our cherished cultural achievements are permeated with it:
music is full of ratios, as is art. Our daily existence involves cooking and shopping, and these are
filled through and through with the usage of ratio. Shadows, which are present with us all through
the day, offer a visual depiction of ratios in action. In mathematics as a subject, the notion of ratio
is embedded into many topics – sometimes in an obvious way, at other times not so. Fractions,
scale drawing, enlargement, trigonometry, tables, linear equations… are all illustrations of this.
Considering that ratio is linked in an essential way to so many concepts in mathematics, it is
important to lay a strong foundation and develop the necessary conceptual base when teaching
this topic. A basic error that often arises while studying ratio is the application of additive thinking
to a context that requires multiplicative thinking. This is an error that students make quite
frequently while solving ratio problems. There is a clear need to point this out in various ways and
by using different examples, so that students clearly see that ratio and proportion are based on a
multiplicative relationship.
Often the problems that students encounter in textbooks are limited and become repetitive.
In addition, lack of exposure to sufficient variety inhibits students from noticing the proportional
situation in varied contexts. They may not develop the ability to distinguish between proportional
and non-proportional situations. Again, most problems involve whole number scale factors
and students get flustered when they come across real life problems that do not fall into that
category; i.e. the scale factor is a fraction. Examples taken up should cover diverse areas and bring
in fractional scale factors as well.
While students encounter the topic in different forms as they go to higher grades, we need to
reiterate for ourselves what we expect the student to achieve during the introductory phase.
Expectations
• Understand the relationship of ratios to fractions: While fractions compare part to the whole,
ratios compare both part-to-part and part-to-whole.
• Understand the importance of order in reading and writing ratios.
• Explore and generate equivalent ratios.
• Understand that a proportion is based on a multiplicative relationship and generalise the
numerical relationship, i.e., if a : b : : c : d, then ad = bc.
• Recognise proportional and non-proportional contexts.
• Understand the usage of unitary method in relation to proportion problems.
• See rate as the ratio of two quantities having different units (rate being a special kind of ratio).
• Divide a number into a given ratio.
• Solve a proportion problem using different strategies and diagrams.
So, where do we begin?
Students have an intuitive understanding of Ratio and Proportion. It may be good to begin there,
to let them use their intuitive understanding and then help them to notice the principles they have
applied, study the principles, experiment with them and articulate them. I shall lay out gradually
what I mean by ‘studying and experimenting’.
Activities 1 to 5 and Game 1 build on this intuitive understanding.
Keywords: Ratio, proportion, fraction, part to part, part to whole, equivalent ratio, unitary method
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ACTIVITY 1
Objective: To draw out intuitive multiplicative thinking through a visual poser.
Materials: Hexagon pieces
Lay out a design pattern for the students as shown
in Figure 1.
If I have 6 blue hexagons, how many brown
hexagons do I need to make designs as in Figure 2?
If I have 12 yellow hexagons, how many blue
hexagons do I need to make designs as in Figure 1?
Figure 2
Figure 1
ACTIVITY 2
Objective: To point out multiplicative thinking used by students.
Pose a problem based on the classroom situation.
Let students use their intuition in responding to it.
‘Mouli solved nine (six more than three) as Tanvi
solved eight (six more than two)’.
Tanvi and Mouli are solving problems. For every two
problems that Tanvi solves, Mouli solves three.
How do the other students respond to such
reasoning? Do they see the flaw in the reasoning?
If Tanvi solved eight problems, how many did Mouli
solve?
Once the discussion is over, the teacher should
reiterate the significance of the words ‘for every’
and then point out to the students the multiplicative
reasoning that they applied.
Let the students think it over for a few minutes and
allow them to come up with their responses. Ask
them to explain their answers.
Some may give the right answer, 12, but it is possible
that some student will respond by saying 9.
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Azim Premji University At Right Angles, July 2019
For every two problems that Tanvi solved, Mouli
solved three. If Tanvi solved eight problems which
is four times 2, then Mouli will solve four times 3,
which is 12.
ACTIVITY 3
Objective: To develop intuitive understanding of proportion through enlargement.
Materials: Square dot paper
Give a drawing of a letter of the alphabet or a numeral, as shown in the figure.
Ask students to double it. Were they able to do it correctly?
Did their drawing seem right to them?
What kind of errors did they make?
What could have caused the errors?
Ask them to triple it. Did they find any difficulty?
Figure 3
ACTIVITY 4
Objective: To develop intuitive understanding of proportion through model building.
Materials: Unifix Cubes
Give the class a 2 × 2 × 2 cube.
Ask them to double it.
How did they approach the problem?
How many cubes will they need? Did
they forget to double any dimension?
Ask the class to triple this shape.
Did they find this challenging?
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ACTIVITY 5
Objective: To spot similar shapes
Materials: Sets of rectangles, isosceles or right-angled triangles and trapeziums (each set to
contain one pair of similar figures). Note that we do not permit use of tools of measurement.
Figure 6
Figure 7
Figure 8
The teacher can prepare these using square grid papers
and mount them on plain card sheet. Each shape can be
labelled using alphabets. We must take care in selecting
the sizes, so that spotting similarity is neither too easy nor
too difficult.
Divide the students into three groups. Give the sets one
at a time to the groups so that they can discuss and select
the pair that looks similar to them. They can use either
the plain side of the shapes or the side with square grid in
deciding about similarity. By rotation, each group looks at
the three sets and notes down the similar pairs in each set.
At the end of the activity, all groups share their findings
and present their reasoning. Now verify the answer using
a ruler.
You could keep the challenge level high by insisting that
they use only the plain side of the shapes.
Follow this up by a worksheet that requires testing for
similarity.
Are the figures in each set similar?
Figure 9
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GAME 1
Objective: To spot equivalence of ratios and group them
Materials: 16 Cards with appropriate pictures to demonstrate equivalent ratios
(for example, four equivalent ratios of 3:1, 1:4, 3:4 and 2:3)
Here are two pictures for equivalent ratios of 1:2
Figure 10
Figure 10.1
Figure 10.2
Divide the class into two teams. Ask each team, one after the other, to sort the cards and group them. If they
manage to group the four sets correctly, then they get four points.
Analyse how they have grouped them. Ask the students the basis of their groupings.
Does everyone agree with the reasoning behind the grouping?
ACTIVITY 6
Objective: To introduce what a ratio is and to show how we read and write ratios.
Materials: Classroom
Point to the furniture arrangement in the classroom.
Figure 11
In this picture, each table has two stools with it.
The ratio of tables to stools is 1:2.
Teach students how the ordering of words affects the way we write a ratio.
For instance, in this picture the ratio of stools to tables is 2:1.
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Get students to describe this cube design using ratios.
What is the ratio of black to grey cubes?
How do the black cubes compare with the white cubes?
Are the students able to use multiplicative thinking?
They can record their findings in sentences.
There are twice as many _____ cubes as _____ cubes.
As a second step, they can write it in the form of ratio statements.
Figure 12
ACTIVITY 7
Objective: To study and experiment with a given ratio.
Materials: Cubes or counters
Here are 2 yellow cubes and 4 black cubes.
The ratio of yellow to black is 2:4.
There are 2 times as many black cubes as yellow cubes.
If you add 2 cubes of each colour, will the black cubes still be 2 times
the yellow cubes? What is the new ratio of yellow to black?
What will happen if you take away 1 cube of each colour from the
original pattern? What is the new ratio of yellow to black?
What will happen if you double each colour? What is the new ratio of
yellow to black?
What will happen if you halve each colour? What is the new ratio of
yellow to black?
Do the students form any conclusions based on this experimentation?
Figure 13
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ACTIVITY 8
Objective: Explore the multiplicative relationship of ratios by generating equivalent ratios.
Materials: Square grid paper
The teacher needs to show the students the
relationship of fractions to ratios. She also needs
to point out the differences. They may not have
recognised before that every fraction is in fact a
ratio.
While studying fractions, we compare the part to
the whole, but while studying ratios we compare
both part-to-part as well as part-to-whole.
Also in the case of fractions, the order in which we
write the numbers is fixed, the numerator is the
part and the denominator is the whole. In the case
of ratios, there is no such fixed order.
The teacher can show the close connection between
equivalent fractions and equivalent ratios through
drawings.
Figure 14
to give 1:3.
8:10 can be divided by 2 on both sides to give 4:5.
This figure shows a part-to-whole ratio.
Discuss part-to-whole and part-to-part ratios in the
same context to make the difference clear.
Students should record the work as shown in this
figure to internalise the fraction, ratio connection.
A class has 18 boys and 12 girls.
Students also need to understand that, just as with
fractions, we can reduce ratios by common factors
of the quantities, to make the simplest form.
The ratio of boys to girls is 18:12 or 3:2. This is a
part-to-part ratio.
The ratio of boys to total number of students is
18:30 or 3:5. This is a part to whole ratio.
Example: 4:12 can be divided by 4 on both sides
ACTIVITY 9
Objective: To show that a proportion is based on a multiplicative relationship
and generalise the numerical relationship.
Materials: Five equivalent ratios of 3:1, 1:4, 3:4 and 2:3 written out on separate cards
Distribute the cards to students. Ask them to
sort the ratio cards and group them. Through
inspection, they will spot the equivalent ratios and
group them. Verify that they are right.
The teacher can now draw their attention to the
multiplicative relationship within each group of
ratios, explain what proportion means and point
out that the ratios in each group are in proportion.
Raise the question: “If we take any two ratios which
are in proportion, how do the numbers relate
to each other? Say, 6:8 to 15:20.” One obvious
relationship is that we can reduce both pairs to a
common ratio. Is there any other relationship? Can
we see a relationship if we write them as fractions?
Help the students notice that 6 times 20 equals
8 times 15. Ask them whether this happens with
every pair of ratios in proportion.
Through discussion, the teacher can generalise the
procedure, i.e., when a:b::c:d, then ad=bc.
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The class can then discuss practical examples
involving proportion like recipes, heartbeat,
physical exercise like runs.
However, at the same time, the teacher must also
expose the students to non-proportional contexts
so that they are able to identify and differentiate
between proportional and non-proportional
contexts.
A simple example of a non-proportional context is
the comparison of the ages of two people.
Aarav is 12 years old. His sister Ami is 6 years old.
How do these numbers compare? Aarav is twice
as old as Ami. What will be the case 6 years later?
Would Aarav be twice as old as Ami?
One day I’ll catch up! My Dad is four times as old as
I am. When will he be three times as old? When will
he be twice as old? When will I catch up?
An interesting exploration to do would be to
examine how the scale factor behaves, as the
daughter and father get older. Can the students
see why this happens?
Reflection: What line of inquiry can the student
now pursue? What tasks can the teacher set?
ACTIVITY 10
Objective: Understand the usage of unitary method in relation to proportion problems.
Materials: Price lists of products, food items etc.
Unitary method, i.e., calculating per unit is a very
common way of solving proportion problems. Many
students find it natural and easy-to-understand
and take to it without any difficulty.
Example. 3 chocolate bars cost Rs. 45. How much
will 7 bars cost?
They may reason it out as, if 3 bars cost Rs. 45, then
1 costs Rs. 15, so 7 bars will cost Rs. 105.
It seems reasonable and efficient to solve it this
way.
Is the unitary method always the best way?
If 500 balloons cost Rs. 2745, what will be the cost
of 800 balloons?
Finding the cost of one chocolate bar from the
price of three bars seems perfectly sensible, but
finding the cost of one balloon from the cost of 500
balloons does not. A far more workable method
here would be to find the cost of 100 balloons and
then to multiply by 8.
Naturally, we need to modify the unitary method
according to the situation.
Figure 15
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ACTIVITY 11
Objective: Aids to solving problems (modelling)
Materials: Square paper
In the initial stages, the teacher can
give problems accompanied by visuals.
In the second stage, the student can
model a given problem.
Modelling is an important tool for
solving word problems.
Ex. In a class, the ratio of girls to boys
is 2:3. If the number of students in the
class is 30 what is the number of boys?
What is the number of girls?
Here is one way of modelling it.
For every 2 girls there are 3 boys in the
class.
Each such group contains 5 students.
How many groups of 5 in 30?
6 such groups.
Therefore, in 6 such groups, the
number of girls is 12 and the number
of boys is 18.
Reflection: An important question
while solving problems is this. What
imagery does the student use to
understand a problem?
Can we teach imagery? Is not imagery
a subjective experience that may be
difficult to articulate?
At the same time, helping students to
find their own imagery requires that
Figure 16
the teacher expose them to visuals,
multiple strategies and approaches to problem solving. Imagery is often closely linked to the way a concept
has been developed. Otherwise, simplifying ratios or finding proportions can become highly mechanical.
Figure 17
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ACTIVITY 12
Objective: Aids to solving problems (rephrasing)
Another important aid in solving ratio problems is
the ability to rephrase a problem in ratio language.
What is ‘ratio language?’ It involves the use of
phrases and words such as ‘for each’, ‘for every’,
‘per’, ‘each time’ and so on.
This makes a group of 7 objects.
How many groups of 7 are there in 84? 12 such
groups.
Therefore, the number of chocolate cones will be
12 times 4, i.e., 48.
In the beginning, it is good practice for students
to rewrite the information of a problem in ratio
language.
Writing Mathematics
Help the students to rephrase the problems.
Students should be encouraged to write out their
reasoning for a solution.
Example: At an ice-cream shop, the ratio of
chocolate cones sold to vanilla cones sold was 4:3.
If the shopkeeper sold 84 ice-cream cones in a day,
how many chocolate cones did he sell?
Example: One-fourth of the class is absent today.
30 students are present today. How many students
are in that class?
For every 4 chocolate cones sold, there were 3
vanilla cones sold.
“If one-fourth of the class is absent, then threefourth of the class is present. Three-fourth stands
for 30. Therefore, one-fourth stands for 10. So the
total number of students is 40.
ACTIVITY 13
Objective: Aids to solving problems (table making)
In a library, the number of non-fiction books is one-fourth the number of fiction books.
How many fiction books would be there if there are 50 non-fiction books?
How many non-fiction books would be there if there are 120 fiction books?
Figure 18
What approaches would you use for these problems?
Aunty Elizabeth decides to give her nephew and niece Rs.1500, to be shared between them. Because the
niece is the older of the two, she decides that she must divide the money between the niece and the nephew
in the ratio 5:3. How much does each child get?
For every 4 kilometres that Jaleel jogged, Kian jogged 3 kilometres. If Jaleel jogged 1 km, how far would Kian
have jogged?
A student finished 8 homework problems in class. If the ratio of problems finished to the problems still left
was 4: 1, how many homework problems did she have in total?
Occasionally, pose an open-ended question.
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Think!
The ratio of two even numbers is 3:7. What are they?
It takes 3 minutes to boil 1 egg. How many minutes does it take to boil 2 eggs?
GAME 2
Objective: Interpret given ratios and construct a feasible design.
Materials: Square grids of 5×5 size
25 tiles of 4 different colours
The teacher constructs a design and shares it with the students using
ratios.
Examples.
The ratio of green to red is 5:4.
The ratio of green to blue is 10:1.
The ratio of red to yellow is 4:3.
Figure 19
The pattern has two lines of symmetry.
Are the students able to construct a design that satisfies all these conditions?
After playing the game a few times, students can form teams and create challenges for one another.
Creating a pattern that lends itself to good clues is an educative and challenging experience in itself,
particularly if the solution is unique.
Extension: We can also use grids of size 4 × 4.
ACTIVITY 14
Objective: Studying ratio applications (maps from an atlas)
Materials: Atlas or map charts
Teacher can help students read a map and observe the way scale is used in the map, the way the scale is
denoted on the map and the need for accuracy in the representation.
Students can apply their knowledge to estimate the distances between pairs of places and the lengths of
rivers. They can also compare the lengths of the coastlines of different states of India. And so on.
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ACTIVITY 15
Objective: Meeting real life ratio challenges (scale drawings)
Materials: Measuring tapes and scales
Divide the students into groups and ask each group
to make a scale drawing of a classroom, library or
the playground.
Each time a teacher draws on the blackboard or
writes on the blackboard, she is implicitly using
a ratio. A 20 cm long pencil when drawn on the
board may become 30 cm long.
Discuss what would be a sensible scale to use.
That is an example of enlargement or multiplication,
by a scale factor of 1½.
In the process of making a scale drawing, they will
encounter fractional ratios and at times need to do
rounding off.
Maps and scale drawings, in contrast, depict
reduction in scale.
ACTIVITY 16
Objective: Highlight the need for a common unit or conversion in a ratio.
Example. Madhu says, “I will take 10 days to complete my project.” Vishal says, “I need 2 weeks.”
Ashok described the ratio of time needed by Madhu and Vishal as 10:2.
What do you think?
Let the students discuss the implication of a 10:2 ratio in this context.
If it is wrong, why is it wrong? What is the right way of expressing it?
ACTIVITY 17
Objective: See rate as the ratio of two quantities having different units
(rate is a special kind of ratio).
Materials: Price list, purchase bills, electricity bills, travel time
What is a rate? A rate compares different kinds of measures, such as rupees per kilogram (cost per unit
bought), kilometres per hour (speed of a vehicle), rupees per day (wages for a worker), heartbeats per
minute (state of health). So many aspects of our life involve some kind of rate.
Plotting of rate in a table as shown demonstrates the multiplicative relationship very clearly.
We can also show this data as a graph.
Time (h)
1
2
3
4
5
Distance (mi)
60
120
180
240
300
Figure 20
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ACTIVITY 18
Objective: Meeting real life ratio challenges (recipes)
Materials: Actual recipes!
No discussion on ratios can end without discussing the use of ratios in recipes, on which our very survival rests!
Strong coffee? Weak coffee?
Teacher can distribute four recipes for making coffee. The
students have to arrange them according to the strength of the
decoction.
If coffee is not welcome, we can substitute it with orange juice!
Ordering four recipes requires comparison of four different
strengths.
However, we may also make comparisons within a recipe.
Here is one such example.
Comparison within one recipe between the quantity of juice and
water.
To make a jug of orange juice, I use two glasses of orange juice
and five glasses of water.
What happens to the strength of the mixture if I add an extra
glass of water?
Figure 21
What happens if I add an extra glass of orange juice?
What happens if I add one of each, two of each, etc.?
ACTIVITY 19
Objective: Solving proportion problems as algebraic equations.
Materials: Word problems
In a bag of red and green sweets, the ratio of red
sweets to green sweets is 3:4. If the bag contains
120 green sweets, how many red sweets are there?
For every 3 red sweets there are 4 green sweets.
Number of red sweets = y
Number of green sweets = 120
Hence, 3:4:: y:120
3 × 120 = 4 × y
4y = 360
y = 90
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Project Ideas
•
Study of shadows
•
Sports data
•
Human body (comparative study)
•
Currency exchange rates
•
Microscope enlargements
History: The ancient Greeks originally posed this problem:
‘How can a line be divided into two parts so that the ratio of the larger part to
the smaller is the same as the ratio of the whole line to the larger part?’
Historically, what kind of ratio problems did people study? What challenges would they have faced which
led to the development of topics like trigonometry? It would be of interest to the students and teachers to
study this question.
To close, a few questions…
•
How does one support and scaffold the students’ learning?
•
Has one exposed the students to various strategies so that we have been able to achieve the
intended goals or learning outcomes?
•
By studying incorrect solutions, can one understand the misconceptions that the students hold and
so address them?
Padmapriya Shirali is currently the Principal of Sahyadri School KFI. She is part of
the Community Math Centre based in Sahyadri School and Rishi Valley School.
In the 1990s, she worked closely with the late Shri P K Srinivasan, well-known
mathematics educator from Chennai. She was part of the team that created the
multigrade elementary learning programme of the Rishi Valley Rural Centre,
known as ‘School in a Box.’ Padmapriya may be contacted at padmapriya.
shirali@gmail.com.
Padmapriya Shirali
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