Cuisenaire rods were invented by Emile Cuisenaire in the early 20th century and popularized by Caleb Gattegno in the 1950s. The rods are integer-length blocks used to teach mathematical concepts like addition, subtraction, multiplication and division visually. The document provides examples of how to use the colored rods to represent and solve math problems involving these operations. It also covers fractions, area, perimeter and symmetry.

2. Learning mathematics by using
Bars
By
Muhammad Yusuf
m.yusuf.edu@gmail.com
aapkayusuf@yahoo.com

3. Emile Georges Cuisenaire, who was born in 1891, came
up with the idea of using integer bars, also called
Cuisenaire Rods. He was a elementary school teacher in
the city of Thuin in the country Belgium in Europe.
Caleb Gattegno, who was born in Egypt, was a professor
at the University of London. He met Cuisenaire in 1953
and realized how good the bars were to teach
mathematics. He helped a lot by talking with many
teachers in many countries about these bars and made
them very famous.
Who invented them

4. Getting Familiar with the Bars
Bars Value Color
1 White
2 Red
3 Light green
4 Purple
5 Yellow
6 Dark Green
7 Black
8 Brown
9 Blue
10 Orange

5. Activity 1 - Building colorful stairs. Now build a 10 step stairs
using a different color bar for each step. You should build 2 sets of
colorful stairs: one with vertical bars and one with horizontal bars.
Make sure that the stairs go up and down. Here are examples of
these.
Some activities to do:
•Draw a large house and make sure that it
contains windows and a door.
•Draw a person
•Any other things which you love to make

6. Addition
Here is an example using both the bars and the
equation.
We added the size 8 bar to show that the answer of 8,
which is the total, is correct because it is the same size
as the train made by the size 5 and size 3 bars above.
Make sure you understand the problem and the
solution.
5 + 3 = 8
8

7. To build a train you connect 2 or more blocks together without
leaving a space between them and without overlapping. We are
going to be using the word trains in the rest of these activities. Here
is an example of a train where we also show the sizes of each of the
bars and the total size:
Building Addition train
Activities
Activity 1 - Make as many trains as you can using only two bars.
The length of the train needs to be 7. Each train needs to be
different. Be sure to write the equation for each train. Here is an
example showing a bar of size 7 and one train:
Now Make train of 12, 15, 17
5 + 4 + 3 = 12

8. Commutative and Associative Rules
Commutative Rule - This means that "5 + 3" is the same as "3 +
5". The order of the numbers that you are adding is not important.
You can add them in any order you want. Here is an example
using bars:
Associative Rule - This means that when you are adding more
than two numbers you can combine them in any way you want.
For example, "4 + 3 + 2" can be added as
(4+3)+2 or 4+(3+2)

9. Combining both commutative and associative rules -
Both rules can be combined when doing addition. Here is an
example:
5+(3+1)
(3 + 1) + 5
3 + (1 + 5)

10. Subtraction
we will start with an example and then You will get opportunity
to do some exercises for you to practice. The example is 8 - 3.
Here are the black and light green bars example:
We start with a 8 bar
Then we align it with the 3
bar that we want to subtract
The next step is to fill in the space
to the right of the green all the way
to the end of the brown. This is also
called the difference. The bar that
fills the space is the yellow bar.
So the answer is 5
8
8
3
8
3 & 5
5

11. Multiplication
Firstly we need to remind us that Multiplication is repeated
addition.
Let us start with 3 x 2
OR
As you can see they add up to 6 units

12. Multiplying more than two numbers - We are going to show
that you can multiply more than two numbers at the same time.
Let's do 2 x 3 x 4. What we are going to do is use 2 size 3 bars to
make a rectangle.
Next make a train using 6 of these rectangles and add them all up.
2
+2
+2
= 6
6 + 6 + 6 + 6 = 24
8
+ 8
+ 8

13. Division
Before going to start, let us remind us, What is division?
Division is the opposite of multiplication. As an example, if you
have 6 pencils to give out to 3 students, how many pencils will
each student get? To solve this, we need to divide the 6 pencils
into 3 equal groups. There are three common symbols to use for
division: "/ ", "÷", and "—". For example, dividing the 6 pencils
into 3 groups can be written in either of the following three
ways:
6/3
6 ÷ 3
6
3

14. We will use integer bars to solve this problem. First we start
with a size 6 or dark green bar that needs to be divided into 3
equal pieces:
Then we have to find 3 equal bars that make up a train that
matches exactly the size 6 bar. In this case, the size 2 or red bars
are the ones that work. Here is the red train:
Since the bars that work turn out to be the red bars, that means
that the answer is 2 which is the size of the red bar. Here is the
equation for the same problem:
6 / 3 = 2
6 divided by 3 equals 2.

15. Remainder - We will introduce the concept of what a remainder
is when doing division. Let's divide 9 pencils among 4 students.
We need to build a train made out of 4 bars to add up to 9. Here is
our try:
As you can see, the light green bars are too long and the red bars
are too short leaving an empty space. To build a train that adds up
to 9 using 4 equal bars, we used 4 red bars and ended up with size
1 leftover or space to complete the 9. The size 1 bar that
completes the train is called the remainder.
So dividing 9 pencils among 4 students, each student gets 2
pencils and we have 1 pencil left over. As an equation this
would be written:
9 / 4 = 2 with a remainder of 1

16. Factors
Firstly we need to remind us that Factors are the numbers
that are multiplied together to get a specific answer.
Using bars to come up with a multiple of two factors we need to end up
with a perfect rectangle. As an example we will use bars to build perfect
rectangles to find all of the factors for the multiple 12.
12 size 1 bars
12 x 1
6 size 2 bars
6 x 2
4 size 3 bars
4 x 3
3 size 4 bars
3 x 4

17. 2 size 6 bars
2 x 6
1 size 12 bars
1 x 2
Activity
1. Find all the perfect rectangles for the multiple 20. This
will give you all the factors for 20.

18. The word fraction means a portion of a whole. For example, if you
have a whole pizza cut into 8 equal slices, then each slice is a fraction
of the whole pizza.
Fraction
Using bars to work with fractions
Let's start with one whole orange bar. The orange bar is 1 unit.
We will show some examples of different fractions of the orange
bar.
Since we are able to fit two yellow bars to match the orange bar,
that means that each yellow bar is one half or 1/2 unit.
In this example five red bars equal one orange bar, therefore
each red bar is one fifth or 1/5 unit.

19. Proportional Fractions
One half can be written as 4/8, 2/4 or 1/2. We say that these
equivalent fractions are proportional fractions because they
are the same portion of the whole. Here are some different
examples of proportional fractions using the bars. All of them
are equivalent.
1/2
2/4
3/6
5/10
7/14

20. More Fractions
The black bar which is size 7 represents 1 unit therefore each
white bar of size 1 is 1/7 unit. As you can see in the following two
pictures what we have is 3/7 which can be represented with three
white bars or one light green bar.
The above example uses the blue bar of size 9 to represent 1 unit. Each
white bar represents 1/9 so the five white bars represent 5/9. The yellow
bar, which is equivalent to the five white bars, also represents 5/9.
Proper Fractions - when the numerator is smaller than the denominator,
as in the previous two examples, it is called a proper fraction.

21. On this above example the dark green bar is defined as 1 unit, so
one red bar is 1/3 unit. The fact that the following picture has four
red bars means that the fraction is 4/3. This is a case where the
numerator is greater than the denominator so it is called an
improper fraction. We know that 3/3 equals 1 unit, therefore an
improper fraction has a fraction value greater than 1, in this case
4/3. The numerator can also be represented by a single brown bar
which is the same size as four red bars.

22. Area - The area of a figure is the total space inside of the
figure which is the size of the surface. The method to
calculate the area of a square or rectangle is to multiply both
sides, the length by the height. We will calculate the areas of
the same shapes as previous slide.
1
3
5
1+3+5+7+5+3+1 = 25
7

23. Perimeter - The perimeter of a figure is the total distance
around the edge of the figure. For example, if we define one unit
as the length of one side of the following square then the square
has a perimeter of 4. A square is a shape that has four equal
sides. We simply add the length of each side as shown below:
1 + 1 + 1 + 1 = 4
1
This next example uses a longer bar. This shape is called a
rectangle where both long sides are the same length and both
short sides are the same length. The long side has a length of 3
and the short side has a length of 1. Add all the sides together to
see that the perimeter is 8 as shown below:
1
3
1 + 3 + 1 + 3 = 8

24. This third example has a complicated shape where each edge, as
in the previous examples, has a length of 1. If you count all the
sides of this shape, what will you find?
the perimeter is 28.

25. Symmetry
First I would like to explain what symmetry means. A
picture where you can draw a line in the middle and
each side looks exactly like the mirror image of the
other side is what is called symmetry. The line in the
middle can be horizontal, vertical, or in any direction as
long as both sides are the mirror image of each other.
The following are examples of symmetrical pictures:

26. Symmetry
As you can see the symmetry line is vertical
and what you see on the left side of the line
is the mirror image of what you see on the
right. You could fold it on the vertical line
and the blocks on the right will fall exactly
on the blocks on the left.
This drawing has a horizontal symmetry line.
The part of the drawing above the line is the
mirror image of the part below the line.
Activity 1 - Make drawings that are symmetrical where the
symmetry line is either vertical, horizontal, or any direction.
Examples of what you can draw are letters, houses, donuts, a
tree, etc.

27. Now It’s Your Turn!