Emile Cuisenaire, a Belgian elementary school teacher, invented Cuisenaire Rods in the early 20th century to help students learn math concepts hands-on. The rods come in different colors representing different values to demonstrate addition, subtraction, multiplication and other math operations visually. Caleb Gattegno later popularized the rods internationally while a professor in London. The document then provides examples of math lessons and activities that can be done using the Cuisenaire Rods to teach concepts like addition, subtraction, fractions and more through visual representations instead of just numbers and symbols.

1. Cuisenaire Rods:
Hands to mind in
early numeracy
Muhammad Yusuf
Pedagogy Expert
Sargodhians’ Institute for
Professional Development

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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.
6 ÷ 3

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
2 size 6 bars 2 x 6
1 size 12 bars 1 x 2

17. 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. How can we find them?
https://www.desertcart.pk/products/26237429-hand-2-mind-plastic-cuisenaire-rods-individual-kit-for-kids-ages-5-13-
hands-on-math-manipulatives-for-kids-to-learn-numbers-fractions-and-ratio-homeschool-supplies-set-of-74
Buy-online
https://www.amazon.com/Learning-Resources-Connecting-Cuisenaire-Introductory/dp/B000F8T9HW
Android Application
https://play.google.com/store/apps/details?id=air.ca.mathies.relationalRods&hl=en
Make your own:

28. Now It’s Your Turn!