HoB icon

Introduction of Cuisenaire Rods

Concrete representations of number values & math concepts

Questioning is the basis of all learning.

Introduction

This page includes information to introduce Cuisenaire Rods to middle level learners so they can become aquainted with how they can represent number values and build fluency to become comfortable enough to use them in hands on concrete activities to represent numbers and conceptualize other mathematical concepts to solve and illustrate solutions to their problems. Examples of other uses are identified in this chart. However, detailed information on how to use Cuisenaire rods to represent other mathematical ideas and operations is beyond the scope of this page. For example, you can find detailed information that explores the multiplication of fractions with Cuisenaire rods on this other page.

What are Cuisenaire rods?

Cuisenaire rods are made from wood or plastic and are shaped as in the figure below with the smallest being 1 cm long and the longest 10 centimeters. If you don't have Cuisenaire rods, there are many sources on-line to purchase them. They are worth the expense and will last for years.

Cuisenaire rods

Cuisenaire rods
Place value stamps

Other manipulatives can be used. Many different manipulatives are also made in centimeter increments and can be used with minor adaptations. One eample, is base ten blocks which can be used for all the place value activities. Stamps are also available for learners to quickly illustrate journals.

Page organization

The page includes pedagogical information and suggestions to assist learners to become familiar with Cuisenaire rods before they are asked to use them to represent problems.

To achieve this, if then, problems, which are listed below and linked to worksheets, can be printed and shared with students to explore the different relationships betwen rods and different combinations of them as they solve the problems, share their solutions, and disscuss them. As they do, they will be learning how to use Cuisenaire rods to represent different number values and develop fluency in reprsenting those number values in different ways; a foundation necessary to be able ro represent mathematical numbers, relationships, and operations concretely. See table for some relationships & operations.

I think you will find, as I did, over the years that learners enjoyed them as puzzles, and will ask for more similar problems.

Suggestions for solving if, then problems.

When solving if, then problems it is important to identify what is being used as the unit, or one. In the first groups of problems, or puzzles, one is identified. Later, learners will have to decide what is one, and when they solve other problems, they will have to decide what to use as one depending on the problem they are wanting to solve. This is especially critical when working problems that have decimal and fractional values as the selection of one unit over another can provide easier solutions and more elegant solution.

When identifying the unit it is good to stress that a white rod does not have to be one. And encourage them to visualize and learn the relationships of the rods to each other rather than relying on only a relationship to the white cube.

For example, it is helpful to know that a red rod is twice a white rod; and a purple rod is twice a red rod; and other possible relationships. For that reason the first worksheet avoids using white as the unit.

Some learners would rather use drawings than the actual rods. I would accept their choice, but when I explain a problem, I would always model my work with the use of the rods before using a sketch or equations.

These introductory activities are organized into the categories of whole numbers, place value, decimals, fractions, and fractional operations. They are linked in this overview.

Mathematical relationships & operations to represent with Cuisenaire Rods

Mathematical term Represented with Cuisenaire Rods as
Patterns

Arrangement of rods in patterns and progressions, repetative colorization

  • R, W, R, W,(ABAB) ...

Red white red white sequence

  • Arithmetic progression is a sequence of numbers in which each number is different from the one before it and after by the same amount (number, quantity).
  • Geometric progression is a sequence of numbers in which each number is different from the one before it and after by the same constant ratio (or multiplying a number in the sequence by a fixed number, called a constant ratio, to find the next number).(Ratio is a comparison between two quantities, expressed as a number.) (The number of times one value contains or is contained within the other.)

Geometric progression

Even numbers

Any number which can be made by a train of red rods.

Or can be represented as any rod or train of rods, that matches a train of red rods.

Even rod
Odd numbers

Any number which can not be made by a train of red rods.

Or can not be represented as any rod or any train of rods, that will not match a train of red rods.

Odd rod
Sequence

Stair cases

Staircase

Inequalities

Collections, trains, Venn diagrams, sets, groups

Inequality

Equalities

Collections, trains, Venn diagrams, sets, groups

Equality

Addition

Trains, regrouping to match another rod or train of rods.

Could also be squares or cubes or groups of any of these.

What added to Red makes Yellow?

R + Y = Y

Red + Green = Yellow

Subtraction

What's missing?

What taken away from Yellow leaves Green?

Y - __ = G

9 - 5 = __ ; What is added to five to get nine?

Nine minus five

Associative property

Associative property of addition: the sum of three or more numbers is the same regardless of how the numbers are grouped (use parentheses) or are added.

Commutative property

Addends can be added in any order.

Commutative property with R + G = G + Y

 

Factors can be multiplied in any order.

redPurpleAndPurpleRed

Multiplication

Repeated addition, squares and rectangles, towers

Four groups of five; 4 × 5 = 20

Four groups of five

Three by four rectangle; 3 × 4 = 12

Three by four rectangle

Five by nine tower; 5 × 9 = 45

Five nine tower

Distributive property

Distributive property of multiplication over addition: Worksheet

For all real numbers a, b, c, 
a × (b + c) = (a × b) + (a × b)

or

(P × R) + (P × G) = P × (R + G)

Distributive property

Division

Repeated subtraction - How many can be taken away? How many groups? How big of groups? Tear down a tower. Place mat division.

Nine divided into threes. How many threes?

9 ÷ 3 = 3

Nine divided by three

How many times can a Red rod by subracted from browN?

Or How many Reds in a browN? Subtract Reds until nothing left to subtract.

N ÷ R = 4

Place value

Use White or any Orange as one and combine groups of them to make decimal numbers.

Grouping game, trading game, place mats with rods, squares and cubes, towers See exponents to do decimal numbers with towers.

Cuisenaire base ten blocks

Fractions

If then equations, puzzles, guess my mind (if red is 2/3 what is one?)

White is 1/2 Red

If Red is two thirds, what is one?

Addition of fractions

Create fractions with identical rods as the denomenator or value of one, then join the rods representing the numerator.

Subtraction of fractions Create fractions with identical rods as denomenators or values of one, then separate the rods representing the numerator.
Multiplication of fractions Repeated addition, four ways to represent multiplication of fractions
Division of fractions

How many groups in __ ?

How many groups can be taken away? (repeated subtraction)

How big of groups? 1/3 divided into three equal groups is 1/9?

4 divided into equal groups of 2/3 = 6 (make train of four light green rods and put red rods(2/3) under to match (6).

7/8 divided by 3/4 use eighths, 7/8 and 6/8, how many 6/8 in 7/8? one and one part of the 6 = 1 1/6.

Decimals

If, then equations, puzzles

Fractional equalities

Rectangles

Make rectangles that are 2 x 4.

Primes

A number that can not be formed by a train of any one color except White.

If W = 1; blacK is 7 and there is no train to match it.

Composite Any number that can be formed by a tower of factors
Greatest common factor

Candy sacks - If all the rods that represent the factors for each number ( 1, 2, 3, 4, 6, 12) were put into a separate sack and each represents a candy, then what would be the largest candy that is in all the sacks?

  • O + R; factors are (W, R, G, P, D, O+R)
  • 12; ( 1, 2, 3, 4, 6, 12)
  • D; factors are (W, R, G, D)
  • 6; (1, 2, 3, 6)
  • Can be done with primes as well as all common factors.
  • Prime factor trees
  • O + R; factor tree is (R × R × G)

  • D; factor tree is (R × G )

  • Common factors are Red & Green.
Least common multiple

Candy sacks - If the primes of multiples are put into separate piles, which piles would have the least number of rods and would exactly match?

  • Multipes of 3 would make these piles or towers ...
  • (G * W) (G * R) (G * G) (G * P) (G * Y) ...
  • Multiples of 5 would make these pilesor towers ...
  • (Y * W) (Y * R) (Y * G)...
  • (G * Y) or (Y * G) are the first identical piles.
Exponents

Towers of rods.

A Dark green square can be represented as a tower of two Dark green rods

D2 =

A Dark green square or a tower of two Dark green rods

 

A tower of two browN rods represents a browN square

brown tower of 2

N2

If w = 1, N2 = 8 × 8 = 64;

 

A tower of three bluE rods rrepresents a bluE cube.

 

Blue tower of 3

E3

If W = 1, E3 = 9 × 9 × 9 = 729;

 

For positive integers start with a White cube as the base.

And

For negative exponents you place a White cube on top of tower.

(3- 1 = 1/3, 3 - 2 = 1/9)

Mean, median, mode

Line up rods to represent data, locate, count, move to balance.

If W = 1, The average of 6, 5, 4, 3, 2 is 4.

Average

 

Introductory getting acquainted problems for representing number values

Introduction to whole number representations with Cuisenaire rods

If red = one, then what?

Materials - Cuisenaire squares, rods, & cubes.

Concepts - Measurements can result in whole and half units. Unknown units can be determined by visualizing how many known units are in it.

Directions - Solve the problems by finding how many red rods in each of the following?


Make red 1. If red = 1, then purple =

 

Make red 1. If red = 1, then red =

 

Make red 1. If red = 1, then brown =

 

Make red 1. If red = 1, then orange + red =

 

Make red 1. If red = 1, then white =

 

Make red 1. If red = 1, then yellow =

 

Make red 1. If red = 1, then light green square =

 

Make red 1. If red = 1, then orange rod + yellow rod =

 

Make red 1. If red = 1, then 3 orange rods =

Answers: 2, 1, 4, 6, 1/2, 2 1/2, 4 1/2, 7 1/2, 15

More If, then Puzzles with smallish numbers

Materials - Cuisenaire rods

Concept - Units of measure can be any size. Unknown units can be determined by visualizing how many known units are in it.

Directions - Solve the problems by finding how many of the unit rods are in each of the following? When you solve the problem, visualize the given unit values (one) as the total value of the unknown rod or set of rods.


Cuisenaire rods

If R = 1, then what is P?

 

If W = 1, then what is N?

 

If G = 1, then what is D?

 

If G = 1, then what is E?

 

If P = 1, then what is N?

 

If W = 1, then what is E?

 

If Y = 1, then what is Capital O with loop ?

 

If W = 1, then what is G?

 

If W = 1, then what is P?

 

If W = 1, then what is Y?

Answers: 2, 8, 2, 3, 2, 9, 2, 3, 4, 5

More If, then Puzzles with larger numbers

Materials - Cuisenaire rods

Concept - Units of measure can be any size. Unknown units can be determined by visualizing how many known units are in it.

Directions - Solve the problems by finding how many rods in each of the following? Solve the problems by visualizing each of the given unit values (one) to determine the total value of the sets. What's one?


If Y = 1, then what is Capital O with loop?

 

If R = 1, then what is Capital O with loop?

 

If Capital O with loop = 1, then what is Capital O with loop + Capital O with loop + Capital O with loop?

 

If G = 1, then what is Capital O with loop + N?

 

If D = 1, then what is Capital O with loop + R ?

 

If N = 2, then what is one?

 

If Y = 1, then what is Capital O with loop + Capital O with loop ?

 

If Y = 1, then what is Capital O with loop + Capital O with loop + Capital O with loop?

 

If Capital O with loop + W = 1, then what is Capital O with loop + Capital O with loop + Capital O with loop + G ?

 

If D = 1, then what is Capital O with loop + Capital O with loop + Capital O with loop + Capital O with loop + N ?

Answers: 2, 5, 3, 6,2, purple, 4, 6, 3, 8

More if, then for - What's My Unit?

Materials - Cuisenaire rods

Concept - Units of measure can be any size. Unknown units can be determined by visualizing how many known units are in it. Units can be thought of as one.

Directions - Solve the problems by visualizing the unit and use it to check the measurement. What's one?


If Capital O with loop= 2, then what is one?

 

If Capital O with loop = 5, then what is one?

 

If Capital O with loop = 10, then what is one?

 

If E = 3, then what is one?

 

If Capital O with loop + R = 2, then what is one?

 

If Capital O with loop + Capital O with loop = 2, then what is one?

 

If N = 2, then what is one?

 

If Capital O with loop + Capital O with loop = 4, then what is one?

 

If 3Y = 3, then what is one?

 

If Capital O with loop + Capital O with loop + Capital O with loop + G = 11, then what is one?

 

If Capital O with loop + Capital O with loop + Capital O with loop + Capital O with loop + N = 8, then what is one?

 

Answers: yellow (Y), red (R), white (W), light green (G), dark green (D), orange (O), purple (P), yellow (Y), yellow (Y), orange and a white (O + W), dark green (D)

Four cubes: p, g,r,w

If, then for Volume - Using a White Cube to Measure Other Squares and Cubes

Materials - Cuisenaire rods, squares, & cubes. Or centicubes

Concept - Volume is a cubic measurement. Volume can be determined by visualizing how many units are in an unknown volume.

Directions - Solve the problems by visualizing the number of white cubes it takes to make it. What's one?

Hint - Think of a cube as a building. How many rooms on a floor? How many floors? How many rooms (white cubes) in each of the following?


red cube

 

light green cube

 

purple square

 

purple cube

 

yellow square

 

yellow cube

 

orange square

 

orange cube

 

brown cube

black cube

blue cube

Answers 8, 27, 16, 64, 25, 125, 100, 1,000, 512

 

Introduction to place value & decimal number representation with Cuisenaire rods

Whole numbers represented from one to one thousand

Orange rod, square, and cube

Visualize the number of white cubes in each.

 

How many white cubes in an orange rod?

How do you know?

 

How many white cubes in an orange square?

How do you know?

How many white cubes in an orange cube?

How do you know?

 

 

If, then puzzles for multiples of tens


How many orange rods in an orange cube?

Prepare an explanation that will convince any skeptic.

Orange rod and orange cube

 

How many orange squares in an orange cube?

Prepare an explanation that will convince any skeptic.

Orange cube with Orange square

More If, then puzzles for multiples of tens

How many white cubes in an orange cube?

Prepare an explanation that will convince any skeptic.

Orange cube and white cube

 

How many orange rods in and orange square?

Prepare an explanation that will convince any skeptic.

Orange cube and orange rod

 

Place Value with Cuisenaire Rods & Cubes

Materials - Cuisenaire rods, squares, & cubes. Or place value sets

Concept - Place values are muliples of ten. Decimal values can be determined by visualizing one and ten parts of one.

Directions - Solve the problems by visualizing each of the given values as part of the unit, (one) to determine the total value of the sets. What's one?


If white = 1

5 orange cubes + 2 orange squares + 4 orange rods + 1 dark green

 

4 orange cubes + 7 orange rods + 4 light green

 

1 orange cube + 4 orange rods

 

3 light green + 4 orange squares + 2 orange cubes + 4 orange rods

 

Make some of your own.

 

Answers 5,246, 4,082, 1,040, 2,449 ...

Place Value of Decimal Numbers to Thousandths

Materials - Cuisenaire rods, squares, & cubes. Or place value sets

Concept - Decimal values are multiples of ten. Decimal values can be determined by visualizing one and ten parts of one. Volume is a cubic measurement.

Directions - Solve the problems by visualizing each of the given values as part of the unit, (one) to determine the total value of the sets.


Orange cube, square, rod, and white cube stacked

Hint - Start by visualizing one and use it as a reference.

If an orange cube = 1, then one orange square =

If an orange cube = 1, then one orange rod =

If an orange cube = 1, then one white cube =

Let an orange cube = 1

8 orange squares, 3 orange rods, and 4 white cubes

 

3 orange cubes and 2 white cubes

 

3 orange cubes, 2 oranges squares, 6 orange rods, and 5 white cubes

 

8 orange squares, 5 orange rods, and 6 white cubes

 

6 orange squares and 7 white cubes

 

9 white cubes

 

4 orange cubes, 2 orange rods, and 5 white cubes

 

3 orange rods, one orange cube, six white cubes, 1 orange square


How many cubes, squares, and rods are needed to represent the following numerals? if an orange cube = 1.

1.23

 

1.25

 

.789

 

1.05

 

.001

Answers .834, 3.002, 3.265, .856, .067, .009, 4.025,

One orange cube, two orange squares, three white cubes; One orange cube, two orange squares, five white cubes; Seven orange squares, eight orange rods, nine white cubes; One orange cube, five orange rods; One white cube.

Decimal Numbers - What Is One?

Materials - Cuisenaire rods, squares, & cubes. Or place value sets

Concept - Decimal values are multiples of ten. Decimal values can be determined by visualizing one and ten parts of one. Volume is a cubic measurement.

Directions - Solve the problems by visualizing the given value as part of the unit, one, to determine one.


Visualize the if ... and then visualize how many of them need to be placed together to make one.

Orange cube, square, rod, and white cube stacked

If an orange rod = .1, then .............................. = 1.

 

If an orange square = .1, then .............................. = 1.

 

If a white cube = .01, then .............................. = 1.

 

If a white cube = .001, then .............................. = 1.

 

If an orange rod = .01, then .............................. = 1.

 

Answers orange square, oragne cube, orange square, orange cube, orange cube

Place value - Units bigger than white
If, then puzzles for decimals

Materials - Cuisenaire rods, squares, & cubes. Or place value sets

Concept - Decimal values are multiples of ten. Decimal values can be determined by visualizing one and ten parts of one. Volume is a cubic measurement.

Directions - Solve the problems by visualizing the relationship of the unit, one, to the unknown.


Let an orange rod = 1

If an orange rod = 1, then one white cube =

If an orange rod = 1, then one orange square =


If an orange rod = 1, then what are the values of these sets?

1 orange square, 2 orange rods, and 4 white cubes

8 orange squares and 3 white cubes

2 orange rods, and 8 white cubes

5 orange cube, 2 orange rods, and 4 white cubes

7 white cubes


Let an orange square = 1

1 orange square, 2 orange rods, and 4 white cubes

8 orange squares and 3 white cubes

2 orange rods, and 8 white cubes

5 orange cube, 2 orange rods, and 4 white cubes

7 white cubes

1 orange square, 3 orange rods, and 2 white cubes

5 orange cubes, 4 orange squares, 6 white rods, and 3 white cubes

How many cubes, squares, and rods are needed to represent the following numerals?

If an orange square = 1

.1

.3

1.5

1.4

.4

.7

12.06

2.05

 

Answers .1, 10,

12.4, 80.3, 2.8, 52.4, .7

1.24, 8.03, .28, 5.24, .07, 1.32, 54.09

one orange rod; three orange rods; one orange square and five orange rods; one orange square and four orange rods; four oragnge rods; seven orange rods; one orange cube, two orange squares, and six white cubes or rods, two orange squares and five white cubes or rods.

Place Value of Decimal Numbers for Orange square = one

Materials - Cuisenaire rods, squares, & cubes. Or place value sets

Concept - Volume is a cubic measurement.

Directions - Solve the problems by visualizing the relationship of the unit (one) to the unknown.


Let one orange square = 1

If an orange square = 1, then one orange rod =

If an orange square = 1, then one white cube =


Solve problems like the following:
If an orange square = 1, then what is the value of each of the following sets?

2 orange rods and 3 white cubes

1 orange rod, and 2 white cubes

3 white cubes

9 orange squares, 4 orange rods, and 5 white cubes

5 orange squares and 7 white cubes

2 orange cubes, 3 orange rods, and 8 white cubes

How many cubes, squares, and rods are needed to represent the following numerals?

1.05

4.01

3.6

10.12

7.13

5

 

Answers .1, .01

.23, .12, .03, 9.45, 5.07, 20.38

one orange square and five white cubes; foiur orange squares, and one white cube; three orange squares, and six orange rods; one orange cube, one orange rod, and two white rods; seven orange squares, one orange rod, and three white cubes; five orange squares

Inroduction of fractional representations with Cuisenaire rods

 

Fractions for Rods Greater than white = one

Materials - Cuisenaire rods. Or centicubes

Concept - Volume is a cubic measurement.

Directions - Solve the problems by using the unit to make it. What's one? How many rods of the unit are in each of the following?


Make light green 1. If G = 1, then black =

 

Make red 1. If R = 1, then yellow =

 

Make yellow 1. If Y = 1, then orange + yellow =

 

Make red 1. If R = 1, then brown =

 

Make white 1. If W = 1, then orange =

 

Make dark green 1. If D = 1, then orange + yellow =

 

Make dark green 1. If D = 1, then orange + dark green =

 

Make black 1. If K = 1, then orange + white =

 

Make light green 1. If G = 1, then black =

 

Answers 2 1/3; 2 1/2, 3, 4, 10, 2 3/6 or 2 1/2, 2 4/6 or 2 2/3, 1 4/7 2 1/3

 

Multiplication of fractions represented with Cuisenaire rods

Multiplication of Fractions As Repeated Addition

Materials - Cuisenaire rods, squares, & cubes. Or centicubes

Concept - Multiplication of fractional numbers can be represented as repeated addition.

Directions - Solve the problems by finding the the nmber of white cubes it takes to make it. What's one?

Background information: A numeral like 1/5 is a fraction. It can be thought of as one out of a group of five. 1/5 of the class has red hair, or one person out of every five has red hair.

Represented with Cuisenaire rods:

If Yellow = one,

then 3 x 1/5 =

1/5 + 1/5 + 1/5 or 3/5

Activity: Illustrate multiplication of a whole number and fractional number by joining the number of fractional parts.

Solve problems like the following:

If you have three opened packs of chewing gum and each has one stick out of five left, then you have 1/5 + 1/5 + 1/5 or 3/5.

Therefore 3 x 1/5 = 1/5 + 1/5 + 1/5 or 3/5

Illustrate solutions for problems like the following.

3 × 1/4

2 × 1/5

4 × 1/6

3 × 1/4

7 × 1/3

5 × 2/3

3 × 4/5

 

 

 

Multiplication of Fractions As Parts of A Group

Materials - Cuisenaire rods, squares, & cubes. Or centicubes

Concept - Multiplication of fractional
numbers can be represented as parts of a group..

Background information - Multiplication of a fractional number with a whole number, may be thought of as a fractional part OF a group. 3/5 OF 10 Can be thought of as how much would be in THREE small groups, if the ten were divided into five equal groups. 1/5 + 1/5 + 1/5 + 1/5 + 1/5 = ONE GROUP (of ten).

3/5 OF 10 represented with Cuisenaire rods:

If white = one, then orange = 10

Think what rod could be used five times to match an orange? red

Use this rod to be 1/5 of ten and then use repeated addition and add two more red rods.

Place them on top of the orange rod to compare them.

These three red rods represent 3/5 of orange (10) or 6 (white).

Activity: Illustrate multiplication of a fractional number and whole number by dividing a group into equal parts and select the fractional number of parts.

Illustrate a solution to problems like the following.

1/3 × 9

1/4 × 12

1/3 × 12

1/5 × 15

2/5 × 10

3/4 × 12

3/5 × 15

1/5 × 5

1/5 × 20

3/5 × 10

1/4 × 12

2/3 × 18

3/4 × 12

3/4 × 16

2/3 × 12

2/3 × 21

 

Multiplication of Fractions As Parts of a Fraction

Materials - Cuisenaire rods. Or centicubes

Concept - A fraction is a division problem which when multiplied can be thought of as dividing the other factor into groups by the denominator and selecting the numerator number of groups.

Directions - Illustrate multiplication of a fractional number and fractional number by dividing a group into equal parts, selecting the fractional number of parts, and repeatedly add the fractional part, and equate it to a fractional number.


Example for 1/2 × 2/3

Let E = 1,

then D = 2/3; and G is 1/2 of 2/3; or 1/3 of E; or 1

One half of two thirds

Select from the following and illustrate a solution.

1/2 × 2/5

1/5 × 5/8

1/2 × 4/5

1/7 × 7/8

1/3 × 3/5

1/5 × 5/6

 

 

 

 

 

Bigger and Better:

Select some and draw a representation for your solutions.

2/3 × 3/5

3/4 × 4/5

3/4 × 8/10

2/3 × 3/4

2/3 × 3/5

3/5 × 5/6

2/3 × 3/7

4/5 × 5/8

2/3 × 1/2

2/3 × 4/5

3/4 × 3/5

3/4 × 1/5

 

 

Resources

White cube Orange cube square rod
Orange cube, square, rod, &
white cube,rod , or square
Orange cube and orange rod
Orange cube & rod,
Orange cube and orane square
Orange cube & square,
Orange cube and white cube
Orange cube & white square,
Orange square and orange rod
Orange square & orange rod,
Dark green square and tower
Dark green square & tower

 

Ligt green cube
Light green cube
Brown tower of two Brown tower of three
Brown towers
Blue tower
Blue towers
R + G = G + R
Commutative property R + G = G + R
GDE stack
If E = 1,
Then G is 1/2 of D
& D is 2/3 of E
∴ G is 1/3 of E

 

 

Top