Math teachers, have you been poking around the math supply cupboards or closets at your school and run across a set of colored rods in ten lengths? They are probably Cuisenaire Rods--a classic mathematics manipulative that provides dozens of hands-on, minds-on opportunities for students in elementary and middle school.
Photo credit Hand2Mind
Hand2Mind used to be called ETA/Cuisenaire but changed their name in 2012.
They are still THE place to buy Cuisenaire Rods. They sell them in all sorts of sets: magnetic, foam, plastic, clings, jumbo and more. Check it out!
First, a little history lesson about the origins of this cool classic. (If you want a long history lesson, read this article by Belgian professor Dirk De Bock who’s an expert on Cuisenaire. Fascinating stuff!)
Georges Cuisenaire (pronouned Kwee-zen-air) was a Belgian teacher who started his career during World War I and in the 1940s founded his own school. Legend holds that he invented his rods after noticing his students were super engaged during music class where they could do something with their hands but were bored stiff in math class where they just listened passively and filled out worksheets. “Hey,” he thought. “We can make math engaging and improve kids’ understanding at the same time. What we need are tools to make it hands-on!” Okay, he probably didn’t say it quite like that—it was probably in French, but you get the idea.
Here is Georges pondering how to make math more about sense making and less about filling in answers to memorized or rote procedures.
He wanted to find a tool that: “Creates visual, muscular, and tactile images that are clear, precise and sustainable” (Cuisenaire, 1952, p. 17).
Big task! But he did it!
Georges set to work creating strips of cardboard in ten lengths, each rod one unit longer than the previous. He colored the rods in families, thinking this would help kids become familiar with the numbers and how they related to each other.
The red family (red, purple and brown, representing 2, 4 and 8 respectively).
The green-blue family (light green, dark green and blue, representing 3, 6 and 9 respectively)
The yellow family (yellow and orange, representing 5 and 10 respectively).
The rods representing 1 and 7 were painted in white and black respectively.
What do you notice about the color families and how the numbers in the families relate to one another?
Pretty clever, right?
Georges implemented the rods in his own school, and the kids and teachers loved ‘em. Soon schools all over Belgium were scrambling to get their hands on the rods, now in wooden form, and a little how-to book called Les Nombres En Couleurs: Nouveau Procédé de Calcul Active Applicable à Tous Les Degrés de L’école Primaire. In English that’s Numbers in Color: New Active Calculation Procedure Applicable at All Levels of Primary School.
It was like Teacher Pay Teacher only it wasn’t just worksheets and coloring pages, it was a tool for making sense of math. And a teacher invented it!
How to Introduce Cuisenaire Rods through Inquiry
Pass out the rods and say nothing. Give kids time for free play, using the rods like any other open material. After about 10 minutes, do a Notice and Wonder inquiry discussion where groups use sticky notes to add their observations and questions to charts like these:
Give kids time to explore with rods and centimeter graph paper to find proportional relationships between rods. After lots of exploration, make a list together.
You may want to set up a shorthand system for labeling rods. Hand2Mind suggests:
W for white R for red G for green P for purple Y for yellow K for black N for brown E for blue O for Orange D for dark green
21 Activities with Cuisenaire Rods
Now you’re ready to start using Cuisenaire Rods as a manipulative for lots of different learning objectives and math concepts. Here are a few I’ve tried and found successful.
1. Odd/Even: Separate the rods into odd and even values. Create as many combinations of 12 as you can with the rods. Record the combinations. For example, G + E = 12. Explain if odd or even numbers can create more combinations of 12.
2. Odd/Even: Explore to find whole number rods by seeing if a rod can be matched with a one-color train of two rods linked end-to-end. For example, 1N = 2P and 4R. Help students discover how even numbers can be split into halves (2 rods of equal length) but odd numbers cannot.
3. Square Numbers: Discover square numbers using rods.
4. Division: Ask “How many yellow rods can an orange rod be divided into?”
O ÷ R or 10 ÷2 = 5. Have students make up division problems for one another. What patterns do you notice?
5. Division: Explore remainders using rods. “One black rod divided by three red rods equals what?” Represent this with 1 black rod with 3 red and 1 white placed below it.
Pose the following problems:
D ÷ P =? (1, R2) O ÷ G =? (3, R1) E ÷ R =? (4, R1) K ÷ P =? (1, R3)
Students can make up their own division sentences and record their answers on a sheet of paper. Have them make up division sentences and share with a friend.
6. Equivalence: Find all the combinations of one-color trains to match the rods. For example: What single color of rod (besides white) could you use to equal 2P? 4R? 8W? Is there a rod that works for all three trains?
7. Decimals: Have the orange rod equal $1 and all the other rods equal to their decimal equivalent to the orange rod. For example, W = $0.10 and Y = $0.50.
8. Decimals: Use the rods to represent decimals, with the orange rod being 1 whole and the other rods equal to their decimal equivalent to the orange rod. The picture above shows this same idea using decimals rather than money values.
9. Guess and Check: Use rods in a magic square. In this example, each side of a 3 x 3 grid must add up to 15. Here is one solution. Are there others?
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10. Area and Perimeter: Draw a design on centimeter grid paper. Color the design using the Cuisenaire rod colors and find the area in square centimeters. Trace the design again on a different sheet of grid paper, but this time make the design using FEWER rods. Repeat by covering the design with MORE rods. The area stays the same but the colorful design changes. Mount all 3 designs on black construction paper. It really looks like art!
11. Area & Perimeter: Make a maze on graph paper using rods. Outline the maze and trade with a friend. Find the area and perimeter of the mazes.
12. Area: Make and record the following shapes with an area of 24 on graph paper:
A one-color train
A two-color train
A three-color train
A four-color train
A rectangle using only one color
A rectangle using two colors
A rectangle using three colors
What do you notice about the rectangles? What patterns can you see?
13. Coordinate System: Use rods to create a design or picture on a grid. For older students, have the creator of the design record the coordinate pairs for the lower left-hand position of each rod. For example, white (1, -1). This could serve as a sort of map for the design. Photograph finished art projects for inclusion in a math portfolio or on a bulletin board.
14. Symmetry: Fold a sheet of graph paper in half lengthwise and then in half again by its width. Use the rods to make a design with the base of the design on the folded edge at the bottom. Outline the design with a pencil. Cut the design out. Open the paper to see a design with four lines of symmetry where the paper is folded.
15. Multiplicative Growth: On centimeter graph paper, build a staircase pattern from shortest to longest rod. Discuss the pattern. Next, make each step of the staircase twice as long with the same color rod of that step. Draw this on graph paper, color, and label with codes above. What do these staircases show? If the staircases were three rods deep, how long would each step be? How many 1-cm squares do you think this graph takes up?
16. Multiplicative Growth: Use rods to make expanding rectangles. What other shapes can you make and expand?
17. Geometry & Measurement: Golden Rectangles are found in nature and often imitated in art and architecture. The ratio of the width to the length in a Golden Rectangle is always approximately 1 to 1.6. (Divide the long side by the short side.)
Make Golden Rectangles by following these steps:
Start with a white rod.
Add another white rod.
Next, add two red rods.
Next, add three light green rods.
Can you see a pattern in rods and numbers? The pattern is 1, 1, 1+1=2, 1+2=3, 2+3=5. The next group of rods to add are five yellow rods + three green rods. The number of rods to add equals their value in centimeters (5 yellow rods because yellow equals 5 cm.)
What comes next? The answer is 8 brown rods from 3+5=8.
Find the next step in the pattern.
18. Area Model for Multiplication: Work with a partner. Each player uses a different 10x10 grid of centimeter paper. Take turns. On your turn, roll the die twice. The first roll tells “how long” a rod to use. The second roll tells “how many” rods to take. Pick up the correct length and number of rods. Arrange the rods into a rectangle. That means they have to be touching and make the shape of a rectangle. The game is over when one of you can’t place your rectangle because there’s not enough room on the grid. To find a winner, count how many of your squares are covered and how many are uncovered. Check each other’s answers. The one with the most squares covered wins.
19. Primary Color Cuisenaire Rod Trains: Measurement comparisons for PreK and Kinder.
20. Race to 50 or Race to 100 Game: For this activity, you'll need Cuisenaire Rods, meter sticks with centimeters shown, dice. Each player takes a turn rolling a die to see which rod to place in a train. For instance, if you roll a 5, you can use a metric ruler to measure the rod you think represents the number 5. You might have to measure a few rods at first to know which one is the right size. Each player takes turns placing corresponding rods on his or her train. After each turn, you can compare the lengths of rod trains. Play until one or both players reach the end of the 50 cm track. When you’re comfortable with Race to 50, try playing Race to 100.
Variations: You can use one die, placing a single rod at a time on the track, or a set of dice to either add or multiply the numbers on the dice. If you’re adding the dice, and you roll a 2 (red rod) and a 3 (light green rod), you can choose to place the red and light green rods or a yellow rod (5) on your track. If you’re multiplying the dice, and you roll a 3 (light green) and a four (purple), that makes 3 fours or 4 threes, so you can place 3 purples or 4 light greens on your track.
21. Race to 0 Game: This is the reverse game of Race to 50 or 100. Keep the original rods from a Race to 50 game and remove one colored rod at a time according to the number you roll. In this way, there’s no trading involved, just identifying the numbers/colored rods to remove.
Variations: For a tougher game, lay out 10 orange rods to make a train. Roll the die and subtract rods till you clear the track and get to zero. Trade rods as necessary in order to facilitate subtraction. This is a game that is best played after many games of Race to 50 or 100 because subtraction concepts can be more abstract to think about while becoming familiar with the rod values.
If you can’t tell, I am in love with Cuisenaire Rods.
Georges Cuisenaire died in 1976, just a few years after I was born. Someday, I’d like to shake his hand and tell him thanks for being proactive in solving a problem in his own school—for recognizing that using manipulatives like his rods “Make calculation sensory, attractive, lively and saves time, while simplifying the task of the teacher” (Cuisenaire, 1952, p. 18).
Let me know what you think of these ideas, add your own suggestions in the comment section, or send me a Tweet. I'd love to hear how parents and math teachers are using Cuisenaire rods in your classroom. Whether you're exploring measurement, number relationships, or operations--in a center or whole group--hands-on learning builds math-positive mindsets.
If you like these ideas, hit subscribe. And find many more tips in my book Math-Positive Mindsets: Growing a Child's Mind without Losing Yours.