 How to Teach Addition with Cuisenaire Rods for Conceptual Understanding - Play Discover Learn 24/7

# How to Teach Addition with Cuisenaire Rods for Conceptual Understanding

Teaching addition with Cuisenaire rods can’t be that complicated, right? It is just addition.

Addition is often the simplest operation to teach.  Who doesn’t love to add?  Adding opens the imagination of more. More candy, more play time, more money, more fun.

Yet, addition is related to all operations, and if the foundation of addition is not laid right, the student may find trouble with all the other operations.

As previous guides on teaching with Cuisenaire rods, this guide focuses on one operation in connection to the other operations.

The hands-on addition guide walks through simple language, meaningful tasks and playful context that allow students to grasp general ideas for addition that also prepares the student for subtraction, division and multiplication. Conceptually-based addition begins with general concepts of addition.  Cuisenaire rods provide hands-on tasks for delving into these general concepts of addition, but what are those general concepts?

In this post, general concepts like equivalency, greater than, less than, odd, even, and transformations (think properties of addition) help students gain an important foundation for number fluency.  Number fluency gives students the ability to streamline math facts, develop problem solving strategies and become master manipulators of numbers and symbols.

## Important Notes to Begin

The simple language and meaningful tasks work together to strengthen the foundation for fluency.  Some tasks appear similar, but the language changes to allow students to see the connection between concepts.  I call these transitional tasks.

Also, these tasks use color rod names to help students to focus in on the concept language.  However, you may have a different philosophy and prefer numbers.  The language remains the same.  Just replace color names for number names.

## Build Trains to Explore Addition

To begin, students are tasked to build trains.  A train is more than one rod laid end to end.  At first, students build trains freely. Below is a picture of 4 individual trains composed of three rods each.  Sometimes the individual rod is referred to as a car in the context of building "trains." Such freedom is important.  Students perceive the rich diversity available in the composition of trains.  Later, this perceived richness translates into the diversity available in mathematical notation.  There is great satisfaction for mathematics for students who taste such diversity.  It touches on the innate creator that exists in us all.

Fun tasks to encourage more exploration of the rods include

• What is the shortest train you can build?
• What is the longest two rod train you can build?
• How do you know this train is the shortest possible train?
• How do you know this train is the longest possible 2 rod train?

"This train is yellow, red, light green and white.  Here is another train of purple, black and light green."

PDL's Math Task Cards and Play mats also have task cards (Module 1.6) for students to practice building and reading trains. Once students confidently read the trains as above, read the trains with a plus sign.  Yellow plus a red plus a light green plus a white. Later, such exercises translate into mathematical notation with abbreviations for color names like y + r + g + w.

If you use Gattegno’s Textbooks, he introduces the plus sign at the beginning of literal work of Gattegno’s Textbook 1 Part 3.  However, the general concepts of this post are from the qualitative exercises of Gattegno’s Textbook 1 Part 2.

I don't think you hinder student's understanding by introducing the plus sign in qualitative learning.  For students already acquainted with math, I would read with the plus sign.

For picture book ideas to incorporate with the concept of building trains with Cuisenaire rods, check out this post here.  The free course also includes a free download for students to journal (draw) their trains.

### Composing Numbers to Explore Addition

The next task is referred to as patterns in Gattegno’s textbook.  These tasks have students explore equivalency of one rod and even exhaust discovery.  The student perceives that the length of each rod may be composed in a variety of ways.  The smaller rods reveal that the composition though is limited.

For example, there is only two ways to compose the length of the red rod.  One is the red rod itself, and the other is two whites.  But the light green provides more diversity and the diversity grows with each rod length. Ask the student to build as many trains as possible for the yellow rod.  Ask the students to place each train side by side as pictured above. This is building a pattern, also known as a mat.

For the smaller rods, students may work independently to exhaust composition.   For larger rods (yellow and longer), students may work in groups.  It also makes a great family activity.  Students may compare their patterns.

"Do you have train white plus white plus white plus red?"
"No, I have white plus red plus white plus white."

In this exchange, students read the trains to one another trying to find the trains that they are missing.  Students strengthen their auditory skills as well as their verbal skills. In such an exchange, students discover the importance of accurate mathematical communication. This may be a hefty task for young students. Students may not accomplish the task the first time for larger rods.  However, it is a good exercise to build the stamina of a student. Bring out this task often but quit before the students tire of it.

Building patterns also helps students to develop organization.  As they work through the it, the greatest question is, “Did you find all the possible compositions?”   Students begin to see the value of organizing their patterns.  This organization does not develop overnight.  It takes some time, so again bring out this activity every couple of months.

Consider using PDL’s Number Building play mats to provide playful context.

### Search and Find Play Mats to Explore Addition

Another way for students to explore equivalent addition statements is through my PDL's Search and Find play mats.  These play mats provide a fun independent activity for students.

It’s a low verbal exercise that can be made as verbal as the educator desires. The student gets an intuitive sense of larger and smaller trains.  Extend the activity by asking the student to record the equivalent trains they found.  Or for the writing resistant student, ask the student to dictate to you the trains they found.

There is a fun game to play with PDL's Search and Find Play Mats.  Students may play blackout. Each student has a pile of rods.  Each student takes turns drawing a rod from their pile and placing it on the mat.  The goal is to use up all your rods before the other player. Consider adding mathematical notation to the game as well.

Purchase Search and Find Play Mats here.

### Greater than, Less than

In the first task, students build trains freely.  Students likely build trains of various lengths.  This task students build trains freely, but then move to order their trains.

Which one is longer?  Which one is shorter?

Task the student to order their trains side by side from shortest to longest.

Next time, add a constraint to the task.  This time ask the student to build only 2 (car) rod trains.

"Which one is longer?  Which one is shorter?
Which train has the greatest length?  Which train is the shortest?"

Consider changing the vocabulary from longest to bigger to greater.  Variety is important to developing depth.

Other ideas include adding the constraint of only solid color trains and 3-rod trains. Compare the trains by observing the lengths of each train.

When mathematical notation is introduced, extend the exercise by introducing the greater than/less than symbol.

Consider using the Comparing Trains Play Mat from PDL’s Math Task Cards and Play mats to offer the student a place to build and sort the trains.

### Build and Find |Missing Addend

The build and find task introduces the missing addend language. It is the same task from a previous guide on subtraction.  The task is used to not only teach addition but to teach how addition is related to subtraction.

The task extends the longer and shorter comparison language from the greater than/less than task above.  A good task always transitions the student from a familiar place to note a new idea and to introduce new language to express the new idea.

Initially, complete this task with the student.   However, these task cards are set up to give students a few minutes of independent work after they are familiar with the exercise.

The task is to find the missing rod.  That missing rod is the rod that makes the train equivalent to other train.

It begins the student’s introduction to the interconnectedness of addition and subtraction. Students use this understanding to later develop strategies for remembering math facts.

The task extends the student’s thinking by asking the student to mix up the rods to build two new trains. After independent work, be sure to connect with student’s thinking.

"What did you notice?  Were all the new trains equal in length?  What rod could be added to make the new trains equivalent in length?"

These task cards are from PDL’s Math Task Cards and Play Mats.

### EquivalencyBuild and Find | Missing Addend

This is another version of build and find.  It continues to build on the missing rod language.  It is simpler in that the student only works with a single rod train and a two rod train.

Ask the student to find two rods of different lengths or use these task cards to provide an independent activity.   The student then finds the rod that makes the shorter rod equal to the longer rod.

It is important to switch between this exercise and the previous one.  In the previous exercise, the student compares the composition of several rods (a train) to another composition of several rods.   In this exercise, the student finds equality between a single rod train and a two-rod train.

Expand these exercise by eliminating different rods of the 3-rod composition.  Then read these eliminations.  For example, if you remove the top single rod train, read it as,

“Something equals red plus yellow.  What is that something?”

If you eliminate the red, you read it as

"Black equals something plus yellow."

Again, if you eliminate the yellow, you read it as,

“Black equals red plus something.  What is that something?”

Create variations in your reading by moving the equal sign or changing the vocabulary. “Red plus something equals black.  What is that something?”  See more tips in this post, 5 Keys to Math Narration for Improved Fluency.

## Exploring and Sorting 2-Rod Trains of Addition

The task begins with a disorder of 2-rod trains and moves towards a composition of two orderly staircases.  This helps students perceive the relationship and order of math facts within a math fact family.

Task the student to create 2 rod trains equal to blue.  Then ask the student to find all the other 2 rod trains equivalent to the first two rod train. The red plus a black is a different train from the black plus a red.  It is important to note that for this exercise.  Then task the students to sort the trains side by side by biggest rod plus the smallest rod as pictured above.  Ask the student,

"What do you notice?"

The students should notice staircases, or they may notice that one side is growing and the other side shrinking.  Ask the student to separate the two staircases and put it back.  If the student doesn't perceive the two staircases, do it for the student.  Then ask the student to do it themselves.

Ask the student, “Would this repeat with another 2-rod train of a different length?”  Offer students the opportunity to test their idea.  Mathematics is a great training ground for cultivating not just self-correction but validating the truthfulness of any idea.   Students may want to test it on several rod lengths.

Try this task on a different day than the previous task.  Students often need to get acquainted with an idea through different experiences.  This task differs from the previous task in that it presorts the 2-rod trains by beginning with a staircase. It also adds in the missing addend language.

Task the student to build a complete staircase (a rod of each color).  Then task the student to find the complement of the staircase. That is, you want the student to make each step of the stair equal to the tallest stair.  In this exercise, students find the missing addend for each stair of the staircase.

The Missing Chicks mats are a fun way to add playful context to this task.  It’s all about finding the missing piece of the ladder, aka the rod, to rescue the chicks.  These play mats are in PDL’s Math Task Cards and Play Mats.

Staircases also let students experience math fact families in their totality rather than on the individual basis.  This general concept of math facts becomes intuitive to the student as they see how a fact family is composed of two staircases.

Later, these experiences serve to help the student streamline math facts.  Check out this post on "Should We Prioritize Math Facts."   Also, check out the free Missing Chick download in the Free Course Section.

### Up and Down the Stairs of Addition

In this task, the language of ordinal numbers is added to mix.  Task students to build a complete staircase.  Tell the student you will name each stair.   “The bottom stair is the “first” stair.  Let’s climb up it.  The next stair is the second stair.”

You could also introduce ordinal numbers with this picture book video, “On the Stairs”   Students need to first get acquainted with using the ordinal language.  Building and naming ordinal stairs is a great activity to add to the weekly mix.

Students may build a variety of staircases from plus one to plus two to random up and down staircases.  Either way talk about the up and down movement of the staircases using the language of addition.

“Let's walk up the stairs and get a book.  This is the first step, now the second step, then the third step."
"Let's go down the stairs.  We start at the fourth step.  Now down to the third, second and finally back to first step."
“Do you like to skip steps? How many steps do you like to skip? Two?  Yes, let's go up the stairs and skip two steps."
"Name the ordinal number of the purple step. The yellow step?"

The staircase provides a new context for addition as well as new vocabulary to express addition.  Too often, the context of adding to a group of beans or bears restricts the student’s perspective for the math around them.   Cuisenaire rods are versatile allowing student and teacher to move through a variety of contexts and vocabulary.

For more ordinal staircase activities, check out PDL's Staircase Task Cards.

In the context of staircases, addition is now seen as an upward movement.  With Cuisenaire rods, the language of addition is expanded beyond the realm of counting objects and into the world of length, distance and upward movement.

In the world of word problems, such variety in language gives students familiarity with numerous mathematical contexts.  See this post on 5 Keys to Math Narration for more tips on creating variety to improve conceptual understanding.

### Staircase Plus One | Exploring Common Addition

Much like common difference in the language of subtraction, plus one staircases provide a context for exploring the language of common addition.

In this task, the student builds a complete staircase (that is a rod of each color).  Then, ask the student to add a white to each stair.  Ask the student to build the new staircase replacing each stair plus a white with one rod of equivalent length.

Students may want to overlay the rods to check their answers, but then move the new staircase off to the side.  Consider using the play math pictured above and below to help the student compare each staircase.  Ask the student,

“What do you notice? What is similar? What is different?”

This play mat is from PDL's Number Building Staircase which is also apart of PDL's Interactive Math Notebook

Repeat the whole task of adding a white to each stair, but this time to the new staircase.  Replace the newest staircase with one rod for each step just like the previous task.  Ask the student the same questions as before, but add the question,

“What do you think will happen if we did it again?”

Students may want to test out their idea or be ready to move on.  Either way, repeat this exercise again in coming weeks.

### Plus-Two Staircases for Odd and Even Exploration

Plus-one staircases only produce one kind of staircase as the exercise above demonstrated. Plus-two staircases though create two staircases, an odd staircase and even staircase.

Task the student to start with a white and create a plus-two staircase.  The first stair is white.  The second stair is light green (white plus two).  Then the third is yellow (light green plus two).  This continues until they have gone through the set of color rods (white to blue).  Leave the staircase to compare with the next plus-two staircase.

This time task the student to start with a red and create a plus-two staircase. That means the second stair is purple. Next is dark green and so on until the student lastly comes to the orange.

Don't use the play mat featured above for this task.  It's just there to show you the odd and even staircase.  This play mat is for the next task.

Have the students compare the staircases. What do they notice?  Students may know that the two staircases added together makes a complete staircase.

### Expand Odd and Even Staircase Comparison

There are two other ways ways to unveil the distinction between odd and even numbers. One is with pairs of same-colored rods.  The other way is with red rods.

Even rods may either be composed by all red rods or by a pair of same-colored rods. Odd rods may be composed of all reds plus one white or by a pair of same-colored rods plus a white. See pictures below.

Use the odd and even comparison play mat for this activity. First, task the student to cover the steps of both the odd and even staircases with red rods.  Ask the student,

"Are you able to cover both staircases?  Which staircase can you not cover completely with red? What rod is missing to complete each step?"

Tell the student that the staircase on the left is made of odd lengths.  The staircase on the right is made of even lengths.

Alternatively, task the student to find a pair of same-colored rods to cover each stair of the two staircases.  Ask the student,

"Can you cover both staircases completely?  Which staircase can you not cover? What rod is missing to complete each step?"

In both tasks, the student experiences and sees the distinction between odd and even.  Name each staircase as odd and even.  Tell the student,

“The odd staircase is made of odd rods.  Odd rods are made up of all red rods plus a white or a pair of same-colored rods plus a white.  Even rods are made up of all red rods or a pair of same-colored rods.”

There is also a game that you can use to teach odd and even. It is called the odd rod out and it is found in PDL's Math Task Cards and Playmats

## Adding Odd and Even Rods

Once students know which rods are odd and even, it is time for students to see what happens when adding these qualities together.

Task the student to compose a 2-rod train of an odd rod and even rod.   Ask the student,

“Is the new length odd or even?”

“Will it always be odd?”

“Is the new length odd or even?”

“Will it always be even?”

Repeat this again but this time with two even rods.  Ask the same series of questions.

Alternatively, use this odd and even comparison play mat from PDL's Number Building Staircases.  Task the student to build each step using two rods.  Ask the student,

"For the odd staircase, are there any steps made of just even numbers? Is it possible to make any?  Are there any steps made of just odd numbers?  Is it possible to make any?  Why not?"
"For the even staircase, are there any steps of just even rods?  Any steps with just odd numbers?  Any steps with both odd and even rods?  Is it possible to make an even step from one odd and one even number?"

The curious question is why is it so? Something to leave for the students to puzzle over.

### Composing and Decomposing Odd and Even Lengths

Here is another activity for exploring the addition of odd and even length.  It helps students to continue puzzling over the nature of adding odd and even lengths and maybe see why it is so.

Use the even/odd staircase play mat pictured.  Task the student to start each step with a white rod.  Then ask the student to complete each step of the staircase with two different colored rods. Start first with the odd staircase.

The two steps of the odd staircase is only able to be composed of whites. The student will find they can not complete the task for these steps.  Let them put just whites.

Ask the student to observe their completed even staircase.

Did you use odd or even rods to complete each step?  Did you use a combination of odd and even lengths?

Leave the staircase to compare with the next task.

Task the student to start each step with a white rod again for the even staircase. Then ask the student to complete each step of the staircase with two different colored rods.  Obviously, first step can only be composed of two whites, so students skip to the next step.

Ask the student to observe their completed odd staircase.

"Did you use just odd or just even rods to complete the step? Did you use a combination of odd and even lengths?"

Have the student compare the two staircases.

"How do they differ in their composition of odd and even lengths?"

Students expand their awareness of how odd and even lengths work to make other odd and even lengths.  Students well-versed in this awareness use the information to later verify if an answer is right or wrong.  The student may think,

“That is adding together two even numbers, so I know the answer must be even.  This eliminates half the available answers.”

It is a vital strategy that plays out in problem solving and in streamlining math fact memorization. Make the time to play with odd and even lengths a lot during the early stages of mathematics.  It pays off for the student.

### Transformations: The Commutative Property

The Commutative Property of addition is so apparent with the rods that it is almost intuitive.  Gattegno calls this general concept a transformation in Gattegno Textbook 1 Part 2.   Students well versed in the manipulation of Cuisenaire rods with lots of free play notice the commutative nature of the rods.

This task emphasizes what may already be intuitively known to the student but maybe not be articulated.   This task requires prep work for the educator to set up the rods to illustrate transformations.  It is why the task cards from PDL’s Math Task cards and Play mats come in handy.

If you don’t have the task cards, set up various configurations of the rods like the ones pictured above.  The idea is for the student to realize that no matter the order of the rods, the sum or equivalency remains the same.

Alternatively, you may ask the student to build a 3 or 4 car train equal to orange. Then remove the orange rod.  Then ask the student to rearrange the order of the rods. “Is the train still equivalent to orange?”
“Will the train be equivalent to orange no matter the arrangement?”

Let the student test their answer by putting the orange rod back. Repeat this activity as needed until the student is confident in their answer.

### The Commutative Race

Alternatively, the PDL’s Fraction Exploration play mats provide a fun context to explore transformations.   Prep the student with 4 piles of the same rods equivalent to the race.  Pictured here is a completed race of 2 yellows and 2 whites.

Tell the student that they will race against you by drawing rods in any order and placing them on the path.

The rules are youngest goes first in the first round. After one rod has been laid down for each path, the next round is played so that the one in last place (shortest total length) goes next.   The game continues as such until everyone completes the race.

As the educator, it is important to race the rods differently than the student, so that the student may observe the transformations.

After the race, observe the trains.  Ask the student,

• “Does everyone have the same number of each color rod?”
• “Is everyone’s race of equal lengths?”
• “What is different between each race?”

The student observes that no matter what order of the rods the equivalency remains the same. That is, the student may commute (move) the rods about in any order and the sum doesn’t change.

Mathematically, it is simply expressed as A + B = B + A.

### Hands-on Substitution Game for exploring Addition

In the simplest version of the substitution game, a rod is substituted for two rods equal in length to the rod being substituted.

This is a staple game in our house.  It has been expanded to mathematical notation now. I have seen adults enjoy this game as it explores the creativity and diversity available in number composition.

Students enjoy it most as they become master symbol manipulators.  See other version of this game here and here.

Use the orange mat from PDL's Math Task cards and Play Mats as the context for the game.  Begin the student with an orange rod.

Ask the student to substitute the orange rod with two rods that are equal in length to the orange rod.

Then ask the student to choose one of the two new rods to substitute for two rods the length of the chosen rod.

For example, the student substitutes orange for black and light green.  Then the student substitutes the black for yellow and red.

The student continues the substitution game until there are no rods to substitute.  That is until there are only white rods.

There are a variety of versions of this game.  When mathematical notation is introduced, the complex creativity available in the game develops a delight for math within most students and educators.

### Inverse Substitution Game to explore the Associate Property

In this version of the substitution game, the student plays this game in reverse.

Use the black mat and task the student to build a three-rod train equal to black.

Ask the student to substitute two rods for one rod of equal length to the two rods.

Ask the student to then substitute the remaining two rods for one rod of equal length which is the black rod.

"If we repeated the game with the same rods but this time substitute a different pair of rods, would the game still end with a black rod?”

Allow the student to repeat the game to test their answer. Through this version of the substitution game, the student experiences the associative property of addition.

Basically, rods or numbers may be added together (grouped) in differing order without changing the sum.   The sandwiching or substituting of rods for one rod of equal length is a concrete experience of the written notation of the parenthesis.

Consider playing this game when you introduce mathematical notation of the parenthesis.

Laziness is always seen as a fault.  But in fact, it is laziness that is the mother of invention.  I know the saying but it really is wrong.  We didn’t invent the wheel out of necessity but out of laziness. We desire to work less and that is a good thing.

This laziness helps us transition students into multiplication.

Task the student to build long solid color trains. Have each student read their trains as addition.

"Red plus red plus red plus red plus red plus red plus red plus red plus red plus red." Next, ask the student to count the number of reds.  The student observes 10 reds. Tell students that 10 reds is another way to read the train.  Ask the student, “Which reading is easier, the first or the second?

The student, of course, will opt to read the shorter way and this is how to transition students into reading solid trains as multiplication.

There is more to be said for multiplication but that is for another post.

### General Ideas to Specific

Charlotte Mason, another educator I appreciate, believed it was better to move students from general ideas to specific ideas.  In this philosophy, Gattegno agrees.

Students rooted in these general ideas of numbers and operations become better able to manipulate numbers and symbols.

The general concepts of addition lay a solid foundation for students to move into subtraction or into multiplication.   When numbers are introduced, students understand the nature of numbers and use those as strategies for problem solving and memorizing math facts.

For a subtraction guide or introducing fractions guide, check out these posts.

A Subtraction Guide

A Fraction Guide

##### Lacy | Play Discover Learn 24/7

Knowing the best kind of learning comes from a highly motivated internal drive, Lacy Coker cultivates tools and resources that help to make learning for young children playful and self-directed.