# 10 Ways to Teach Subtraction using Cuisenaire Rods

Even if you don't use Cuisenaire rods to teach subtraction, these language tips will set you up for success. In this guide of 10 Ways to Teach Subtraction with Cuisenaire, I lay a clear path for teaching subtraction that even makes for an easy transition into division later.

I use simple progressive language, meaningful tasks and playful context that helps move students from the concrete to the abstract.

Language and tasks are two areas that offer flexibility in teaching subtraction to students. In this guide, I use those two areas to provide a variety of ways to teach subtraction with Cuisenaire Rods. This variety provides deep conceptual understanding of subtraction and its relationship to addition and division.

Before we look at this progression for teaching subtraction with Cuisenaire rods, I need to explain why I use rod color names versus the number names.

### General to Specific

There is one reason to use color names: to give students room to observe and focus on general concepts about subtraction to deepen conceptual understanding.

A student steeped in conceptual understanding of broad math concepts is able to streamline math facts and gains a strong foundation for division and fractions. Basically, it prepares them to delve into more specific relationships like math facts with greater ease.

The following tasks provide ample opportunity for students to explore general concepts such as the relationship between addition and subtraction and the shared composition and decomposition of numbers.

However, the language and tasks below convert easily to numbers, so no matter which way you choose to teach, this guide is still useful.

Let's dive into the language of teaching subtraction with Cuisenaire rods.

**Language of Subtraction **

Subtraction is just take-away, right? Take away is the common language used when presenting subtraction to young children. It is easy default to this language because we can easily provide context for the student to see and observe the" taking away" of objects from a group.

I do not staunchly oppose this method. However, there are other ways to present subtraction that provide students with context for general mathematical ideas.

The following tasks connect students to the relationship between addition and subtraction. There are two key pieces of language to present this idea, missing addend and the difference between numbers.

**Missing Addend**

When teaching subtraction with Cuisenaire Rods, the focus is on length. It is the missing length that makes train one equivalent to train two. It is the missing addend.

Set up your Cuisenaire Rods with two trains side by side. "Train one" is two rods long and "train two" is one rod long equal in length to "train one." Read the trains to the student. For instance, red plus dark green is the same length as tan.

Then remove the dark green. Ask the child which rod has been removed. After the student responds, read the trains. Red plus “the missing rod” is the same length as tan. Ask the student, “What is that missing rod?”

Repeat this exercise a few times or over a course of a week before introducing the next task. There are several variations of this exercise below to keep this exercise interesting. Come back to this task even after introducing the subsequent tasks.

For more missing rod activities, check out Math Task Cards and Play Mats for Module 1.

**How much more? **

The next step is to introduce the language of “how much more?” Set up two rods of different lengths side by side (not end to end). Ask the student, “how much more do I need to make the smaller length the same as the larger length?”

Repeat this activity changing out the rods. Let students choose the rods for themselves. **Choice is a key factor in internal drive and puts responsibility on the student to take charge of their education.**

The language of “how much more” with the use of the Cuisenaire rods will give them added context to understand the relationship between subtraction and addition. While we are emphasizing the missing addend, the student becomes aware that the missing rod can be described in more than one way.

**What’s the difference?**

The difference is the next way to teach subtraction using Cuisenaire Rods. The task remains the same as before, but the language changes. Set up two rods of different lengths side by side (not end to end).

Tell the student the task today is to find the difference between the two rods. What is the difference in the two lengths? Explain to the student that the difference is the space between the larger rod and the smaller rod that could be filled with a missing rod.

When the student finds the rods, describe the rod. The dark green rod is the difference of between tan and red. As you repeat the task, change the language every so often. The difference between tan and red is dark green. The language is moving the equal sign around to develop a fluent understanding of mathematical notation. Read more about it here.

**The Difference of Language**

The language of difference builds upon the context and language of “how much more.” However, this time the language changes the mathematical notation. The previous language (missing rod and how much more) emphasizes addition (2 + __ = 8).

Looking for the difference paves the way for the written notation of subtraction. (t – r = d or 8 – 2 = 6) The difference of tan and red is dark green. A predictor for mathematical success lies in reading meaning into mathematical notation.

Using the task of missing addend but changing the language helps students see the connection between addition and subtraction. Students use that general idea to streamline the memorization of math facts.

For those who dread the kill and drill method, you are right to dread it. It is inefficient use of energy to memorize all those facts. A student with a strong understanding of important mathematical relationships have less to memorize.

Stand-alone facts are simply harder to maintain. Whereas a conceptual understanding steeped in interrelated ideas creates more connections in the mind providing not only for memorization of math facts, but deeper number fluency.

### Subtraction with Cuisenaire Rods

Once the student perceives the relationship between subtraction and addition, the student is ready to add the word subtraction to their vocabulary. The teacher may introduce it as take away and then add the terminology of subtraction and minus.

Ask the student to build a two rod train equivalent to black. If the student needs the black rod to ensure their train is equivalent, let them have it. However, remove it once they are sure. Then read the train, "Yellow plus red equals the length of black." Then remove (subtract) a red and read, "The length of black minus (subtract/take away) a red equals a yellow."

Repeat this activity. Then let the student create their own subtraction sentences.

### Subtraction to Division

Okay, I have one more language tip for teaching subtraction with Cuisenaire rods. Often you hear of repeated addition, but rarely do you hear repeated subtraction. Yet, this piece of language paves the way for division.

Set up the task with one orange rod. Ask the student, "how many times can a light green be subtracted from the orange rod?" The student uses the rods to see. The student may build a train underneath or overlay the orange rod with as many light greens that fit. When the student is finished, ask, "What is left over?" or "What rod fits in the remaining space?"

With a hands-on manipulative like Cuisenaire rods and the right language, it is easy to transition from understanding the relationship between addition and subtraction to understanding the relationship between subtraction and division.

Progression in language and tasks makes this simple. True fluency though means a student transverses between the language and tasks with ease. Don't be afraid to go back and forth within the progression. This will help the student to deepen their fluency and understanding.

Let’s look at a variety of meaningful tasks and playful context for the simple language of subtraction.

**The Missing Chicks**

Okay, so I may be a fan of Peg plus Cat and those cute little chicks. In this exercise, the students start with staircases and find the missing rods to complete the staircase to reach those mischievous chicks.

The language for this task is the missing addend, but we layer in a story to capture student imagination.

Story is a great bridge for taking the concrete presentation of the rods and moving the student to the abstract. Through narration, the student discovers that finding that right size Lego to fill in their latest wall to their new hobbit hole is a form of subtraction and addition.

You can use both the language of the missing rod or “how much more” interchangeable for this activity.

This activity is in Math Task Cards and Play Mats for Module 1.

**Lock and Key **

The mind is hardwired for story. This activity connects student’s hunger for story with subtraction. Students build locks and the find a key to unlock the lock.

The lock is composed of 3 rods. Two rods are the same length. One is the top of the lock and the other is the bottom. The middle rod is shorter these two other rods. The key fits inside to make the shorter rod of the lock the length of the rest of the lock.

The language used is the missing addend. It’s a great alternative to missing chicks and simple missing addend tasks. Read more about this activity here.

Pick up PDL's Lock and Key Subtraction activity HERE.

**Exploring and Building Staircases**

You can also explore the language of difference through staircases. Task students to build staircases with difference of a red rod. This emphasizes that the difference of two is found amongst many numbers.

Students note that both 8 – 6 and 9 – 7 have a difference of two. The student realizes subtraction points to a relationship between numbers and that more than one number has this relationship. Understanding common difference prepares the student to manipulate numbers to make difficult problems easier to solve.

Mathematical competency lies in fluency. Deep fluency relies on students perceiving these general ideas shared amongst numbers. Staircases emphasize the shared relationship of common difference amongst number more than any other task.

For more staircase activities that explore difference, check out PDL’s Staircase Task cards. For more staircase activities, check out this post here.

**Structure Studies**

Cuisenaire rods naturally make squares and rectangular shapes. Students enjoy variety. Structure studies provide this variety.

Task students with building 2 rectangles using Cuisenaire Rods. Explore the difference of those structures. Uncover which rods are needed to make the smaller structure the same as the first.

I encourage layering structures on top of one another to compare difference whenever possible. Other times it might not be possible.

Working back and forth between the language of difference and missing addend solidifies that relationship between addition and subtraction. Don’t be afraid to change up the language.

For more structure studies, check out my Math Journal pages. There are a lot of opportunity to explore subtraction with Cuisenaire rods and these pages give variety and context for exploration.

**A Side Note on the Benefits of Laziness**

Large structure comparisons are great for moving kids to the abstract, so let students build BIG! It can be tedious and cumbersome to manipulate the rods for finding differences.

This forces the student to record their structures using symbols and numbers. Let students figure out on their own how to order the information.

If students struggle with this, remind them to use what they know. If they lack confidence, remind the student about a similar instance. Maybe narrate that event to them to see if it reminds them of anything that maybe useful in ordering the information. Or if you are keeping a math journal, flip open to that page.

An important note about the brain: it is mass consumer of energy. Therefore, it is always working towards efficiency. This sometimes appears to be laziness. However, this desire to do less work is an excellent quality if harnessed for efficiency.

It means that your student’s work appears at first to take the longest possible way. Overtime, that student looks for opportunities to do the problem in a way that requires less effort. Trust the laziness of your student to work towards efficiency.

It isn’t necessary to hand every algorithm to the student from the beginning. Their laziness leads to their own algorithms. Students have a deeper conceptual understanding of algorithms that they create for themselves.

And if you do choose to eventually show them an algorithm, their previous struggles give them a stronger appreciation for it. So at least let them struggle for a little bit.

**Measure and Compare**

Another activity for teaching subtraction with Cuisenaire rods is connecting the student’s world to math through measuring. Students find their world interesting. Use Cuisenaire rods to measure their interesting world.

There is no need for me to inspire you. Just ask your students to measure for the length of whatever they like with the rods. A toy, a book, a chair, a wall, maps, each other…. the ideas are endless.

Don’t forget to get out that balance scale. How many Cuisenaire rods do you need to make the scale balance with Larry Boy? Record results and explore the differences.

**Living Mathematics**

All these activities develop curiosity in students for the math that is available in the world around them. Cuisenaire rods, stories and simple tasks make math a tangible and useful language to explore and manipulate.

Math becomes a conversation worth having and this approach is vital to cultivating a lifelong love for math. Providing plain language and enjoyable context paves the way for students to find mathematics useful and interesting.

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