# Teaching Addition and Subtraction, Part One

After writing my last blog post, What’s the Difference?, I thought it would be helpful if I blogged about all the different addition and subtraction situations.

This post will talk about one of three addition and subtraction situations: the Change Plus/Change Minus situation. What I find fascinating is how researchers have found that Pre-K children can understand the concept of adding to and taking away from at the age of 3 or 4 (Clements and Sarama, 2004). But, when I think about it, my kids learned the sign language for and concept of “more” at 18 months when they were eating. And when I “took something away,” they knew that item wasn’t there any more.

What are the concepts kids need to understand? Kids need to understand three things: that a quantity can be represented by a number, the action of addition or subtraction, and the symbols “+” and “-.” Notice: I am not using the equal sign. THERE IS A REASON!! Too many students have the misconception that the equal sign means “the answer.” I would not show students an equal sign until they can understand quantitative sameness.

Let’s talk about a Change Plus/Change Minus situation. Here is a video explaining how to explore it using a number machine.

Seeing the pattern of Start, Change, and Result would represent the Mathematical Practice Standard 7 which is look for and make use of structure. When kids create fact families, they think it is magic that all the numbers can build different equations. BUT, it is the relationship of addition and subtraction that allows fact families to work. Understanding these situations will allow students to understand fact families and not just memorize them.

So as you explain it to your students, emphasize that whether we add or subtract, we always have a start, a change and a result. As your students see this, they will build a folder in their brain called addition and subtraction. Intuitively they will know there is a relationship. You can download the PDF worksheet below to help show your students what happens in the number machine.

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### 15 Responses to Teaching Addition and Subtraction, Part One

1. […] I’m going to use the concept of decomposing and composing as we journey from the concrete to the abstract. Decomposing and composing is new language in the common core, but the idea has been around in mathematics for a long time. In fact in Table 1 of the Common Core Standards they use the language of put together and take apart as one of three addition and subtraction situations. (If you want more info on those situations, check out my blog series on Teaching Addition and Subtraction.) […]

2. […] to help them solve multiplication problems. This is similar to when students use counting to solve Change Plus/Change Minus Situations in Addition and Subtraction. A simple way to build on students’ prior knowledge of counting is to count to solve a […]

3. […] Addition and Subtraction. Now I would like to help her see how three situations are in that file: Change Plus/Change Minus, Decomposing/Composing, and Comparison. (If you want to explore these categories in more depth, you […]

4. Gloria says:

With Common Core, I feel it is important that we understand what the standards truly mean and this blog as helped me see the importance!

• Beverly Ford says:

I so glad that it allowed you to see the importance. Addition and Subtraction is so foundational, and it is taught at the beginning of a students educational experience. Not everyone will love math, but everyone can understand it and have a positive disposition towards it! The connections across concepts in math are beautifully intertwined. A great foundation in addition and subtraction will give students something to connect with when they multiply, divide, and probably the most difficult, adding and subtracting integers and fractions.

5. […] blog post will be the final in a three-part series on teaching addition and subtraction. Part One talks about the Change Plus/Change Minus, and Part Two talks about Composing/Decomposing. Our last situation is […]

6. […] relationship to the two addends. Definitely a challenge for some students! In the last situation, Change Plus/Change Minus, students rely on their counting strategies to supply them with the result. Now they have to see a […]

7. Katie Schiyer says:

I agree… It amazes me to observe students as young as K look for ” the right answer” without trying to grasp the true manipulation of numbers. I have never thought of or heard of leaving out the equal sign…. Cannot wait to use this strategy! Thank you for this thoughtful, researched based idea. AIMS participants will benefit from this discussion and activity as well.

• Beverly Ford says:

Thanks, it is unfortunate it happens to students at such a young age, but I already see my daughter filling in the blanks without realizing what the equal sign really means. We just don’t realize that we are communicating this message because we forget that students build concepts based on what they EXPERIENCE and what we say. When most of the problems are formatted like this 2+5 = then students construct meaning for the symbol based on their experience. Therefore the meaning must be answer. Given that logical assumption, I especially think we need to stay away from an equal sign during this situation.

8. Barbara Gonzales says:

Love this model that ignores the equal sign! Places student focus on what is happening with the numbers and away from the “correct answer” to the problem.

• Beverly Ford says:

I think we are on the same page when it comes to “correct answers”. As a kid I was very anxious to get the answers and be done. I needed a teacher to raise the bar and expect me to understand why I got the correct answer. My experience at Fresno Pacific gets most of the credit for my content knowledge. It is so easy to forget the numbers are symbols of some important concepts. If we rush into the world of symbols, the kids forget what they mean. Since most of our tests don’t ask them what +, -, or = means, kids don’t think it is important.

9. Judy Leinius says:

I think the idea of not including the equal sign is powerful. As soon as the equal sign is included, teachers are quickly moving to algorithims for addition and subtraction, taking away to opportunity for young students to really conceptually understand what adding and subtracting is.

• Beverly Ford says:

Exactly, lets not give them too many symbols at a time to understand. We can focus on + and – and how they are connected. The equal sign is really misunderstood by most kids, which becomes a real challenge during Algebra. Richard Thiessen and I have had a few conversations about this. An activity or maybe a blog may come soon to develop the concept of the equal sign. I pinned a lego activity on Pinterest that I would tweak. You have a stack of 7 legos and a stack of 3 legos. Then ask, “How many more do I need to add to my stack of 3 to make 7?” You could put a NOT EQUAL symbol in between the two stacks, and after you make them equal change the symbol to an equal sign. I think it would be very concrete for young children.

10. What a wonderful concrete model for fact families. When my kids were younger, I watched them mindlessly fill in the blanks on fact family worksheets (they’d fill in all the 3’s, then the 4’s, etc.). The number machine makes this concept both tangible and fun; it also promotes algebraic thinking at an early age. Thanks for the idea!

• Beverly Ford says:

Thanks for the reminder of how this situation prepares students for algebra! I am highly motivated to create concrete ways for students to understand these essential math concepts. When I do homework with my kids it often gets me thinking of new ideas. Watching my kids do things robotically, drives me crazy as an educator.