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Modeling Addition and Subtraction Structures

I am finishing out the school year with my first and second graders by trying to give them a solid foundation with the different structures for addition and subtraction problems. I’ve been using a marvelous free resource called Thinking Blocks. They offer a variety of free iPad apps, but I’ve been using the web version on my interactive whiteboard because I wanted to bring in some concrete learning.

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The Thinking Blocks program allows you to choose the type of problem to work with, and we began with part/whole problems with two parts. I made part/whole mats for the kiddos to use (on the backside is a part/whole mat with three parts) because I wanted them to be able to act out the models. My first graders used cubes while my second graders used base-10 blocks to find their solutions.


The problems cover all the structures the students need to be familiar with. All too often, we only expose them to join problems where the parts are known and they are asked to find the sum. But they also need to be able to solve missing part problems. Thinking Blocks mixes up the problem types, which I really like.

For each problem, we went through a series of steps that I hope will become a habit for the students as they work on problems independently. First, we read the problem straight through. When we finished reading, I asked them to tell me what the story was about with one word (pizzas, shoes, pages, etc.). This helps to focus them on what we’re counting in the story. Next, I asked them to summarize the story. This was really tough! I was looking for them to just retell the story without numbers or details. For example: Maria had some marbles. She gave some to her brother and she kept some.

Next, the Thinking Blocks program asks students to place the labels in the model. I love that this is the first step. Too often, students focus only on the numbers and lose the meaning of the problem. We re-read each sentence and discussed if it was a part or the whole and a student placed the labels in the model. After the labels are placed, the program offers a check feature.

Now it’s time to add numbers to the model. Again, we re-read each sentence to make sure we got the numbers in the correct place on the model. A question mark represents the missing number. Once the numbers are placed, the program offers a check feature and also re-sizes the blocks. That allows for some great discussions about what the size of the block tells us about the missing number.

After we had the completed model in Thinking Blocks, I had the students show the problem on their part/whole mat and use the manipulatives to find the missing number (sometimes the whole and sometimes a part).

After working with 2- and 3-part part/whole problems, we tackled comparison problems. This time, the kiddos used two colors of cubes to build trains to match the model. We used blue cubes for the difference and yellow cubes for the two quantities being compared. Again, sometimes the difference is known and one of the quantities being compared is missing, and sometimes the difference is missing.


We are working to the point when we will mix all the problem types. We’ve had such great discussions, and I’m excited to see their mathematical understanding grow each day!

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  1. This is my focus for this summer to make students understand the structure of word problem. The pictures you shared are going to be very helpful. The website I am sharing actually gives a table for the different types of word problems. Thanks!

  2. My first graders are about to be taught comparison with our new district math program, which is heavily into passive, picture driven lessons, ugghhh. I love how the difference is shown in blue! In the past, for comparison, I represented the two groups being compared with respective different colors. This model you shared will allow for a visual to see the difference and will be readily incorporated into the part, part, whole idea.
    When I teach word problems, I constantly ask my first graders to name what they are counting…numbers are the adjectives, what they are counting are the nouns….so important for them to keep that focus!

  3. Hi Donna,
    I love thinking blocks too and I love using bar models with my kids. I’m a math interventionist grades 1-6. My question is: How do you sustain all the great strategies you’re using with your kids, like the work with Thinking Blocks, if they’re not also seeing it in their classroom?
    I see my kids for 30 minutes every other day. Go Math, the program my district uses, presents multiple strategies (sometimes within 1 lesson). Sometimes the strategies I feel are the most crucial for my population (like bar models or number lines) just become one of myriad others, and it confuses my kids more than helps them.
    How do you make that work for your kids?

  4. I also love Thinking Blocks! Another wonderful site I use as Math Coach is http://www.gregtangmath.com word problem generator. We can select the problem type (or choose random), choose what we want to be the unknown, and the number range. You can do one at a time or print out 10. Plus when you click on hint there is the tape diagram model that works for all add/sub problem types. Love it!

  5. You could screen shot the problems and post on an interactive white board or in a pwp. Wouldn’t have the utility of being in the program Thinking Blocks but it would be a work around.

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