This year, our school has chosen to work on addition and subtraction word problem types in our grades 3-5. You may think that it seems strange to work on add/sub in the upper grades, but I assure you it is quite a complicated endeavor. Read my previous post on the various problem types here: Word problems with addition and subtraction are easy, right?
Our initiative exposes all of our students to the variety of addition/subtraction word problem types (then we’ll go on to mult/div) as based on the research in the book Cognitively Guided Instruction (and described in the CCSS) and have used the tape diagram as the way to model these situations. We have followed the rectangular tape diagram as found on Greg Tang’s math website within his word problem generator program. (If you haven’t checked out this part of his site, I would encourage you to do so. You can select the problem type, the unknown, and the number range. Awesome resource to create differentiated Guided Math workstations). There is a large rectangle on top and beneath it are two smaller boxes whose size equals the large rectangle and hence the values of the two smaller boxes must equal the value in the top box. I find it an incredibly powerful model that allows us to teach many mathematical concepts. Here is an example from the site at http://www.gregtangmath.com.
First, the tape diagram provides us a visual model to teach the relationship between addition and subtraction since the values of both boxes (and eventually more when there are more addends introduced) must equal the value of the large box. I love using Cuisenaire rods to model this using numbers within 10 – even for our upper elementary students since our goal initially is not about the computational skills to get the numerical answer, but the underlying structure of the situation. If the top box is unknown, students will need to add the two smaller boxes. If one of the smaller boxes is unknown students have a choice of adding OR subtracting. There is no dictate in terms of how they can find the numerical answer. The problem may be an “Add to” problem type, but they could use subtraction to solve it if one of the addends is unknown.
Secondly, the tape diagram makes complicated-sounding word problems much more manageable…
For example, there were 5 more sheep than cows on a farm. There were 7 sheep. How many cows were there?
Before my work with CGI and the tape diagramming it would take me a bit to figure out exactly what is going on. Now, though, I know that there is a bigger amount and a smaller amount being compared as well as a difference between those two. I actually see the tape diagram in my head. After I read the first sentence, I know one of the smaller boxes is the difference of 5 and I can even label the small box cows the larger box sheep since I know there are more sheep than cows. Once I read that there are 7 sheep, I would put that value in the big box and then be left to my own decision as to how I want to compute the answer of how many cows there are. For example, I may want to use 7-5 or I may want to think 5 + ? = 7. That’s where the student choice comes in. When numbers become larger, students would then choose if they want to use an open numberline, act out with blocks, use a 100’s chart, draw a picture, etc. Total freedom to figure out the numerical answer, yet the model is the same.
Thirdly, these problem types will follow our students throughout many years of their math journey. It’s just that the numbers will get larger or become fractions and decimals. Yet the underlying structure is always there. If we teach the tape diagram from the earliest years, I feel like we are providing them with a tool in understanding the structure for years to come. They will then bring to the table their own inventive strategies for calculating the answers.
Finally, having learned this particular model for the add/sub problem types, students will have developed the mathematical thinking to find answers where they may not need to find a numerical answer at all. I’ve seen many questions on our district standardized testing measure NWEA as well as SBAC where it gives the students the situation and then asks them to choose an equation that could be used to solve it. There’s no number answer at all. Inherent in the given equation choices are understandings of the commutative property, the equal sign as meaning “the same as”, the inverse relationships between addition and subtraction, etc. All powerful foundational mathematics concepts we are able to discuss while working on this…and the exact same model is used for all the add/sub problem types!
So, while I’m not in favor usually of mandating things, we are mandating the model including labeling the diagrams with words from the word problem, but in no way are we mandating the computational methods. In fact, in many instances we don’t find the answers at all. We have students determine which equations, of many given, could be used to solve the problem. We put the answers on the left side of the equal sign, we change the order of the addends, we have some with the inverse operation, and we even have some with the smaller number minus the larger number to discuss that we COULD do that math but we’d end up in negative numbers and not find the answer we are looking for (we live in a cold area so weather below zero is very familiar to them). Students are doing a lot of discussing about why given equations will solve the situation as well as why some will not. I get the chills listening to them support their answers.
I have struggled with the idea of “mandating” as well, but I feel like the tape diagram specifically provides such rich mathematical concepts that it is worth it. I’m not seeing nearly as much as I used to of students mindlessly plucking numbers from word problems and finding a key word that would suggest an operation with no thought to the context in the problem. I think that right there is a win.
I’m hoping that in this process, not only are we setting the foundation of the structure of addition/subtraction word problems but we are encouraging the thinking process needed to diagram word problem situations that we can then apply to more difficult problem types they will encounter down the road like multiplication and division of whole numbers, decimals and fractions.