There’s a lot to be learned from the data that comes from state testing if we use it correctly. Oftentimes, we’ll take a problem that students, as a whole, perform poorly on and we create a bunch of problems just like that one and “teach” them how to work that type of problem. In the end, that’s not an effective strategy, because they’ll probably never see a problem exactly like that one again. A more effective way to use the data is to analyze the wrong answers to determine underlying misconceptions that resulted in the wrong answers and provide students with strategies and tools to improve their overall mathematical reasoning. As far as strategies go, drawing strip diagrams is one of the most powerful strategies students can have in their toolbox. And I have some super clear examples to persuade you!

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The test items in this post come from the 2021 State of Texas Assessments of Academic Readiness (STAAR) test. A wonderful organization called lead4ward analyzes the test each year and provides error analysis statistics.

Analyzing errors

Let’s get started! As you can see from this test item, only roughly half of the students in the state got the correct answer. Look at the most common wrong answer, choice H. Can you see the error that the students who chose that answer made? Take a minute to figure it out before you scroll down.

Look closely at the order of the numbers in the problem. The smaller number comes first in the problem. Students who chose H realized it was a subtraction problem, but took the numbers in order from the problem and subtracted them. So they did 379 – 514. Nine minus 4 is 5. Seven minus 1 is 6. They couldn’t do 3 minus 5, so they did 5 minus 3. The answer they got was 265. Your first thought might be, I need to make sure my students always know to subtract the smaller number from the bigger number. Except that’s not true. If the temperature is 18 degrees and the temperature drops 29 degrees… We need to be extremely careful to not teach “rules” that expire.

The students who chose H lacked an understanding of the meaning of the numbers in the problem. The way we help them be more successful with problems like this is to give them tools to improve their comprehension of word problems and the numbers they contain. Enter strip diagrams.

Using drawings to describe problems

Drawing strip diagrams is a process that actually begins in Kindergarten and 1st grade when students should be drawing pictures to represent word problems. Students should understand that the numbers in a problem represent something—pizzas, soccer players, apples, money saved—and their drawings should include labels identifying what the numbers represent. At this point, the drawings don’t need to resemble strip diagrams. What’s important is the labeling. It could look something like what you see here.

Notice a couple of things. First, the boys and girls are represented by circles. Easy to draw and count. Students need to understand that these are math pictures, not art pictures. Next, and of critical importance, are the labels.

Beginning in 2nd grade, students can begin to draw more formal strip diagrams. Strip diagrams, also called tape diagrams, are often associated with Singapore Math. Char Forsten’s Step-by-Step Model Drawing is the book I learned model drawing from. Another great resource is Math Playground’s Thinking Blocks.

Modeling how to draw a strip diagram

Now let’s get back to that released test item I started this post with and see what the model-drawing process might look like.

This problem is a comparison subtraction problem. We always want students to draw the model with labels first. they will add the numbers in the next step.

Teacher: [Reads problem out loud] What is this story about? (lions) How many lions? (2) What does the problem tell us about the lions? (their weight) Do we know their weights? (yes) Which lion weighs more? (the older lion) What is the problem asking us to find? (the difference in their weights) Huh, what does that mean? (The older lion weighs more than the younger lion. The problem is asking how much more.)

NOTE: Notice that we didn’t talk about the numbers at all! The point of this discussion is to help students make sense of the numbers in the problem and verbalize what the problem is asking them to find.

Teacher: Drawing a model really helps me understand what math I need to do to solve a problem. Let’s draw a model to represent this problem. We know that the older lion weights more, so his bar should be longer, right? [draws and labels the older lion’s bar] That means the younger lion’s bar should be shorter. [draws and labels younger lion’s bar] And you guys told me the problem is asking for the difference. [adds the difference with a question mark]

Now we can plug in the numbers from the problem.

Notice that what I’ve described is very scripted. I want students to hear my mathematical thinking, and I’m teaching them the mechanics of drawing the model. But it’s important to let students use the tool to solve problems. Think how the models would look for these variations of the problem:

There are two lions at the zoo. The weight of the younger lion is 379 pounds. That’s 135 less than the weight of the older lion. How much does the older lion weigh?

There are two lions at the zoo. The weight of the older lion is 514. That’s 135 more than the weight of the younger lion. How much does the younger lion weigh?

Remember, our goal is for students to be able to use strip diagrams to solve new types of problems, so once they understand and can use the model, we have to give them new types of problems to solve without scripted instruction.

More examples of strip diagrams

Let’s take a look at a few more problems from the same test. Each of these problems had pretty dismal results.

This first problem is what we in Texas call a gridable. That means it’s not multiple choice—students have to write and bubble in their answers. A lot of times students will miss gridables due to calculation errors. But I’m pretty sure that’s not the case here. I doubt they miscalculated 4 x 5. What that means is that 38% of the 3rd-grade students in Texas did not recognize this as a multiplication problem. They likely added 4 + 5. Teaching keywords could be the culprit. Students see the word total in the problem, and they’ve been taught that total means addition. Teaching keywords basically gives students permission to not read and understand the problem—just find the keyword and plug the numbers into the operation. Not a sound problem-solving strategy. Instead, we see how a strip diagram could be drawn to represent the problem.

Here’s another multiplication problem. You can see from the error analysis that only 52% of the 3rd-graders correctly answered this problem. Answers B and D are calculation errors. Can you figure out the error these students made? Doing so can help you prevent these types of errors by addressing them in your instruction. Answer choice C results from adding the two numbers, not multiplying. Again, we see the keyword total in the question. Drawing a model would not only help the students visualize the problem as multiplication, but it might also prevent calculation errors. Students who are not confident with the standard algorithm could solve the problem with repeated addition.

Last one and it’s a doozy! Look at that error distribution. When it’s spread out like that, it usually means the kids just didn’t have a clue and guessed. There’s a lot going on here. How could we help students tackle a problem like this?

First, of course, is drawing a model. We see that this is a part/whole problem with three parts, one of which is missing. If you looked carefully at the wrong answers, answer choice F was adding all three numbers. Pretty hard to look at this model and think you’re supposed to add all three numbers.

Aside from model drawing, however, students should learn to write equations to match their models. That’s really the other thing that was hard about this problem. They weren’t asked to solve the problem, just find the correct way to solve it.

Final thoughts

Strip diagrams have to be presented to students as a problem-solving tool and they have to be used consistently. Yes, it takes longer for students to draw strip diagrams to represent their problems, but it should be an expectation. That means we probably need to assign fewer problems to allow students the time to draw their strip diagrams.

The labels are a must! If students can’t label their numbers, it’s a huge red flag. Work with those students in small groups to help them develop comprehension skills.

Students who say they know the answer without drawing a strip diagram should be reminded that we draw models when the problems are easy so we can use them as a problem-solving tool when the problems get harder. And if a large percentage of students can really solve the problems without drawing models are we challenging them enough?

So there you have it. Have I sold you on having students draw strip diagrams? Do you have tips of your own to share? I hope you’ll sound off in the comments.