I have always encouraged my students to model, model, model when solving story problems. Modeling is a way for students to make sense of a problem and catch mistakes before they happen. Sometimes, what seems like a good answer just doesn't work out once it's drawn as a model.
Until recently, I really didn't care what kind of model my kids used. Drawings, tally marks, symbols: anything was fair game. But as we've worked our way through multiplication and into division, I've found myself returning over and over again to the tape diagram. It's a nearly fail-proof way to work through a story problem and my students have become problem solving masters - the best I've ever had the pleasure to teach, in fact!
So, what exactly is a tape diagram? It looks like this...
To use a tape diagram, students must first ask, "Do I know the whole amount?" If supplied by the problem, fill it in. If not, put a question mark at the bottom of the diagram. Then look for other information and fill that in. The question mark always represents whatever piece of information is missing.
Students will quickly begin to see a pattern... the bottom number is always the product of the top two numbers. Therein lies the beauty of this model. 1) It can just as easily be used for multiplication problems and 2) it is self-correcting. If the top two numbers, when multiplied, do not equal the bottom, you've done something wrong.
When teaching with this model, a good idea is to have students write all four possible equations. So in the above example, we would also write ? x 4 = 32 and 32 ÷ ? = 4. This is important because it reinforces the concept of inverse operations and fact families. It also gives students a tool to use when the encounter problems with a missing factor or divisor.
Once students are able to interpret story problems and solve using tape diagrams, I teach them to analyze a tape diagram and write their own story problems to go with it. This requires high-order thinking and really develops their mathematical minds.
To use tape diagrams with your students, I suggest following this instructional sequence...
- Teacher supplies the story problem, draws the tape diagram, and models how to solve.
- Teacher supplies the story problem, draws the tape diagram, and students help solve.
- Teacher supplies the story problem, students help draw the diagram, and students solve on their own.
- Teacher supplies the story problem, students draw and solve alone.
- Teacher draws a tape diagram and students create a story problem to go with it.
Along the way, you will want to move from including one equation that represents the problem to showing all four possible equations. This process, from introduction to proficiency, may take several weeks depending on the skills of your students. But it will pay off in the long run when your students become experts at deciphering and solving story problems.
If you're interested in teaching this strategy to your students, you may want to check out this resource:
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