Thursday, April 1, 2010

M*P5: Tape diagrams

M*P5. Students will represent mathematics in multiple ways.

In the recently released draft of the Common Core Standards, there is a noticeable emphasis on linear models such as number lines. I have discussed how many Japanese textbooks use double number lines to discuss multiplication and division of fractions, as well as some proportional problems. However, the Common Core Standards also include a model that is called "tape diagram." In their glossary, "tape diagrams" is defined as follows:
Drawings that look like a segment of tape, used to illustrate number relationships. Also known as strip diagrams, bar models or graphs, fraction strips, or length models."
In an earlier post, I discussed how a tape diagram may help children represent addition and subtraction situations. The primary purpose of such diagrams is to help students decide the appropriate operation, that is addition or subtraction. However, Japanese textbooks also use tape diagrams, or segment diagrams, to deal with problems in upper grades, too.

Consider a problem like this one:
A fifth grade class counted the number of cars that went by the front entrance of the school between 9 o'clock and 10 o'clock. The total number of cars counted were 156. There were 3 times as many passenger cars as trucks. How many passenger cars and how many trucks were counted?
For this problem, you can use a diagram like the following:

From this diagram, we can see that the total number of cars are made up of 4 equal segments, one of which is equal to the number of trucks and the other three are equal to the number of cars. Since the four segments are equal, we can divide 156 by 4 to find out how many cars each segment represent.

Here is another problem:
There are 3.5 times as many fifth graders at School A as School B. There are 115 more students at School A than at School B. How many students are there at School A and at School B?
This problem can be represented as follows:

From this diagram, students can determine that 115 is made up of 5 equal segments since the last short segment is a half of the other segments, each of which is equal to the number of students at School B. So, 115÷5=23 represents a half of School B. Thus, the number of students at School B is 46 students. The number of students at School A is 23x7=161.

You might notice that these problems can be easily solved if we use algebra, but having diagrams such as tape/segment diagrams, students can develop the foundation for solving these problems algebraically.

There are other types of problems for which tape/segment diagram may be useful. Consider this problem:
At Jimmy's school, there were 475 students last year. This year, there are 24% fewer students. How many students are at Jimmy's school this year?
You can represent this problem using a tape/segment diagram like this:

From this diagram, we can tell that the number of students this year should be 100-24=76% of last year's student population, 475. Thus, we can find the answer by multiplying 475 by 0.76. Alternately, we can subtract 475x0.24 from 475, too. [Click here for a discussion on how double number lines may be used with problems involving percents.]

What we need to keep in mind about these representations is that they are supposed to be students' thinking tools, not just teachers' explanation tools. In order to help make these representations as their own thinking tools, these representations have to be carefully taught. In the Japanese textbooks, they start building these linear models starting in Grade 2 and help students experience increasingly more complicated representations gradually and systematically. I believe the emphasis on linear models in the Common Core Standards is important, but just showing these models to students will not automatically produce positive results.

Nigel Nisbet said...

thanks for this. I was reading the Common Core standards and also came across these descriptors and while I could make some educated guesses about what the double number line was referring to, I was not sure what the "Tape diagram" was.

Your blog helped me nail this down (basically it's a schema building tool, visual modeling etc.)

I think your last comment is extremely well made. These are primarily designed to students build their own methods to "get inside" real world problems, not just another tool for teachers to tell them how to do it. Must get the kids doing this!!!

I think we should always keep thinking about how to help students make diagrams they draw into their own thinking tools.

Anonymous said...

I found this very informative and useful. I too wanted a clarification of the term "Tape diagram". I think there is a misprint in the question about the two schools. School A should have 161 students.

Thanks for pointing out the error.

Dick Stanley said...

I like the use of tape diagrams you illustrate. I like especially the cases where the strip is divided into two parts by a horizontal line. The top part and the bottom part are different representations of the same quantity. This illustrates nicely Polya's succinct statement from his book Mathematical Discovery 1:
"In order to obtain an equation we have to express the same quantity in two different ways".

Anonymous said...

I saw this in the common core standards too and I could not find any other explanations for this concept. Thank you so much for the detailed discriptions. I appreciate you sharing.

WCanfield said...

Anyone know the history of using bar/tape diagrams?

Ward Canfield
Assoc. Professor
Dept. of Mathematics
National Louis University

Anonymous said...

I would love to know how you set up the : "School A School B" problem using algebra.

Probably a typical "algebra" solution goes something like this:

Let x be the number of students at School B.

Then, the number of students at School A is 3.5x.

The difference, then, is 3.5x - x = 2.5x. But, this is also equal to 115. Therefore, 2.5x = 115. Solving this equation for x, we get x = 46.

Therefore, the number of students at School B is 46, and the number of students at School A is 161.