## Saturday, July 21, 2007

### M1N3 - Diagrams for Addition & Subtraction

[I just want to apologize for the sizes of some of the pictures - I am still trying to figure out how to do this better.]

M1N3. Students will add and subtract numbers less than 100 as well as understand and use the inverse relationship between addition and subtraction.

Problem 1
Cathy had some candies. She gave 5 to her brother, and she now has 7 candies left. How many candies did Cathy have at first?

Problems like this one is a very difficult one for some children. In the previous post, I discussed different meanings of addition and subtraction. According to the GPS, there are 3 situations in which addition and subtraction are used: combine, separate, and compare. This particular problem is a separate situation – a set (subtrahend) is separated from another set (minuend) to result in another set (difference). The take-away subtraction is used to find the number in the resulting set given the numbers for the first 2 sets, that is, (minuend) – (subtrahend) = (difference). However, in the problem above, what is not known is the minuend. To find the minuend, we have to add the subtrahend and the difference.

Similarly, a combine problems like the ones below require subtraction to find the answer.

Problem 2
Juan had some marbles. Salvador gave him 7 more and Juan now has 12 marbles. How many marbles did Juan have at first?

Problem 3
Kim bought 7 books with her allowance. Her grandmother gave her some more books for her birthday, and Kim now has 12 books altogether. How many books did Kim receive from her grandmother?

One thing you might notice is that these problems show that the “key words” do not work – in fact, those children who depend on key words are much more likely to miss these problems. Therefore, our instruction should not emphasize key words in word problems. Rather, what we want children to think about is how the quantities in a problem relate to each other.

One potentially useful tool to help children visualize the relationship among the quantities in a problem is to use a diagram. In many Japanese elementary mathematics textbooks, a linear model called “tape diagram” is often used. Using this diagram, the relationship among the quantities in Problem 3 may be represented like this:

From this diagram, we can tell that, in order to find the number of books Kim received from her grandmother, we will have to subtract 7 from 12. Can you draw tape diagrams for the other 2 problems?

So, how can we help our students make this diagram as their tool? We need keep in mind that it will take some time before children can make these representations as their thinking tools. However, it has to start some time toward the end of students’ initial study of addition. As we have students model addition (most likely combine problems) using manipulatives (counters), we can intentionally arrange them in a straight line. For example, suppose we are working on the following word problem: Cameron had 5 apples. His mom gave him 7 more. How many apples does Cameron have now? We can model this problem (5 + 7) this way,

At some point, we might even want to place boxes around the counters like this,

You may even want to label “Apples Cameron had at first” and “Apples Cameron’s mom gave him.” We can introduce similar diagrams as students study subtraction. In the context of take away subtraction, students will be introduced to the idea that an empty “tape” may stand for a number, which is not a trivial idea.

We should introduce a new diagram only after students are reasonably comfortable with the operation the diagram is supposed to represent. Japanese teachers believe that we teach a new concept using a familiar diagram and a new diagram with a familiar concept. We should not try to teach both a new concept and a new diagram simultaneously.

Here is a tape diagram template for a comparison problem:

I encourage you to represent problems you find in your textbooks until you feel comfortable with these tape diagram representations. Then, start thinking about how you might introduce the diagram to your students.