# Using tape diagrams to solve fraction problems

Read the following problem and think about how a student might solve it.

There are 250 students in the fifth grade.  Three-fifths of them speak two languages.  How many speak two languages?

As I’ve mentioned before, I’ve been introducing tape diagrams, or bar models, to students and teachers recently as a way to solve word problems. These models have been popular in Singapore and Japan and are slowly working their way into some US programs. Essentially, the models use bars of different lengths to represent the size and relationships between different quantities.

Students who worked with tape diagrams previously solved the problem in the following way:

First, they drew a bar to represent the 250 students.

Next, they partitioned it into 5 equal parts to represent the fifths.

Then, they tried to figure out what number would go in each box. What was interesting at this stage was that students used different strategies depending on their abilities. Students who were fluent with division immediately divided 250 by 5. Students who were less comfortable with division tried to find what number times 5 would equal 250. They used guess and check until they arrived at 50. Other students used repeated addition and tried to find what number could be added five times to equal 250.

They then added the number to the diagram:

Finally they either multiplied 50 times 3 or added 50+50+50 until they arrived at the answer.

What’s neat about using this model is that students can approach the problem at different levels. Lower level students can use guess and check or repeated addition until they build up their multiplication and division skills.

Furthermore, I think it helps students make sense of the problem and the mathematical operations they are using to solve it. After enough practice drawing the models, students can be asked to visualize what they would draw in their head.  This will help them to focus on what operations they are performing. It can also help them generalize about what they are doing to solve these types of problems.

Eventually, students will be able to solve the problems without drawing the models. However, starting with the models allows them to develop meaning for why they divide the set by the denominator and multiply by the numerator.

I’ll talk more about what the research says about these models next time, but I’d love to hear what your experience has been.

# “But I learned it that way”: The case for student-invented strategies

Recently, I have been receiving a lot of push back from parents and teaches who don’t think it is worthwhile to expose students to methods other than traditional algorithms. They tell me how they memorized the procedures in math class and they don’t understand why their students or children can’t do the same. They complain about all these diagrams I encourage students to use and don’t understand why students should invent their own strategies to solve problems.

I held this belief when I first started teaching as well. However, I soon realized that my students weren’t learning math by memorizing procedures. Maybe they could do the problems in class that day, but ask them to do a mixed set of problems a couple weeks later and they were lost. When I began to encourage students to invent their own strategies, two things happened:  1. They got the problems right more often. 2. They started to like math better and become more confident.

Last week, I was working with 4th graders on the following problem:

Ms. Placa bought 15 packets of pencils.  Each packet had 12 pencils in it.  How many pencils did she buy?

Students were able to represent the problem with a diagram or explain why they knew they needed to multiply 15 times 12.  Awesome, right? Yes. Until they tried to multiply 15 times 12 using the traditional algorithm. Over half the class got the wrong answer.

I asked them whether 30 was a reasonable answer. They knew it wasn’t, but kept going back to the algorithm. Then, one student pointed to the picture she drew–15 boxes with the number 12 written in each one. She suggested counting by 2’s 15 times and then counting by 10’s 15 times and then adding them together. She could even explain to the class why that worked. Later, I learned she was one of the “weaker” math students in the class.

I tell the story because it is amazing what students can do when we allow them to invent strategies that make sense to them.

The research supports my experience.  The benefits of student-invented strategies include:

1. Students make fewer errors than when using standard algorithms that they do not understand.

2. There is less of a need for remediation later on if students understand what they are doing and make connections on their own.

3. In many cases, using an invented strategy can be faster than using a standard algorithm.

Standard algorithms that are taught with meaning can have their place as well, but I think the case for allowing students to make sense of math needs to be made as well.

Want to know more? Check out:

Fuson, K.C. (2003). Developing mathematical power in whole number operations. In J. Kilpatrick, WG Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 68-94). Reston, VA: NCTM

Gravemeijer, K., & van Galen, F. (2003). Facts and algorithms as products of students’ own
mathematical activity. In J. Kilpatrick, WG Martin, & D. Schifter (Eds.), A research companion to
principles and standards for school mathematics (pp. 114-122). Reston, VA: NCTM

# Solving Multi-Step Word Problems

Recently, I was working with fourth-graders on the following word problem from the 2013 New York State test:

Ms. Turner drove 825 miles in March. She drove 3 times as many miles in March as she did in January. She drove 4 times as many miles in February as she did in January. What was the total number of miles Ms. Turner drove in February?

Let’s put aside for a minute what we think about all of this testing or the common core state standards. Let’s ignore the issue of whether or not this is a “real-world” problem or whether solving these types of problems will make students career or college ready.

Instead, let’s pretend our goal is to help students solve these types of multi-step word problems. What would you do?

When I pose that question to teachers, I often hear them tell me how they would break down the steps and explain each one to the student. That’s one strategy, but I’m going to guess that if the student sees a problem that is a little bit different, they are going to be at a loss as to what to do.

One thing I try to do when I am approaching problems like this with students is to make the numbers a little friendlier at first so I can see if the students are getting stuck because they don’t know how to think about the problem or because they are struggling with the computation. In this case, I changed 825 to 900.

Another thing I do is to encourage them to use diagrams. This problem is a great example of why I think bar models are a good tool for students.

Here’s how a student used a bar model to solve this problem.

I think that’s a pretty neat way to solve the problem. Of course, drawing and using this type of diagram is not spontaneous. Students need to have exposure to it and understand how to use it. But once they have it as a tool, they can use it to model a variety of situations.

We can talk more about how to introduce these types of models at another time and we can also talk about what the limitations are, but first I’d love to know how you would approach this problem with your students.

# One Strategy for Attacking Word Problems

They know the math but they can’t do word problems.

They can do all the problems on the page correctly, but they skip the word problems at the end.

It’s a reading comprehension problem. They don’t understand what they are reading.

The dreaded word problem. In many classes I taught, students weren’t used to being asked to read in math class and they certainly weren’t used to thinking about what they were supposed to do. Math was about following a set of procedures and getting the right answer. Math wasn’t about thinking about what types of calculations might help you make sense of a situation.

Early in my career, I fell into the trap of teaching them to circle the key words.  That became a problem very quickly.

Then I tried teaching them Polya’s four step model: 1. Understand the problem 2. Make a plan 3. Carry out the plan 4. Look back

This didn’t help either.  If students managed to understand the problem and explain it in their own words, they didn’t know how to make a plan.

What finally worked, particularly with my struggling students, was encouraging them to use pictures. At first, students created elaborate drawings of the situations that often focused on superficial aspects of the story. Then, I guided them to use simple diagrams to represent the story– diagrams that helped organize the information given and what was missing. I found that after a bit of practice, students were able to solve a variety of word problems, including complicated multi-step ones, with a picture.

Recently, I’ve been working with teachers and students on drawing diagrams to solve word problem and I figured it would make sense to go to the research and see if there was evidence that supported my experiences.  It turns out there is.

The benefits of using diagrams for story problems include:

• Reducing the memory demands
• Assisting in unpacking the situation
• Helping identify important information
• Focusing students on the quantities involved in the situation and the relationships between them
•  Flexibility  Diagrams can be used across grade levels and for solving both routine and non-routine problems

It is important to mention that these studies don’t encourage the use of any pictorial representation, but rather ones that highlight the structure of the problem. Bar models or part-whole diagrams are some of the representations I’ve found successful.

I’ll talk more about these particular models later, but I’d love to know about your experience with word problems. What have you found to be successful?

Want to know more? Check out:

Diezmann, C., and L. English. (2001). Promoting the use of diagrams as tools for thinking. In A.A. Cuoco and F. R. Curcio (Eds.), The role of representation in school mathematics. Reston, VA: National Council of Teachers of Mathematics, 77-89.

Yancey, A. V., C. S. Thompson, and J. S. Yancey. (1989). Children must learn to draw diagrams. Arithmetic Teacher, 36 (7), 15–23.

Teaching Mathematics to Middle School Students with Learning Difficulties by Marjorie Montague and Asha K. Jitendra