The ability to solve word problems ranks high on any math teacher’s list of goals. How can I teach my students to reason their way through math problems? I must help my students develop the ability to translate “real world” situations into mathematical language.

In a previous post, I analyzed two problem-solving tools we can teach our students: *algebra* and *bar diagrams*. These tools help our students organize the information in a word problem and translate it into a mathematical calculation.

Now I want to demonstrate these problem-solving tools in action with a series of 2nd grade problems, based on the Singapore Primary Math series, level 2A. ~~For your reading pleasure, I have translated the problems into the universe of one of our family’s favorite read-aloud books, Mr. Popper’s Penguins.~~

UPDATE: Problems have been genericized to avoid copyright issues.

## Teaching Tips

**Teaching algebra**

When using algebra with young children, keep the abstraction to a minimum. Do not introduce generic variables like *x* and *y*. Instead, use significant words from the story, like the names of the characters or their initials, or use words like *Total *and *Leftover *that name the relationship between quantities.

And when you write or read an equation, emphasize the connection between the math and the story by *saying* the whole word, even if all you write is the initial.

**Teaching block diagrams**

Bar diagrams (also called “models”) are normally drawn as long rectangles — imagine Lego blocks or Cuisenaire rods. Numbers or words may be written inside to label each block. When introducing the diagrams, help your students recognize the meaning of the bar by saying, “Let’s imagine all the books/fish/ snowballs set out in a row…”

If your student has trouble figuring out where the numbers go in the diagram, you might ask, “Which is the big amount, the whole thing? What are the parts it is made of?”

## Try These Problems

We read 34 books about Antarctica. Then we read 5 books about penguins. How many books did we read in all?

**Using algebra**

We will let the name of each topic stand for the number of books we read about that topic.

*Antarctica *= 34

*Penguin *= 5

*Total *= *Antarctica *+ *Penguin *= 34 + 5 = 39 books.

**Using a bar diagram**

*Whole *= 34 + 5 = 39 books.

We had 78 fish. The penguins ate 40 of them. How many fish did we have left?

**Using algebra**

*Total *= 78

*Eaten *= 40

*Leftovers *= *Total *– *Eaten *= 78 – 40 = 38 fish.

**Using a bar diagram**

*Part *= 78 – 40 = 38 fish were left.

When we opened the windows and let snow come into the living room, the children made snowballs. Janie made 18 snowballs. Bill made 14 more than Janie did. How many snowballs did Bill make? How many snowballs did the children make altogether?

**Using algebra**

In this problem, we will let each child’s name stand for the number of snowballs he or she made.

*Janie *= 18

*Bill *= *Janie *+ 14 = 18 + 14 = 32 snowballs.

*Total *= *Bill *+ *Janie *= 32 + 18 = 50 snowballs altogether.

**Using a bar diagram**

In this story, we introduce a more complex form of bar diagram: the *comparison*. Many word problems involve comparing two quantities that are related to each other somehow. In this case, Bill made *more *snowballs than Janie.

The dotted lines connecting two blocks indicate those blocks are the same size. Bill’s bar is the same length as Janie’s bar, plus an extra amount. The bracket on the right-hand side of the diagram shows that we need to find the total of all the bars.

*Bill *= 18 + 14 = 32 snowballs.

*Total *= 18 + 18 + 14 = 50 snowballs altogether.

Mama had a ribbon 90 cm long. She had 35 cm of it left after making a bow for Janie. How many cm of ribbon did Mama use to make the bow?

**Using algebra**

*Ribbon *= 90

*Leftover *= 35

*Bow *= *Ribbon *– *Leftover *= 90 – 35 = 55 cm.

**Using a bar diagram**

*Part *= 90 – 35 = 55 cm of ribbon.

The penguins did theater shows for 2 weeks. They performed 4 shows every week. How many shows did the penguins perform?

**Using algebra**

Here we have the introduction of what is called a “this per that” quantity, also known as a *rate*. In most problems, a “this per that” quantity will require multiplication or division.

How can the student know whether to multiply or divide? Good question! In simple problems, it should be easy to see which operation is needed. (In this problem, most students will not even notice the multiplication — they will simply add 4 + 4 = 8.) We will come back to this topic at a deeper level in middle school or junior high.

*Weeks *= 2

*Shows per week* = 4

*Total *= *Weeks *x *Shows per week* = 2 x 4 = 8 shows performed.

**Using a bar diagram**

In a bar diagram, two or more parts that are the same size are called *units*. In this case, each unit is one week’s worth of theater shows performed by the penguins.

*Whole *= 2 x 4 = 8 shows.

Papa put a leash on his penguin, and took him for a walk. They climbed up 3 flights of stairs. There were 10 steps in each flight. Then the penguin flopped onto his stomach and slid down all the stairs. He pulled Papa with him all the way. How many steps did Papa fall down?

**Using algebra**

*Flights *of stairs = 3

*Steps per flight* = 10

*Total *= *Flights *x *Steps per flight* = 3 x 10 = 30

**Using a bar diagram**

Here, each flight of stairs is one unit. When there are several same-size units in a diagram, we often write the quantity only on the first unit.

*Unit *= 10

3 *units *= 3 x 10 = 30 steps.

## Which Approach Is Best?

As I worked through these examples, I noticed that the algebra approach required me to recognize on my own which operation was needed to solve the problem. It offered an efficient way to write down my steps, but it gave little guidance on whether to add or subtract in a given situation.

Drawing bar diagrams, however, forced me to analyze the relationship between the numbers in each problem. While algebra may work well for students with strong reading and reasoning skills, I think bar diagrams offer more help for students who struggle with the question, “What do I do?”

Bar diagrams clearly demonstrate the interconnections between basic arithmetic operations. Addition and subtraction are inverse operations. Therefore, the same basic diagram can represent either situation. Multiplication is repeated addition, so a multiplication diagram is simply an addition diagram with several same-size units.

**My conclusion:** Bar diagrams help 2nd-grade students avoid confusion and develop a stronger foundation for future studies in mathematics.

For more practice creating bar diagrams, your students may enjoy this online tutorial:

In your diagram for Janie and Bill, I noticed that you didn’t write the word “snowballs” next to their names. Do you find this to be a significant detail? I’ve started using this method with my 9th grad remedial math students, and I am forcing them to write both the “who” and the “what” each time. I’m doing this so that the bar can always represent the exact quantity described verbally to the left. I want them to associate the bar with a specific variable quantity. I’ve seen this problem happen a lot in algebra – students will say “let x = Janie” instead of “let x = # of Janie’s snowballs”. For some problems, this doesn’t matter, but when the problems get more complicated, students get lost and forget what their variables actually represent: “wait, was x the number of tickets or the price of the tickets?”. What do you think about this?

We’ve used Primary Maths as a curriculum for 6 years. One thing that is required of all third grade students learning the bar method is the final sentence to sum up the problem. “Mr. Popper fell down 30 steps” “Bill made 32 snowballs” “They made 50 snowballs altogether” This helps a student check for reasonability. There is a big difference in answers between “Jill weighed 3 kg” and Jill weighed 3 kg more”.

I let my young students get away with a minimum of writing. We often put only the initial next to the bar, but we will talk as we are working: “Okay, here are all of Janie’s snowballs lined up in a row. And now we want to line up Bill’s snowballs next to them. What will his row look like?” etc. This is the freedom of homeschooling and of small co-op classes, I guess, that we can rely so much on oral and mental work.

As the problems get more complicated, of course, students do have to write down more—and yes, mine often get confused just like you describe, Dan. Some of them are naturally organized, and writing out a solution comes easily to them. (I have had one or two like that.) But for most of the kids, breaking a problem down into logical steps, defining variables, and putting everything in order…well, it might be easier to train them to fly a bicycle. You have more teaching experience than I do, in number of students though not in years. If you have found that sort of detail to be significant, then I guess I’d better keep an eye on it this year.

[The problem tends to persist into college. See, for example, Algebraic Word Problems (pdf 86KB) by Jerome Dancis.]

Would they do better at it if I made them write whole sentences from early on? I don’t know. Cassyt, do you know how those third graders do when they get to algebra?

These diagrams are largely used in Russia and Australia. Why they are not so popular in North America?

I don’t know why, Elena, but I think the use of such diagrams is growing, thanks to the influence of Singapore Primary Math. Still, it is hard for busy people (and most teachers are

verybusy) to get their minds around something new. So until the big textbook publishers start using these, most teachers won’t bother with it.send me an email and ask me to solve my algebra problem soon as soon as possible please i am in a very difficult situation.

There are several algebra help websites on my free resource page.

i used the penguin methoud with my ks3 students and they really do understand the algebra much more now, thanks.

I realy don’t know why the school need to teach the 2nd grade a complicated Math-Algebra like this:

1.) triangle + 7 = 18 – 10

2.) 4 + 9 = circle + triangle

You have to derive it to get the answer but the problem is… the school didn’t give any textbook or any example for the kids to let them know how to answer the solving problem. Instead the parents are the one who’s learning those all homework!

This type of problem is designed to build flexibility in thinking about numbers. And judging from my experience with my own kids, the teacher probably did give instructions and examples, but they slipped back out of your child’s mind by the time he or she got home. That’s one reason why I don’t like the idea of “homework” for such young students.

Wow! This discussion is pretty amazing. I am not a teacher but I do mind the learning of my kids specially in advance math. I hope you can send me the basic formula to teach math effectively. Thanks in advance.

Hi Denise,

I like you explanations, they are very simple and straightforward. I’ve been an early childhood educator for 20 years and never taught this method to preschoolers, myself not being fantastic at Math ;P However, I have three sons and help them in Math and am working one to one with several preschoolers in their math problem solving. So whatever you have illustrated and explained above, has helped me formulate how I break it down and make it even simpler for my students.

Your article, and explanations are very much appreciated.

Keep up your good work!!!!