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
I’ve put the word problems from my elementary problem solving series into printable worksheets:
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?”
Mr. Popper’s Penguins
During the winter, Mr. Popper read 34 books about Antarctica. Then he read 5 books about penguins. How many books did Mr. Popper read in all?
Using algebra
We will let the name of each topic stand for the number of books Mr. Popper read about that topic.
Antarctica = 34
Penguin = 5
Total = Antarctica + Penguin = 34 + 5 = 39 books.
Using a bar diagram
Whole = 34 + 5 = 39 books.
Mr. Popper had 78 fish. The penguins ate 40 of them. How many fish did Mr. Popper 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 Mr. Popper opened the windows and let snow come into the living room, his 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.
Mrs. Popper 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 Mrs. Popper 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.
Popper’s Performing 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.
Mr. Popper put a leash on his penguin, Captain Cook, and took him for a walk. They climbed up 3 flights of stairs. There were 10 steps in each flight. Then Captain Cook flopped onto his stomach and slid down all the stairs. He pulled Mr. Popper with him all the way. How many steps did Mr. Popper 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 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:
Don’t miss any of “Let’s Play Math!”: 
Subscribe in a reader, or get updates by Email.
Have more fun on Let’s Play Math! blog:
- Elementary Problem Solving: The Tools
- Ben Franklin Math: Elementary Problem Solving 3rd Grade
- Word Problems in Russia and America
- Solving Complex Story Problems
- Solving Complex Story Problems II
