*[Photo by armigeress.]*

In 4th grade, math problems take a large step up on the difficulty scale. Students are more mature and can read and follow more complex stories. Multi-step word problems become the new norm, and proportional relationships (like “three times as many”) show up frequently. As the year progresses, fractions grow to be a dominant theme.

As a math teacher, one of my top goals is that my students learn to solve word problems. Arithmetic is (relatively) easy, but many children struggle in applying it to “real world” situations.

In previous posts, I introduced the problem-solving tools of *word algebra* and *bar diagrams*, either of which can help students organize the information in a word problem and translate it into a mathematical calculation. The earlier posts in this series are:

- Elementary Problem Solving: The Early Years
- Elementary Problem Solving: The Tools
- Penguin Math: Elementary Problem Solving 2nd Grade
- Ben Franklin Math: Elementary Problem Solving 3rd Grade

In this installment, I will continue to demonstrate the problem-solving tool of bar diagrams through a series of ten 4th grade problems based on the Singapore Primary Math series, level 4A. For your reading pleasure, I have translated the problems into the universe of a family-favorite story by C. S. Lewis, The Lion, the Witch and the Wardrobe.

## Update

I’ve put the word problems from my elementary problem solving series into printable worksheets:

## Tea with Mr. Tumnus

Mr. Tumnus told Lucy about a midnight party of fauns and Dryads in the forest of Narnia. 35 fauns came to the party. There were 3 times as many Dryads as fauns. How many creatures were at the party?

If the whole group split equally into 5 large circles for dancing, how many were in each circle?

For students who are not used to bar diagrams, the natural tendency is to multiply creatures at the party. A quickly sketched diagram will show their error:

There are four units in all, and .

So 140 creatures were at the party.

The second part of the problem is straight-forward. A diagram may not be needed, but here it is anyway:

.

There were 28 dancers in each circle.

## Back on This Side of the Door

The Professor had 486 books at his house, some in the library room and some in his study. There were 50 books more in the library than in the Professor’s study. How many books were in the study?

In my experience, students who have not learned to think in bar diagrams usually divide the books in half and then subtract 50 from that number to get their answer. The diagram shows the correct path to a solution:

First, we need to subtract the 50 extra books:

.

Then we split the rest of the books evenly between the two rooms:

.

There were 218 books in the study.

## In the Witch’s House

The White Witch had 300 servants at her house, which was really a small castle. There were 10 more wolves than Red Dwarfs. The number of Red Dwarfs was twice the number of Black Dwarfs. How many Black Dwarfs worked at the Witch’s house?

For most students, a problem like this is easiest to work backwards, because “twice the number” is a familiar diagram:

Next we add the wolves, which match the Red Dwarfs plus 10 extra:

Now we can see that we have five of our unknown unit (which is the number of Black Dwarfs), plus 10 more, to make a total of 300:

.

.

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There were 58 Black Dwarfs.

## A Day with the Beavers

Mrs. Beaver baked a “great and gloriously sticky marmalade roll” for dessert. She cut 1/6 of the roll for her and Mr. Beaver to share, and then she sliced up 4/6 of the roll for the children. What fraction of the marmalade roll was left?

Fraction problems start very simple, but don’t let the easy numbers make you lazy. Draw those bars! They will teach your students an intuitive understanding of the connections between math concepts: fractions are intimately related to multiplication and division.

The Beavers’ 1/6 of the roll, plus the children’s 4/6, make a total of 5/6 of the roll that gets eaten. So 1/6 of the marmalade roll remains.

## What Happened After Dinner

Mrs. Beaver had a pitcher of milk. She poured 1/2 of it into glasses for the children to drink with dinner. Then she poured 1/8 of the pitcher into their cups of after-dinner tea. How much of the pitcher of milk did Mrs. Beaver use?

The fractions in this problem are a little more difficult than the last one, having different (but related) denominators. The first part of the story is easy to draw:

But how can we show the extra 1/8 of the pitcher that gets poured into the tea? We must go back and sub-divide the bar so that each half of it becomes 4/8. As we create the equivalent fraction, can you see why the numerator goes up in direct proportion with the denominator? As the size of the pieces gets smaller, the number of pieces increases by the same factor.

Now we can mark the extra 1/8 used:

Mrs. Beaver used 5/8 of the pitcher of milk.

## The Spell Begins to Break

The Witch’s sledge got stuck in the mud and slush 24 times before she gave up and decided to walk. 2/3 of those times, the Witch made Edmund get out and help push. How many times did Edmund have to push the sledge?

Bar diagrams are almost like magic for fraction (and later, percent) problems, because the “whole” bar can represent ANY amount. The process of cutting the whole amount into fractional pieces is clearly related to division, and the picture of the bar communicates what is happening more clearly than an abstract equation like ever could:

The whole 24 is split into 3 units:

.

We need 2 units:

.

Edmund pushed the sledge 16 times.

## Peter’s First Battle I

2/5 of the creatures waiting with Aslan at his pavilion beside the Stone Table were Dryads and Naiads. There were 20 Dryads and Naiads in all. How many creatures were waiting with Aslan at his pavilion?

With problems like this, we begin to reap the full benefit of our work on learning bar diagrams. Instead of struggling to understand the algebraic reasoning required to solve something like , our students can draw:

.

.

There were 50 creatures with Aslan at his pavilion.

## Peter’s First Battle II

Aslan sent 20 of the swiftest creatures to follow the wolf and rescue Edmund. 2/5 of these creatures were eagles, griffins, and other flying fighters. The rest were centaurs, leopards, and other fast-running beasts. How many of the creatures that Aslan sent could not fly?

This problem uses the same numbers and the same basic diagram as the last problem, but students must read and understand the story to know how the numbers relate to the diagram. Also, students need to be aware of how little, easy-to-miss words like “not” can change their answer.

.

.

So 12 of the creatures could not fly.

## The Triumph of the Witch

The White Witch’s evil minions used 4 2/5 m of rope to bind Aslan’s legs together. They used 3/10 m less of rope to tie him tightly to the Stone Table. How many meters of rope did the wicked creatures use in all?

Measurements lead naturally to mixed-number problems. We again have different (but related) denominators — this time, 5ths and 10ths. Start by drawing one bar that is 5 units long. Each unit will represent one meter, and the last unit will be divided into 5ths to show the mixed number:

By making an equivalent fraction, splitting the 5ths into 10ths, we can easily see how long the second rope must be:

.

.

Always remember to put your answer into simplest form! They used 8 1/2 m of rope.

## At Cair Paravel by the Sea

4/5 of the sea people who sang and played music for the coronation party were mermen. If there were 8 mermaids, how many sea people came to the party?

The end-of-workbook review takes the students through several multi-step fraction problems. In algebra, this calculation would look like . With a bar diagram, it’s easy:

There were sea people at the coronation party.

## Conclusion

In the beginning, bar diagrams often take up more space and require more pencil-to-the-paper work from the student than other approaches to solving simple problems. Many simple word problems can be solved mentally, which makes drawing a bar seem like tedious busy work. But as word problems become more complex, the bar diagrams offer significant help for students who struggle with the question, “What do I do?” Diagrams make visible the abstract relationships between numbers, enabling the student to decide which arithmetical operation makes sense in the context of the problem.

One clear advantage of bar diagrams, in my opinion, is how well they lead students to understand fractions, a topic which will continue to haunt our students in problems of ever-increasing difficulty through 5th and 6th grade. Multiplication and division problems will also grow more challenging as students progress, until they become the dreaded ratios and proportions of algebra. These topics are notoriously difficult for students [JSTOR access required, or try this article instead], but I believe the bar diagrams provide a much better foundation for understanding than any other method I have seen.

To get more practice creating bar diagrams, your students may enjoy these online tutorials:

- Thinking Blocks Addition and Subtraction Word Problems
- Thinking Blocks Multiplication and Division Word Problems

And for some fun practice with equivalent fractions:

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Excellent explanation. You have inspired me to give bar diagrams a try. I also love the way you linked the word problems with Narnia. I’ll have to remember to be more creative when making up word problems. Thanks for the tips.

What happens when a kid works in one place with bars, and in another without? Does this more likely lead to confusion? Or to using the bars even when they are not explicitly required?

The foreshadowing of variables is very strong here, even while the direct links are all arithmetic. I’m just wondering how this fits in in a world where not every teacher/course/year is on the same page.

Jonathan

I don’t know how a student would handle it if one teacher used bars and another did not, but my guess is that it would be like any other problem-solving method. If the student

the bars when the teacher used them, then the drawings would simply make those particular problems clearer — but unless the student got plenty of practice creating bars himself, to the point that they were incorporated into his mental toolbox, he wouldn’t use them outside that teacher’s class.understoodMy high school son, who grew up using the bars, still sketches them when they seem helpful to him (ratio problems, for instance), but if he can see how to understand the problem without them, he will not bother. My 5th grade daughter hasn’t had as much practice (oops! gotta add that to the “to do” list), so she doesn’t think to make a sketch on her own, but she has no trouble understanding the ones I make. In fact, she did a proofreading check for me on this article, to make sure the bars I drew fit the story problems.

And you’re right: these bars make an easy connection to beginning algebra problems. The only difficulty I’ve had there was that many of the problems in my son’s algebra 1 textbook were so easy to do with bars and simple mental arithmetic that he didn’t see any point in writing down equations.

This post is in this week’s Carnival of Homeschooling! Please help promote the carnival, and thanks for your participation!

Oops, meant to tell you that in this week’s carnival, there were two entries I liked the best, and this was one of them.=)

I love this page and explanation (and the theme :)). We’ve been working to get Utah to adopt a Singapore math pilot program (http://www.utahsmathfuture.com) and I’m going to forward your page to some legislators to see an excellent explanation of how the bar graph models work. Also, I agree that 4th grade is a step-up in difficulty with mastery of basic facts taking on a new dimension. I’d like to suggest you take a look at Mathino, a math card game (full disclosure: I’m producing it for the creator of the game who is unable to). If you read his story at the Mathino math game site and watch the videos you’ll see why I think this is a fantastic math game for young children to adults. It really helps you quickly see where they’re having some breakdowns in their skills and lets you give immediate intervention to teach while you play.

That Mathino card game does look good. I’d like to try it with Kitten sometime.

This blog is so right in many ways.

Coming from Singapore, I say, indeed algebra and bar models go hand in hand. Don’t need to go overboard with bar models. Although with bar models, it is sometimes fun to crack seemingly hard problems mentally because you could visualize the solution. The kids may get the “Aha” moment.

For higher order problems, bar models can sometimes get complex and “non standard,” so if your child mastered algebra at an early stage, good for him/her.

I say attack the problems with whatever it takes, it brings understanding and fun.

Cheers

LionMath.Com

“Although with bar models, it is sometimes fun to crack seemingly hard problems mentally because you could visualize the solution.”My son does this. It is amazing (to me, at least) the problems he can work in his head, problems for which I would generally use algebra. I don’t know whether he visualizes the bars themselves or just recognizes the relationships between numbers, but I am sure that working with the bar models in elementary and middle school helped build this ability.That is because your son has a good math teacher : ) And he is bright to have made that connection. Not all children can do that.

For your readers …

An oil drum holds 120L of oil when it is 3/8 full. What is the capacity of the oil drum?

Here is a grade 3 child solution with bar models from Singapore :

http://www.lionmath.com/forum/viewtopic.php?f=40&t=464

Bar Models does make it clearer for the child and even for me.

Algebra fill very complex problems

But the explanation much easier and excellent