[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: Problems have been genericized to avoid copyright issues.
There was 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.
A 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.
The 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:
There were 58 Black Dwarfs.
The lady beaver baked a “great and gloriously sticky marmalade roll” for dessert. She cut 1/6 of the roll for her and her husband 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.
Mama 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 Mama 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:
Mama Beaver used 5/8 of the pitcher of milk.
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, she made her prisoner get out and help push. How many times did the witch’s prisoner 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:
The prisoner pushed the sledge 16 times.
2/5 of the creatures waiting with the lion at his pavilion were Dryads and Naiads. There were 20 Dryads and Naiads in all. How many creatures were waiting at the 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 at the pavilion.
The lion sent 20 of the swiftest creatures to follow the wolf and rescue the prisoner. 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 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 witch’s evil minions used 4 2/5 m of rope to bind the lion’s legs together. They used 3/10 m less of rope to tie him tightly to the 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.
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.
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:
And for some fun practice with equivalent fractions: