# The Craft of Word Problems

## Using Algebra Word Problems to Explore Problem Space

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## Word Problem Sets

This unit develops a “multiple-meaning” approach to problems in an attempt to draw students in from any of several directions as they solve problems and explore mathematical ideas Since I teach primarily Algebra and Geometry, I developed problems and models that work within these subjects but which also are accessible for my students with less than grade level achievement. Thus, I hope my unit will also be useful for teachers and vertical teams seeking ways to build greater meaning in the progressive study of mathematics. For each set of problems I provide a brief overview discussion of the particular problem space and a sample or two of a complete solution to a problem.

### 1. Piling Up Blocks--Multiple Solutionss

During our seminar, my initial set of problems (see “Functions of
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n
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” below) designed to demonstrate the use of multiple solutions (the “5-way problem page”) was too difficult for teachers in elementary grades to apply in their classes. With input from second and third grade teaching colleagues, I created a set of algebraic word problems, which could be solved by elementary students in at least one or more of the corners of the multiple solutions template. Algebra and geometry students should be able to solve these in all four of the template corners and in addition, if given one of the template corners completed, should be able to write a corresponding verbal description. It is not necessary to require that students “fill the page” with all four solution methods, but a great deal of inter-related learning can be discovered by having small teams of students take on the task of completing the page. Some students are very adept at pattern visualization, others at table building, while others may leap to define variables and set up equations. All middle and high school students should practice the skills of going from table to graph and from graph to table. Thus, while I list this problem set as a series of word problems (verbal descriptions), each question could be presented as a graph, table, equation or pattern sketch.

Including two different types of block/step patterns enlarges the problem space--relationships between “numbers of blocks” and “number of rows high”. Each pattern has its own block-to-row relationship, but a comparison of the two relationships may yield more insights for students. (Two lines on the graph, two sets of equations, etc. are useful analytical tools for algebra students). The final problems ask students to generalize a formula (equation) for both of the patterns and then to make a comparison of the two patterns’ results for a project involving 200 blocks. Given enough blocks or LegosTM, any student could construct these steps, but even younger students may see the value in solving mathematically when block counts get quite large. (Problem 1 below is solved for demonstration purposes on the template in Figure 1 of the appendix.)

- 1. Amir is arranging blocks. He puts one in the top row, 2 in the 2nd, 3 in the 3rd, etc. How many total blocks does he need to build 5 rows?
- 2. Amir is using blocks to build steps. He has a total of 55 blocks. If each block sits on two others, how many rows high will his steps be?
- 3. Amir wants to build block steps 10 rows high. How many blocks does he need if each block rests on two others?
- 4. Michelle decides to build steps similar to Amir’s. She starts with a row 10 blocks long, how many blocks will she need if she wants her last row to have just one block?
- 5. Sara wants to build steps that go up one side and down the other. She decides to put each block directly on top of another. How many rows high will her steps be if she uses 10 blocks?
- 6. Sara wants to build block steps 5 rows high with just one block on the top. She will put each block directly on top of another. How many blocks does she need in all?
- 7. Alfred borrows Amir’s 55 blocks and builds steps using Sara’s pattern. How many rows high will his steps be? (Will he have any extra blocks?)
- 8. How many total blocks will Amir need to build n rows with his step pattern?
- 9. How many total blocks will Sara need to build
nrows with her pattern?- 10. If Amir and Sara each have 200 blocks to work with, what is the ratio of the number of rows that Amir will complete to the number Sara will complete?

### 2. “Go East Young Mathematicians”

In our seminar we discussed the generally poor performance of American students on international mathematics tests. Along with reading several books by educators who place great emphasis on development of algebraic understanding from an early age (see bibliography for suggestions), I looked at curricular materials from Singapore and Japan--two countries with track records of high success among their students on international math exams. The following two sections provide more information about these curricula and illustrative problem sets.

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A. Singapore Approach: Concrete->Pictorial->Abstract
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The problem solving approach in the Singapore Curriculum involves not so much a “strategy” as a graphical way of relating numbers to solve problems visually and logically. Algebraically, the level of abstraction (in the sixth grade texts studied) does not include emphasis on naming variables and writing equations. Since my ninth and tenth grade students at Cross Annex often have great gaps in their educational development, this approach may have real appeal as well as value in building up some of the missing foundation of numeric understanding. A pair of problems inspired by the Singapore series will illustrate the approach.

Sample #1: Jose had twice as much money as Ted. But after Jose spent $50 and Ted spent $20, they each had the same amount of money. How much money did each have at first?

Discussion: No formal equations are used in this problem solving approach. The information in the word problem is blocked out in units or bars to represent the money Ted and Joe each had at the beginning of the problem and after their spending. The key idea is to encourage students to look at the relationships between the boys’ amounts of money both before and after their spending. Solving requires identifying in the diagram (see Figure 3 in the appendix) the value of “1 unit” of money as just the difference between Jose’s spending and Ted’s spending (50 - 20 = 30) as shown in Figure 2. Ted started with one “unit” or $30 and Jose started with 2 “units” or $60.

Sample #2: Sam earned $140 doing yard work after school. His sister Kim earned $60 babysitting. When their grandmother gave them each an equal amount of money for new school clothes, Sam had twice as much money as Kim. How much money did their grandmother give each of them?

Discussion: using a “working backward” strategy and the graphic representation of the two students’ money AFTER the grandmother’s gift and BEFORE the gift can solve this problem. Again, the key is in looking at the relationship between the amounts of money the two students had at those times. Careful attention to the information in the problem is a must--as with all word problems. Visually representing the money amounts may keep students focused on the details. Although, students could define variables and solve this problem with equations, in this Singapore approach, equations are not necessary. Understanding what is going on with the numbers is the goal. Figure 4 in the appendix illustrates the solution to this problem using unit bars and shows that $140 - &60 -$60 = $20, the amount of the grandmotherr’s gift.

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- 1. Jose had twice as much money as Ted. But after Jose sent $50 and Ted spent $20, they each had the same amount of money. How much money did each have at first?
- 2. Sam earned $140 doing yard work after school. His sister Kim earned $60 babysitting. When their grandmother gave them each an equal amount of money for new school clothes, Sam had twice as much money as Kim. How much money did their grandmother give each of them?
- 3. Matt and Nigel each had the same amount of money. Nigel spent $18 each week on lunch and Matt spent $24 on bus fares and lunch. When all of Matt’s money was spent, Nigel still had $120. How much money did each of the boys start with?
- 4. Mike and Tom were trading baseball cards. Mike had 40 more cards than Tom. After Tom gave Mike 12 cards, Mike had twice as many cards as Tom. How many cards did they have altogether?
- 5. Javon and Phyllis compare their music collections. Javon has three times as many CD’s as Phyllis and six more besides. If Phyllis has 14 CD’s, how many does Javon have?
- 6. Sheila had three times as much money as Donaya. After Donaya made $6 and Sheila spent $17, they each had the same amount of money. How much did each have at first?
- 7. Ashlee had $35 and Jackie had $80. After they each earned an equal amount of money at their jobs, Jackie had twice as much money as Ashlee. How much money did each earn?
- 8. Mimi had $55 and Jackie had $80. How much money must Jackie give Mimi so the girls have the same amount of money?
- 9. Jackie had $80 and Mimi has $55. The girls went to the mall and each spent the same amount of money. On the way home they compare their money and find that Jackie now has twice as much as Mimi. How much did each girl spend?
- 10. Tony had $45 and Jon had $75. They go into business selling hotdogs agreeing to share all costs and profits equally. At the end of the first day, Jon has twice as much money as Tony. How much did they each make? How much does each now have?

The set above provides an opportunity to tackle word problems with relatively small amounts of data, but with some interesting relationships among the data. The Singapore graphics technique will work well on all of these problems, but students may just as easily discover other approaches (defining variables and writing equations; graphing, etc.) Problems 6-10 should give an opportunity to explore interesting aspects of comparing quantities--they require adding equal amounts, subtracting equal amounts, or adding and subtracting different amounts in search of solutions. A word of caution: problem 10 introduces the possibility that an answer might be negative (Tony and Jon suffer a loss of $15 each on their first day of business. Not all ideas are instant moneymakers!) It may be important to ask students how they can explain their answers and if they think their answers are the only possible ones for the problem conditions. Stretching studentss’ grasp of the problem space will benefit from individuals or small groups sharing solutions with the class and reflecting together on the outcomes.

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B. A Japanese Curriculum--Algebra Problems on the Internett
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For those who have access to the Internet in their math labs or classrooms, there is a collection of Japanese algebra word problems online at: www.japanese-online.com/math. The problems presented are translated from a Junior High School math placement test of 225 problems that are given to 12 year olds. Students in Japan are given a time limit ranging from 1 to 5 minutes for each problem. To complete the 225 problems within the time provided requires over 8 hours of testing!

There are about 20 different types of word problems, so some selecting to arrange a set of cohesive problems addressing concepts would be needed by a teacher. However, this site could be used as randomly arranged to challenge students and provide some insight into the nature of math work students are doing in Japan. Japanese students regularly place among the top 3 countries worldwide in student math abilities. (According to information on this website, the United States was recently ranked #14 in international math placement.)

Each problem on the site is presented with a clear (often colored) graphic and verbal description along with the minutes within which Japanese students are expected to solve the problem; most have multiple choice answers. In addition, each problem offers a “hint” button and an “explanation” button that provides the answer (most often with defined variables and equations). The site also offers a free download of World Math Challenge volume1. “The math problems contained within this site are for free use by individuals and can be copied for use by teachers within their classrooms.”

This website provides such an accessible resource that all algebra teachers should check into using it for math tutorials or labs. It is limited to the defined variable and equation approach to solving problems, but very useful for students who need practice with reading and solving key types of problems. There is even a button to have the problem read aloud and a button to have the answer read aloud--this may make it possible for slower readers to do more challenging math. One problem from the website is included as Figure 5 in the appendix, to illustrate the arrangement visible on the screen when solving problems..

### 3. Lines, Diagonals, Handshakes and Sums as a Function of n items

I began this problem set with a question similar to one posed on a past CAPT math test: “how many diagonals can be drawn in a closed figure with
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n
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sides?” In working on this problem with the seminar group, it became clear that a whole series of problems, though seemingly unrelated, have somewhat similar mathematical structure--in other words, they share the same “problem space.”

- 1. How many diagonals are there in a convex n-gon?
- 2. If there are n points in a plane (no 3 on the same line), How many lines can be drawn between all pairs of the points?
- 3. If there are n lines in a plane (no 3 through the same point), how many regions are formed?
- 4. If n people in a room all shake hands with each other, how many handshakes occur in total?
- 5. What is the sum of the first n numbers? (i.e., 1+2+3+4+5 . . .)
- 6. What is the sum of the first n odd numbers? (1=3+5+7+9 . . .)
- 7. How many ways can we express a positive number as the sum of two positive numbers if order is relevant? (i.e., for 3: 1+2 and 2+1 both count)
- 8. How many ways can we express a positive number as a sum of 2 positive numbers if order is irrelevant? (i.e., for 3: 1+2 is the same as 2+1)

A variety of approaches can be taken to solving these problems, but in keeping with the general theme of this unit--exploring the problem space--the most comparative approaches are best. By building a table of values for each problem simultaneously, patterns, similarities and differences emerge. (See Figure 6 in the appendix.) )

These problems are ideal for “multiple solution” examination because they have similar progressions of numbers as functions of n. Students should be encouraged to sketch and determine values for at least n = 0 to 10 for each problem before attempting equation solutions. Encourage them to look at the patterns as they emerge. Some are so alike that students might gain new insights. By looking at the numbers for “lines” (problem2) and “handshakes” (problem 4), it readily is apparent that handshakes all around is the same as making sure each person (point) is connected to all others in a group. For a closed figure this amounts to all the sides plus all the diagonals. To determine the number of diagonals in closed figures (problem 1), students should look at the first order differences of numbers running down each column. The numbers for diagonals and lines both increase in a 2,3,4,5,6,7… sequence, but with different starting points of n. Graphing the tables of data would be another useful exercise for students to look at how these problems play out. Graphing calculators or using computer software with this set of problems is recommended.

It may be quite a leap to get to the equations for these problems, but that is the ultimate goal for students advancing in algebra. Sums of the first n odd integers (problem 6) should be not difficult at all. Discovering the similarities among the other seemingly different cases will be a challenging task requiring a good deal of time with the data. One strategy that we explored in our seminar involves representing the problems as stacks of blocks or grids. For example, problem 5, the sum of the first n numbers can be drawn as a rectangular grid of width n and length (n+1). Shading in half the blocks on this grid shows the table sequence (1,3,6,10, . . . ) and demonstrates why the sum is just half of n times (n+1).

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### 4. Unitary Rates and Mixing Problems

The problems in this set are intended to take students from simple, one step calculations to more complex problem solving involving unitary rates (e.g., price per pound) and mixing of rates. Each problem can be analyzed using a Vee Chart and using a horizontal unit conversion graphic organizer may assist solutions. (These solution techniques are demonstrated below.) Again, however, it is important to note that just finding the answer is less relevant to the lesson than the manipulating of the parts of these challenges to gain understanding of the nature of the concepts imbedded in mixed rate problems.

My students often find problems such as the Japanese Curriculum “hose and bucket” rate problem above logically possible to explain (“obviously the two hoses will fill the bucket faster than either one hose”), but mathematically they are at a loss at how to put the data together to get a concrete solution.

Sample solution for the “hose and bucket” problem:

The first step is to read carefully for details of the problem. Then these details need to be organized and examined for further less obvious information. For this I recommend a Vee - Chart technique, which is just a graphic organizer format for the problem solving analysis popularized classically by the mathematician George Polya.

(Chart available in print form)

Using the organizer for this problem hopefully will help students stretch their thinking from just the data presented in the problem to mathematical elaborations on that data--in this case forming unitary rates: the fraction of the total bucket that a hose can fill in one time unit (minute in this problem)..

Since the two hoses (A+B) can fill 1/45 + 1/30 of the bucket in one minute, the problem is one of finding a common denominator for 45 and 30. With minimal trial and error, most students should find that 90 works in this data. 1/45 = 2/90 and 1/30 = 3/90, so 1/45 + 1/30 = 2/90 + 3/90 = 5/90. 5/90 = 1/18, so the two hoses together can fill 1/18th of the bucket in 1 minute. Therefore, it takes 18 minutes to fill the whole bucket with both hoses. (A Japanese 12 year old is expected to solve this problem in just one minute!)

Similar strategies can be used with the problems below. Students need to see a variety of problems to gain understanding that unitary rates can be amount per 1 minute,

per 1 dollar, per 1 pound, etc., depending on the data in the problem

- 1. Cashews sell for $ 3.50 per pound (rate: $3.50/lb). How much will it cost to buy five pounds?
- 2. Almonds sell for $5.00 per pound (rate: $5.00/lb). How many pounds can be bought for $ 12.00?
- 3. What will it cost to buy two pounds of cashews (at $3.50/lb) plus one and one half pounds of almonds (at $5.00/lb)?
- 4. If three pounds of cashews are mixed with three pounds of almonds, what should be the total cost of the whole mixture?
- 5. What is the mixture price per pound for the mixture in problem 4?
- 6. If fifteen pounds of an almond/cashew mixture contain five pounds of almonds (at $5/lb), how many pounds of cashews (at $3.50/lb) does the mix contain?
- 7. What would be the total price for the fifteen-pound mixture in problem 6?
- 8. Equal amounts of cashews ($3.50/lb) and almonds ($5.00/lb) are mixed, what should be the price per pound for this mixture?
- 9. Mary wants to sell fifteen pounds of a cashew/almond mixture at the school fair for $4.00 per pound. How many pounds of each type of nut should be used?
- 10. James thinks they can sell more pounds of nut mix if they make a mixture that sells for $3.75 per pound. If they want to take in a total of $100 on the nut sale, how many pounds of mixture must be sold at $3.75/lb and what amount of each type of nut should be in the mix? Round answers to the nearest hundredth or penny.

### 5. Speed, Time, Distance Problems

In the Singapore 6A curriculum, algebra and speed problems form first and last chapters. Visual and graphic representations can be useful in solving these problems, but students may also be comfortable with tabular, graph or equation solutions (using the speed formula: speed = distance/time). In fact, since the speed, distance, time relationships are introduced at in elementary grades, this type of problems are ideal for vertical teams to consider and for students to apply to the “5-part problem-solving template.”

The following Problem set starts with problems appropriate for middle school math and leads to problems that can challenge Algebra I students at the high school level. Data from the first problems is needed for later problems, so these should be solved in order.

- 1. In one hour, Anne’s boat floats three miles down the Quinnipiac River. How fast is the river flowing?
- 2. If Anne can row 2 miles in one half hour on a still lake, what is her rowing speed in miles per hour?
- 3. Anne leaves her dock and tries rowing her boat upstream against the river current for one hour. How far upstream does she go in one hour?
- 4. When Anne turns and rows downstream with the current, how fast is she going?
- 5. How much time does it take Anne to row back to her dock from her farthest point upstream?
- 6. If Anne leaves her dock and rows downstream for one hour, how far will she then be from her dock?
- 7. How long will it take Anne to row back upstream to her dock this time?
- 8. Aaron puts a small motor on Anne’s boat and goes six miles upstream in one hour. How fast would his motorboat go in still water?
- 9. If Aaron motors downstream for one hour, how far will he travel?
- 10. After going downstream for one hour, how long does it take Aaron to return upstream to his starting point?
- 11. Write the following ratios:
- a. Anne’s speed in still water to the speed of the river current
- b. Aaron’s boat speed in still water to the River current
- c. Anne’s upstream rowing speed to her downstream rowing speed
- d. Aaron’s downstream speed to his upstream speed
- 12. Aaron and Anne borrow a friend’s canoe and paddle together on the West River. If it took an hour for them to go six miles upstream and only forty five minutes to return to their starting point, what is the speed of their boat in still water and what is the speed of the current in the West River?

### 6. Algebra Problems that use the Addition Table

“If this is algebra, why are we looking at the addition table?”

This will likely be the first response of your students to a suggestion of exploring problem space with the addition table. Students usually are convinced that addition and addition tables are only for the very young, even though they have rarely taken a look at the table with an eye to discover patterns and problem solving relationships. I include this topic here because we spent a few enjoyable hours in our seminar thinking mathematically about the addition (and multiplication) tables as launch pads for investigation into “big ideas” of math. For purposes of illustration, I include examples of problems which can be solved with tables and hope other math teachers will reconnect their students with these tools for thought and exploration.

Example 1. A student was tossing two 6-number cubes (dice) and wanted to know what chance she had of getting a number greater than nine.

By looking at a portion of the addition table (6 by 6) it is quite easy to see that there are 36 possible outcomes for the tossing of the dice and that 6 of those outcomes are greater than nine. Since the probability is a ratio of desired outcomes to possible outcomes for an event, her answer is 6/36 or 1/6.

A geometric grid aspect of this solution (see Figure 7 in the appendix) would lend itself to a variety of two dimensional problems involving various types of number cubes--all of which rely on the addition table format for “possible” outcomes versus “desired” outcomes.

Example 2. John was tossing a 4-sided number cube numbered 1-4 and a 7-sided number cube numbered 1-7 (such cubes are available in math supply catalogues). He wondered what his chances were of getting a number less than seven. By using the table in figure 8, determine his probability.

Again, the solution (1/2 or 50%) can quickly be determined from the portion of the addition table appropriate for this problem, or by using the corresponding grid (see Figure 8 in the appendix) to determine the ratio of desired outcomes (14) to total possible outcomes (4 x 7 = 28).

Any combination of two dimensions can be used (2x9, 3x3, 10x5, etc) to quickly work out the probable outcomes for a desired event.

### 7. A Political and Economic Problem?

As a final illustration of the remarkable possibilities for exploring problem space and mathematical concepts, I include a problem our seminar tackled which opens the door to interdisciplinary study of history, social studies, economics, ethics, and justice issues. The data in this hypothetical problem echo very real data recently debated in the popular press, which may give the mathematical understanding a bit more urgency and interest.

Problem I: A group forms 10 % of a larger (whole) group. The members of this group receive 60% of the income of the whole. How much richer, on average, are the members of the 10% group than the average of the rest?

A visual graphic representation of the problem (Appendix, figure 9) helps lead to the solution:

a. For the small population group:

Income % / Population % = 60% / 10% = 6

(each 1% of population in this group gets 6% of whole $)

b. For the majority:

Income % / Population % = 40% / 90% = 4/9

(each 1% of population in this group gets 4/9% of whole $)

c. To compare a & b:

6 / 4/9 = 54 / 4 = 13.5

So, the 10% group, on average, gets 13.5x the income of the 90% population group. Or, to be more dramatic numerically, each small group member gets, on average, 1350% of the average income of each large group member.

Problem II: Let’s suppose that the “Whole Population “ of our first problem represents just 5% of the World Population; and lets suppose the “Whole Income” of the first problem is 40% of the Worldwide Income.

The original 10% of the whole Population now is (10% of 5%) = .5% of the World Population and this group gets (605 of 40%) = 24% of the Whole World Income.

How much richer, on average, are members of the original “10% group” than the average for the rest of the world?

a. % of World Income/ % of World Population = 24%/ .5% = 48

b. For the rest of the world: % of World Income / % of World Population = 76% / 99.5% = .76

c. To compare a & b:

48/.76 = 63

Each member of the small group gets 63 times (6300%) of the average Worldwide person’s income. Also, the members of the 90% group in problem I receive 3.6 times or 360% of the average worldwide person’s income

The problem space just beginning to be explored in this case is the enormous issue of wealth and income distribution within a rich country and for that country in comparison to the rest of the world. As we enter a hot political season in this country, a great deal will be said about our economy and our world relations. Mathematical understandings will be needed to help cooler heads prevail in the midst of hot rhetoric. For those who have the opportunity to teach interdisciplinary courses, this problem may point the way to explore some critical issues in social studies with mathematical insights.