Keeping the Meaning in Mathematics: The Craft of Word Problems

Appendix

Picture Problem

The perimeter of a rectangle is 100ft. If the length is five feet more than twice the width, find the length and width.

• What language is telling you what to do? (five feet more that twice the width)
• Can you draw a picture, sketch, line to show this problem situation? ( ____ +____ + __________ + __________ = 100 )
• Tell me which one of the pieces do you know nothing about? ( I know something about the length, but I know nothing about the width - lets call the width X )
• Therefore X + X + (2X + 5) + (2X + 5) = 100 and by combining like terms we can now say 6X + 10 = 100. Since we are only interested in the value of the X we can eliminate the positive 10 by adding a negative 10 to both sides of the equation or 6X = 90. Now the X is being multiplied by six and division by six will result in X = 15.
• Now we are ready to answer the question, remember we were asked to find the length and the width of the rectangle so the answer is not 15, the width is 15 and the length is 35. We can check to see if 2 width and 2 lengths equal 100 or 2(15) + 2(35) = 100 or 30 +70 = 100.

Equal Problems

The New Bill Phone Company charges $ 30.00 sign-up fee and 12 cent per minute. The Best Buy Phone Company charges $ 10.00 sign-up fee and 15 cents per minute. How many minutes can I talk to make sure my bills will be the same?

• What is the question? (When will both phone plans be the same price)
• How do you represent New Bill Phone Company in an equation? (.12X + 30)
• How do you represent Best Buy Phone Company in an equation? (.15X + 10)
• Can you rewrite the question in symbols? (.12X + 30 = .15X + 10) or .03X = 20 or X = 666.67.
• Answer the question. I can talk 667 minutes and it won't matter which company I choose because my bill will be the same - $ 110.04.
• Nice Job - pat yourself on the back!!!!

• What is the question? (When will Janell have the same amount of money as April.)
• How do you represent Janelle's money in an equation? (6X + 20)
• How do you represent April's money in an equation? (-4X + 150)
• Can you rewrite the question in symbols? (6X + 20 = -4X + 150) or 10X = 130 or X = 13.
• Answer the question. In 13 weeks Janell and April will have the same amount of money in their accounts.
• Nice Job - pat yourself on the back.!!!!

A box of fruit has four more apples than oranges. Together there are 52 pieces of fruit. How many of each fruit are there?

• What language is telling you what to do? (Four more than)
• Can you draw a picture, sketch, line to show this problem situation? ____ + ____ = 52
• Tell me which one of those do you know nothing about. ( I know something about the apples, but I don't know anything about the oranges - let call the oranges X)
• Therefore apple plus oranges can be represented by (X + 4) + ( X) = 52, by combining like terms our equation becomes 2X + 4 = 52. Since we are only interested in the value of X, we need to get all the other numbers away from X, so starting with the number that is furthest away we need to eliminate the positive four by adding a negative four to both sides of the equation, or 2X = 48 Finally we need to eliminate the two and since the two is multiplying the X we can "undo" multiplication of two by dividing by two by or X = 24.
• Lets make sure we have answered the question - How many of each fruit is the question - the answer is not 24, rather there are 28 apples and 24 oranges or 28 + 24 = 52
• All our statements are true. Nice Job - pat yourself on the back!!!!

Two Type Problems

Cashews cost $ 5.50 per pound and peanuts cost $ 2.00 per pound. I want a 7 pound bag with a mixture of both nuts but I only want to spend $ 28.00. How many pound of each kind of nuts can I buy?

- Lets look at the numbers we are given and what they represent.

5.5 represents the price per pound of the cashews (pounds)

2 represents the price per pound of the peanuts (pounds)

7 represents the number of pounds I need to buy (pounds)

28 represents the money I want to spend

• What two things do these numbers represent? (money/pounds)
• Lets make two separate equations that make true statements.

equal seven pounds (C + P = 7) and the cashews cost $ 5.50 per pound, while

the cost of the peanuts is $ 2.00 per pound and I want a combination of both

with a final cost of $ 28.00. I can combine this information into a equation

stating (5.5C + 2P = 28).

- Lets make sure our equations match our sentences.

- Since we now have two equations, we can make a system and solve.

5.5C + 2P = 28

By multiplying the first equation through by -2 we can eliminate one of the variables.

-2C + -2P = -14

5.5C + 2P = 28

This will result in 3.5C = 14 which becomes C = 4 and since C represent the pounds of cashews we know that the number of pounds of peanuts I can buy must be 3.

- Lets check to see if we are correct.

4(5.5) + 3(2) = 28 True

• Please note that this is not the only way to solve this problem and student will be encourage to use whatever method they are most comfortable with.
• Example problems unsolved

Two Equation Method

The two equation method is a more advanced algebraic solution to the picture problem. I have included these examples for the benefit of the reader, however my students are not ready for this approach when they enter my classroom. This unit leads my students in that direction.

X + Y = 126

2Y = X

- Use substation to make one equation.

- Combine like terms.

- Divide both side of the equation by three to solve for Y.

- Plug the Y value into the equation.

- Check for accuracy

In the football game, Rockey gained three times as many yards as Bullwinkle. Rockey also gained 10 more yards than Boris. The three players gained a total of 410 yards. How many yards did Boris gain?

In science class, Hector decided to measure the mass of his math, science and Spanish textbooks. He discovered that the mass of his Spanish text is five grams more that four times the mass so his math book, while the mass of his science text is 110 grams less than his Spanish book. Combined, the mass of the texts is 1,178 grams. Find the mass of each book.

West High School's population is 250 students less than twice the population of East High School. Combined they serve 2,858 students. How many students attend West High School?

The perimeter of a triangle is 76 centimeters. The second side is twice as long as the first side. The third side is four centimeters shorter than the second side. How long is each side?

The length of a rectangle is three centimeters more than twice the width. The perimeter is 60 centimeters. How long and how wide is the rectangle?

The length of a rectangle is three times the width. The perimeter is 16 feet. What is the width of the rectangle?

The notebook costs $ 0.15 more than the pen. The total cost of the pen and the notebook is $ 2.25. How much does the pen cost?

An apple tree produces three times as much fruit as a pear tree. During one season, two apple trees and three pear trees produced a total of 126 pieces of fruit. How many apples does an apple tree produce in one season?

In Harry's backyard, his peach tree produces two pieces of fruit for every three pieces that grow on his apricot tree. If the two trees have a total of 265 pieces of fruit on them, how many pieces of fruit does each tree have?

It is now 7:51 pm. The movie that you have waited three weeks to see starts at 8:15 pm. Standing in line for the movie, you count 146 people ahead of you. Nine people can buy their tickets in 70 seconds. Will you be able to buy your ticket before the movie starts?

The length of a rectangle is 4 less than twice the width. If the perimeter is 244, find the length, width and area.

Valentine's Day is quickly approaching and Pablo is arranging flowers for his girlfriend. As a rule, he always uses twice as many roses as carnations and three more tulips than roses. If the arrangement he's designing will take 98 flowers, how many of each flower will he need?

The number of wild horses at the Lazy Z Dude Ranch could be found by counting them, but when Hank visits the ranch, he suggests that 23 fewer than five times the number of horses at the ranch is the same as 58 more than twice the number of horses on the property. If hank is right, how many horses does the Lazy Z have?

Juan and his friends are going to an amusement park and discover that they have two ticket options. One option is to buy an admission ticket for $ 5.00 and then pay .25 for each ride. The other option sis to buy an admission ticket for $ 2.00 and then pay .75 a ride. What do you think Juan should do?

Two car rental agencies are competing for business. Deluxe Driving charges $ 35 a day and .15 a mile while Rent-It-Cheap charges $ 20 a day and .35 a mile. Which agency should you choose?

Highland has a population of 12,200. Its population has been increasing at a rate of 300 people per year. Lowville has a population of 21,000 which is declining by 250 people per year. Assuming that the rates do not change, how many years will the population be equal?

It is the end of the semester and the clubs art school are recording their profits. The Science Club started out with $ 20 and has increased its balance by an average of $ 10 per week. The Computer Club saved $ 5 per week, but started out with $ 50 at the beginning of the semester. The Math Club was just formed this year and managed to raise $ 15 per week. Which club is the wealthiest at the end of the 12th week?

Future Problems

Students will progress to the following type of problems throughout the year. The skills that they have mastered will make this progression easy for them. I hope that they will be able to see the similarities to the examples we have gone over.

Three times a number increased by three is equal to 30. Find the number.

If two times a number minus six is equal to 20, find the number.

A person has eight coins consisting of quarters and dimes. If the total amount of this change is $ 1.25, how many of each kind of coin are there?

In a bank, there are three times as many quarters as there a half dollar and six more dimes than half dollars. If the total amount of the money in the bank is $ 4.65, find the number of each type of coin in the bank.

Bill is 8 years older than his brother. In 3 years, Bill will be twice as old as his brother. Find the present age.

A craftsman has two alloys of silver. The first one is 70% pure silver and the second one is 50% silver. How many ounces of each must be mixed to have 12 ounces of alloy which is 65% silver?