# The Craft of Word Problems

## CONTENTS OF CURRICULUM UNIT 04.05.09

- Introduction
- Rationale
- Mathematics Standards
- Strategies/ Role of the Teacher
- Percentage Review
- The Percentage Formula(s)
- Additional Ways to use the Formula (1)
- Extensions of percent
- Conclusion
- Reading List: Electronic Resources
- Print Resources: Teacher
- Print Resources: Student
- Mathematics Standards Appendix
- Notes

### Unit Guide

## Do the Math 100%

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## Conclusion

When working with percent problems, the real world situations that can be presented are abundant. Students can relate to the story of the problem due to the prior experience and knowledge they have with so many of the occurrences. They can enjoy creating their own problems based on the world in which they, as teenage consumers, are surrounded by. Their mathematical skill level with the many aspects of working with percentages will be tested and challenged. The key to effective teaching in today’s world is to help children learn to think, reason, and solve problems. Students must be able to demonstrate and communicate understanding. Critical thinking skills are the hardest and most important skills to teach. The learners’ ability to interpret and communicate mathematics is the most important for them to learn. The design of the word problems will stimulate the students’ mathematical thinking and enhance their capability to meet success with the future performance tasks that lie ahead for them.

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Activities in the Classroom

All activities will follow the mathematics standards located in the index of this unit. The materials needed include the chalkboard or overhead projector; handouts; grid paper; and calculators. At the discrimination of the teacher, students may do activities in pairs or small groups.

Activity one:

Review of conversion of percents to fractions-numerical and word problem practice. One class period is needed followed by homework and continual checking for understanding in subsequent classes. Handouts will be used for class and homework.

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Performance Task
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Students will convert a percent to a fraction in lowest terms.

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Objectives:
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Students will review changing a percent to a fraction. Students will review simplifying a fraction to lowest terms. Students will recognize the magnitude of a fraction and compare that fraction to one closest to it. Students will read and solve a word problem changing a percent to a fraction.

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Procedure
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:

Following a short demonstration that when you have a percent it converts to a fraction by putting the whole number percent over 100, have students shade a hundreds grid paper with the whole number of the percent shaded. Also ask students what percent is unshaded. Students can then find the lowest terms of the fraction through the use of the grid paper, a calculator or manually depending on the students’ abilities. Model a sample of each for parts 1, 2, and 3 on the chalkboard or overhead projector.

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Handout:

Conversion of percent to fraction.

1. For the following percents, shade the hundreds grid to represent the percent. Then change the percent to fraction. Simplify the fraction to lowest terms. Change the percent that is unshaded to a fraction in lowest terms.

a) 10% b) 25% c) 50% d) 75% e) 30%

2. For the following percents, find the fraction in lowest terms which is equal to the percent.

a) 15% b) 20% c) 35% d) 40% e) 55% f) 60% g) 70% h) 80%

i) 90% j) 98%

3. For the following percents, find the fraction with denominator 10 or less which is closest to it.

a) 10% b) 12% c) 15% d) 16% e) 20% f) 25% g) 42% h) 45%

4. For the following word problems, find the fraction in lowest terms which is equal to the percent.

a) The entire class is present in school today. What percent of the class is

present? What percent are absent?

b) Twenty percent of the class collects baseball cards. What fraction of the class is

baseball card collectors?

c) 75% of the students at Troup eat lunch. What fraction of the students do not eat

lunch?

d) If 50% of the students at Troup are walkers, what fraction of the students is

walkers? What fraction is bus students or gets a ride to school?

e) 40% of the class is wearing tee shirts. What fraction of the class is wearing a tee

shirt?

f) The eighth grade consists of 30% of the students at Troup. What fraction of the

students at Troup are eighth graders?

g) 10% of the class participates in football after school. What fraction of the class is

involved in football?

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Handout
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: This student homework sheet provides similar examples for the students.

1. For the following percents, find the fraction in lowest terms which is equal to the percent. a) 85% b) 12% c) 31%

2. For the following percents, find the fraction with denominator 10 or less which is closest to it. a) 30% b) 50% c) 70%

3. For the following word problems, find the fraction in lowest terms which is equal to the percent.

a) Deanna spends 80% of her money. What fraction of her money does she not spend?

b) 10% of the students are absent. What fraction is not absent?

c) If 5% of the students were tardy and 10% of the students were absent, what fraction of the students was in school on time?

d) John ate 50% of the pizza and Jeremy ate 25% of the pizza. What fraction of the pizza did I eat?

e) Mom spends 30% of her budget on food, 40% on the bills, and 5% for miscellaneous expenses. What fraction is left for rent?

Activity two:

Review of conversion of fractions to percents-Numerical and word problem practice. One class period for initial review is needed with subsequent follow-up.

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Performance Task
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Students will convert a fraction to a percent.

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Objectives
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Students will review how to change a fraction to a percent. Students will compare different fractions that equal the same percent. Students will round the percent to the nearest whole number.

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Procedure
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Model for the class the numerical calculation of dividing the numerator by the denominator (a/b), and multiplying the result by 100 (a/b x 100) to get the equivalent percent. Also, review with the students that the expression 5 out of 6 is written as the fraction 5/6.

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Handout
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Conversion of fractions to percents with numerical and problem practice.

1. For the following fractions, find the whole number percent, p. Indicate which if any of the fractions are equal to the same percent.

a) 1/10 b) 1/9 c) 1/8 d) 1/7 e) 1/6 f) 1/4 g) 1/3 h) 1/2 i) 2/3 j) ¾

k) 5/6 l) 6/7 m) 2/10 n) 1/5 o) 10/100 p) 5/10 q) 6/8 r) 12/16

2. For the following words problems, change the fraction into a percent.

a) One fourth of the class is left handed. What percent of students in this class are left handed? What percent of the students are right handed in this class
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b) Alexis made three fifths of her shots from the free throw line. What is her field goal percentage?

c) John ate one third of the pizza. What percent of the pizza is remaining?

3. Write a fraction and convert the fraction into a percent.

a) 90 of the 500 seats at the movies are empty. What percent of the seats are empty?

b) 14 out of 20 students brought their books to class. What percent of the students brought their books to class?

c) 10 out of 25 students wear glasses in the class. What percent of the students wear glasses?

d) 45 is what percent of 135? e) 17 out of 20 is what percent?

f) 6 is what percent of 24? g) 10 out of 10 is what percent?

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Handouts
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This student homework sheet includes similar examples for the students.

1. For the following fractions a/b, find the whole number percent, p. Indicate which if any of the fractions are equal to the same percent.

a) 19/20 b) 3/10 c) 4/8 d) 7/14 e) 5/20 f) 3/5 g) 2/8 h) 18/30

2. Express the fraction as a percent. About 3/20 of the total land surface of Earth is covered by deserts. What percent is this?

3. Write a fraction and convert it to a percent.

a) Four out of five students brought their pencil to class. What percent is that?

b) Out of 25 students in the class, 24 will be going on our school trip. What percent of the class will be going?

c) Five out of 25 students made high honors this marking period. What percent is that?

Activity three:

The three main types of percent equations-numerical practice. One or two class periods for practice and reinforcement is suggested.

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Performance Task
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Students will solve an equation for the part or quantity, the whole number or the percent.

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Objectives:
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Students will interpret an equation and write it into formula format. Students will identify which of the items- quantity, whole or percent is needed to be found. Students will solve the formula for the correct missing item.

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Procedure
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The teacher will introduce the formula (1) as explained in the unit. Students will use a calculator to solve the formula to find the quantity. Once the teacher is comfortable that the students can work the formula to satisfaction, introduction of solving for percent and whole amount is modeled. Be sure the students understand the vocabulary which identifies the part that is needed to be solved for. Then mixed numerical problems such as in the handout can be worked on in small groups. A review of rounding to nearest percent, tenth, and hundredth (penny) should be included.

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Handout
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Solving for the quantity, q; solving for percent, p and whole, w.

1. Find the quantity, q, using the formula: q = (p x w) /100.

a) 15% of 60 b) 25% of 40 c) 60% of 90 d) 65% of 300

e) 50% of 322 f) 5% of 40 g) 6% of 100 h) 2% of 50

2. Find the quantity, q, using the formula: q = (p x w) /100. Round your answer to nearest hundredth, if necessary.

a) 10% of 39 b) 2% of 43 c) 75% of 102 d) 15% of 30

e) 50% of 116 f) 65% of 93 g) 12% of 80 h) 33 1/3 of 240

i) 4% of $10.40 j) 5% of $20 k) 6% of $42 l) 5.5% of $30

3. Solve for the quantity using the formula q = (p x w) /100.

a) 10% of 12. b) 10% of 12 is what number? c) What is 10% of 12?

d) Find 6% of 50. e) 6% of 50 is what number? f) What is 6% of 50?

g) Find 50% of 91. h) 50% of 91 is what number? i) What is 50% of 91?

4. Solve for the quantity, percent or whole amount. If necessary round your answer to the nearest whole number.

a) 19 out of 20 is what percent? b) 20% of 140 is what number?

c) 66 is 10% of what number? d) 45 is what percent of 60?

e) 38 out of 40 is what percent? f) What number is 34% of 50?

g) 35% of 70 is what amount? h) 80% of what number is 48?

i) What is 6% tax of $60? j) 30 is 75% of what number?

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Handout:
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The student homework sheet includes similar examples for the students.

1. Find the quantity, q, using the formula: q = (p x w) /100.

a) 15% of 90 b) 25% of 60 c) 50% of 322 d) 5% of 40

2. Solve for the quantity using the formula q = (p x w) /100.

a) What is 10% of 32? b) Find 6% of 50. c) 80% of 40 is what number?

d) What is 50% of 50? e) 50% of 91 is what number? f) What is 25% of 50% of 320?

3. Solve for the quantity, percent or whole amount. If necessary round your answer to the nearest whole number.

a) 10% of 120 is what number? b) 30 is 75% of what number?

c) 5 is what percent of 10? d) 38 out of 40 is what percent?

e) 80 is what percent of 40? f) 30% of 75 is what amount?

h) 75 is 30% of what? i) What is 20% of 15/22? j) 1006 is what % of 1001?

4. Try these. Solve each problem for the part. Round to nearest penny, if necessary.

a) There are 240 students in the eighth grade. About 20% of the students made the honor roll. Approximately how many students made the honor roll?

b) Of the 240 students in the eighth grade, 10% made high honors. How many students made high honors?

c) During a recent fundraiser in the 8th grade, students raised $2000. 70% of the $2000 will be used for school trips. How much money will be used for school trips?

d) The other 30% of the $2000 raised will be used for a special graduation breakfast for the parents and the graduates. How much money will be used for this breakfast?

Activity four:

Mixed Percentage Problems. Two class periods for word problems.

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Performance Task
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Students will solve mixed word problems for quantity (part), whole and percent.

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Objectives:
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Students will determine what part of the word problems is missing. Students will solve word problems involving mixed percentages.
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Procedure
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Review type of problem using the previous day’s homework. Ask students to discuss what type of problem (part, whole or percent) it is before attempting to solve.

After a certain time allowance, go over the problems with the whole class.

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Handout
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Classwork day 1. Solve these mixed percentage problems. First determine what you need to solve for -quantity, percent or whole.

1. John deposited $850 in the bank. The bank paid 2% interest per year. How much money did he receive as interest after one year? 1b) How much money did John have in the bank after the one year?

2. There are 12 rooms at my school use by eighth graders. 20% of the rooms are for the eighth grade students. How many total rooms are in the school? 2b) How many rooms are not used by eighth graders?

3. I bought my CD player for $20 less than the original price. It was on sale at 15% off. What was the original price?

4. Sally went shopping with $100. She spent 25% of the money on new shoes and 20% on a new jacket. She spent the remainder of the money on new sneakers. How much did she spend on sneakers?

5. Troy and Kevin shot 60 baskets each at the gym. Troy made 45 of his shots. What percent of the shots did Troy make?

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Handouts
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Homework day 1. This includes similar word problems for the students.

Read the word problem and identify the part that needs to be solved for. Then solve the problem.

1. Mike has saved $90. He spent 30% of his savings at the mall. How much did he spend?

2. James saved $120 for his shopping trip to the mall. He spent 30% of his savings. Predict who spent more, Mike or James. How much did James spend?

3. Last month Mrs. Jones spent $150 on utilities. Her electric bill was $75. What percent of the total was her electric bill? Mrs. Jones’s phone bill was 10% of her monthly utility cost. How much was her phone bill?

4. Miguel has 250 baseball cards in his collection. He has traded 20% of his cards. How many cards has Miguel
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NOT
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traded?

5. Of the cards that Miguel has not traded, he has 80% of them in a card collecting book. How many cards does he have in the book?

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Handout
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Classwork day 2. Solve these mixed percentage problems. First determine what you need to solve for -quantity, percent or whole.

1. In the class of 25 students, 20% are left handed. How many students are right handed?

2. In another class of students, 20% are left handed. Six students are left handed. How many students are in the class?

3. Shelia’s savings account earns interest per year. She has $460 in her account now. If she leaves it there for a year, she earns $23. What is her interest rate (percent)?

4. The football team has 55 eighth graders on it. The volleyball team has 24 eighth grade players. No one plays on both teams. If there are a total of 169 eighth graders what percent of the class does football or volleyball? Round your answer to the nearest whole percent.

5. 48 students will attend the Philadelphia trip. Out of the 169 students in the 8th grade, approximately what percent will attend the trip?

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Handouts
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Homework day 2. This includes similar word problems for the students.

1. A new fitness center has opened up in town. There is room for 500 members. Find the number of members if 75% of the 500 openings are full.

2. The number of apartments rented in a building is 40. If 80% of the apartments have been rented, how many apartments are there in total?

3. 5 of the 50 apartments need painting. That is what percent?

4. Find the percent of students who take a bus to school if about 200 out of 800 students take the bus to school.

5. What is the amount of students who get a ride to school if 10% of the 800 get rides?

Activity five:

Finding sales tax, discount, tipping, interest, and other forms of percent change. Two class periods recommended with homework.

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Performance Task
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Students will solve problems involving real world situations with percent change. Be sure students justify and show work to support their response.

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Objectives:
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Students will solve consumer percentage problems.

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Procedure
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Introduce to the class this type of problem: I have saved $20 to buy a new DVD. The one I want to buy just came out for $18.99. Do I have enough money to buy the movie including 6% tax? Brainstorm with the students their thoughts on how to solve this problem. Ask how it is different from previous days’ problems.

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Handout
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Where can I get the better buy? There is a 25% discount on the $80 basketball jersey that I want to buy at Sam’s Super Sports Store. In Connecticut I know that I will have to pay a 6% tax also. On the internet, I found an NBA website where the same jersey will cost me $62. I do not have to pay tax but I need to pay a shipping and handling fee of 5% for all items costing $50-$99.99.

Do the math to find out where I can buy that jersey at the least possible price.

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Handout:
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Ray bought some running shoes for $89.99 and a jersey for $39.99 at Foot Locker. There is a 25% discount today, with 6% sales tax, estimate what will the total cost be?

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Handout
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Action Sports is selling the skateboard that I want for $21.80, plus 6% tax.

The same skateboard costs $25 at Sports Authority, but this week they have a 20% discount on all their sports equipment with wheels. Of course I still have to pay the 6% tax. Compare the cost of the skateboard at both stores and help me choose which store has the better buy. Please show and label your work in order to support your decision. Buy the way which sport on wheels do you like the best?