# 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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## Extensions of percent

Consider word problems of this type: House of Pizza has increased their prices by 15%. What will the new cost of my favorite $10 small pizza be?

They could use their formula (1) to find the quantity of increase and then add it to $10. They may choose to add 100% + 15% =115% and then use formula (1) to get $11.50. After calculating the answer, you can ask the students what the new costs of all the items at House of Pizza would be giving them a menu format or list of the costs.

To challenge a more advanced learner, this is a good time to work with matrices and percent increase. A matrix is a rectangular arrangement of numbers (data) in rows and columns. They are written within brackets [ ], not large parentheses. When you multiply a matrix by a number it is called scalar multiplication. Every number (element) within the matrix is multiplied. Consider the pizza costs:

small medium large

Cheese and tomato [10 12 16]

Pepperoni [12 14 18]

Supreme [14 16 20]

The manager at House of Pizza wants to increase all of his prices by 15%. Use scalar multiplication to do this for him.

[10 12 16 ] [11.50 13.80 18.40]

1.15 [12 14 18 ] = [13.80 16.10 20.70]

[14 16 20 ] [16.10 18.40 23.00]

The application of matrices to business is very important. By combining percent increase (or decrease) with scalar multiplication you can put this all together for the student. You can also have discussion with students about if rounding the prices would be suitable here. Graphing calculators will also assist in this teaching.

Consider this performance task. Mom went to Party City for supplies for my graduation party. She spent $5.99 on paper plates, $3.49 each on 2 packages of cups, and $4.99 on forks, spoons and knives. She also bought a graduation balloon that cost $1.50. Tax is 6% on these items.

From this story a variety of questions can be asked, each of which should require a skill that has been previously learned by the students. Depending on the level or levels of the students within your classroom at that time, you can ask a sequence of questions that will guide individuals of various ability ranges to answer. Alternatively, you can ask a question that will require the student sequentially to find answers leading up to the bigger question without guiding them. Prior to giving this real world problem as a task, several skills have been previously learned and mastered to varying degrees. The task may ask:

- - Estimate what Mom has spent at Party City without tax.
- - What was Mom’s actual total cost at Party City without tax? Write one or two sentences comparing your estimation to the actual total.
- - What is the amount of tax on Mom’s items purchased with a rate of 6%?
- - What was Mom’s final cost including 6% tax?
- - A new skill-How much change did Mom receive from $25?

Or go directly to the bigger questions:

- - What was Mom’s final cost including 6% tax?
- - How much change did Mom receive from $20?

Asking students to estimate is a question that is important because it focuses on what consumers really need to do at the store. The students often have difficulty with rounding and estimating, but not as frequently when money is involved. There may be different strategies of estimating used by the class. Students need to be prepared to defend their answers.

Students will most often meet success solving addition problems with and without a calculator. The terminology ‘how much spent’ is familiar due to both real world exposure and previous practice. Often in cooperative learning groups, they will discuss their strategy for solving. You can encourage this discussion by requiring all the students in the group to do their own work but then come up with an answer agreed upon by all members. They will then have one answer they can all defend as a group when called upon. If, after group discussion they cannot come up with one common answer, they may ask the teacher for assistance. Using group activity with problem solving, correction of minor addition errors are fixed, the order of operations can be addressed, and entering the numbers correctly into a calculator can be checked. Now you want students to compare the exact cost to the estimation.

For finding tax, students should be able to calculate using formula (1). For final cost, students should be familiar with adding the tax to the exact amount. To find the change, this should be one small subtraction step that some students will have difficulty remembering. Groups or pairs will help those students. Some students are ready to skip the sequential questions and go directly to what Mom spent or what her change is. That is a decision best left to the teacher of each particular class.

By adding in some additional information you can create a new set of challenges for the student: Party City is having a special graduation sale: 10% discount on all items!

Students may not be quite sure where to put in the 10% portion of the problem. Items for discussion within groups will include where does the new information fit in to the solution in finding the final cost, and how do we work with discount. You may give this problem on its own without the previous one and let them work to discover the final cost or guide them to the answer with a group of questions. Again, your determination of which way to question the students will depend on your classes’ ability to complete the word problem. This is a good example of differential instruction in adding a challenge to the story for those in the class who are up to the task. Further questions for individual groups to consider is to ask if there is a 10% discount and a 6% tax, is it better to take off the 10% first, then pay the tax? Or decide to add the tax and get the discount on the total? Also, if a store raises its prices by 10%, then offers a 10% sale, what is the relation between the original price and the now sale price?

Follow up with this task next to assign the students. You and your friend go to Electronic Boutique. Estimate the cost of buying a video game for $49.99 and 2 controllers for your system at $19.99 each. Then find the exact cost, including 6% tax. (Optional: EB is having a 20% off sale on all video games)

After doing the performance task problem involving Party City, you would like the students to see a similar relationship in attacking this new problem. They should be familiar with the terms and plan for solving. Using cooperative learning groups let the students strategize ways of solving this cost and question each other. They will explain their methods to each other. A sample homework problem may include:

I went to Sports Authority and bought a beginner’s tennis racket for $39.99 and two packs of tennis balls for $1.99 each. Sports Authority is having a 20% off sale. Tax on all purchases was 6%.

You can make up a question(s) about the story or you can ask the students to make up one or more questions about the story and answer their own questions at home. If some of your students do not have calculators at home, you may opt to let them just write their questions at home and estimate a solution. They can then take a few minutes at the beginning of class to solve or check their solution. Students can share their questions with the class or their group the next day or you can collect them and go over selected questions for the class to solve. They enjoy having input in a lesson and as the teacher; you can access their level of understanding of the mathematical computational skill as well as the mathematical understanding of the concepts. Another sample homework assignment to follow up is to have students write their own story giving them the numbers to use. This is an extension of types of assignments asked on the CMT. Consider as a homework assignment.

Write a story about the following equation: 25% of 80 = q Solve your equation. Or:

You are going to the mall shopping. You have $100 to spend. Make up a story about what you purchased and the costs. Do not forget to include any sales (discounts) you came upon while shopping and you cannot forget some items may include 6% tax. Show your total purchases and if you have any change left. Have fun!