## Rising and Falling Fractions

### Sharla ShultsBay District Schools

#### Description

Why are announcements stating that a particular stock is "up 1 and 5/8ths" or "down 2 and 3/16ths" now history? Students explore outcomes from the conversion – fractions to decimals – in stock market quotes and explore fluctuating interest rates using an amortization calculator.

#### Objectives

The student refines vocabulary for interpersonal, academic, and workplace situations, including figurative, idiomatic, and technical meanings.

The student locates, gathers, analyzes, and evaluates written information for a variety of purposes, including research projects, real-world tasks, and self-improvement.

Understands the relative size of integers, rational numbers, irrational numbers, real numbers and complex numbers.

Understands concrete and symbolic representations of real and complex numbers in real-world situations.

#### Materials

-Computer with data display panel for teacher demonstration (See Teacher Preparation Step #2)
-Student calculators – FCAT style
-Transparency copies found in the associated file (See Teacher Preparation Step #4)
-Copies of documents found in the associated file prepared for discussion and formative assessment (See Teacher Preparation Step #5 for list.)

#### Preparations

1. Have folders ready for distribution at the beginning of class.

2. Set up teacher computer with data display panel or Smart Board, bookmark and minimize the [Loan Amortization Calculator] using the following Website:
http://nt.mortgage101.com/partner-scripts/1007.asp?p=mtg101&pw=600 or
http://ray.met.fsu.edu/~bret/amortize.html
(NOTE: These can be accessed through Google search engine and the key words [loan amortization calculator] or see Weblinks below for links to the Websites.)

3. Set up overhead projector for class discussion.

4. Download and make transparencies of Diagnostic Problem #4 and Diagnostic Problem #3 found in the associated file.

5. Download and duplicate the following documents found in the associated file:
a) Warm-Up Activity #1 – one per student
b) Warm-Up Activity #1 Answer Key – one teacher’s copy
c) General Approach to Problem Solving handout – one per student
d) Handout on Vehicle Makes, Models, and Prices – one per student
e) Rising and Falling Fractions Problem Solving practice sheet – one per student
f) Rising and Falling Fractions Problem Solving Answer Key – one teacher's copy
g) Student Problem Solving Skills Checklist – one teacher’s copy

6. Secure a set of classroom calculators, FCAT style. Scientific calculators are not necessary.

7. Prepare extra worksheets as needed for students having difficulty with basic arithmetic skills. These reflect areas of difficulty as demonstrated in problem solving with fractions.

#### Procedures

Day 6 of the unit, -The Math Connection.-

This lesson addresses only those parts of the standards involving problem solving with fractions.
Irrational and complex numbers are not addressed or assessed in this lesson.

NOTE 1: Steps #1, #2, #3, and #15 are for teachers using the Unit Plan. For a stand-alone lesson of fractions in the real world, please begin with Step #4 and end with Step #16.

NOTE 2 LESSON PLAN: Prior to teaching this lesson, it is helpful to visit the Websites referenced ahead of time to become familiar with information before presenting to students.

NOTE 3 UNIT PLAN: Letters should be arriving during this week. Since the letters will be coming in to the school, don’t forget to make a teacher’s copy to keep in a separate folder. This is a backup in the event students’ folders or materials are misplaced. Also, keep a checklist as the letters arrive. Distribute copies of the Rough Draft Comments Sheets to students to complete as they receive their letters. This document is found in the associated file of the lesson To Whom It May Concern to be duplicated as needed. Instruct students to write thank you notes to the contact persons for taking the time to answer the letters and make copies for the portfolio.

1. At the beginning of this lesson distribute student folders and allow time for students to check for inclusion of the following documents:

Student Daily Log Day 1*
Notes and chart from the class discussion Day 1*
Copy of The Ten Commandments of Math
Student Daily Log Day 2: Notes from the Web*
Paragraphs – Draft*
Student Peer Editing Rubric (Day 2)
Student Daily Log Day 3 – Notes on Bad Boy Business Letter*
Eight Essential Parts of a Business Letter
Requirements for Body of Business Letter
Hand-written edited rough draft of business letters
Student Peer Editing Rubric (Day 3)
Student Daily Log Day 4 – Common Errors*
Typed edited rough draft of business letters
Student Peer Editing Rubric (Day 4)
Computer disks with saved letters

*If students are writing in journals, these documents are not included.

2. When are we ever going to USE this MATH? Refer to the Math Scavenger Hunt and the math connection question, Why should I study math? Ask students to participate in a class discussion about their findings from the essays they chose to skim. (Note to teacher: Review essays by visiting http://www.maa.org/careers/index.html. See Weblinks below for link to this Website.) Prompt student discussion with questions such as: What occupation or field of work is the individual employed? Did the individual have positive or negative feelings about mathematics? Is math directly related to the occupation? Did the math required directly relate to problem solving?

3. Inform students that the next four days encompass interactions between the study of math and the world of work. This involves extensive problem solving utilizing basic mathematical skills dealing with fractions, decimals, percents, and scientific notation set in real-world context. Today’s lesson is on fractions.

4. Distribute the Warm-Up Activity #1 and tell students to work with partners in answering general questions about fractions they encounter in the real world. Hold a class discussion for about 10 minutes covering the information gathered.

5. Up an eight, down a quarter? What does that represent? What happened to a rise of 1½% or a fall of 2¼%? What problems did investors and brokers face with rising and falling fractions in the real world? Prompt a class discussion about why the stock market abandoned complicated fractions in favor of pennies. (Note: For more information on this change, see Weblinks below for link to Stock Market Decimalization.)
Note: Possible background information to share with students might include:
Stock quotes are dollar amounts for a single share of the stock. If a stock is currently trading for 15, you can buy one share of the stock for \$15 plus the commission on the transaction. Simple math tells you that 100 shares equal \$1,500. But stock quotes were often expressed in fractions of a dollar — as small as 1/16 (sometimes called one teenie). How much is 1/16 of one dollar? How much is 3/16? To find the decimal value of 1/16, divide \$1.00 by 16. Multiply by 3 to find the decimal value of 3/16.

6. Display the overhead transparency of Problem #4 from the Diagnostic Assessment. (See Associated File.) Identify all of the different arithmetic skills that are necessary to solve this problem. Focus especially on the answer in complete decimal notation and why the number is not rounded. (This helps with the explanation why the stock market converted to decimals.) Discuss the importance of knowing and refining mathematical vocabulary terms. Distribute General Approach to Problem Solving (see Associated File) and go over problem-solving strategies reminding students that careful reading is essential to insure the correct response to the question asked.

7. Display and discuss the overhead transparency of Problem #3 from the Diagnostic Assessment. (See Associated File.) Follow the same procedure as in Step #5 above. Explain to students that in the advertising business it is very important for the ads to “fall” properly on the page to maximize space and minimize waste.

8. How much does it really cost to buy an automobile or a house? The following demonstrations are real eye openers for students! Using the teacher computer connected to a data display panel, Smart Board, or similar technology tool, explore the areas of rising and falling fractions relative to interest rates regarding buying an automobile and a house.

9. Maximize the loan amortization calculator Website previously bookmarked and minimized. (See Teacher Preparation Step #2.)

10. First, explore buying an automobile. Distribute the handout on Vehicle Makes, Models, and Prices to students. (See Associated File.) Have each student select a type of vehicle from the list that they would buy. Tell students to allow \$500 for a down payment. Be sure students have a clear understanding of the meaning of the term down payment. Call on students to share their choices and calculate the balances financed over 3 years (36 payments) at interest rates rising from 6.5% (read as 6½%) interest to 9.25% (read as 9¼%)using the computer.

11. Next, look at houses. Since this is really a very complicated area, explain to students that the exploration will be very basic. Examine prices without the benefit of a down payment to keep the steps down to a minimum. Explain to students the meaning of the terms principal payments and interest payments*. Discuss amortization*. Demonstrating with the computer and data display panel, begin with an example of a house loan of \$100,000 for 30 years (360 monthly payments) at 7.5% (aka, 7½%) interest, then lower the interest rate to 6%, and finally down to 5%. Point out to students that at 5% the total interest payment is finally lower than the total principal payment!
*Be precise in explanations so students do not get lost in terminology. For example:
a) Principal – the amount borrowed or financed
b) Interest – a charge for borrowed money generally a percentage of the amount borrowed
c) Amortization – path to complete ownership (month by month detail of principal and interest payments)

12. Now, let’s practice! Distribute the Rising and Falling Fractions Problem Solving practice sheets. (See Associated File.) Allow students to use calculators in the problem solving, but be careful that a clear understanding of the arithmetic skills is already in place. Use teacher discretion whether to utilize this as an independent activity or perform with a partner or in groups. If students are working in groups, create a way that they can be partnered or grouped effectively.

13. Turn off the data display connection to the computer and allow students to come to the teacher station and use the loan amortization calculator. If students are working in groups, select one student from each group to work at the computer station. That student is responsible for delivering the information to the rest of the group complete with explanation. If enough computers are not available, elect to complete this activity as a whole group.

14. As students are engaged in the problem solving, continue to monitor students’ progress with arithmetic skill and lend assistance as needed. This provides the necessary feedback for determining the additional practice needs regarding problem solving with fractions.

15. As students are completing the problem solving, collect students’ folders. The added documents of today’s lesson include the Warm-Up Activity #1, handout on Vehicle Makes, Models, and Prices and the General Approach to Problem Solving handout.

16. Collect students’ Rising and Falling Fractions Problem Solving at the end of the class period to be returned the following day. Check students’ problem solving skills using the Rising and Falling Fractions Problem Solving Answer Key found in the associated file. Use the Student Problem Solving Skills Checklist* in the associated file to record the results. Identify any students struggling with the problem solving due to the inadequacy of understanding fractions. Attach additional practice work sheets as needed to the fractions problem solving activity sheets and return to students the next day to take home for extra practice.

*If deemed necessary, adapt this checklist to use in the upcoming lessons on decimals, percent, and scientific notation.

NOTE: Remember during each day of the prior week, students have practiced the needed basic arithmetic skills as demonstrated on the diagnostic assessment. This time is devoted to problem solving and the math connection with the real world, not reteaching of arithmetic skills.

#### Assessments

This assessment aligns with the part of the standard only dealing with fraction representation in real-world situations. Vocabulary is embedded throughout the lesson plan allowing students to refine vocabulary for interpersonal, academic and workplace situations.

Students demonstrate the understanding of the relative size of rational numbers, and concrete and symbolic representations of rational numbers on the Rising and Falling Fractions’ Problem Solving practice sheet. (See Associated file.)

Students analyze and evaluate the information from the text within the real world math problems presented for solving. Students demonstrate understanding by correctly solving the problems.

Students are formatively assessed on their problem-solving skills demonstrated on the practice sheet using the provided answer key. (See Associated File.) The checklist in the associated file provides a means by which students receive feedback and can be grouped for peer mentoring. Students checking low in multiple areas require additional practice in problem solving techniques.

Standards addressed will be modeled and formatively assessed throughout the unit and will be summatively assessed at the completion of the unit.

#### Extensions

This is Lesson 6 of 10 in the unit The Math Connection addressing [When Are We Ever Going to USE This MATH?]

The Beacon Unit Plan associated with this lesson can be viewed by clicking on the link located at the top of this page or by using the following URL: http://www.beaconlearningcenter.com/search/details.asp?item=2947. Once you select the unit’s link, scroll to the bottom of the unit plan page to find the section, “Associated Files.” This section contains links to the Unit Plan Overview, Diagnostic and Summative Assessments, and other associated files (if any).

Why Should I Study Math?
Career Profiles

Loan Amortization

Calculate almost any data field on this form
Amortization Calculator

What You Need to Know About: Mathematics
Mathematics Tutorials

Basic Aspects of Fractions and Operations
Math Tutorials: Fractions

From Definition to Practice to Quiz
Math.com: Fractions

When Are We Ever Going to USE This Math?
The Math Connection

Stock Market Goes Decimal: Complicated fractions abandoned in favor of pennies
Stock Market Decimalization