It is a very big decision to become a first-time homeowner. There are many things that need to be considered. Three important factors are the price of a home, mortgage interest rates and a person’s salary. In this project we will look up current values for these three factors and compare them across the United States. We will see how current interest rates affect the mortgage a person can afford. Finally we will look up data about how the cost of a home has changed over the past years and try to predict future costs.

Part I, "Comparing Home Prices" is suitable for an Applied Math course or above. It can also be used in an economics or business course. Part II "Mortgages" can be used in beginning math courses or in business or economics courses if the TI-83 or similar calculator with built-in financial functions is available. If financial formulas must be used, it is appropriate for students with Algebra II background. The primary audience for Part III "How the Cost of a Home has Changed" is students in Algebra II or above.

Part I: students should be able to find Internet sites; they should know the meaning of "median" and "mean". Part II: familiarity with the TI-83 financial functions or ability to evaluate complex algebraic formulas; they should know the meanings of "mortgage", "interest" and "interest rate". Part III: understanding and familiarity with fitting curves to data using either a calculator such as the TI-83 or software such as Excel, Graphical Analysis or Curve Expert.

To use the Internet to find current home prices, mortgage rates and salaries in different parts of the United States.

To compare median and mean prices of homes.

To find the monthly payment needed to purchase a home.

To use the Internet to find historical data about home prices.

To create a graph showing how the cost to purchase an average priced home has changed over the past thirty years.

To create a best-fit curve for the yearly change in purchase price and predict the cost of an average priced home in the future.

Students will search the Internet for the required information and complete the attached Activity Sheets. If desired, they may share information about housing and create a class table or graph.

See points on the Activity Sheets. 

Internet access. TI-83 or similar calculator or software to evaluate financial functions and perform curve-fitting recommended.

Students can share results and make a class histogram showing median price of a home for the different cities or states they chose. Students may wish to find average salaries for the cities they studied and decide if home prices are comparable. They can pick an occupation and find out what the average salary is, determine what price house they could afford, see if they can find an available house for this price and create a budget. Students may see how monthly mortgage payments vary as the interest rate changes and how much house they can afford.

Teacher Notes.

!!!! Warning !!!! The Web sites given in this lesson may have changed! Before using this lesson with your students, be sure to check if the sites are still working or if you must find another site. Sometimes the sites still have the relevant data but you may need to change the directions to access the data.

The site http://homeadvisor.msn.com gives information on demographics including median income, household size, median age, etc. The site http://www.homefair.com compares cost-of-living among U. S. cities. An excellent curve-fitting computer program Curve Expert can be downloaded from http://www.ebicom.net/~dhyamo/cmain.htm

In Part I when students search for houses in a certain city, they may have a choice of neighborhoods rather than choosing the whole city. Direct students to choose one of more neighborhoods, but the more neighborhoods chosen, the longer it will take to display all results. Up to about 5 neighborhoods is usually a reasonable search.

Part III requires students to fit a linear and an exponential curve to data. The spreadsheet Excel can do this using the LINEST (for a linear fit) and LOGEST (for an exponential fit) functions. Also the TREND (linear) and GROWTH (exponential) functions will predict future values.

If you need more background about the statistical concepts involved in the lesson, some good sites to check are:

http://davidmlane.com/hyperstat/index.html

http://www.anu.edu.au/nceph/surfstat/surfstat-home/surfstat.html

http://www.math.unb.ca/~maureen/SSCEdCom/basicstats/basicstats.html

http://www.math.unb.ca/~knight/BasicStat/$content.htm

http://www.bbns.org/us/math/ap_stats

http://www.grad.cgs.edu/wise/linksf.shtml

http://www.cvgs.k12.va.us/DIGSTATS

http://www.statsoft.com/textbook/stathome.html

http://www.stats.gla.ac.uk/steps/glossary/index.html

http://www.crpc.rice.edu/CRPC/GT/sboone/Lessons/lptitle.html

http://forum.swarthmore.edu/library/topics/statistics

http://www.psychstat.smsu.edu/introbook/skb00.htm

TI-83 instructions:

http://www.ti.com/calc/docs/act/koehler001.htm

http://www.wku.edu/~neal/manual/ti83.html

The Calculator website at the Mathematics Department of the University of Wisconsin-La Crosse will perform basic statistical calculations. If you do not have access to a simple statistical computer package or calculators with statistics options, your students may access http://www.compute.uwlax.edu/stats_htdocs/newmenu.html to perform statistical computations on-line.

In order to print out just a copy of the student worksheet, highlight this section, then copy and paste it into your word processor. You may then revise the worksheet if you wish.

Worksheets. 

1. Open Excel and enter the following data in columns A and B.

Typical Heating Expenses for 1980 to 1990

2. Now make a graph showing how all these expenses changed between 1980 and 1992.

Highlight cells A3 through B15 and click on the graph icon. Choose XY(Scatter) and then click Next.

Click on the Series Tab and type in the name "Expenses". Click Next.

Click on the Titles tab. Type "Heating Expenses" for the Title. Type "Year (1980 = Year 1)" for the Value(X)Axis and "Dollars" for the Value(Y)Axis. Click Next.

Click on As Object in and then click Finish. You can move or resize your graph as you wish. By clicking or double-clicking on any part of your graph, you can change the size, font, placement, etc. of that part. Your final graph should look similar to the following graph.

3. To find the equation of the best fitting line or exponential curve, click on the graph. Then click on Chart on the top bar. Click on Add Trendline. Click on Linear. Then click on the Options Tab and check Display Equation on Chart. Click OK. To fit an exponential cure to the data repeat as above except click on Exponential type instead of Linear. Your final graph should look something like this.

Graphing Data with the TI-83 

1. Type in the following data in lists L₁ and L₂.

2. To make a scattergram of this data, press 2^nd Y= (STAT PLOT). Turn off all the Plots by highlighting 4: PlotsOff. Press ENTER and ENTER again. Next press 2^nd Y= again and press ENTER to choose Plot1. The press ENTER to highlight ON. Cursor to Type and press ENTER to choose the first icon (scatterplot). Press L₁ for the Xlist and L₂ for the Ylist. Then press ZOOM and choose 9:ZoomStat. Your graph will look something like this.

3. To find the equation of a best-fit line, press STAT, highlight CALC, then cursor down to 4:LinReg(ax+b). Press ENTER. Then type L₁, L₂, Y₁ and press ENTER. (You can find the symbol Y₁ by pressing VARS, then Y-VARS, then 1:Function, then 1:Y₁.) To find the equation of the best-fit exponential curve, press STAT, highlight CALC, then cursor down to 0:ExpReg. Press ENTER, then type L₁, L₂, Y₂, then press ENTER. If you press GRAPH, you will see the scatterplot with the graphs of these two equations.

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1. (10 points) You will choose three different areas of the United States and find the mean and median costs of a single family home for those areas.

Access the site http://new.realtor.com Click on "Find a Home". A map of the United States will be displayed. Click on a state that you are interested in, then click on a region in that state. Finally highlight a city or section of a city among those listed. Check to see that this region has "Neighborhood Information" available. If not, find a close region that does.

What city or section did you chose? ____________________________

A screen requesting your "General Search Criteria" will be displayed. Click on "Single Family Home." Choose the "Price Range" to be $0 to No Maximum. Select "any" as the "Minimum Number of Bedrooms", "Minimum Number of Baths" and the "Minimum Square Footage". Finally "Display Homes in Groups" of 12. Now click on Find Homes.

The search results will tell you how many properties were found. Houses will be listed in order of price, from least expensive to most expensive.

How many homes were found in your search? ____________

To find the median price, approximately which home in your list must you find? ____________

(Hint: Divide the total number by 2.)

You can find the median home quickly by choosing the right page to display. How can you decide which page to look on? (Hint: Remember there are 12 homes per page.) ____________

What is the median price of homes in the area you chose? _____________

If there is a button called "Neighborhood Information" click on it and scroll down to "Average Home Qualities". What is the "Average Home Cost"? (This is the mean price) _____________

Copy the data you found above in the table below.

2. (10 points) Choose two other states and complete the following table. Click on "Find a Home" at the top of the page in the left hand corner to start a new search.

3. (5 points) Name at least five factors that impact the cost of a home. (Hint: What type of information is given about a house?)

4. (5 points) Were the median or mean costs of homes larger for the states you chose? Give an explanation for this disparity.

5. (5 points) In the past, homes in the West of the United States were the most expensive, followed by homes in the Northeast, then the South and finally the Midwest. Do you find this to be true for the data you found? You may wish to share your data with a few classmates to find home prices in these four parts of the United States.

Median Prices

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1. (12 points) Use the Internet to find the current average mortgage rates for the three states you used above. Access the site http://www.bankrate.com Click on "Mortgages" in the list given on the left hand side, then scroll down to the section titled "All About Mortgages" and finally click on "National and State Averages". Fill in the states you chose in Part I in the table below. Find the average interest rates for these states by using the "Today’s Averages For" box. Write down the average interest rate for a 30-year fixed mortgage in the following table. Also copy the median home price that you found in Part I for that state.

2. (5 points) Suppose the bank requires a 15% down payment on your house. How much money will you have to borrow? Fill in this amount in the table above under "Mortgage Amount". Hint: Take 15% of the median home price and subtract from that price (or find 85% of the home price).

3. (5 points) Calculate your monthly payment to take out a 30-year fixed rate mortgage for this amount and fill in this amount in the above table.

4. (5 points) A good estimate for how much a person can afford to pay for housing is often approximated by 25% or 1/4 of one’s income. How much would a person have to earn in a month to be able to purchase the above median homes? (Hint: if you know your monthly payment, how much should you multiply this payment by to find your salary?) Fill in this salary in the table above.

5. (10 points) What type of job do you think could support the above homes? Pick three different occupations and find out what the median salary is for them. Access the site http://stats.bls.gov/ocohome.htm (which gives on-line access to the Occupational Outlook Handbook) to find these salaries. Click on "Index to the Handbook", select the occupation you are interested in, then click on "Earnings".

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1. (15 points) Various government agencies keep records on home prices in the United States. Access the site http://www.huduser.org/periodicals/ushmc/fall97/histdat2.html Choose either the data for "New Single-Family Home Prices" or "Existing Single-Family Home Prices". Compare the median price for the entire U.S. with the mean ("average") price for the U.S. for various years. Which is larger? Give an explanation why the mean home price should be larger than the median price.

2. (15 points) Choose one of the columns of data (for example, median Midwest prices or U.S. average) and make or print out a graph showing year versus price below. (Designate 1963 as Year 1.)

3. (5 points) Does the cost of a home seem to increase by a constant additional amount from year to year? For example, does the cost increase every year by, say, $2000? Or does the cost of a home seem to increase by a constant multiple. For example, does the cost increase by, say 5% each year (that is, we would multiply by 1.05)? If you had to choose between these two possibilities, which would you choose? Why?

4. (10 points) If the cost of a home increased by approximately a constant additional amount each year, then your graph should look like a straight line. Use a calculator or appropriate software to find the best-fitting linear fit to your data. What is the equation of this line? Use this equation to predict the cost of a home in the years 2015, 2025 and 2050.

Equation: _______________________________

Cost in 2015: ________________ 2025: _______________ 2050: ________________

5. (10 points) If the cost of a home increased by approximately a constant multiple each year, then your graph should curve upward. Use a calculator or appropriate software to find the best-fitting exponential fit to your data. What is the equation of this curve? Use this equation to predict the cost of a home in the years 2015, 2025 and 2050.

Equation: _______________________________

Cost in 2015: ________________ 2025: _______________ 2050: ________________

6. (10 points) Which model, the linear or the exponential fit, do you think fits the data better? Explain why. (Hint. Graph the two equations and the data on one graph. Which curve seems to approximate the data points more closely?)

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1. (10 points) You will choose three different areas of the United States and find the mean and median costs of a single family home for those areas.

Access the site http://new.realtor.com Click on "Find a Home". A map of the United States will be displayed. Click on a state that you are interested in, then click on a region in that state. Finally highlight a city or section of a city among those listed. Check to see that this region has "Neighborhood Information" available. If not, find a close region that does.

What city or section did you chose? ____La Crosse 54601, 54602_____

A screen requesting your "General Search Criteria" will be displayed. Click on "Single Family Home." Choose the "Price Range" to be $0 to No Maximum. Select "any" as the "Minimum Number of Bedrooms", "Minimum Number of Baths" and the "Minimum Square Footage". Finally "Display Homes in Groups" of 12. Now click on Find Homes.

The search results will tell you how many properties were found. Houses will be listed in order of price, from least expensive to most expensive.

How many homes were found in your search? ___ 205 ____

To find the median price, approximately which home in your list must you find? ___ 103 ___

(Hint: Divide the total number by 2.)

You can find the median home quickly by choosing the right page to display. How can you decide which page to look on? (Hint: Remember there are 12 homes per page.) ___ page 9 ___

Divide by 12 and round up.

What is the median price of homes in the area you chose? ___$104,900 ___

If there is a button called "Neighborhood Information" click on it and scroll down to "Average Home Qualities". What is the "Average Home Cost"? (This is the mean price) __ $118,155___

Copy the data you found above in the table below.

2. (10 points) Choose two other states and complete the following table. Click on "Find a Home" at the top of the page in the left hand corner to start a new search.

3. (5 points) Name at least five factors that impact the cost of a home. (Hint: What type of information is given about a house?)

Age

Lot size

Number of rooms, number of bedrooms, number of bathrooms

Garage size

Other amenities such as deck, patio, swimming pool

Building materials

Location such as urban/suburban/country, proximity to schools, crime risk

4. (5 points) Were the median or mean costs of homes larger for the states you chose? Give an explanation for this disparity.

The mean was higher because there is not upper limit on the price of a house. If there are a few very, very expensive houses, the mean will be higher. However, the median is not affected by the actual price of these expensive homes.

5. (5 points) In the past, homes in the West of the United States were the most expensive, followed by homes in the Northeast, then the South and finally the Midwest. Do you find this to be true for the data you found? You may wish to share your data with a few classmates to find home prices in these four parts of the United States.

etc.

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1. (10 points) Use the Internet to find the current average mortgage rates for the three states you used above. Access the site http://www.bankrate.com Click on "Mortgages" in the list given on the left hand side, then scroll down to the section titled "All About Mortgages" and finally click on "National and State Averages". Fill in the states you chose in Part I in the table below. Find the average interest rates for these states by using the "Today’s Averages For" box. Write down the average interest rate for a 30-year fixed mortgage in the following table. Also copy the median home price that you found in Part I for that state.

2. (5 points) Suppose the bank requires a 15% down payment on your house. How much money will you have to borrow? Fill in this amount in the table above under "Mortgage Amount". Hint: Take 15% of the median home price and subtract from that price (or find 85% of the home price).

3. (5 points) Calculate your monthly payment to take out a 30-year fixed rate mortgage for this amount and fill in this amount in the above table.

4. (5 points) A good estimate for how much a person can afford to pay for housing is often approximated by 25% or 1/4 of one’s income. How much would a person have to earn in a month to be able to purchase the above median homes? (Hint: if you know your monthly payment, how much should you multiply this payment by to find your salary?) Fill in this salary in the table above.

5. (10 points) What type of job do you think could support the above homes? Pick three different occupations and find out what the median salary is for them. Access the site http://stats.bls.gov/ocohome.htm (which gives on-line access to the Occupational Outlook Handbook) to find these salaries. Click on "Index to the Handbook", select the occupation you are interested in, then click on "Earnings".

These are beginning salaries. Most people just starting out need to buy a house whose price is below the median cost.

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1. (5 points) Various government agencies keep records on home prices in the United States. Access the site http://www.huduser.org/periodicals/ushmc/fall97/histdat2.html Choose either the data for "New Single-Family Home Prices" or "Existing Single-Family Home Prices". Compare the median price for the entire U.S. with the mean ("average") price for the U.S. for various years. Which is larger? Give an explanation why the mean home price should be larger than the median price.

The mean is larger because there can be some very, very expensive homes which will bring the mean higher.

2. (15 points) Choose one of the columns of data (for example, median Midwest prices or U.S. average) and make or print out a graph showing year versus price below. (Designate 1963 as Year 1.)

3. (5 points) Does the cost of a home seem to increase by a constant additional amount from year to year? For example, does the cost increase every year by, say, $2000? Or does the cost of a home seem to increase by a constant multiple. For example, does the cost increase by, say 5% each year (that is, we would multiply by 1.05)? If you had to choose between these two possibilities, which would you choose? Why?

The cost doesn’t seem to change by a constant additional amount because in the 1960’s the prices increased by about $1000 each year, but in the 1970’s, prices increased by about $2000-$4000 each year. It seems the cost of a house increased about 4% to 7% each year except in the last 10 years, when it decreased by much less.

4. (10 points) If the cost of a home increased by approximately a constant additional amount each year, then your graph should look like a straight line. Use a calculator or appropriate software to find the best-fitting linear fit to your data. What is the equation of this line? Use this equation to predict the cost of a home in the years 2015, 2025 and 2050.

Equation: y = 4099.3x - 3710.7

Cost in 2015: $213,552

Cost in 2025: $254,545

Cost in 2050: $344,729

5. (10 points) If the cost of a home increased by approximately a constant multiple each year, then your graph should curve upward. Use a calculator or appropriate software to find the best-fitting exponential fit to your data. What is the equation of this curve? Use this equation to predict the cost of a home in the years 2015, 2025 and 2050.

Equation: y = 16289e ^0.0693x

Cost in 2015: $641,231

Cost in 2025: $1,282,273

Cost in 2050: $5,889,874

6. (10 points) Which model, the linear or the exponential fit, do you think fits the data better? Explain why. (Hint. Graph the two equations and the data on one graph. Which curve seems to approximate the data points more closely?)

The linear model seems to fit better overall. The exponential model fits better from 1963 to about 1988 but then rises much too quickly.