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EDITED Resources. The resources on this page have been aligned with the revised K-12 mathematics TEKS. Necessary updates to the resources are in progress and will be completed Fall 2006. These revised TEKS were adopted by the Texas State Board of Education in 2005–06, with full implementation scheduled for 2006–07.

(M.1) The student uses a variety of strategies and approaches to solve both routine and non-routine problems.

(M.1.A) The student uses a variety of strategies and approaches to solve both routine and non-routine problems. The student is expected to compare and analyze various methods for solving a real-life problem.

Clarifying Activity

Students investigate questions connected to real-world situations such as the following:

Doctors sometimes need to know the concentration of a certain substance, such as medication or poison, in a patient's bloodstream after a given amount of time. For example, when a person gets a 500 mg injection of penicillin, the amount of penicillin in the person's bloodstream decreases by about 20% per hour. How can we determine how much penicillin remains in a person's bloodstream after several hours?

Students discuss the initial amount of penicillin present in the bloodstream (a = 500 mg) and the rate of decrease per hour in order to determine the decay factor, the factor that tells the percent of penicillin left in the bloodstream after one hour (b = 1 - 0.20 = 0.80).

In groups of 2–4, students create two different representations to model the situation. Groups post and explain their representations. Possible models may include a picture, graph, calculator program, calculator iterations, symbolic model, etc. (Refer to the Additional Clarifying Activity for M.8.A for sample representations.) Students then compare and analyze their models to understand decaying behavior.

Additional Clarifying Activity

Students predict probabilities from multistage experiments in which each stage has the same two equally-likely outcomes and compare the results to the binomial expansion model (see the Clarifying Activity in M.4.B). Collecting, organizing, and interpreting the data from the multistage experiments creates a statistical mathematical model in which data analysis is used to make conclusions and draw inferences. The binomial expansion is an example of a structural mathematical model created directly from the structure of the situation, without the need for data collection or data analysis. Students discuss the advantages and disadvantages of the two methods.

(M.1.B) The student uses a variety of strategies and approaches to solve both routine and non-routine problems. The student is expected to use multiple approaches (algebraic, graphical, and geometric methods) to solve problems from a variety of disciplines.

Clarifying Activity

Students use a variety of approaches to solve the following problem:

As a new building contractor, you have carefully planned a patio project and ordered a specific amount of sand to fill between the bricks. When the sand is delivered, however, you look at the pile and question whether or not you received as much as you actually ordered. Realizing that reloading and remeasuring the volume of sand is impractical, how can you verify the amount you actually received? Also, it would be very practical for you to derive a solution to this problem that would provide you with a way to answer this question for future projects of different sizes.

Students can approach the problem geometrically using the assumption that the pile of sand approximates a cone. The formula for the volume of the cone whose height and base radius is equal to that of the pile of sand, V = 1/3pr²h, would be a good model for determining the amount of sand. The students pour seven samples of known volumes of sand into a pile and calculate the radius and height of each sample. The data is recorded in the following table.

Based on the data, students find that the computed values for the volume using the assumption that the pile of sand is in the shape of a cone are less than the actual volumes. Students use graphing calculators to make a scatterplot and do a linear regression on the actual volume and computed volume to find that the actual volume is 1.08 times the computed value. Thus, the current model for the volume of the pile of sand is V = 1.08x(1/3)pr²h. Realizing that the circumference of the base of the pile of sand is easy to calculate, but the height is not, the students assume a direct proportional relationship between the height of the cone and the radius of the base, or h = kr where k is the constant of proportionality. Calculating the height-to-radius ratio of the data in the table, the students adjust the model to V = 0.36pr²(0.60r) = 0.216pr³.

(M.1.C) The student uses a variety of strategies and approaches to solve both routine and non-routine problems. The student is expected to select a method to solve a problem, defend the method, and justify the reasonableness of the result.

Clarifying Activity

Students select and defend a method to solve the following problem:

There are 12 different certificates that are randomly and evenly distributed in boxes of Sweet Pops cereal. If you get all 12 certificates, you can send them in to get a free CD. How many boxes of cereal should you expect to have to buy to get all 12 certificates?

(Possible methods include doing many trials of rolling a 12-sided die and counting the number of rolls needed to get all 12 numbers, doing many trials of spinning a spinner with 12 sections and counting the number of spins needed to get all 12 numbers, or doing many trials of drawing numbered tiles one at a time with replacement from a bag of 12 tiles until all 12 numbers have been drawn.)

Students record their data, justify why many trials must be done, and determine how to interpret the results of the trials to solve the problem. Students discuss how their method of solving the problem would be changed if the situation was changed from 12 to 15 certificates, or if there were only half as many of one of the certificates as there were of all the others.

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(M.2) The student uses graphical and numerical techniques to study patterns and analyze data.

(M.2.A) The student uses graphical and numerical techniques to study patterns and analyze data. The student is expected to interpret information from various graphs, including line graphs, bar graphs, circle graphs, histograms, scatterplots, line plots, stem and leaf plots, and box and whisker plots to draw conclusions from the data.

Clarifying Activity

Students make a bar graph (or stem-and-leaf plot) of the results of the simulation trials in Clarifying Activity M.1.C (e.g., the number of rolls needed to get all 12 numbers) and use the graph to conclude what a reasonable number of boxes would be that they would expect to buy to get all 12 certificates.

Additional Clarifying Activity

Students get data from The Age of Female Oscar Winners (from the Exploring Data website) and use the list capabilities of a graphing calculator to make a scatterplot comparing the ages of female Oscar winners to the year that they won. Students observe trends in the scatterplot to conclude whether winning females have been getting consistently older or younger.

(M.2.B) The student uses graphical and numerical techniques to study patterns and analyze data. The student is expected to analyze numerical data using measures of central tendency, variability, and correlation in order to make inferences.

Clarifying Activity

In Clarifying Activity M.1.C, students use measures of central tendency (mean, median and mode) and measures of variability (range, variance, and standard deviation) to infer the number of boxes of cereal they would expect to have to buy to get all 12 certificates. Students use the characteristics of each measure of central tendency to determine which one gives the most reasonable solution.

Additional Clarifying Activity

Students use the list capabilities of a graphing calculator to find the correlation coefficient describing the correlation of the age of female Oscar winners to the year they won and compare the correlation results to their observations of the data in the scatterplot in Clarifying Activity M.2.A.

(M.2.C) The student uses graphical and numerical techniques to study patterns and analyze data. The student is expected to analyze graphs from journals, newspapers, and other sources to determine the validity of the stated arguments.

Clarifying Activity

Students collect examples of graphs from the local newspaper and give mathematical justifications for misconceptions that might occur from the graph. For example, in the top graph below the median cost of renting an apartment in Bryan-College Station, Texas is 34% of average personal income and in Bloomington, Indiana is 32.2% of average personal income. However, the heights of the symbols indicate that an apartment in Bryan-College Station costs more than twice the percent of personal income that an apartment in Bloomington, Indiana does.

Additional Clarifying Activity

Students compare the data presented in the following graph from the March 29, 1998 Houston Chronicle article "Hopes for a Healthy Harvest" to the claim in the article that spring onions in 1998 in Texas were planted on 7% fewer acres than in 1997 and 25% fewer than in 1996 and that the expected onion crop harvest in 1998 is 10,800 acres.

Students determine whether or not the data presented in the graph supports the claim made and justify their decision.

(M.2.D) The student uses graphical and numerical techniques to study patterns and analyze data. The student is expected to use regression methods available through technology to describe various models for data such as linear, quadratic, exponential, etc., select the most appropriate model, and use the model to interpret information.

Clarifying Activity

Students review the distinguishing characteristics of linear, quadratic, and exponential functions, as shown in the graphs below:

Students use a motion detector and electronic data-collection device to collect data from the following three ball experiments, predict what family of functions the data belongs to, and use the regression methods available on a graphing calculator to identify an appropriate mathematical model for and answer questions about each one.

Experiment 1: Drop the Ball

Place the motion detector on the floor, facing up. Place a chair next to the detector, stand on the chair, and drop a foil-covered Nerf ball onto the detector from as high as you can reach. Determine for how long (in seconds) you should collect data and an appropriate range (in meters) for your experiment. Record a data point every 0.1 second (and save your data). Make a sketch of a graph of the data points and predict what type of function best fits the data. Estimate a mathematical model for your data. Then use the regression capabilities on the graphing calculator to find a mathematical model for the data. Compare the two with questions such as:

• How are the two models alike? How are they different?
• What do each of the numbers in the models represent?
• What do each of the variables in the models represent?
• According to your estimated model, how long did the ball stay in the air?
• According to the model you found with the calculator, how long did the ball stay in the air?
• According to your estimated model, how high was the ball when it was dropped?
• According to the model you found with the calculator, how high was the ball when it was dropped?
• Are these reasonable answers? Why or why not?

Experiment 2: Throw the Ball

This time, set the motion detector level on a table or desk, step back a comfortable distance, and throw the foil-covered Nerf ball at the detector. Determine for how long (in seconds) you should collect data and an appropriate range (in meters) for your experiment. Record a data point every 0.1 second (and save your data). Make a sketch of a graph of the data points and predict what type of function best fits the data. Estimate a mathematical model for your data. Then use the regression capabilities on the graphing calculator to find a mathematical model for the data. Compare the two with questions such as:

• How are the two models alike? How are they different?
• What do each of the numbers in the models represent?
• What do each of the variables in the models represent?
• According to your estimated model, how far from the detector was the ball when it was thrown?
• According to the model you found with the calculator, how far from the detector was the ball when it was thrown?
• According to your estimated model, how long did it take the ball to hit the detector?
• According to the model you found with the calculator, how long did it take the ball to hit the detector?
• According to your estimated model, what was the average velocity of the ball?
• According to the model you found with the calculator, what was the average velocity of the ball?
• Are these reasonable answers? Why or why not?

Experiment 3: Bounce the Ball

Use duct tape to fix the motion detector so that it faces downward from a height of about 6 feet, over a smooth, hard surface. Since the Nerf ball doesn't bounce well, use a golf ball. Hold the golf ball against the detector itself and then let it go, allowing it to bounce at least 6 times. Determine for how long (in seconds) you should collect data and an appropriate range (in meters) for your experiment. Record a data point every 0.1 second (and save your data). Make a sketch of a graph of the data points. Then, consider the graph of only the set of high points to which the ball bounced before dropping again and predict what type of function best fits that set of points. Estimate a mathematical model for the set of high points. Then use the regression capabilities on the graphing calculator to find a mathematical model for the set of high points. Compare the two with questions such as:

• How are the two models alike? How are they different?
• What do each of the numbers in the models represent?
• What do each of the variables in the models represent?
• According to your estimated model, from what height was the ball dropped?
• According to the model you found with the calculator, from what height was the ball dropped?
• According to your estimated model, how high would the ball bounce on the 8th bounce?
• According to the model you found with the calculator, how high would the ball bounce on the 8th bounce?
• Are these reasonable answers? Why or why not?

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(M.3) The student develops a plan for collecting and analyzing data in order to make predictions.

(M.3.A) The student develops a plan for collecting and analyzing data in order to make predictions. The student is expected to formulate a meaningful question, determine the data needed to answer the question, gather the appropriate data, and draw a reasonable conclusion.

Clarifying Activity

Students consider having $200 per month to invest in a choice of stocks, bonds, or annuities. Students use the Internet to find information about the requirements and returns on each of these kinds of investments. For example,

Students use a table or spreadsheet to create graphs to compare the different types of investments and answer questions. For example,

• How would the graph change if the initial investment increased? Decreased?
• How would the graph change if the percent of return increased? Decreased?

• How would the graph change if the initial investment increased? Decreased?
• How would the graph change if the percent of return increased? Decreased?

• How would the graph change if the initial investment increased? Decreased?
• How would the graph change if the percent of return increased? Decreased?

• Which is the best option after 5 years? 10 years? 20 years? 30 years? How do you know?
• What type of function is each one: linear, quadratic, exponential, or other? How do you know?
• What do the points of intersection represent?
• If you could invest $250.00 per month at the above rates, how would the graphs change? How would the answers to the previous questions change?
• How would each graph change if the rate of return increased? Decreased?
• How would your answers to the first question change if the rates of return increased? Decreased?

(M.3.B) The student develops a plan for collecting and analyzing data in order to make predictions. The student is expected to communicate methods used, analyses conducted, and conclusions drawn for a data-analysis project by written report, visual display, oral report, or multi-media presentation.

Clarifying Activity

Students collect data to study the fairness of different models of decision-making (voting) and use ratio, proportion, percent, linear equations, and linear inequalities to present their findings to the class. For example:

A group of students are planning an end-of-year party. Parents and friends are being invited. The organizer decides to take a vote to see what kind of musical entertainment they will have. In the first vote taken, everyone gives one vote to their favorite choice, and the results are as follows:

One member of the committee opposed to classical music suggests a "fairer" vote where each student ranks his or her choices from 1 to 6, with 1 representing the favorite. In this vote, the results are as follows:

A third type of vote was taken, in which each student voted for as many of the choices as he or she found acceptable, with the results as follows:

Students prepare a written report and a visual display on the "fairness" of each vote and make an oral presentation to the class. The report and presentation should address questions such as:

• What percentage of voters selected the winner in each model? What percentage of voters would have preferred an alternate choice?
• What is the minimum number of votes needed in each model for the winner to win by a majority? By a plurality?
• Suppose only two voters changed their vote in each model. In which models could it make a difference in the winner? How?
• What are the advantages and disadvantages of each of the three models of decision-making?
• Which model is most fair in this situation? Why?
• In what types of situations might the other models be most fair? Why?

(M.3.C) The student develops a plan for collecting and analyzing data in order to make predictions. The student is expected to determine the appropriateness of a model for making predictions from a given set of data.

Clarifying Activity

Students compare different models for predicting growth and determine which model is best for given situations. For example:

In an Addition Model for representing growth, we assume that the population we are observing increases by the addition of a constant quantity each time period. Use the constant operation feature on your calculator to make a table that shows the population over time of a colony of 75,000 bacteria that, with combinations of divisions and deaths, increases by 12,000 organisms each hour. Identify the common difference between the populations at consecutive time periods and write a function that represents the relationship of the values in the table [y = 75,000 + (x)(12,000)] where y is the population of bacteria, x is the number of hours, and 12,000 is the common difference between consecutive time periods. Use a graphing calculator to make a graph of the Addition Model.

In the Multiplication Model for representing growth, we assume that the population increases at the same rate each time period. In other words, the population at the beginning of one time period is multiplied by a constant factor, say 1.16, to determine the population at the beginning of the next time period. Use the constant operation feature on your calculator to make a table that shows the population over time of a colony of 75,000 bacteria that, with combinations of divisions and deaths, increases by 16% each hour. Identify the common ratio between the populations at consecutive time periods and write a function that represents the relationship of the values in the table [y = (75,000)(1.16)^x] where y is the population of bacteria at each time period, x is the number of hours, and 1.16 is the ratio of the population at each time period to the population of the previous time period. Use a graphing calculator to make a graph of the Multiplication Model on the same coordinate plane with the graph of the Addition Model.

Students compare and contrast the general shapes of the models and analyze each model by responding to questions such as:

• What kind of function (linear, quadratic, or exponential) is the Addition Model? the Multiplication Model? How do you know?
• Based on your understanding of the shapes of the graphs, which model predicts a greater population after 10 hours?
• In which model does the population seem to be increasing faster?
• Describe a situation in which the Addition Model would be most appropriate.
• Describe a situation in which the Multiplication would be most appropriate.
• What kind of situation would combine the two models, for example, growth described by the model y = (75,000)(1.16)^x + 12,000x? [Possible Student Response: You could use this model to find the total population (y) after a certain number of hours (x) if you start with a population that includes a group of 75,000 bacteria that grows at its normal rate of 16% per hour (birth rate) and every hour 12,000 new bacteria enter the system (in migration).]

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(M.4) The student uses probability models to describe everyday situations involving chance.

(M.4.A) The student uses probability models to describe everyday situations involving chance. The student is expected to compare theoretical and empirical probability.

Clarifying Activity

Students discuss the theoretical probability for getting heads when a coin is tossed:

Students determine the empirical results of tossing a single coin by performing 20 trials and completing the data table below:

Students compare the two ratios. Students then combine their empirical results to form a class ratio for the empirical results and compare the class's empirical results to the theoretical probability. Students apply this concept to the following genetics problem:

The gender of mammals is determined by the union of an X chromosome from the mother and either an X or a Y chromosome from the father. Two X chromosomes, XX, determine a female and XY chromosomes determine a male offspring. This biological phenomenon can be modeled by simple probability. Use a spinner or a graphing calculator's random number generator to design an experiment that models the probability of having a male offspring, assuming that an X chromosome and a Y chromosome from the father each have an equal chance of joining an X chromosome from the mother.

Students compare the results from their experiment to the theoretical probability of having male offspring. Students research statistical records for actual birth rates of male vs. female children in a population. Students summarize their results and explain possible differences between the theoretical and empirical probabilities of having male children and actual birth rates of male children.

(M.4.B) The student uses probability models to describe everyday situations involving chance. The student is expected to use experiments to determine the reasonableness of a theoretical model such as binomial, geometric, etc.

Clarifying Activity

Students predict probabilities from multistage experiments in which each stage has the same two equally-likely outcomes and compare the results to the binomial model. Students toss two coins 20 times to determine the empirical results of 0, 1, or 2 heads occurring. Students record the results in a table like the one below and calculate the ratio describing the experimental results of each event:

Students compare the results to the expansion of the binomial:

(H + T)² = 1HH + 2HT + 1TT

Students repeat the experiment with 40 tosses and 80 tosses and recalculate the experimental results for P(0), P(1), and P(2). Students explain how the experimental probabilities for 40 trials and 80 trials compare to the predicted probabilities from the binomial expansion:

• P(0 heads out of 2) = 1/4
• P(1 head out of 2) = 2/4
• P(2 heads out of 2) = 1/4

Students extend this concept to the following genetics problem:

A couple plan to have three children. What is the probability that the couple will have 0 girls? 1 girl? 2 girls? 3 girls? Design an experiment using coins that will predict the probabilities for each event. Assume that boys and girls have equal chances of being born. Compare the results to the binomial expansion:

(B + G)³ = 1BBB + 3BBG + 3GGB + 1GGG

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(M.5) The student uses functional relationships to solve problems related to personal income.

(M.5.A) The student uses functional relationships to solve problems related to personal income. The student is expected to use rates, linear functions, and direct variation to solve problems involving personal finance and budgeting, including compensations and deductions.

Clarifying Activity

Students develop a personal budget, identifying percentages they want to allot to housing, food, clothing, entertainment, personal care, miscellaneous. Students use rates to describe their allotments and compute monthly amounts based on given yearly incomes. Students look up some data on average salaries and living expenses in a given area to see if their percentages are appropriate or need to be adjusted.

(M.5.B) The student uses functional relationships to solve problems related to personal income. The student is expected to solve problems involving personal taxes.

Clarifying Activity

Students investigate solutions to the following problem:

You have the option of buying stereo speakers from two different stores. One store is located down the street and charges 8.25% sales tax on all purchases. You can purchase from the other store through a catalog where no tax is charged, but shipping and handling fees are applied to every purchase, as shown in the following chart:

Where should you buy the speakers?

To compare the two options, students illustrate each option using a table or spreadsheet and a graph. For example,

Graph of T_A and T_B:

Click here to see a bigger version of this graph.

Students explore the graphs and mathematical models for each option:

• T_A = C(1.0825) where C is the cost of the item, and
• T_B = C + S where C is the cost of the item and S is the S & H cost

answering questions such as:

• How are the two graphs alike? How are they different?
• What type of function is represented in situation A? How do you know?
• What type of function is represented in situation B? How do you know? (Note: Situation B is actually a step function. Is each piece of the function linear? How can you tell?)
• Does the difference between the two totals remain constant? How can you tell from the table? How can you tell from the graph of the difference?
• What do the points where the two graphs intersect represent? What would that information look like in the table?
• Make a graph of the difference vs. the cost of the item. Where on this graph would the two options be equal?

(M.5.C) The student uses functional relationships to solve problems related to personal income. The student is expected to analyze and make decisions about banking.

Clarifying Activity

Students investigate possible solutions to the following problem:

You decide to open up a checking account at your local bank. The bank offers you several options:

• Regular checking: There is a $4.00 monthly service charge plus $0.25 per check.
• Value checking: The first 5 checks are free. If you write more than 5 checks, there is a monthly service charge of $6.00, plus $0.20 per check over the first 5.

Which is the best option for writing 5 checks? 15 checks? 30 checks?

Students use a table or a spreadsheet to represent the monthly charge for each account for given numbers of checks written:

and compare the table to the mathematical models of the monthly charge for each option:

Students graph the functions and answer questions such as:

• What types of functions are these? How do you know?
• What is the independent variable?
• What does the point of intersection at (20, 9) represent?
• What does the point at x = 5.5 for each graph represent?
• What are an appropriate domain and range for these functions?
• If the bank offered a third option—4% interest per month and $0.75 per check—how would it compare to the other two options if you maintain a balance of $100?

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(M.6) The student uses algebraic formulas, graphs, and amortization models to solve problems involving credit.

(M.6.A) The student uses algebraic formulas, graphs, and amortization models to solve problems involving credit. The student is expected to analyze methods of payment available in retail purchasing and compare relative advantages and disadvantages of each option.

Clarifying Activity

Students investigate possible solutions to a problem similar to the following:

Foley's is having a sale on all of their shoes. You have the option of saving 75% off the original price or taking an additional 25% off the reduced price of 50% off.

Which of these options is better? Why? Does the better option change as the price of the shoes increases or decreases? Why or why not? Identify mathematical models and create data that help you answer these questions and justify your answers.

For example, students might create the algebraic formulas

• N_A = P - 0.75P to model the new price with 75% off, and
• N_B = P - 0.5P - 0.25(P - 0.5P) to model the new price with 25% off of the 50% reduction.

Students use the models to create either a table or a spreadsheet to compare the two options.

Students explore the mathematical models with questions such as:

• What is the relationship between the original price and the difference between the two options? (For each increase of $5, there is a difference of $0.62 or $0.63; or for each increase of $10.00, there is a difference of $1.25)
• For situation A, the sales clerk is told to ring up P x 0.75. How is this model like the one you used for N_A?
• For situation B, the sales clerk is told to ring up P x 0.5 x 0.25. How is this model like the one you used for N_B?
• Graph the two options. What is the independent variable? What is the dependent variable? What is an appropriate viewing window on the graphing calculator? Are these functions linear, quadratic, or exponential? How do you know?
• What is the total percent off of the original price for situation B?
• How much should the additional percentage off of the 50% reduced price be to make it equal to 75% off?

Additional Clarifying Activity

Students use algebraic formulas and spreadsheets to compare the costs associated with two credit cards with different minimum payments, grace periods, annual fees, etc. and determine which credit card is most appropriate for which personal finance situations.

(M.6.B) The student uses algebraic formulas, graphs, and amortization models to solve problems involving credit. The student is expected to use amortization models to investigate home financing and compare buying and renting a home.

Clarifying Activity

Students use amortization models (such as the calculation tables found at www.kiplinger.com on the Internet) to answer questions connected to the following situation:

The Gonzales family is moving to Austin. They are trying to make a decision about their housing options in Austin. They found an apartment they can rent for $700 per month, not including utilities. They are also considering purchasing a home that costs $110,000.

How large would their down payment have to be in order to have monthly payments of $700 if the annual interest rate compounded monthly is 7.5% on a 30-year loan?

What if they wanted to finance it for only 15 years?

Use a graphing calculator with financial function capabilities to graph the amount of principal versus the amount of interest in each payment.

Based on this information, do you think the Gonzales family should rent the apartment or buy the house? Why? (Include advantages and disadvantages of each to justify your conclusion.)

(M.6.C) The student uses algebraic formulas, graphs, and amortization models to solve problems involving credit. The student is expected to use amortization models to investigate automobile financing and compare buying and leasing a vehicle.

Clarifying Activity

Students use tables and formulas to answer questions related to situations such as the following:

Laura wants to buy a new car. She needs to borrow $7000, and the annual interest rate for her car loan is 8.5% of the unpaid balance, compounded monthly.

If she makes monthly payments of $225, how long will it take her to pay off the loan?

What is the total amount she will have paid back over time?

Write a recursive routine to create a table to answer these questions.

Students use the algebraic formulas I = B(r/12) (where I is the interest, B is the unpaid balance, and r is the yearly interest rate); P - I = N (where P is the payment amount, I is the interest, and N is the note reduction); and B_n-1 - N_n = B_n as a recursive process to create a table or spreadsheet such as the following:

Students compare their solution to how many monthly payments it will take to pay off the $7000 (using the recursive routine) to results using the mathematical model for paying off a loan:

where M is the amount of the monthly payments, P is the original loan amount, r is the interest rate per year, and n is the number of monthly payments.

Students use their recursive routine and the mathematical model to answer questions such as:

• What happens to the monthly payment if the interest rate goes up one quarter of a point?
• If Laura could afford $25.00 per month more, how would that affect the time to pay off the loan?
• If Laura wanted to prepay the remainder of the loan after 1 year, how much would it cost?
• If you can afford to pay $300 per month for 5 years, how much can you borrow at 8.5% annual interest? 8.0%? 9.0%?

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(M.7) The student uses algebraic formulas, numerical techniques, and graphs to solve problems related to financial planning.

(M.7.A) The student uses algebraic formulas, numerical techniques, and graphs to solve problems related to financial planning. The student is expected to analyze types of savings options involving simple and compound interest and compare relative advantages of these options.

Clarifying Activity

Students are presented with the following question, "How long does it take to save enough money for a car (or a college education, etc.)?" and use various rates and mathematical models to investigate the possible responses.

For example, students compare saving $2400 yearly without interest (keeping it in a safe place without depositing it into a bank) to saving the same amount yearly in an account with an annual interest rate of 7%, compounded yearly. Students use spreadsheets to make the comparison.

Students compare the mathematical models for representing the total amount saved under each set of conditions

• T_a = 2400 x n (a direct variation where n is the number of years)
• T_b = 2400[(1 + 0.07)ⁿ - 1] — 0.07 (where n is the number of years)

by looking at the data in the spreadsheet and answering questions such as:

• How are the total amounts saved each year determined in each situation?
• Is situation (a) a linear, quadratic, or exponential function? How do you know?
• Is situation (b) a linear, quadratic, or exponential function? How do you know?
• In situation (b), how is the amount of interest earned for each year determined?
• In situation (b), why does the amount of interest earned each year change?
• How would the mathematical model change if you contributed only $1000 each year?
• How would the mathematical model change if the interest rate was 8% instead of 7%?
• How much would you have after 10 years if the interest rate was 7%? How does that compare to 8%?

Students then answer similar questions to investigate the mathematical model for a monthly contribution of $200 at an annual percentage rate of 7% compounded monthly:

T3 = $200[(1 - 0.07/12)¹²ⁿ - 1] — 0.07/12 (where n represents the number of years)

and use a spreadsheet to compare these results to the previous results of making a yearly contribution of $2400 and earning interest yearly.

Students then use the model to find combinations of monthly contribution, interest rate, and length of time to save $10,000 for a car (or $40,000 for college expenses, etc.).

(M.7.B) The student uses algebraic formulas, numerical techniques, and graphs to solve problems related to financial planning. The student is expected to analyze and compare coverage options and rates in insurance.

Clarifying Activity

Students investigate solutions to the following problem:

A car owner must decide if he should take out a $100.00 deductible collision policy in addition to his liability insurance policy. Records show that each year, in his area, 8% of the drivers have an accident that is their fault or for which no fault is assigned and that the average cost of repairs for these types of accidents is $1,000.00. If the $100.00 deductible collision policy costs $100.00 per year, would he save money in the long run by buying the insurance or should he just take a chance? In other words, what are the expected costs if he has the policy and if he doesn't have the policy?

Students write algebraic equations to represent the expected values and make a conclusion:

Without collision insurance, his expected cost per year is $1000(0.08) = $80. With insurance, his expected cost per year is $100 + $100(0.08) = $108.

Thus, he could expect to save money without collision insurance if he is as good as the average driver.

Students generalize their algebraic equations to model the expected values for different policy and repair costs, different probabilities, and different deductibles. For example:

Expected cost without insurance = pr, where p is the probability of a driver having an accident and r is the average cost of repair.

Expected cost with insurance = I + dp, where I is the insurance premium, d is the deductible on the policy and p is the probability of a driver having an accident.

Students use these models to investigate questions such as:

• If he wanted a $250.00 deductible collision policy, what might the yearly cost be for it to be comparable to the $100.00 deductible collision policy?
• What if the percentage of drivers in his area that have had an accident that was their fault or where no fault was assigned increases or decreases? How does this affect his decision of whether or not to have a collision policy?

(M.7.C) The student uses algebraic formulas, numerical techniques, and graphs to solve problems related to financial planning. The student is expected to investigate and compare investment options including stocks, bonds, annuities, and retirement plans.

Clarifying Activity

See the Clarifying Activity at M.3.A.

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(M.8) The student uses algebraic and geometric models to describe situations and solve problems.

(M.8.A) The student uses algebraic and geometric models to describe situations and solve problems. The student is expected to use geometric models available through technology to model growth and decay in areas such as population, biology, and ecology.

Clarifying Activity

See the Clarifying Activity in M.3.C.

Additional Clarifying Activity

Students investigate questions connected to real-world situations such as the following:

Doctors sometimes need to know the concentration of a certain substance, such as medication or poison, in a patient's bloodstream after a given amount of time. For example, when a person gets a 500 mg injection of penicillin, the amount of penicillin in the person's bloodstream decreases by about 20% per hour. How can we determine how much penicillin remains in a person's bloodstream after several hours?

Students discuss the initial amount of penicillin present in the bloodstream (a = 500 mg) and the rate of decrease per hour in order to determine the decay factor, the factor that tells the percent of penicillin left in the bloodstream after one hour (b = 1 - 0.20 = 0.80).

Students represent the decay situation with a large piece of paper representing the initial 500 milligrams of penicillin. Students cut off one fifth of the piece of paper to show the amount of penicillin left after one hour and label it with the amount of penicillin it represents (500 mg x 0.80 = 400 mg). Students continue to cut off 20% of the piece of paper left each time until they understand the decaying action and the mathematics that goes with each step (the new concentration = 0.80 x the previous concentration).

From their understanding of the decaying behavior (((a x b) x b) x b) . . ., students create the function y = ab^t to find the amount of penicillin (y) in milligrams in a person's bloodstream a certain amount of time (t hours) after getting a 500 mg injection.

Students use a graphing calculator to graph the function and use the graph to predict how much penicillin is left in the bloodstream 2 hours, 4 hours, and 8 hours after getting a 500 mg injection. Students use the mathematical model to calculate these concentrations and compare them to their predictions.

Students also make predictions about what happens to the concentrations over time if the initial dosage is increased or decreased, or if the rate of decay is increased or decreased, and adjust the mathematical model to test their predictions.

Students connect their results to the importance of carefully following directions in taking medication.

(M.8.B) The student uses algebraic and geometric models to describe situations and solve problems. The student is expected to use trigonometric ratios and functions available through technology to calculate distances and model periodic motion.

Clarifying Activity

Students discuss examples of objects that move in a circular path, such as a bicycle wheel or Ferris wheel. Students predict what the graph of an object's position might look like if the object moves in a circular pattern, where the x axis represents time and the y axis represents height above the ground. Students draw a sketch of their prediction on a coordinate plane. Students compare their predictions with the graphs of the sine and cosine functions.

Students connect their observations to periodic motion in a real-world situation such as the following:

A Ferris wheel has a radius of 25 feet. The center of the wheel is 30 feet above the ground, and the wheel rotates at a constant speed of 20ƒ per second. The height h, in feet, of a passenger above the ground is given as a function of time t, in seconds, with the mathematical model h = 30 + 25 sin(20ƒ x t).

Students use technology to evaluate the function to complete the data table below.

Students use graphing calculators to graph the function. Students discuss which part of the graph models the problem, determine an appropriate viewing window to isolate that portion of the graph, and answer the following questions:

• What is the height of the passenger at the top of the Ferris wheel? How do you know?
• When does the passenger first reach this height? How do you know?
• What is the height of the passenger at t = 36 sec? at t = 40 sec? at t = 60 sec?
• How long will it take the passenger to travel one complete turn of the Ferris wheel? How do you know?
• How many times will a passenger go around if the Ferris wheel ride lasts 5 minutes?
• What if the Ferris wheel is adjusted to go slower? How will the function change? How will the graph change?
• What if the Ferris wheel is adjusted to go faster? How will the functions change? How will the graph change?

(M.8.C) The student uses algebraic and geometric models to describe situations and solve problems. The student is expected to use direct and inverse variation to describe physical laws such as Hooke's, Newton's, and Boyle's laws.

Clarifying Activity

Students collect data from an experiment exhibiting the direct variation involved in Hooke's Law.

Attach a paper cup to a rubber band with paper clips and suspend the cup from the edge of a table. Place an increasing number of pennies in the cup, measuring the length of the rubber band with each increase of pennies. Record the data in a table like the one below.

Draw a scatter plot of the data points and use a graphing calculator to determine the line of best fit. From the line of best fit, write a linear equation that expresses the relationship between the number of pennies, x, and the length of the stretch, y.

Students use the model to predict how much the rubber band will stretch if 25, 50, or 100 pennies are placed in the cup. Students perform the experiment with 25, 50 and 100 pennies and compare their results to the model's predictions.

Students use the model to predict how many pennies it will take to stretch the rubber band to a length of 5 cm, 10 cm, or 100 cm. Students perform the experiment to test the model's predictions.

Students predict how the scatter plots and mathematical models would change with thicker or thinner, longer or shorter rubber bands and perform the experiment with new rubber bands to test their predictions.

Additional Clarifying Activity

Given Boyle's Law involving the relationship among three properties of a gas in a container:

PV = kT where P is the pressure of the gas in force per area, V is the volume of the container, T is the temperature of the gas, and k is a constant value determined by the units used for the other quantities

students analyze the model and connect it to real-world situations, through questions such as:

• If the temperature of a gas in a container increases and the volume stays the same, what happens to the pressure? When might a situation like this occur?
• If the volume of a gas-filled container increases and the temperature stays the same, what happens to the pressure? When might a situation like this occur?
• If the air pressure in a room drops, but the temperature remains the same, what happens to a filled balloon?
• What happens to the model if you measure the temperature on a Fahrenheit scale? A Celsius scale? A Kelvin scale?

Students could use a CBL as described on the Internet in the guide to exploring Boyle's model to verify the values of k for various units.

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(M.9) The student uses algebraic and geometric models to represent patterns and structures.

(M.9.A) The student uses algebraic and geometric models to represent patterns and structures. The student is expected to use geometric transformations, symmetry, and perspective drawings to describe mathematical patterns and structure in art and architecture.

Clarifying Activity

Students explore the relationship between size transformations, scale drawings, the window on the graphing calculator, and distorted presentations of data.

Students are given a picture that is exactly square. They are then to redraw the picture on a rectangular piece of paper. Before they draw the picture, they are to predict how they think the picture will be different and explain why.

Students use ratios to describe the relationship between the pictures.

Students are then to graph y = cos x on their graphing calculator. Based on what they have learned from the grid drawing above, they are to predict what will happen when they change the window. Have the students change the y and the x min and max separately and record what happens to the graph.

Students discuss how changing the values of the window makes the same graph look like different graphs. Students draw conclusions as to how the graph varies in relation to how the window is changed.

Students discuss how advertisers and researchers use various types of graphs and make conclusions as to how the same data may appear to be different based on the scales used. Students discuss questions such as, "Why would people want to represent the same data differently?"

Extensions:

• The Earth is circular and yet some maps of the Earth are flat. How does a flat map distort the dimensions of different countries?
• Compare various types of maps and see if there is any type of pattern or ratio of changes.

Additional Clarifying Activity

Students use geometric transformations to create and describe the seven line symmetry groups and the seventeen plane symmetry groups (or "wallpaper patterns"). For example, the linear pattern

is described by the repetition of a linear translation, T, of a basic design, whereas the linear pattern

is described by the repetition of a 180-degree rotation, Ro, of a basic design.

Students can identify the basic line and plane symmetry groups in famous art, such as Escher designs or Moorish tilings from the Alhambra in Spain. (See The Surface Plane, Book Two of The Golden Relationship: Art, Math and Nature by Martha Boles and Rochelle Newman, published by Pythagorean Press.)

Additional Clarifying Activity

Students use geometry exploration software to create a two-point perspective drawing of a rectangular box. Students use the capabilities of the software to explore how the perspective changes with different horizon lines and different vanishing points.

(M.9.B) The student uses algebraic and geometric models to represent patterns and structures. The student is expected to use geometric transformations, proportions, and periodic motion to describe mathematical patterns and structure in music.

Clarifying Activity

Students bring in sheet music or CDs that they enjoy and use geometric transformations to describe the musical transformations that they see or hear. For example, translations can be used to describe repeats and transpositions. Reflections can be used to describe retrogressions (horizontal reflections) and inversions (vertical reflections). Students who play an instrument can demonstrate some of these for their classmates.

Additional Clarifying Activity

Students use proportions to find the frequencies of notes in an octave. For example,

The note named A above middle C on a piano is called a 440 A because its frequency is 440 hertz (number of cycles per second). The A one octave above that has a frequency of 880 Hz. There are 12 half steps from the 440 A to the 880 A. (See the table below.) There is a constant ratio between the frequencies of each pair of consecutive notes. What are the frequencies of the notes between these two As?

Students use the idea that there is a constant ratio between each pair of consecutive half-steps to find the other frequencies. Students begin applying the idea of a constant ratio with simpler situations. For example:

If the two As were next to each other (one half step apart), then their proportional relationship could be expressed by 2 x 440 = 880 and the constant of proportionality would be 2.

If there was one note between the two As (i.e., they were two half steps apart), then there would exist some constant of proportionality, p, such that p x 440 = f (the frequency of the hypothesized note between the two As) and p x f = 880. Then, by substitution, p x p x 440 = 880, and p x p = 2. Therefore, the constant of proportionality if the two A's were two half steps apart is 2^1/2, or the square root of 2.

If there were two notes between the two As (i.e., they were three half-steps apart), then there would exist some constant of proportionality, p, such that p x 440 = f₁ and p x f₁ = f₂, and p x f₂ = 880 (where f₁ and f₂ are the frequencies of the two hypothesized notes between the two As). By substitution, p x p x p x 440 = 880, and p x p x p = 2. Therefore, the constant of proportionality if the two As were three half steps apart is 2^1/3, or the cube root of 2.

Following this pattern, students determine that the constant of proportionality between consecutive half steps in the given octave is 2^1/12, the twelfth root of 2. Students use this constant of proportionality and technology to compute the frequencies for each note in the table. Students can test the results of their mathematical model by playing the appropriate notes on a musical instrument and measuring the frequencies with a CBL.

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