Parametric/Projectile Motion Simulated in Desmos - A Common Core Activity for Algebra/Precalculus

[Updated using folders to reduce amount of visible text. Click on the arrow next to the Folder icon to see the frames below. Thanks to Desmos team for this helpful hint!]

CLICK ON GRAPH TO ACTIVATE DESMOS...

The Desmos activity above is both an investigation of parametric representation and a tutorial for more advanced use of this remarkable WebApp. The The text in the side frames begins with a detailed background of the activity for the instructor and how Desmos can be used to demonstrate projectile motion using both parametric and rectangular coordinates. Some of the uses of slider 'variables' are demonstrated including animation, a powerful feature of Desmos.

In addition to showing how to use parameters in Desmos, the activity itself asks students to compare two different trajectories, representing an object dropped from some initial height, then a 2nd object two seconds later. The horizontal translation of the first graph is juxtaposed against the algebraic representations of these graphs using both system of coordinates.

The student activity starts about halfway down. There is a series of questions and actions the student needs to take in Desmos.

I'm hoping this will prove useful for both the instructor and the student.  Desmos is powerful but, in my opinion, some of the illustrative examples provided by Desmos do not flesh out the ideas behind the various uses of slider 'variables'. I'm hoping this will fill in some of those gaps.  I'm still a novice here so I'm sure more advanced users will be able to improve upon this...

Your comments and reactions are very helpful to me...







If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back and DETAILED SOLUTIONS/STRATEGIES for the 1st 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Common Core Assessments, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Price is $9.95. Secured pdf will be emailed when purchase is verified.

Reciprocals, Square Roots and Iteration -- The gift that keeps on giving!

OVERVIEW

SEASONS GREETINGS!

While gifting and regifting this holiday season, here's my gift to all my faithful readers without whom I'd have no reason to put finger to touch screen...

The following series of problems does not on its surface involve anything more than basic algebra, but it is intended to provoke students to reflect on the interconnectedness of number and algebra.

The extension at the bottom goes beyond what might be expected from the beginning of this exploration.

Math educators can adapt this for Algebra 1 through AP Calculus students...

THE PROBLEMS

What are the number(s) described in the following?

1. A number equals its reciprocal.

2. A number equals 25% of its reciprocal.

3.  A number equals twice its reciprocal.

4.  A number equals the opposite of its reciprocal.

5.  A number equals k times it's reciprocal. Restrictions on k? Cases?

Answers:
1. 1,-1
2. 1/2,-1/2
3,  √2,-√2
4. i,-i
5. k>0: √k,-√k; k<0: i√k,-i√k; k=0:undefined

OVERVIEW and much more...

• So why don't we just solve the equation x^2=k? See extension below for one reason.
• Why not ask the students what the graphs of, say, y=x and y=2/x have to do with #3. They might find it interesting how the intersection of a line and a rectangular hyperbola can be used to find the square root of a number!
• Extension to Iteration
Ask students to explore the following iterative formula for square roots:

(*) New = (Old + k/Old)/2

Have them try a few iterations for k=2:
x1=1 (choose any pos # for initial or start value; I chose 1 as it's an approximation for √2 but any other value is OK!)
x2=(1+2/1)/2=3/2=1.5
x3=(1.5+2/1.5)/2=17/12≈1.417 Note how rapidly we are approaching √2)
x4= etc
[Note: Plug in √2 into the iteration formula (*) to give you a feel for how this works!]

Students may want to explore further and they might be curious about where this formula came from, how it's related to Euler, Newton, Calculus and Computer Science. For example, they could  implement this on their graphing calculator or program the algorithm themselves!

Solve x^2 - 10000x - 10000 = 0 without a calculator! A Precalculus Investigation

BACKGROUND/OVERVIEW
No, there's no mistake in the constant term in that equation. Imagine giving this to your Precalculus/Math Analysis/Adv Math students! Actually, 'solving' it with the TI-84 requires some effort using Solver since one needs to make an approximate guess or adjust the lower bound so that the positive root is obtained. Using the graph is no 'walk in the park' either! The TI-89 or Mathematica would have much less difficulty in displaying the exact radical form or a suitable decimal approximation but they may not be within reach. Perhaps an important issue here is that sometimes technology gives us unexpected or even inaccurate results. That's when students need some understanding of theory to recognize the limitations of the technology and adjust accordingly.

Here's the point of all this. The given quadratic is not factorable over the integers, however we can replace it with a 'nicer' quadratic that is. The roots of the desired quadratic can be shown to be approximately the same as the 'nice' quadratic and we can show that the absolute error is less than two ten-thousanths (and a much much smaller relative or % error)! Does this 'numerical analysis' have any practical value? Why approximate roots when powerful technology can produce exact answers? Do professionals who need to apply mathematics to the solution of 'real' problems ever use such approximation techniques? Could it be that theory actually provides practical application!

THE INVESTIGATION
(1) Show that the roots of the x²-10000x-10001 = 0 are 10001 and -1 by factoring.

(2) Show that the roots of x²-10000x-10000=0 can be approximated by 10001 and -1 with an error of less than 0.0001.
(a) By direct calculation: Using the quadratic formula and, yes, you may use the calculator!

(b) (Challenging) By comparing, in general,
(*) the roots of x²-bx-(b+1)=0 and
(**) the roots of x²-bx-b=0.Here we are assuming that b > 0.
(i) First show by factoring that the roots of equation (*) are b+1 and -1.
(ii) Then use the quadratic formula to express the roots of (**) in terms of b.
(iii) Compare the positive roots of these equations by subtracting them and (after algebraic manipulation and simplication), show that the absolute value of the difference is less than 1/(b+1).
Note: For b=10000, this error is therefore less than 0.0001.

(c) Explain intuitively why the roots of the original equation and the 'approximating' equation are virtually the 'same' for 'large' values of b. One possibility here is to consider how the graphs of the associated quadratic functions are related. What do they have in common? How are they different?
Note: Subtle point here for students. Even though the difference of the function values (i.e., y-values) is always 1, this is not true of the difference between their zeros! This may be the essence of the numerical analysis in this investigation.

EXTENSION/PRACTICE
Ok, now "solve" x² - (googol)x - googol = 0 without a calculator.

RELATED PROBLEM:
Without your calculator show that √(10001) - √(10000) is less than 0.005.
Does this provide us with an effective method of approximating the square root of some large numbers or is it limited and impractical?

For Calculus students: How does this compare to using linearization to approximate the square root?

For more advanced calculus students: Newton's Method? The Binomial Formula (using fractional exponents)? A Taylor Polynomial approximation? All equivalent?

Setting the Tone in Precalculus - Another Coordinate Investigation

Note: Read the first comment I posted which suggests a purely Euclidean geometry approach to this problem...

Don't forget our MathAnagram for Aug-Sept. Thus far we have received a couple of correct responses. You are encouraged to make a conjecture!
Look here for directions. Here is the anagram again:

PRINCE? NAH! E-ROI!

Tangent problems are usually the domain of calculus but we can keep them within the reach of geometry and algebra if we restrict our attention to circles. The calculus student spends a considerable amount of time solving a wide variety of "tangent to the curve" exercises. As any calculus instructor will tell you, many of the harder problems ask students to determine the equations of the tangent to a curve from a point not on the curve. The issue there is not the calculus. It's all about an understanding of the interface between the algebra and geometry, the essence of coordinate methods. I developed this investigation specifically to address this issue before students enter calculus. Might be another "fun" problem to start the year off with. If nothing else, it will establish the rigor of your precalculus course early on!

Part I of Investigation
Determine the coordinates of the points of tangency for the tangent lines to the unit circle from the point (0,2).
Note: Unit circle refers to the circle of radius 1, center (0,0).

The remaining parts will all refer to this same circle.

Part II of Investigation
Repeat part I for (0,3) and (0,4).
Write your observations, conjectures.

Part III of Investigation
Show that the y-coordinate of the points of tangency for the tangent lines to the unit circle from the point (0,k) is 1/k, where k ≥ 1.

Notes, Comments...
(1) The result of Part III suggests that as k increases, the y-coordinate of the point of tangency decreases (inverse ratio). Ask students what happens as k approaches 1.
Students should make sense of this visually by sketching tangent lines from various points on the y-axis above the circle.
(2) There are several effective methods for solving the above parts, however, one needs to know the fundamental relationship between a tangent line and the radius drawn to the point of tangency. From that point on, one can represent the slope in two ways or represent the y-coordinate of the point of tangent in two ways. This requires strong understanding of coordinates, graphs and algebraic relationships. You may find other methods -- share them! BTW, one could also use trig methods.
(3) I chose the unit circle and a point on the y-axis for simplicity so that the student could focus on essential ideas. However, one could generalize the result to any circle and any point outside. Have fun with that!
(4) Anyone mildly surprised by the reciprocal relationship between the y-coordinate of the point on the y-axis and the y-coordinate of the point of tangency? Can anyone make sense of that?

When Curves Collide Part II - Quadratic Systems Re-Explored!

One of MathNotations more popular posts (hundreds of views) was published one year ago this week: When Curves Collide.

Here's a variation to review the essential ideas or to use as an assessment problem or just to challenge yourself. Parts (a) thru (d) require some theoretical analysis and algebraic skill. Part (e) is the main challenge...

An Investigation for Algebra 2/Precalculus
Consider the quadratic-quadratic system:

x² + y² = 1
y = ax² -1, a>0

(a) Show that (0,-1) is always a solution to this system.

(b) For what values of the parameter 'a' will there be 3 distinct solutions to the system?
Coordinate Interpretation: For what values of 'a' will the parabola and circle intersect in 3 distinct points?

(c) For what value(s) of the parameter 'a' will two of the points of intersection be above the x-axis? Below the x-axis (in addition to (0,-1))? On the x-axis?

(d) For the case that there are 3 distinct solutions, determine the two solutions, other than (0,-1), in terms of 'a'.

(e) Now for the main problem:

Assume the graph of our system has three points of intersection: P, Q and R(0,-1). If the area of ΔPQR is 32/25, determine the coordinates of P and Q and the value of 'a'.

(f) Can you think of an even more clever variation!

Multiple Representations (Rule of 4) in Algebra 2 or Precalculus

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar.  175 problems divided into 35 quizzes with answers at back. Suitable for SAT/Math Contest/Math I/II Subject Tests and Daily/Weekly Problems of the Day. Includes both multiple choice and constructed response items.
Price is $9.95. Secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL (dmarain "at gmail dot com") FIRST SO THAT I CAN SEND THE ATTACHMENT!

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Did you overlook our Mystery Mathematician for May? I've received two correct responses thus far, but submissions can still be emailed until the 15th of the month. Don't forget to include the info requested in a previous post.



If (A+3) ÷ (B+5) ≥ 10 and B ≥ 7,
what is the least possible value of A?
DISCUSSION

The use of multiple representations of a concept or procedure in mathematics is highly recommended by NCTM and other math education experts. Also known as the Rule of Four, it suggests that instructors use some or all of the following, when introducing a new concept. This requires careful planning and considerable thought on the part of the teacher. Over time and with experience, it will flow. However, it does help to see many models of this heuristic for geometry, algebra, etc.

The Rule of Four suggests that a concept be presented
(a) Using natural language (words)
(b) Numerically (concrete examples, 'plugging in', use of data tables, etc.)
(c) Visually (e.g., using graphs, charts, concrete models)
(d) Symbolically (algebraical mode)

From my experience, many students will approach the problem at the top by ignoring the inequalities and simply plug in 7 for B. They've learned that this strategy usually works on standardized tests. It is our role as educators to challenge them to think more deeply. Create disequilibrium by provoking them with a question like,
"But to make a fraction small, don't you need to make the denominator as large as possible?" Of course this statement does not apply to this problem, but I'll wager that it would cause some to reconsider their initial answer!

Do you think that most students would quickly recognize that the relationship between A and B can be described by a linear inequality, which can be then be approached both algebraically and graphically? Do you think I need strong medication for asking you that question!

To deepen their understanding, one could ask:
How would you have to change the above problem so that one could ask for the greatest possible value of A?

I plan on posting further examples of the Rule of Four. I am aware that I have not fully demonstrated this technique for the problem above. I'm only hinting at it. More will likely come out in the comments...

Video Mini-Lesson: Cone in the Sphere Problem

As a result of the numerous views of a calculus problem I published in November, I decided to present the following video mini-lesson. As before, I had to break it up into parts to control the file size for uploading. I hope this has some value for those who were looking for a more detailed discussion of this question. Much of this is highly appropriate for precalculus students.

Note: Before playing the videos below, a correction and comments:
(1) In error, I referred to the cross-section of the cone as an isosceles right triangle. Make that isosceles only!
(2) The video and audio quality is far from perfect. Bear with me on this!
(3) I didn't discuss the case where the height of the cone is less than or equal to the radius. This will not produce maximum volume but should have been noted. I will have more to say about this later.
(4) There is so much more to discuss about this question, in particular, the result that the cone of maximum volume has height equal to (4/3)R or that the center of the sphere divides the altitude into a 3:1 ratio. These may be discussed in upcoming videos. In particular, as suggested in the videos below, there will be a treatment of the 2-dimensional analogue of this problem, namely, the isosceles triangle in the circle problem.
(5) These video 'mini' lessons are designed for the university or secondary calculus student (probably comes too late for the college final exam) or for anyone wanting a refresher. Beyond my personal style of presentation, there are pedagogical issues (instructional tips) that arise in the videos that might be of interest to someone teaching calculus for the first time.

If you're getting bored of watching the same chalkboard and my same drab outfit, well, it is a low-budget video! I hope you will let me know if this proves helpful and if you'd like me to continue these. As mentioned previously, I will also be employing other technologies for demonstration purposes.

Happy Holidays!

The Classic Cone Inscribed in the Sphere Problem: Developing Relationships Before Calculus

Update: View the series of videos here explaining the procedure for solving the cone in the sphere problem below as well as related questions.

Many Algebra 2 and Precalculus textbooks have begun to include those challenging 3-dimensional geometry questions involving 2 or more variables and/or constants. However, we know from the difficulty that calculus students continue to have with these, that we need to do more before students do their first optimization problems in calculus. You know the kind: Determine the radius of the __________ of maximum volume that can be inscribed in a _________ of radius R. These problems have fallen out of favor somewhat with the AP Development Committee, perhaps because they lack that real-world flavor or perhaps because they had become predictable or perhaps too hard. I would argue they have been part of the rites of passage for calc students for many generations for a reason - they blended the spatial reasoning of geometry with the need to identify variable relationships and reduce the number of conditions down to one function of one variable if possible. In other words, they help to develop mathematical sophistication. I 'cut my teeth' on these -- did you? Any calculus teachers reaching this topic yet in AP Calc?

Anyway here's an activity for you Algebra 2 or Precalculus students to prepare them for these challenges. As usual we proceed from the concrete (i.e., given numerical dimensions) to the abstract. Rather than attempt to draw the diagram, which is fairly challenging for me given the tools I have, I will describe the problem verbally. Good luck!

STUDENT ACTIVITY

(1) A right circular cone of height 32 is inscribed in a sphere of diameter 40.
Note: Students need to learn how to make a diagram of this problem situation.

(a) Determine the radius of the cone.
(b) Determine the volume of the cone. [Imagine asking students to memorize the formula!]
(c) Keep the diameter of the sphere at 40. This time, determine both the radius and volume of the inscribed cone whose height is 80/3. The numbers are messy but try to work in exact form (fractions, radicals) before rushing to the calculator to convert everything to decimals. Oh well, we all know what will happen here!
(d) Try another value for the height of the cone, keeping the diameter of the sphere at 40. See if you can produce a volume greater than in (c). Any conjectures?

(2) We could throw in an intermediate step by using a parameter R to denote the radius of the sphere, and use numerical values for different possible heights of the cone, but I'll leave that to the instructor. Instead, we'll jump to the abstract generalization:

A right circular cone of height h is inscribed in a sphere of radius R.
(a) Express the radius, r, of the cone in terms of R and h.
(b) Express the volume, V, of the cone as function of h alone (R is a constant here).
(c) Use your expression for r and your function for V to verify your results in (1).
(d) Calculus Students: You know what the question will be! Oh, alright:
Determine the dimensions and volume of the right circular cone of maximum volume that can be inscribed in a sphere of radius R. Anything strike you as interesting in this result?

Another View of sin(A-B) for a Special Case: An Investigation

[As always, don't forget to give proper attribution when using this in the classroom or elsewhere as indicated in the sidebar]

In a standard trig unit, students learn those wonderful formulas for the sin and cos of the sum and difference of angles. Many creative methods have been developed to derive these formulas and, depending on the ability of the group and teacher preference, these are demonstrated or not. Students are typically shown various mnemonics for recalling them on the big test, but, in this investigation, students will derive sin 15° using only 30-60-90 triangle ratios and the Pythagorean Theorem. We will then compare the result to that obtained by the traditional formulas for sin(45°-30°) or sin(60°-45°) and show equivalence by algebraic methods using radicals. This is not an attempt to develop a general approach to deriving sum/difference formulas, although readers are invited to try a generalization. You may recall other posts on this blog of a similar nature.

THE PROBLEM/INVESTIGATION
Refer to the triangle above. If the print is too small, click on the image to magnify.
∠A = 75° and ∠B = 15°

(a) In the triangle above, locate point D on side BC such that ∠CAD = 60° . Express the lengths of the sides of triangle CAD in terms of a.
[Note: We could avoid the variable a altogether and assign a value of 1 since this is a ratio problem.]
(b) Show that CB = a √3 +2a.
(c) Use the Pythagorean Theorem to show that AB = a √(8+4 √ 3)
(d) Verify the identity (√ 6 + √ 2)² = 8+4 √ 3. Use this to rewrite AB.
(e) Use above results to obtain an expression for sin 15°.
(f) Use the standard trig formula for sin(45°-30°) to obtain an expression for sin 15°.
(g) Show your results in (e) and (f) are equivalent.

An Instructional Aside
When introducing the formula for sin(A+B), for example, teachers sometimes motivate it effectively using numerical values or considering the special case A=B. Here's an alternative:
Consider the special case A+B = 90°
Ask students to verify the formula for sin(A+B) in this special case. Simple, but at least it's something slightly different to pique their curiosity.

Standard Disclaimer:
This investigation is not copied from some other source. As it is original and has not been edited by others, there's always the possibility of error. Please feel free to suggest corrections/edits/extensions...
You know I welcome your comments!

Mortgages - Third installment

[Update as of 6-17-07: At the bottom you will now see 3 screenshots from the TI-84 showing all of the formulas used for this series of mortgage activities and the input screen for the built-in Finance Application on the TI-84 that can be used to determine the monthly mortgage payment. The first 2 screens overlap, i.e., the 2nd screen contains part of the first screen and the 4th function, Y4. You will need to refer to the index of variables below to make sense of all this. There are more details below.]

The following is the 3rd and possibly the last in this particular series of classroom activities. All three should be assigned for complete effect:
Part I: Taking the Magic Out of Mortgages
Part II: Puff the Magic Mortgage

Thought I forgot to finish this activity?
Well, with the school year over for some and ending for others, here's Mortgages Part III to think about as we look forward to making our monthly payments during the summer and plan enrichment classroom activities for the fall and spring. Part III is more ambitious and requires more sophistication on the part of the Algebra 2, Advanced Algebra or Precalculus student. As always, I am attempting to provide a completely developed enrichment lesson ready to use or modify as needed. You may want to bookmark this and return to it when teaching this unit next year.
The goals here are:
(a) Providing a more challenging application of exponential functions and their relation to geometric sequences and series
(b) Systematic development of the formulas for the equalized monthly mortgage payment as well as the portion of the monthly payment that goes toward paying off the principal, etc.

This is an activity that is particularly suited for block scheduling. If begun in a 40-45 minute period, the lesson will probably run over two periods or the last few parts can be assigned for homework. Another effective approach is to give this as a long-term individual or group project. In this case, I would recommend combining all three Mortgage activities.

STUDENT ACTIVITY
In the previous activity, you should have observed that the sequence of data values in the Y₁ column formed a geometric sequence with common ratio 1+I, where I was the interest rate per payment period (decimal form). It's time to derive this mathematically and see how the other columns were generated and how some of those famous mortgage formulas came to be. Did you figure out that Y₁ contained the amounts labeled P_x below?

The following is an index of the variables we will use . I'm using uppercase variables and X for ease of entry when instructed to enter these formulas into your graphing calculator. Note that the discussion below answers the questions from the previous activity regarding the meanings of the Y-columns in the calculator.

P = Original amount of Loan (remember, it was $100 in the previous activity)
I = Rate of interest per payment (expressed as a decimal)
Note: E.g., if the bank is charging 6% annual rate on your loan, I = 6/12% or 1/2% = 0.005 per month!
Z = 1 + I (to make formulas easier to write and enter into the calculator, since 1+I appears frequently when doing compound interest)
N = number of payments (e.g., N = 360 for 12 payments a year over 30 years)
X = the index used for the xth payment
P_x = Amount of the xth monthly payment that goes toward reducing the principal
I_x = Monthly interest payment
A = Level (equal) monthly payment
U_x = Amount of debt (Unpaid amount) remaining after Xth payment


(1) Explain the meaning of the equation: P₁ + PI = P₂ + (P-P₁)I.
(2) Show that P₂ = P₁(1+I) by solving the equation in (1) for P₂.
(3) Explain why P₁ + PI = P₃ + (P - P₁ - P₂)I
(4) Show that P₃ = P₁(1+I)² by solving the equation in (3) for P₃ (after substituting for P₂ from (2)).

The results in questions (2) and (4) suggest the following general formula which can be verified by mathematical induction:
(**) P_x = P₁(1+I)^X-1.
Recall that P_x denotes the amount of the Xth payment that goes toward paying off the original loan amount P.

The next few parts require that you recall the formula for the sum of the first N terms of a geometric sequence. If you have forgotten it, research it or your instructor will review it.

(**) shows that the sequence P_x is a geometric sequence with first term P₁ and common ratio, 1+I (or Z).
(5) Explain why P = P₁ + P₂ + P₃ + ... + P_N
(6) Using (5) and the formula for the sum of the first N terms of a geometric sequence, show that P₁ = PI/((1+I)^N-1) = PI/(Z^N-1) where Z = 1+I.
(7) Use (6) to explain why A = PI/((1+I)^N-1) + PI.
(8) Simplify the result of (7) to derive:
A = PI(1+I)^N/((1+I)^N-1) = PIZ^N/(Z^N-1)
[Again, Z = 1+I]

(9) STORE the following values from the Home screen:
100 STO P
.1/12 STO I [10% annual rate divided by the number of payments during the year]
1+I STO Z
12 STO N

Note: If you haven't used the ALPHA key before, you will now! Remember: The variables listed above will store these constant values until you or some program changes them. Clearing the screen has no effect on stored variables.

(10) Enter the last formula for A (Z-form) from (8) into Y₁ in your graphing calculator. You may have to modify it slightly for entry purposes. The * symbol for multiplication is not necessary for most graphing calculators. Try it!

(11) Start a TABLE from X = 1 and display your TABLE. If entered correctly, the values for
X = 1 through 12 should all be the same. Why? Which column was this in Part II of the Mortgage Activity?

(12) Using ** and the formula for P₁ from (6) (the one in Z-form), write a formula for P_x in terms of P, I, Z, X and N. Enter this into Y₂. Display the TABLE starting from X = 1. Which column was this in Part II of the Mortgage Activity?

(13) Derive a formula for I_x using preceding results. Again, express it in terms of P, I, Z, X and N and enter this into Y₃. Which column was this in Part II of the Mortgage Activity? Explain why these values are decreasing.

(14) Derive a formula for U_x using preceding results. Again, express it in terms of P, I, Z, X and N and enter this into Y₄. Which column was this in Part II of the Mortgage Activity? Explain why these values are decreasing.

NEW!!
Below you will find 3 screenshots from the TI-84. The first 2 show the actual functions used to compute the 4 key quantities used for mortgage repayments. The 3rd screenshot shows the finance application screen (APPS, Finance, TVM Solver) used to input the actual data values used in this activity. Students will need to refer to the index of variables above to make sense of these functions. PMT (the monthly mortgage payment) was obtained by pressing ALPHA ENTER (SOLVE). One of the main goals of this series of activities was to show students how they could obtain the formulas that are hidden behind this 'cool' application. Ask your students to explain why PMT is displayed as a negative amount!

Y1 = The payment toward principal function, i.e., the portion of the xth monthly payment that is applied to the loan principal (increasing function)
Y2 = The monthly interest payment (decreasing function)
Y3 = The fixed monthly mortgage payment (constant function, thus the variable x does not appear)
Y4 = The debt function, i.e., the amount still owed on the principal after the xth payment (decreasing function)

Investigating an SAT Algebra Problem - Going Beyond the SAT Strategy for Deeper Meaning

[Update as of 6-2-07: Solutions to most parts of the problem are now posted in the Comments.]

The following problem is typical of a somewhat difficult Algebra 2/Precalculus type of SAT problem. Students who have been shown the 'quick and easy way' to solve this definitely have an advantage when taking the test. This has been developed into an activity for Algebra 2 students who might be taking the SAT on June 2nd. However, the activity explores more than just a strategy for solving the problem efficiently to get an answer for the test. Students will be asked to solve the problem the traditional way as well and analyze why this question should not appear on the test. Two separate graphical interpretations are included to deepen student understanding, one for Algebra 2 students and a more sophisticated one for the Precalculus class. Is it worth spending 25 minutes or more on one problem in class? I'll let you judge...

Consider the system of equations:
1/x + 1/y = 1/4
1/(x+y) = 1/3

(a) (SAT-type of thinking): Show that xy = 12 (without solving for x and y).
Note: I'm giving the 'answer' here so that students will focus on the method; also for part (f).

(b) (Algebra 2): Solve the system (i.e., find all possible solutions for x and y). Write your answer(s) as ordered pairs.

(c) Verify, algebraically, that your solutions satisfy the original equations.

(d) Explain why this exact question should not appear on the SATs (although this type of question has appeared frequently).

(e) (Precalculus extension): Graph the system:
1/x + 1/y = 1/4
1/(x+y) = 1/3
Notes/Hints:
You must graph each equation separately. Do not replace the first equation by xy = 12.
Hint: Solve each equation for y.

(i) Explain clearly why the graph of the first equation has both vertical and horizontal asymptotes. Label these in your graph.
Note: The customary way is to solve for y first. See if you can also determine the asymptotes without changing the form of the equation!

(ii) Does your graph of the first equation contain the origin? Explain why or why not.

(iii) How many solutions of the system are evident from the graph? Explain the reason for this.

(f) (Algebra 2): Now solve the system
x+y = 3
xy = 12

(i) In what way is this system equivalent to the original system?
(ii) How many solutions of this system are evident from the graph? Explain the reason for this.

Taking the Magic out of Mortgages Part I: Exponential Functions and Geometric Sequences to the Rescue

[Note: For an exceptionally clear and definitive exposition of all things financial, the best resource I have found is MoneyChimp. There are interactive calculators to thoroughly understand the concepts in this post and much much more. More importantly, for math nerds like me, the formulas are explained and, in some cases, derived. The mathematics is accurate and the analysis is excellent. Enjoy it!]

In Algebra 2 and Precalculus (or whatever it may be entitled in your local schools), students often do compound interest problems. Typically, the author of the text and/or the instructor will derive the formula for what your original investment of P dollars will be worth in t years, if interest is compounded n times per year for t years at an annual rate given by r (a decimal for this discussion):
Compound Interest Formula: A(t) = P(1+r/n)^nt.

This is a nice practical application of exponential functions, exponential growth in particular. A similar, but more sophisticated, concept applies to annuities and amortization of a mortgage (paying off debt over time in n equal payments). In both an annuity and a mortgage, the original amount of money (whether it's the amount invested or the debt you owe) generally decreases over time. In an annuity, you receive a fixed amount at the end of each period, whereas, in a mortgage, you pay a fixed amount. In an annuity, your original investment is earning (accruing) interest (it may be possible to 'live off' the interest and not touch the principal), while you are receiving periodic equal payments that are deducted from your account. A central concept in both annuities and mortgages is that that interest is applied before receiving an annuity payment or before making a mortgage payment.

The following is the first part of an activity introducing students to the mysteries of mortgage calculations. The fact that the formulas for monthly payments or the decreasing amount of debt seem very intricate lead many to believe that this topic is too sophisticated for most secondary math students. Just give them the formulas, mention that it is related to exponential functions and let them plug it all into their graphing calculators. We know most adults, other than those in the business of lending, punch the numbers into the computer and read the results. Before calculators, bankers would look it up in those mortgage tables on some well-worn-out card. This activity may demystify a little bit of this. Students need good algebra skills, knowledge of exponential properties and functions in particular, a basic knowledge of compound interest and background in geometric sequences and series (later on). I am well-aware that sensitivity is needed here for students whose parents do not own a home, however, all students can benefit from these ideas since these principles apply to far more than a monthly mortgage payment.


STUDENT ACTIVITY
You can find many excellent web resources for mortgage calculations. You can also find the actual formulas for all of this either in your text or in other sources. Most of us would probably use the built-in applications typically found on a graphing calculator or more likely use those free mortgage calculators all over the web. In this activity you will take an active role in the process of borrowing and lending and see what lies behind those sophisticated formulas.

In actual practice, mortgages can range from less than a hundred thousand into the millions of dollars. Therefore, these loans are typically repaid over 5, 10, 15, 20, 25 or 30 years to make the monthly payments more manageable. In this activity you will be borrowing a small amount and considering an oversimplified form of repayment, leading up to more general considerations.

You will be borrowing $100 from a reputable lender, Stan, The Mortgage Man.
Stan is charging the going rate at the moment, which is 10% compounded annually.

(a) If you repay the loan in one year, explain why your single payment would be $110.
(b) If you agree to repay the loan at the end of two years, in one single payment, explain why that single payment would be $121?

Discussion: Parts (a) and (b) should remind you of the compound interest formula you've learned:
One year re-payment: 100(1+0.1)¹
Two year re-payment: 100(1+0.1)²

Surely, increasing the payment schedule to TWO payments over two years or one year, cannot be that much more difficult? Let's find out...

(c) This time you will make two equal payments over two years. Stan gives you the repayment schedule: Two equal payments of $60. He explains it as follows: The loan (your debt) of $100 is divided into two equal payments of $50 each. The interest charges are $10 (10% of the amount you borrowed) on each payment. Explain the mistake that Stan is making (or is he trying to take advantage of an unsuspecting borrower who didn't pay attention in algebra?). We're not asking you to correct Stan's error here - just explain why his calculation is either wrong or unfair.

(d) Now that you figured out that the two equal annual payments should not be $60, we will tell you what the actual payments would be according to mortgage formulas:
Each annual payment is $57.62 (rounded to the nearest penny).
Show why these two payments correctly repay the loan of $100 and the interest that is due on each payment. Show method clearly. Use calculator as needed.

(e) Do you think you could figure out an algebraic way to determine those equal payments of $57.62? You are about to...
Let A represent the equal annual payments you will make.
At the end of the first year, before you make your payment, you owe $100(1.1) = $110.00. Now the fun begins:
(i) Represent, in terms of A, the amount of debt (loan + interest) you will owe AFTER you make your first payment.
(ii) Represent, in terms of A, the interest you will be charged by the end of year 2 BEFORE you make your final payment.
(iii) Represent, in terms of A, your debt, AFTER making your final payment.
(iv) What should be the numerical value of your debt AFTER making your final payment? Now, write an equation and solve for A. You should come up with $57.62!

(f) Are you up to the challenge of solving for the general formula for A given an original loan of $P at an annual interest rate of i (expressed as a decimal)? Of course, you are! For now, you only need to do this for TWO payments, just as we did in (e). Of course, your formula for A should be in terms of P and i.

More to follow...