A Recurring Problem for SATs (Functions)

SAT "Grid-In" Type
Level of Difficulty: 5 (High)
Content: Algebra 2, precalculus

The function F satisfies the condition
F(N + 6) = F(N) + 8, for all integers N.
If F(7) = -2, what is the value of F(25)?

Click on Read more to see the answer, solution, discussion.


Answer: 22

Suggested Solution:

Replace N by 7 since F(7) is known:
Therefore, F(7 + 6) = F(7) + 8 or F(13) = -2 + 8 = 6

Next, replace N by 13 since F(13) is known:
F(19) = F(13) + 8 = 6 + 8 = 14

Finally, F(25) = F(19) + 8 = 14 + 8 = 22.

Comments
(1) Too difficult for the SATs? Not really! A similar problem recently appeared. There aren't that many "hard" questions (Level 5) on the SAT but, if a student wants to score over 700 they will need exposure to these types in practice.

(2) Consider writing some variations of these function-type problems for additional practice. At first, change the constants, then consider changing the operations (from addition to multiplication for example). One could raise the bar even higher by asking the question in reverse:
If F(25) = 22, what is the value of F(7)?

(3) There is considerable advanced theory in functional equations and recurrence relations underlying these problems. However, the student needs only to feel comfortable with the function symbolism (or should I call it "Math Notations!"). Starting by "plugging in' N = 7 seems simple in retrospect but most students are too intimidated to consider it. Even the precalculus student may be able to get started, but, without experience, they will often get lost. This is all about exposure, but isn't it always?

(4) One could rewrite this problem using sequence notation:
a_N+6 = a_N + 8. By expressing the problem in the context of the Nth term of a sequence, students may grasp it a bit better, but, in the end, it's all about interpreting function notation.

...Read more

Those "Function" Questions on the SATs - Practice, Tips

PLS NOTE THE EDIT TO THE PROBLEM BELOW. THE ORIGINAL WORDING WAS INACCURATE.

The following is not a classic function question even though it uses function notation. This is an original problem I wrote but it is the kind of question that might appear. The level of difficulty would be medium. The math content is middle school level but the wording and notation are the challenge for most students. Beyond preparing students for a test like the SATs, my strong belief is that such questions should be included in textbooks from middle school on (even with that function notation!). This question reviews basic math concepts (primes, factors, gcf) and can also be used as a springboard for discussion of the concept of "relatively prime", Euler's phi function, π(x) and other number-theoretic topics.
Note: The "For example" hint may or may not be included in the question. It certainly makes the notational issue less formidable.

If n is a positive integer greater than 1, then the sets F(n) and P(n) consist only of positive integers and are defined as follows:

A positive integer, k, belongs to the set F(n) if k ≤ n and the greatest common factor of k and n equals 1.
A positive integer, k, belongs to the set P(n) if k ≤ n and k is prime.
For example, F(6) contains the numbers 1 and 5 and therefore has two elements. P(6) contains the numbers 2, 3 and 5 and therefore has three elements.

What is the ratio of the number of elements in F(20) to the number of elements in P(20)?

Click Read more below to see answer (suggested solution will be posted later).

Answer: 1
Explanation: Not yet...

...Read more

A 'Simple' Traversal through a Number Grid -- Patterns, Functions, Algebra Investigation Part I

Here is an activity for Prealgebra and Algebra students. This introductory activity is not meant to be a conundrum for our crack problem-solvers out there, but the extensions below may prove more challenging.



Target Audience: Grades 6-9 (Prealgebra through Algebra 1)

Major Standards/Objectives:
(1) Representing numerical relationships and patterns algebraically
(2) Recognizing, interpreting and developing function notation
(3) Applying remainder concepts

A 2-column number matrix (grid) is shown above and assumed to continue indefinitely. We will be visiting (traversing) the numbers in the grid starting in the upper left corner with 1. Following the arrows we see that the tour proceeds right, then down, followed by left, then down and repeats.

First, some examples of the function notation we will be using to describe this traversal:
T(1) = 1 denotes that the 1st cell visited contains the number 1.
T(4) = 3 denotes that the 4th cell visited contains the number 3.
Similarly, T(6) = 6.

STUDENT/READER ACTIVITY/INVESTIGATION

(a) Determine T(1), T(5), T(9), T(13), T(17).
(b) 1, 5, 9, 13, 17, ... all leave a remainder of ___ when divided by 4. (Fill in the blank)
Therefore, these numbers can be represented algebraically as 4n + 1, n = 0,1,2,3,...
(c) Based on (a) and (b), it appears that T(4n+1) = _______, where n = 0,1,2,3...
(d) Determine T(2), T(6), T(10), T(14)
(e) 2,6,10,14,... all leave a remainder of ___ when divided by 4. Therefore, these numbers can be represented algebraically as ______, n = _________ (Fill in blanks)
(f) Based on (d) and (e), it appears that T( _____ ) = _____, n = __________.

Note: The instructor may choose to start n from zero or one throughout this activity. I will vary it depending on our needs. It is important for students to see how restrictions (domain of a variable) is critical for an accurate description and that more than one set of restrictions is possible (provided they are equivalent).

Since T(3) = 4 and T(4) = 3, we cannot say that T(n) = n for all n. The numbers 3 and 4 leave remainders of 3 and 0 respectively when divided by 4. We will need a different rule for these kinds of numbers. Let's collect some more data:

(g) By extending the table, determine T(7) and T(8); T(11) and T(12); T(15) and T(16)
(h) Without extending the table, make a conjecture about the values of T(35) and T(36).
(i) Numbers such as 4,8,12,16,... can be represented algebraically as ____, n= 1,2,3,...
(j) Numbers such as 3,7,11,15,... can be represented algebraically as ____, n = 1,2,3,...

Note: Again, the instructor may not like varying the restrictions here. Adjust as needed.

(h) Ok, so you're an expert now. Well, prove it:
T(100) = ______; T(153) = _____; T(999) = ______
Show or explain your method.

EXTENSIONS

Surely, a 3-column number grid or even a 5-column number grid can't be that much more difficult to solve using the same kind of traversal (move to the right until you come to the end, go down, move left until you come to the end, move down, lather, rinse, repeat...). ENJOY!

Ok, for our experts: Try an n x n grid!

DISCLAIMER: As with all of the investigations I publish, these are essentially original creations and therefore have not been proofread or edited by others. You are the 'others!'. You may not only find errors but alternate and perhaps superior ways to present these ideas.
Also, please adhere to the Guidelines for Attribution in the sidebar.

Absolute Zero Part II: Applying Piecewise Function Approach for Algebra 2

[Update: Read Eric Jablow's profound comments on this post and some general discussion of graphing calculators...]

As promised, here's another installment of a piecewise function development of the absolute value function, suitable for advanced Algebra 1 students but more appropriate for Algebra 2. You may not agree with the target audience or the approach, but I have used it with mixed effectiveness. Of course you can redesign it to meet the needs of your students but the key ingredient is the use of function tables. Do you see the Rule of Four being utilized? I apologixe in advance for the klutzy formatting of the tables and the inequality symbols. I will eventually clean this up.

1. Consider the functions, f(x) = |x|, g(x) = x, and h(x) = -x.

(a) Complete the following function table.

x ............... Y1=f(x)=|x|...............Y2=g(x)=x...............Y3=h(x)=-x
-3.............. 3 ................................ -3 .......................... 3
-2............... ___ ............................ ___ ..................... ___
-1............... ___ ............................ ___ .................... ___
0............... ___ ........................... ___ .................... ___
1................. ___ ........................... ___ ................... ___
2................ ___ ........................... ___ ................... ___
3................ ___ ........................... ___ ................... ___

(b) Sketch the graphs of f(x), g(x) and h(x) on the same set of axes in THREE different colors on the domain [-3,3].

(c) Answer the following based on the table and graphs:
f(x) = g(x) when x is _________
f(x) = h(x) when x is _________
Now, rewrite this symbolically as:
|x| = x when x is __________ and
|x| = -x when x is _________.

[Note: This could easily have been handled on a graphing calculator, which is why the functions are labeled Y1 and Y2. This is one of the best uses of this technology. However, I'm a believer in doing it by hand the first time around - your choice! Also, note the heuristic of repeating the function on each line rather than the standard braces used for piecewise definition. Later on the student can abbreviate the format. ]

2. Consider the function f(x) = |x| - x
(a) Complete the table:

x..............Y1=f(x)=|x|-x
-3
-2
-1
0
1
2
3

(b) Sketch the graph of f(x) on the domain [-3,3].
(c) From the table and/or the graph we conclude that
f(x) = _____ for x < 0;
f(x) = _____ for x ≥ 0

3. [More difficult] Consider the function f(x) = |x-2| + |x-4| + |x-6|
(a) Make a table of values for f using the ten integer values from x = -2 to x = 7 inclusive.
(b) Sketch the graph of f.
(c) Define f piecewise, similar to 2(c).
(d) Determine the coordinates of the minimum point of f. Justify.

4. [The Generalization] Consider the function f(x) = |x-a| + |x-b| + |x-c|,
where a < b < c
(a) Define f piecewise as in 3(c).
(b) Determine the coordinates of the minimum point of f. Justify.

Absolute Zero Part I: Do Absolute Values Leave Some Out in the Cold?

[If you absolutely can't wait for a challenge problem, go to the bottom, but it might be worth reading through this first...]

Absolute value equations and inequalities, in particular, are notoriously difficult for most students. If you are a math educator, is this a topic you relish?

So what method of solution usually works best for the student? What method of presentation is most effective for the instructor? If the equations are straightforward such as |x+3| = 7, most students seem comfortable with expressing the equation as a disjunction: x+3 = 7 or
x+3 = -7. Some instructors, in preparing students for the technical definition (using cases), require the student to express this as x+3 = 7 or -x-3 = 7. Rarely have I observed instructors introduce the full-blown piecewise definition using cases early on in algebra:
x+3 = 7 if x>=-3
OR
-x-3= 7 if x<-3. This is generally believed to be too sophisticated for an introductory treatment. Motivating the technical definition of |x| usually comes later on in Algebra 2. However, I have always been a bit uncomfortable teaching the traditional algorithm for absolute value inequality problems such as |x+3| "<" 7 which leads to the conjunction x+3 ">" -7 AND x+3 "<" 7 or, in combined form, -7 "<" x+3 "<" 7. Math instructors devise creative mnemonics to help students recall the procedure. This all begins when prealgebra students are exposed to the verbal description of the piecewise definition of the absolute value function:
The absolute value of a positive number is that number and the absolute value of a negative number is its opposite. The absolute value of zero is zero.

This is immediately followed by a number of numerical examples, guided practice with some more complicated variations involving mixed operations and an assignment. The student sees this as another example of something they're supposed to learn in math without much meaning attached. Most catch on to the idea by repetition and errors are generally caused by weaknesses with signed numbers or order of operations.

However, some educators prefer the 'distance' interpretation of absolute value to make this notion more meaningful. Technically one has to distinguish between the real number, x, and the graph of x on a number line, but this distinction is often sacrificed for clarity:

The absolute value of x is its distance from zero.

Thus, both 8 and -8 have the same absolute value because they are the same distance from zero on a number line. Since distance cannot be negative, we have a powerful visual model that students can use to make sense of this idea, although it seems limited when solving more complicated equations or inequalities later on.

Stay tuned for more on this topic (including a 2-dimensional graphing approach using functions), but, since some readers are disappointed if there is no challenge problem, here's one for you. It's not that difficult, a version of this has recently been tested on the SAT and it can be approached in a variety of ways (formal algorithm, guess-test, etc.), but I will challenge you to solve it using the distance model! Don't hesitate to take strong exception to my comments above!

For how many positive integer values of x is
|x-2007| > x?

Motivating Zero, Negative and Fractional Exponents in Middle School and 1st year Algebra

Algebra teachers, like myself, are always looking for ways to help students make sense of exponents. We look through copies of the Mathematics Teacher, we go to the Math Forum and now we Google, Google ad infinitum (or some other search engine to be fair!). Here's an approach that I have found helpful. I assume the student has had some basic introduction to exponents and their properties. I call it the exponential function approach which sounds too challenging for middle schoolers but you decide if they can handle this. Students will use pattern-based thinking and graphs to make conjectures about extending powers of 2 to include zero, negative and even fractional exponents. Properties of exponents will then be used to 'justify' the conjectures. The juxtaposition of the numerical, symbolic, graphical and verbal descriptions are consistent with the Rule of Four that is now regarded as the most powerful heuristic in teaching mathematics.

Begin by making an x-y table - this is the critical piece.

Exponent (x)...........................Power (y = 2^x)
3 ..................................................2³ = 8
2 ..................................................2² = 4
1 ..................................................2¹ = 2
0 .................................................2⁰ = ??

The instructor of course is prompting the students for the powers while they are taking careful notes. At the same time the instructor is plotting these results as ordered pairs and the students do likewise. It might be helpful to let 2 or 4 boxes represent one unit on the y-axis since, at some point, the y-values will be fractional. Similarly for the x-axis (play with it first).
At this point, the instructor asks a key verbal question (you may phrase it much differently depending on the level of the group and your preference):
[While pointing to the left and right columns]
"When the exponent decreases from 3 to 2, the corresponding power of 2 is divided by ___.
Repeat this phrase a couple of more times until you reach an exponent of 0, then -1 and voila! Keep going until x = -3, plot the points and the students are seeing an exponential curve in grade 7? 8? 9?
Motivating zero and negative exponents using a function model (tables!) seems to make sense to me because it begins to create a 'function' mind-set that can be carried through all subsequent math courses. It may also help students to 'see' that the range of the function consists of positive real numbers. If you're wondering why I didn't mention turning on the graphing calculators to make the TABLE and GRAPH, I hope you can guess why. It was important for me to have students do this by hand first, then I will turn on the overhead viewscreen and we can explore with technology. Just my opinion of course but students in my classes seem to make sense of this. Of course, I don't kid myself that this approach will lead to better grades on tests of this unit! Facility with the properties of exponents only comes from considerable skill practice with paper and pencil.
For fractional exponents, I'll begin the discussion but I will have to explore further on another post or leave it to your imagination. "Ok, boys and girls, if mathematicians believed exponents could be zero or negative integers, would you be surprised if they wondered about 2^1/2? From the table and the graph, 2^1/2 should fall between ___ and ___? Do you think it will be exactly 1.5? Why or why not?
I know many of you use the exponent properties to develop this topic, but I wanted to suggest an alternative. I usually follow this discussion with arguments like: " Hmmm, I wonder what
2^1/2 times 2^1/2 would be?" etc...

Another Quadratic Function Problem 2-16-07 through 2-20-07

You may want to read the comments for this post. Answers and possible solutions are discussed. There is also considerable discussion about teaching techniques for f(x-h).

The parabola problem from 2-15-07 generated some interesting discussion. I haven't had a chance to see it implemented with our Algebra 2 classes yet but I'll let you know if and when...
Today's problem is along the same lines. I'm trying to provide some problems that are exclusively high school math content for this time of year. There are dozens of outstanding problem-solving sites for MathCounts and similar middle-school competitions but there appears to be a dearth of secondary math problem-solving sites (or I haven't found them yet!). Again, how might one use the problem below? As a bonus or an extended in-class activity or a performance assessment or ??? How many would regard this question as suitable only for honors or accelerated students? My take is that if students are exposed to higher levels of thinking and know they are expected to learn how to do these and held accountable on an assessment, they will adjust. Not all will experience equal success but that's ok too! Many should be able to do part (a) or are my expectations way too high?

(a) Consider the quadratic function f(x) = 4(x+4)².
The graph intersects the line y = k, k>0, in 2 distinct points B and C.
The rectangle whose base is on the x-axis and 2 of whose vertices are B and C
has area 64. Determine the value of k. Show method clearly.

(b) Now let's generalize the result of (a).
Consider the quadratic function f(x) = a(x-h)², a>0.
The line y = k, k>0, intersects the graph of f in two distinct points B and C. The rectangle whose base is on the x-axis and two of whose vertices are B and C has area R.
(i) Explain graphically (not algebraically) why the area, R, of this rectangle is independent of h.
(ii) Express k in terms of a and R. Check that your formula for k gives the value you obtained from part (a).

Algebra 2 Challenge for 2-15-07

The following is an open-ended problem for Algebra 2 students...
Enjoy it but I'd really like to hear how you might implement this in the classroom. Part of homework? A bonus? An open-ended activity in class? Students working independently or in pairs? Part of an assessment? At what point would you use this? At the end of the chapter on quadratic functions?

(a) Consider the function f(x) = 6x - x².
If P and Q are the points of intersection of the graph of f with the x-axis and R is a point on the portion of the graph above the x-axis, what is the maximum area of triangle PQR?


(b) Consider the quadratic function whose x-intercepts are the nonzero numbers p and q, p > q, and whose y-intercept is -pq.

(i) Explain carefully why the graph of this function has a maximum point no matter what the signs of p and q are.
(ii) Write an expression for the y-coordinate of the vertex of the graph of this function in terms of p and q (simplified).
Note: This appears to be a standard problem using the formula -b/2a, but there are other approaches and the result may surprise you.