Developing Conceptual Reasoning in Geometry Via Standardized Test Questions

Ok, here's a fairly typical SAT-type of question that requires application of fundamental ideas in geometry. Some students 'see' a way to find the value of x in less than 15 seconds. These students have strong conceptual ability and are confident of their knowledge and reasoning. They are not thrown by a question that is somewhat different from the textbook problems they've seen.
Some of the issues as I see them are:
(1) How do we raise the knowledge/experiential base and confidence of those students who cannot seem to find a solution path and inevitably give up?
(2) How do we extend the thinking of those talented students who solve the problem in short order and just sit there complacently?

Perhaps, beyond these considerations and the instructional strategies employed is the bigger issue of PROVIDING FREQUENT CHALLENGES FOR OUR STUDENTS THAT GO BEYOND NORMAL TEXTBOOK FARE.
Where does one normally find these types of challenges in school curricula? Embedded in a natural way in the regular set of textbook exercises that can be routinely assigned? OR are they labeled as Standardized Test Practice at the end of a section or in a separate part of the chapter or text? OR are they found primarily in ancillary materials provided by the publisher? Can you guess where I think they should be and if they should be labeled? Should they be stand-alone multiple-choice questions or more open-ended with several parts that go beyond the 'answer?'

Before I suggest an activity based on this innocent-looking problem, I invite readers to consider a variety of methods of solution (so far I've observed about 6-7 'different' approaches) and how one might go beyond this standardized test question to deepen student reasoning. To remove the element of surprise and focus attention on process, the 'answer' is 90... (sorry!).

Have fun but if you 'see' the answer in 15 seconds or less, don't stop there!

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I can't believe I'm sinking to these depths but it is summer and this may be the time for silliness. Two of my students shared the first couple (I'm doing it from memory so they're not exactly the originals) and the third I saw somewhere in my travels across the web (I forget the site to give proper attribution):

1. Who designed King Arthur's RoundTable?
Sir Cumference, of course. (ugh, groan,...)

2. What did pi say to i?
Get real!

3. Number two said to number three: "Boy, number 6 is really bad news. He's always in trouble."
Number three replied, "What do you expect. He's a product of our times!"

Ok, let me have it!

The 12th Carnival of Math on Friday the 13th!

Of course, there would be more harmony in the universe if this had been the 13th Carnival but enjoy the 12th Carnival nevertheless over at Vedic's Math Forum.
Vedic has selected an interesting assortment of posts covering a broad range of math. For example, Knot Homology in Algebraic Topology, Key Concepts in PerCents, Math Mnemonics, Observing Objects Near the Speed of Light, Sudoku and Graph Theory, An Interesting Way to Calculate the Square Root of 2, a discussion about finding Next Gen Math Teachers and a couple of recent posts from this blog. Enjoy! A great way to beat the summer doldrums!

The next Carnival (#13!) will be hosted on 7-27 over at Polymathematics.

Two Coordinate Problems: SATs and Geometry Enrichment

Two of the opposite vertices of square PQRS have coordinates P(-1,-1) and R(4,2). (a) SAT-Type: Find the area of PQRS.
SAT Level of difficulty: 4-5 (i.e., moderately difficult to difficult).
Note: For standardized tests, in particular, students are encouraged to learn the special formula for the area of a square in terms of its diagonal.


Now for a more significant challenge that can be used to extend and enrich. Students can work individually or in small groups:

(b) Explain why there is only one possible square with the given pair of opposite vertices. Use theorems to justify your reasoning. Would this also be true if a pair of consecutive vertices were given? How many rectangles, in general, are determined if 2 vertices are given (opposite or consecutive)?

(c) Determine the coordinates of Q and S, the other pair of vertices of the square.
Note: There are many many approaches here. Students often get hung up on the distance formula leading to messy algebra (with 2 variables). There are simpler coordinate methods. You may want to provide a toolkit for students here: Graph paper, Geometer's Sketchpad, etc. Students who estimate or 'guess' the coordinates must verify (PROVE) that these vertices do in fact form a square. Students who quickly 'solve' the problem should be encouraged to find more than one method. This is the only way they will expand their thinking!

And now another coordinate problem that can be solved by a variety of methods...

In right triangle PQR, with right angle at Q, the coordinates of the vertices are:
P(-p,0)
Q(0,8)
R(r,0)
Determine the value of the area of the triangle. Assume p and r are positive.

Notes: Students should again be encouraged to try both synthetic (Euclidean) and analytical (algebraic, coordinates) methods.

General Comment: Students often forget how powerful slopes can be when solving geometry problems by coordinate methods!

x = 3: It's a Point! No, It's a Line! No, It's a Plane! It's... It's SuperPlot!

Sorry for the silly title but when the temperature approaches 100, I start becoming delusional!
Algebra teachers know that the equations of horizontal and vertical lines (the coordinate axes in particular) are stumbling blocks for students and creative educators and desperate students often resort to clever mnemonics and other memory aids to recall these. I invite readers to share their favorite. The teachers in my department (former department that is -- be kind, I'm adjusting to retirement) became enamored of HOY-VUX. I'm not sure who originated it but this person deserves credit! The name is silly (reminiscent of horcruxes from Harry Potter), the students laugh at it, but when the test comes around, they write it at the top of their paper. Here's how it works:
HOY: Horizontal, slope 0, Y=...)
VUX: Vertical, Undefined slope, X=...)

Now I know that others out there have their favorite ways of teaching this so PLEASE SHARE!

Believe it or not, the above was not the intent of this post but it's probably more interesting than the technical stuff to follow! This discussion is intended for Algebra 2 students and beyond. A full treatment requires some vector analysis but I will avoid that for now. Unlike most of my offerings, I did not set this up as a worksheet but you'll get the idea. You may want to bookmark this and save it for when 3-dimensional graphing comes up in the curriculum.

Start with a horizontal number line: <------------------|---------------->x
Ask students to plot x = 3 on the line. No ambiguity here, right!?!
Thus, the 1-dimensional graph of x = 3 is a POINT! Easy, so far.

Now draw both coordinate axes. Plot the point at 3 on the x-axis, ask a student for both of its coordinates and ask if it satisfies the equation 1x + 0y = 3.
Confirm this: 1(3) + 0(0) = 3.
Students generally treat x = 3 as an exceptional case of the equation of a line, but having both variables may help them see it isn't that special (other than its slope of course!).

Ask students to verify that (3,1), (3,2), (3,-1), (3,-2) all satisfy the equation 1x + 0y = 3.
Have them plot these points.
Ask the class (I didn't feel like writing this in the form of a worksheet today) to verify that (3,k) satisfies this equation for any real value of k. Students need to understand this significance of the zero coefficient of y.
This should help them to recognize that the graph of x = 3 is a vertical line. Don't get me wrong. Understanding this does not necessarily lead to getting it right on a test! They still need survival gear (mnemonics) for that! HOY-VUX to the rescue!

BUT THERE'S MUCH MORE TO THIS!
Point out that in the equation 1x + 0y = 3, we see that the resulting line is PERPENDICULAR to the axis with the non-zero coefficient (x-axis here) and PARALLEL to the axis whose coefficient is zero (the y-axis in this case). We're not proving anything here or explaining why this is true, just making an observation that we will generalize later.


Ok, so where's the plane in all of this?
In 3-dimensional space, we examine the equation 1x + 0y + 0z = 3. We can still graph the point corresponding to 3 on the x-axis, the vertical line graphed above (y can be chosen arbitrarily) and now z can be any real number. Corresponding to each variable whose coefficient is zero, the graph will now be a plane PARALLEL to that axis and PERPENDICULAR to the axis whose coefficient is not zero. Thus, our graph is now a plane parallel to the y- and z-axes (therefore parallel to the yz-plane determined by these axes) and perpendicular to the x-axis. Of course, software like Mathematica, Derive, or even freeware available on the web will help students visualize this better. Cardboard or Styrofoam models are also highly effective here. The more the students construct these models and label the axes and planes, the better they will be able to make sense of all this.

So what is the graph of x = 3? All of the above!
Now have your students analyze the equation y = 3 following this model!

7-7-07: Your Lucky Day?

Of course, July 7th, 2077 or 7-7-77 will be fun too for those around to enjoy it 70 years from now!

Now you all know the story of the inveterate gambler who waited until the 7th day of the 7th month to bet $777 on horse #7 in the 7th race at 7-1 odds. Of course, the horse finished 7th! Sorry, I couldn't resist telling this groaner...

If the difference of 2 numbers is less than the sum of the 2 numbers... Developing logical reasoning in our students

If the difference of 2 numbers is less than the sum of the 2 numbers, which of the following must be true?

(A) Exactly one of the numbers is positive
(B) At least one of the numbers is positive
(C) Both numbers are positive
(D) At least one of the numbers is negative

The answer given in the original source was (B). Do you agree? See notes below for further discussion of the wording of the question (before you react!).

This SAT-type question was posted about a year ago in my discussion group, MathShare (which is still extant but possibly being phased out). On that forum, I discussed how students struggled with the subtleties of logic in their analyses of the problem. That online discussion led to a meaningful debate (involving some exceptionally thoughtful educators) about how and when logical thinking needs to be developed in our students. All agreed that critical thinking and logical reasoning must begin when children enter school, long before the formalism of an axiomatic approach. Do you believe this is currently happening in most elementary schools? What materials are being used by those districts or teachers who are infusing critical thinking and logic? If we move toward a more standardized curriculum nationally, how important is this? I'm sure you know how I feel!

Other Notes about the problem above:
(1) Is the question ambiguous or flawed because the term difference fails to specify in what order the numbers are subtracted? Should the domain of numbers be specified (would integers be better?). On the SAT, it is understood that the domain is always real numbers.
(2) Why do you think so many students (these were strong SAT prep students) struggled with this and had great difficulty accepting that (B) was the correct choice? Do you think phrases like at least one and exactly one are problematic for many students?
(3) What methods do you think were used by students? There were several approaches as I recall.
(4) Do you think students should be encouraged to use an algebraic approach here rather than plugging in numbers and testing various cases?
(5) Would restating the question in its contrapositive form make it easier to grasp? (how many students remember this from geometry?!?)
(6) Would this question lead to a richer discussion if it were open-ended, i.e., no choices given?
(7) Could this question be given to middle-schoolers after they have learned the rules of integers or do you believe they do not have sufficient maturity to handle the logic?

Your thoughts....

The Multiplication Principle - Developing SAT/Math Contest Skills

How many even 4-digit positive integers greater than 6000 are multiples of 5?

Students who have had many experiences with problems like this have a huge advantage on Math Contests and SATs. To level the playing field, you might want to consider giving your middle and secondary students a daily SAT/Math Contest Problem of the Day like the one above. Questions like this require:
(a) Careful reading skills (encourage underlining or circling of keywords
(b) Knowledge of the Fundamental Principle of Counting (most often termed the Multiplication Principle)
(c) Clear thinking
(d) Careful reasoning

The answer is 399 (pls correct this if you feel I erred!). The process is equally important. Students should be encouraged to list a few examples (preferably the first 2-3) of numbers satisfying the conditions. The challenge is to recognize HOW MANY conditions are subtly embedded in the dozen or so words in the question ! Some students will prefer to make an organized list and count by grouping, which is fine, but, as they develop, they should recognize that is the basis for the Multiplication Principle (and later on, permutations).