Math: My Solutions to mathhombre’s Problems

Part of the job of a teacher is to model her or his Subject-specific thinking to help students understand the subject techniques and the Subject processes, patterns and metacontent. To this end, I dedicate this post to modelling how I tackled the math problems, created by John Golden as part of a math final for preservice teachers, that I mentioned in my last post.

mathhombre's Twitter avatar I enjoy these types of problems. They are chuck full of math — and Math — from geometry and trigonometry to algebra and arithmetic. But even more so, they require mathematical thought, imagination, creativity, logic and perseverance to solve them. Had John expanded the questions to ask for metacognitive reflection and recording, at least the trigonometry question could easily pass as a Math lab.

Once again I invite you to visit his post and take a crack at his problems before reading on.

The remainder of this post contains my solutions to these problems. (I went overboard with the trigonometric problem. I blame it on play and slightly improved health.)

Trig Problem 2

The first problem John offers is a circle geometry / trigonometry problem. He presents a diagram and simply asks the test taker to “Figure out some of the missing information in the diagram”. The only information the student is sure of from this task is that some information — more than one piece — is missing. Interpretation, logic and problem solving are all the tools the student has to continue with the question.

As I mentioned in my last post, what I really like about this question — and the next one — is its openness. The student isn’t asked for all possible missing information. He can choose what she wants to explore and discover.

I began my activity to this task by stating two assumptions on which the rest of my work rests. I assumed that point C is the center of the big circle. I also assumed that point A is the center of the small one. You might think this obvious given the diagram, but I like stating these assumptions right away. That way everyone knows where I stand and the rest of my work can not be faulted based on ambiguity on these points.

I then stated the explicit givens, that angle ADE is 30°, angles AED and ACD are 90°, and segment AC is 3 units long. (All of this and the rest of the original problem diagram is in light blue in the answer diagram below.)

Here is a diagram of my answer.

 

 

Click to enlarge or visit the GeoGebra construct from which it derives.

Joy, Pythagoras

Based on the assumptions that C is the center of the big circle and A is the center of the small one, I calculated and labelled the radii for the big and small circles. So CD, CI, CL and CM are each 6 units long and AC, AG, AH, AI, AJ and AK are each 3 units long.

This made apparent that triangle ADC is a right triangle with legs CD of 6 units and AC of 3 units, allowing (through Pythagoras’s Theorem) hypotenuse AD to be calculated as square root 45 or 3√5, which is also the hypotenuse of triangle AED.

Making angles

Equally obvious to the use of Pythagoras’s Theorem to calculate the common hypotenuse of triangles ADC and AED, is the use of the 180° (or π) “rule” for use in triangle AED.

This concept is so well known, it has almost become pneumonic. Yet, think on how profound the notion that the internal angles of all triangles always add up to a common constant, and that this constant, 180°, is also found in: the angles that accumulate to make a line, the conversion from degrees into radians and the command that sends us back the way we came.

In triangle AED, angle DAE is 60° because angle AED is 90° and angle EDA is 30° and these add up to 180°.

Attempting a similar algorithm in triangle ADC requires some labelling. I set the measure of angle ADC to unknown x, which makes the measure of angle CAD to 90-x by the 180° rule. But the fun does not stop there. Since the angles touching line EF on one side at common point A also add up to 180°, angle FAC is 30+x (60+(90-x)+(30+x)=180).

One could circle point A applying this rule to label all angles surrounding A, but another way is to apply the Vertically Opposite Angle Rule (yet another pneumonic one can ignore since the 180° rule produces it) to determine that angle JAK is 60°, IAJ is 90-x and HAI is 30+x.

Similar rules can be used to determine that all the angle measurements about points C and E are 90°, but also to calculate that angles CFA, LFN, ABE and VBW are 60-x in measure. (Oh, by the way, I added points B, L, N, M, P, Q, R, V, W, S, T and U to the diagram. The last five are hidden off the edge, but are used solely to allow labelling of angles VBW, UDT, TDS and LFN vertically opposite of and equal in measure to angles ABE, EDA, ADC and CFA respectively.)

The value of x, and consequently of the other angles, is calculated applying the Law of Cosines in triangle ADC.

AC² = AD² + CD² – 2(AD)(CD)cos(x)
3² = (3√5)² + 6² – 2(3√5)(6)cos(x)
9 = 45 + 36 – (36√5)cos(x)
cos(x) = [(-72)/(-36)](1/√5)
cos(x) = 2/√5
x = 26.57°

From this, 90-x = 63.43°, 30+x = 56.57° and 60-x = 33.43°. Using the 180° rule to back check these values confirms them.

These calculations take care of all the obvious angles that the problem suggests are missing.

Measuring sides

Observation of the diagram to this point reveals that quadrilateral AEDC is not a kite since AC ≠ AE (AH = AC and AE = AH + EH), angle DAE (at 60°) ≠ CAD (at 63.43°) and angle EDA (at 30°) ≠ ADC (at 26.57°).

The lengths of AE and DE need to be calculated.

I used the Law of Sines to calculate the lengths of AE and DE. This is fairly easy since the sines of 30°, 60° and 90° produce easy ratios, 1/2, √3/2 and 1 respectively. The calculation looks like this.

sin(30°)/AE = sin(60°)/DE = sin(90°)/(3√5)
1/(2AE) = √3/(2DE) = 1/(3√5)
AE = 3√5/2, DE = 3√15/2

I repeated this process to calculate the lengths of AF, CF, BA and BE.

sin(33.43°)/3 = sin(90°)/(AF) = sin(56.57°)/(CF)
AF = 3sin(90°)/sin(33.43°) = 5.45 units
CF = 3sin(56.57°)/sin(33.34°) = 4.54 units

sin(33.43°)/(3√5/2) = sin(90°)/(BA) = sin(56.57°)/(BE)
BA = 3√5sin(90°)/2sin(33.43°) = 6.09 units
BE = 3√5sin(56.57°)/2sin(33.43°) = 5.08 units

It did not escape me that triangles ABE and ACF, with equivalent interior angles, are mathematically similar.

Having calculated the length of CF as 4.54 units and the length of CL as 6 units, it holds that FL = CL – CF = 1.46 units.

Unfortunately, the same can not be said for calculating the lengths of EQ, EP and FN, which remain undetermined. Perhaps there is a trick to calculating them. I was considering the power of a point using secants BD (through Q) and BM (through I) to calculate EQ, but the lengths of DQ, BQ, MI and BI are unknown. Unfortunately, the length of “chord” RQ, subtended by angle EDA and creating triangle QDR, also can not be determined because the length of DQ is again unknown. With point A and angle DAE mathematically set, the length of chord PN might be calculable, paving the way to calculating the lengths of EP and FN. I have not yet figured out how to do this. This leaves EQ, EP and FN unmeasured.

Measuring JR

The length of JR is 1.02 units. But I cheated to determine this!

Whereas all the other determined values described in this post are calculable from the initial information provided in this post and John’s original problem, calculating the length of JR required mechanical measurement of another length, mainly the length of DO. GeoGebra did the measuring, and had it not done so, the length of JR would be undetermined like those of EQ, EP and FN.

Wait! So why didn’t I measure the lengths of EQ, EP and FN? For the lengths of these segments, I have no theory to back my measurements with. For calculating the length of JR, I do.

The length of JR can be calculated using the Side-Splitter Theorem. (Funny, I used to call this the Parallel Projection Theorem. The Side-Splitter Theorem sounds like a comedy act.)

Segment AC splits triangle DOR parallel to leg OR of that triangle. The proportions of triangle DCA are already known to be CD:AC:AD = 6:3:3√5. So knowing the length of DO or OR can provide the length of DR in triangle DOR.

Here is where the mechanical measurement comes in. I used GeoGebra to measure (not calculate) the length of DO at 9.60 units. From this, using proportional scaling of similar triangles DCA and DOR, I calculate the lengths of OR and DR as 4.80 units and 10.73 units respectively.

The length of JR = DR – DA – AJ = 10.73 – 3√5 – 3 = 1.02 units.

Still if this were a real exam and I did not have access to GeoGebra, I would not be able to determine JR’s length.

Why stop there?

That essentially covers the basic values of John’s original problem that any preservice teacher would reasonably be expected to determine on a test. But why stop there?

I am under no time pressure and I saw right from the start that I can calculate values associated with those circles. I am sure John did not expect such a step, but here I am noticing patterns and recognizing opportunities. So, here John are my bonus marks.

The circumference (2π*radius) of the small circle is 6π since its radius is 3 units. That of the big circle is 12π from its radius of 6 units.

The central angles of these circles produce isosceles triangles with radial legs of either 3 units (small circle) or 6 units (big circle) and chords subtended by these angles.

In particular, the small circle’s central angles GAH and JAK at 60° suggest equilateral triangles AHG and AKJ. They also suggest that arcs GH and JK have arclengths of π. Since these triangles are equilateral, GH and JK are 3 units long just like the radius of the circle. This implies an arclength of one radian or π. Independently, 60° = (60°/180°)π = π/3, while the arclength of an arc of a circle is calculated as the measure of its central angle times the radius of the circle (angle*radius). In this case angle*radius = (π/3)*(3) = π.

Since the other central angles of the small circle are not 60°, their angles times three are used to calculate the arclengths of their arcs and the Law of Cosines is used to calculate the lengths of their chords.

The length of CG, for instance, is 3.15 units as calculated below.

(CG)² = (AC)² + (AG)² – 2(AC)(AG)cos(angle CAG)
(CG)² = 3² + 3² – 2(3)(3)cos(63.43°)
CG = 3.15 units

Arc CG, corresponding to this chord and similarly subtended by central angle CAG, has arclength 1.06π units ((63.43°/180°)*3 = 1.06).

For the big circle, there are four isosceles triangles with 6 unit-long radial legs and 6√2 unit-long chords (calculated using Pythagoras’s Theorem since the central angles around C are 90°). The corresponding arcs are all 3π in arclength.

I used the 180° rule for triangles, used in the Making Angles section, to calculate the other angles of each of these isosceles triangles.

For triangle CAG, angles AGC and GCA are each equal to 180° – angle CAG = 180° – 64.43° = 58.29° in measure.

And that exhausts the values I calculated.

What I missed

I noticed upon reviewing my answer diagram that I forgot to calculate the “exterior angles” around B, D and F, that is angles WBI, EBV, NFC, KFL, CDU and SDE. However, I did calculate these in the GeoGebra construct using the 180° rule for lines.

Angle WBI = angle EBV = 180° – 33.43° = 146.57°
Angle NFC = angle KFL = 180° – 33.43° = 146.57°

Angle SDE = angle CDU
180° – angle TDS – angle EDA
180° – angle ADC – angle UDT
180° – 26.57° – 30°
Angle SDE = angle CDU = 123.43°

Geometry Problem 1

Relative to the circle geometry / trigonometry problem above, John’s line geometry problem is trivial. There is not much to do with it except fill in the missing angles following the given information.

I started out by assuming that lines JK and LM were parallel to each other. This was a pretty good assumption, given that John was testing student-teacher knowledge of parallel lines, perpendicular lines, transversal lines and associated angles.

Here is my answer. I will leave the explanation to you.

 

 

Click to enlarge or visit the GeoGebra construct from which it derives.

Benefitting Our Students

Solving an unfamiliar problem in front of, or better yet, with our students can greatly benefit them. They get to see how an unfamiliar problem is solved. They get to see how we approach and think on a problem. They get clearer — or messier — explanations as we think aloud. They even get to see us make mistakes and how we recognize and resolve them.

Sometimes modelling mathematical activity when faced when an unfamiliar task or challenge can flop on us and backfire on our kids. But how we deal with these flops is also informative.

Most of the time, our flops are minor — and often silly (yes, 1 + 2 = 3, not 4) — and our modelling, after correcting the error, leads to a correct answer.

But what our students gain is our thinking, reasoning and problem solving expertise. They learn the difference between solving and solution, between proving and proof. They learn to be — and how to be — creative, logical, persistent and progressive in our tackling of a problem. They learn to touch Math, instead of just math.

And that is where we want our students to be.

Gone are the days where the answer to a particular problem is satisfactory. Now thinking is where it’s at. We have to model our thinking and discuss solving with our students. We have to teach them to think, so they can excel beyond our meager abilities.

Our students deserve better and modelling is one key way (another would be rich math problems and labs) to ensure they get the meta out of their math.

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Foil the Fool: The Vertical on Polynomial Multiplication

What we are teaching

Last night I had a great “nightowl #mathchat” — to quote @mathheadinc — with @davidwees, @malynmawby, @mathheadinc and @reflectivemaths about how to teach math. From the tweets I have since received, others have continued this conversation. Its unresolved gist is that we teach a curriculum full of insular skills (little–m math) at the expense of teaching, and even exploring, overarching problem solving schema and Math processes and patterns (big–M Math) often presented in cover stories authentic to students and their cultures. (I talk about little–m and big–M in the Math Labs section of my Teaching Resources website and in another post.)

 

 

 
 
 
Though until recently I never owned the terminology to describe the difference, since I first entered junior high I realized the distinction between solving and solution, and proving and proof. It seems obvious that the solution and the proof are the cleaned-up, teacher-readied products of solving and proving processes.

The solution and the proof are what we teach to precision and grade our students on. They are obviously what we value. Just ask your students. And why not? After all, if you can not communicate mathematically, what is the point?

Yet the solving and the proving are where the Math lives! It is through these that problems are solved. You can follow formulae and instructions to produce solutions and proofs, but someone has to create those formulae and instructions, and program those calculators and computers. Someone has to learn and work with the Math and realize its underlying ideas. The whole point of the changing curriculum is to produce leaders and creators, thinkers, not followers.

It is in the rough work where the interesting Math lurks — the messy scribbles, the barely geometric doodles, the scratches and cross-outs and circles and AHAs! It is in the inked or graphite sweat scrawled in lines and blocks and columns on toss-away paper. It is in the work, in what the teacher and the consumer do not see, where Math guides and learning grows. It is (in) this that we should assess. For it is in this that problem solving and true demonstration of problem solving reside.

 

 

 
 
The solution and the proof summarize the Math. They are not the Math, just our way of communicating its product. If we want to teach Math, we need to change how we teach, value and grade math.

When accountability trumps Math

Which brings us to interesting mathematical (little-m) facts such as 1+2+3+…+(n-2)+(n-1)+n equals (n2/2)+(n/2). Do you know how (big-M)?

 

 

 
 
Did you know that if the longer series has an odd number of terms it simplifies to n(n+1)/2 for reasons different than why the series does so if it has an even number of terms? What insights and information, patterns and processes, lie within the differences between the oddly and evenly termed series?

Just going with my statement that the two expressions are equivalent, and without considering the Math that equates the two, you lose the following information.

    Pattern

  1. How each series is structured.
  2. How the first series when odd-termed compares and contrasts to itself when it is even-termed.
  3. How the two series are related, or if they even are.
  4. How the two series differ and congrue in their structure and relationship when the first series is odd, then even.

    Process

  1. How to approach the first series to simplify it into the second one, in both the even and odd cases.
  2. How to use and apply this Math to solve problems that are different from the original.

    History

  1. How the series and their relationship can be (usability) used and are (applicability) used.
  2. How the series and their relationship can be recognized as relevant to different problems.
  3. How the series and their relationship can be tailored to suit different problems.
  4. How the series and their relationship were first discovered, by whom, and with what related, back and implication stories.

With increased need to account for skill and knowledge in a packed curriculum, math, the minimalist calculation, takes front stage over Math, the engine of engagement and problem solving. Almost all Math is lost in received wisdom, to coin Hewitt (1999).

New wisdom can not form when the Math is forgotten and ignored.

And that is my problem with FOIL.

Through FOIL illiteracy

FOIL is an acrostic mnemonic (standing for First Outside Inside Last) designed to help students apply the distributive law during the specific case of double binomial multiplication. Only it is often taught as a general method of polynomial multiplication, rather than as a learning aid specific to multiplication of a pair of binomials. This is because most curricula only require students be able to multiply binomials. In fact, foiling is used as a verb erroneously synonymous with polynomial multiplication (Dictionary.com 2011).

Problems involving polynomials with more than two terms in one of the factors are not often encountered in schools, so the practice of distributive multiplication beyond FOIL is uncommon. Students I have subbed rarely encounter the limitations of FOIL and the underlying Math and methods of distributive multiplication. In fact FOIL impedes the learning of this Math ensuring continued illiteracy and closing problem solving opportunities for students.

Let’s foil 12 groups of 36 and 23 groups of 456, then move on to algebraic polynomials. Let’s also multiply polynomials vertically, a method of distributive multiplication alternative to foiling that most students I teach understand and favor.

 


 

Many students have trouble understanding, and retaining learning of, algebraic polynomials. A lot of this has to do with uncertainty with algebra itself. But additionally many students who learn how to multiply polynomials using FOIL have a problem recognizing and understanding FOIL as a method of distributive multiplication. This clouds student comprehension, leading to Mathematical illiteracy.

The restrictions of FOIL, tabulated in the Prezi, further confuse students when they try to use this case-specific mnemonic to multiply general, non-binomial polynomials together, as they must in the exploratory Math Lab, Strange Dice.

Strange Dice

Strange Dice is a challenging recreational inquiry designed to engage students in the Mathematics of several aspects and levels of polynomial multiplication. The students use technology, imagination, math and Math to:

  1. find how traditional Western and Chinese dice mathematically function, and
  2. create new dice with the same probability distribution as these traditional dice, within given constraints.

Ultimately, the students explore the patterns and processes of the Mathematics underlying the math of probability and polynomial multiplication — how for instance multiplying and factoring polynomials can construct and analyze different and congruent probability distributions and what each polynomial factor contributes to dice probability distribution and die face value.

Additional polynomial multiplication exercises can be found in Polynomial Multiplication of Urban Teaching Resources.

References

Dictionary.com. (2011). Foil. Dictionary.com.

Hewitt, Dave. (1999). Arbitrary and necessary part 1: A way of viewing the mathematics curriculum. For the Learning of Mathematics 19(3): 2-9.

Urban, Shawn. (2005). Math lab 5: Multiplying polynomials with strange dice. Urban Teaching Resources.

Prezi images (in order of appearance)

Dawson, Fred. (2007). Ball and Chain.

beast love. (2007). Fox Hounds.

Dunn, Natasha C.. (2009). laundry {post}.

Hanchanahal, Nagaraju. (2009). Morning Fog, Nandi Hills, Karnataka.

Wallis, Caro. (2010). Garlic Bread.

Tanaka, Hisako. (2007). Maple.

Arutemu. (2008). Rapier Guard.

Gimpert, Adam. (2006). Genius.

Frangipani Photograph. (2008). Matryoshka Doll.

Orlando, Giovanni. (2008). NEW “Hay Bale” Version.

S., Rishi. (2008). Friends.

Ballez, Romain. (2010). Stairway to Heaven.

Post inspired by

Urban, Shawn. (2005). Math lab 5: Multiplying polynomials with strange dice. Urban Teaching Resources.

Wees, David. (2011). Flipping fractions. Reflections of a Math Teacher Candidate.

Math Lab: Revisiting Technology and Imagination

I have to tell you about an incredible activity that I engaged in a few days ago.

This was one of those rare days of late when I just took the day off – no teaching, no marking, no tweeting, no blogging.

I relaxed. And may I say relaxing is highly underrated? You never know you need to until you do it.

On this day, I reworked a math lab called Sherlock Holmes offered by Dr. David Pimm of the University of Alberta. It is a 5 x 4 tangram containing seven pieces with a minimum of side lengths and angle widths given. The task has two parts. First, find the length of each side of each piece. Second, prove that the tangram can never be arranged into a square.


Sherlock Holmes” by Dr. David Pimm

The fun begins when an additional task is given to you: while engaged in solving the tangram, describe and justify either the math processes or the math content you use at each step in your activity. Now you have to pay attention to the task, the Math and what you are doing. School math does not get any richer! Nor does it come closer to the activities professional mathematicians engage in.

Or does it?

Most real-world math problems are open ended; they do not provide specific solution schemes or even all the information necessary to solve the task set before the mathematician. The mathematician must make assumptions in order to address the math task and engage in the Math activity. The Sherlock Holmes task does this too. The student or mathematician must determine what assumptions (there are more than one of them) must be made to solve the task. So, the task is open to interpretation, solution and answer. Now that is math worthy of a Math lab.


Out of the Box

I first worked on this problem about five years back, and I got it wrong! I received high marks because I was meticulous with my activity, but my solution was messy, took several steps and never did solve the task.

I did everything from drawing to geometry to calculations by hand, which was fun. I endorse math-by-hand-first, at least when dealing with unfamiliar math. Too many students do not know Math nor how to solve math unless they have a calculator, and when asked they have no idea why the calculator spits out the correct answer. With each lost comprehension or misunderstanding, these students fall further behind in their Math reasoning and soon learn to hate math and Math.

This is so sad. I love Math and working on rich math tasks. Math to me is a series of puzzles and riddles and games. It is

a way of thinking: of thinking things through imaginatively, logically, thoughtfully and step by step. … It is the process of solving and creating puzzles; that is, of recognizing, understanding, manipulating and describing patterns. Mathematics, then, is about patterns and school mathematics is about riddles, challenges designed to encourage the exploration of [overarching and] underlying patterns.Me (2005); I credited Geri Lorway, but these are my words.

What other class in elementary and secondary school is premised on playing with puzzles and games?

So, a few days ago I took a day off and I returned to the “Sherlock Holmes” lab. And I solved it! This time, I used GeoGebra, a visual mathematics software, much like Geometer’s Sketchpad except that it is open-source and free. I use GeoGebra a lot and provide several free resources for teachers to use.

I actually used GeoGebra to create the puzzle image above, but I decided, while I was at that, to use the program to also solve the problem.


“Sherlock Holmes” changed

GeoGebra did a superb job. With it I saw more clearly what aspects of the task were fixed and what aspects were free to manipulate. This clarified what assumptions needed to be made and allowed quick experimentation of these assumptions to solve the task and check the answers I came up with.

The best part is I could see the puzzle moving as I manipulated it and I identified the (one silly, arithmetic) error I made five years ago which prevented me from solving the puzzle. I solved the puzzle in three steps using the program. But the activity was not as fun as doing it by hand. There is a certain “magic” and flow in using one’s imagination to problem solve.

And there is something fun and relaxing about playing a challenging game.

So, why don’t you take some time and choose a math lab and play a game? I would love to hear what lab, of mine or others’, you chose, how you fared and what you experienced. Play, enjoy, learn and leave a comment.

This post was inspired by The Rascal Triangle which I read a couple of days after engaging in this activity.