Never ASS-U-ME in Geometry: A Triangle Problem to Get Them Thinking!

Not quite back to school for most but the problem above might prove interesting to review some geometric/deductive reasoning.

For new geometry students, replace 'a' by a value, say 40, and ask them to fill in all the missing angles. Most should deduce that angle 5 = 50, but my educated guess is that many will assume b = 40, so
angle 5= angle 6 = 50 and angle 3 = angle 4 = 40. From there to angle 1 = angle 2 = 50, so
angle 2 + angle 3 = 90. QED!  Not quite...

Well, the '90' is correct but the reasoning is another story! So this is all about justifying, checking validity of mathematical arguments, sorta' like some of the Eight Mathematical Practices of the Common Core!

In fact, you might ask them to redraw the diagram, keeping the given conditions but making it clear that b does not have to be 40 and that Angles 3&4 also do not have to be 40!

Desmos Common Core Activity Linking Circles, Tangents and Linear-Quadr Systems

Detailed investigation with extensive background notes for instructor and step by step outline for students to follow. Students will be asked to use a slider to approximate the position of a tangent line of slope -1 to a circle centered at (0,0). The tangent line, x+y=k, requires use of a parameter.

Students will begin with a particular radius, 3, then solve a linear-quadratic system to determine the exact equation of one of the tangent lines. They will also be asked to enter an expression for the other tangent line of slope -1 using the same parameter k. After approximating the locations of similar tangent lines for other radii, they will be asked to solve a general system using radius r.

There are different systems offered to the instructor, depending on the sophistication of the student. Finally, a geometric solution is suggested using 45-45-90 triangles.



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New video tutorials uploaded to MathNotationsVids YouTube channel

Haven't posted in awhile and haven't uploaded videos to YouTube recently. I now have the technology to make short screencasts. Here's a link to the latest video I posted, a typical medium- level standardized test question relating to a non- convex quadrilateral. As aways these tutorials are intended for both students and educators.

https://www.youtube.com/watch?v=OuPkf9bqaPA&feature=youtube_gdata_player

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Three congruent isosceles right triangles walked into a bar...

OVERVIEW
Silly title but you might want to try the following problem with your high school geometry students or with middle schoolers doing a unit on right triangles. Furthermore, elementary school children need many hands-on experiences with pattern blocks, tangrams, pentominos and the like to develop their innate spatial sense. They should also be allowed to experiment with two such triangular pieces to make a square, a parallelogram, a larger isosceles triangle, etc. Then have them work with the 3 triangles to make different polygons including the trapezoid. They don't need to consider the area or the 2nd part of the question.

THE PROBLEM
Three congruent isosceles right triangles are joined to form an isosceles trapezoid having an area of 3 sq units.

(a) Draw a possible diagram.
(b) Determine the perimeter of the trapezoid.

Answer: (b) 6+2√2

REFLECTIONS
•How much time would you allow for a discussion of this problem! 10 min? 15? 20? Guess it depends on whether you see this as just an exercise or as an activity.
• How much difficulty do you think most middle and secondary students would have with drawing an appropriate diagram?
•Do you think most will need to draw several figures before arriving at the isosceles trapezoid? Do you think some will come up with a trapezoid which is not isosceles and think they're finished? Can you anticipate that some will miss one of the key words like isosceles (which occurs TWICE!).
• Do you think the spatial "puzzle pieces" part of the problem is more significant than the numerical part or about equal?
• Do you expect some students to hit a wall and express something like "I forgot the formula for the area of a trapezoid!" We should make this a teachable moment -- "WE DON'T NEED TO RECALL THAT FORMULA! WHY!"
•Do you see benefits from students working in pairs here? Would you have them work independently then come together after a few minutes? My view is the stronger spatial student will "see" the correct figure more rapidly and influence the other who may give up and wait for his/her partner to draw it. So I might ask them to draw a few figures on their own for a couple of minutes.
•Do you think any of the older students need manipulatives?
• What is our role here? Catchphrases like"guide on the side" do not tell us what interventions we should actually use? Part of knowing what to do/say comes from our experience and part from instinct but my rule of thumb was "less is more". Allowing them to struggle for awhile is critical or, to put it another way, "without irritation there would never be a pearl!"
• How would you solve this problem? When planning do you feel it's important to think of alternate solutions or let this flow from the students?
•Finally, I think it's important to identify which of the  Mathematical Practice Standards are brought to play in this investigation. All of them? A couple? Guess that depends on you...

I typically get few if any comments from these detailed investigations. That's ok. Just planting seeds I guess...

Two overlapping circles of radius r... - A Common Core Geometry Problem

OVERVIEW
Intersecting circle problems are always interesting and often challenging whether you find them in the text, on SATs or on math contests. The general case involves trig and formulas can be found online.

The objectives of the problem below include:

• Drawing a diagram from verbal description
• Dissecting or subdividing an unknown region into more common parts
• Applying circle theorems and area formulas
• Solving a multistep problem (developing organizational skills, attention to detail)

THE PROBLEM
Two circles of radius r intersect in two points in such a way that the overlap is bounded by two 90° arcs. If the area of the common region is kr^2, determine the value of k.

Answer: (Pi-2)/2
Note: Please verify!

REFLECTIONS FOR MATH TEACHERS
[Note: These are discussion points --- not short answer questions with simple answers!]

• Should the diagram have been given to eliminate confusion?
• Does this problem appear to have any practical application?
• Have you seen a similar problem in your geometry texts? On standardized tests like SATs?
• In similar problems, were the arcs 60° or 90°?
• How would you introduce this problem? Is it worth the time to have students cut out congruent paper or cardboard circular disks, keep one fixed and move the other until it approximates 90° arcs?
Better to use geometry software?
• Assign this for homework? As a group activity in or out of class? As a demo problem with a detailed explanation provided by you?
• How much time would be needed for classroom discussion of this problem?
• Would you plan on providing extensions/generalizations?
• Too ambitious for "regular" classes? Appropriate only for Honors?
• So what makes this a Common Core activity? Are you guided by the Mathematical Practice Standards?

How (m^2)/(n^2)=(m/n)^2 is Fundamental to Geometry!

OVERVIEW
The Common Core stresses the importance of students developing a deeper understanding of fundamental concepts and to discover/uncover the interrelatedness of mathematics. The discussion below can be used to demonstrate how a basic law of exponents is tied to the geometry of similar figures.

THE PROBLEM/INVESTIGATION
1) If the sides of 2 squares are in the ratio 2:1, show that their areas are in the ratio 4:1
(a) visually
(b) numerically by examining particular cases
(b)  algebraically

2) If the sides are in the ratio 3:1, do you think the areas will be in the ratio 6:1 or 9:1? Now do parts a-c as in 1).

3) If the ratio of the sides is 3:2 show algebraically that the ratio of the areas is 9:4.

4) Show algebraically that if the ratio of the sides of 2 squares is m:n then the ratio of their areas is (m/n)^2.

Note: How does this result connect to the idea that the area of a square varies directly as the square of its side length?

4) If squares are replaced by circles using radii or diameters in place of "sides" show that the results of questions 1-4 are the same.

How does this result connect to the idea that the area of a circle varies directly as the square of its radius or diameter (or circumference)?

REFLECTIONS
• Squares and circles are of course special cases of similar figures. Beyond this investigation lies the BIG IDEA:

The areas of 2-dim similar figures are proportional to the squares of their linear dimensions.

Note: In 3 dim, we can replace 'area' by what?

• Do you see this as one of the fundamental theorems of Euclidean geometry? Is it sufficiently stressed in textbooks and in the standards? Of course you may not feel as I do about all this!

• So what is the geometry connection to
(m/n)^3 = (m^3)/(n^3)...

'.

All Tied Up - a Geometry Classic Challenge

For exercise, a prisoner was chained  to one corner (lower) of a 10 ft concrete cube located in the center of the yard. If the chain was 16 ft long and was not obstructed except for the cube, over how many sq ft of ground could he roam?

Ans: 210π sq ft



1.  Give the students the diagram or have them draw it themselves?
2.  Have them work individually or in groups?
3. How much time would you give them to work on this in class?
4. After discussion, how would you know if they 'got' it? Assessment?
5. Makes more sense to give them a variant of the problem for HW or ask them to design their own and solve it?



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GEOMETRY: When is a cone half full...

Ever wonder about practical applications of those 'some liquid is being drained from a conical tank' calculus problems?

Well, they do manufacture storage tanks with cylindrical tops and cone-shaped bottoms. Ask your students why, then share the following  excerpt 'borrowed' from the website of a company which makes these:

"Cone bottoms provide for quick and complete drainage."

Alright already - enough motivation for a geometry  problem! No calculus needed!

A conical storage tank with a maximum depth of 10 feet  is completely filled with a chemical solution. Some of its contents are then drained from the bottom.

Ask your students:

(a)  When depth of liquid falls to 5 ft, explain intuitively (no calculations) why much more than half the contents has drained out.

(b) Now for the geometry application...
What % of the total liquid has been drained when depth drops to 5 ft?

Ans: 87.5%

(c) (More challenging) What should depth be for tank to be half full? Give both one place approx and 'exact' answer.

Ans: approx 7.9 ft
I'll leave exact answer to my astute readers!

Note for instructor: You may want to explore different depths like 6', 7', 8' first to see how close we can come to half full.

QUESTIONS FOR THE INSTRUCTOR
WHAT ARE THE BIG IDEAS HERE?
DO YOU BELIEVE THIS CONCEPT IS ASSESSED ON SATs?
GIVE PRECISE WORDING OF THIS OBJECTIVE IN THE CORE CURRICULUM.
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13-14-15 triangle as special as 3-4-5

Show that the area of a 13-14-15 triangle is 84. Compute mentally - 30 seconds tick tick tick...

I'm being silly with the ticking clock but it is possible to do this if you choose the "right" base!  Unless of course you can mentally apply Heron's formula which is doable! Ok, so there's more than one way as always!

So what makes it special!? Somebody out there knows...

If you like these challenges consider purchasing my new Math Challenge Problem/Quiz Book - 175 questions - SAT format - with answers. Go to top of right sidebar to order.

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SAT GEOMETRY REVIEW Is it a Rectangle or a Triangle...

A diagonal of length x of a rectangle makes a 30° angle with the base.

(a) Show that the area of the rectangle is
(x^2)√3/4.

(b) The formula in (a) is also the area of an equilateral triangle of side length x.  What triangle is this the area of? Explain!

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Geometry in the Tiling Patterns All Around Us

I took this picture of a section the floor of the hospital where I volunteer and fortunately I wasn't dragged to the psych ward. Students see tiling patterns every day yet rarely think of applying their knowledge of geometry.

Assume each white square has side length 2 and that the shaded square is obtained by rotating one of the white squares 45 degrees.

Show that the overlap is a regular octagon of side length 2√2 - 2.


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(E) Cannot be determined...

I've posted the following geometry classic before but it seems relevant now with SATs and other standardized tests looming.

Given 2 concentric circles, segment AB is a chord of one and a tangent segm of the other. If AB=10, show that the pos difference of the areas of the circles CAN BE DETERMINED!
Explain.

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The Third Wheel...

Two wheels with  diameters 18 and 8 are touching and are on level ground. Show that the diameter of a 3rd wheel on the ground which touches the other 2 is 2.88.


For info on my  NEW MATH CHALLENGE QUIZ/PROBLEM BOOK check top of right sidebar!

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Investigation for "Squares"

CHALLENGE YOUR GEOM STUDENTS OR YOUR MATH TEAM

In square ABCD of side 1, E is the point on diagonal AC such that AE=1.

(a) Explain without numerical calculation why
√2 < BE + DE < 2
(b) Show that BE+DE = 2(√(2-√2)) ≈ 1.531 without using Law of Cosines
(c)  Be a math researcher! How might you generalize this?

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GEOM CHALLENGE 2-7-12

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 I, Math I/II Subject Tests, Math Contests and Daily/Weekly Problems of the Day. Includes multiple choice, cases I/II/III type 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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DON'T FORGET TO VISIT ME ON TWITTER AT twitter.com/dmarain

TODAY'S TWITTER PROBLEM - A CLASSIC GEOMETRY CHALLENGE
A regular octagon is formed by cutting congruent isosceles right triangles from the corners of
a square of side 1. What is the length of a side of the octagon?

[Ans: ≈ 0.414; also give "exact" answer!]

If interested in purchasing my new 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 practice or Problems of the Day/Week.

Price is $9.99 and secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL FIRST SO THAT I CAN SEND THE ATTACHMENT!

"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur SchoDONTpenhauer (1778-1860) You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific

Challenging Geometry Assumptions: Review for SAT I/II

The video below presents a more challenging 3-dimensional geometry problem which would be at the upper end of SAT I or SAT II - Subject Tests (Math I/II). The key here is to challenge students' assumptions about a quadrilateral being a square because it has 4 congruent sides, a common error. This question will also review a considerable amount of geometry: Pythagorean Theorem, Volume of cube, spatial reasoning, 45-45-90 triangles, area of a rhombus, etc.

As always, the focus is on the art of questioning, suggested instructional strategies and pedagogy, although this problem may be interesting enough to capture the attention of some students who are preparing for upcoming standardized tests. For students who need help with spatial visualization, a model could be provided or have enough empty boxes available (they don't have to be cubes!).  

I strongly urge using learning partners or pairs for the discussion. 

Benefits include:

(1) Students feel less tentative when offering ideas to one other person or in a small group.

(2) Instead of posing conceptual questions to individuals, receiving little or no response except from the most confident or capable, you can pose a question to a learning pair: "Julie and Jason, what is needed to insure that ABCD is a square?" They should be given a few moments to think and confer before responding. The stronger student will usually explain it to the other. If neither can respond, they can say, "Pass!"

(3) The biggest advantage of student dialog is that often our explanations simply don't click with several students, but they do make sense to others. Those who "get it" can usually explain it in terms that their peers understand better, a benefit to both the "explainer" and the "explainee"!



By the way, the question posed near the end of the video is worth pursuing if time permits:

"Without calculating the areas, is the area of the non-square rhombus less than or greater than the area of the square?"

The answer is less for many reasons, but we would hope they would recall the base x height formula for a rhombus. The height is maximized when the angle between the sides is 90°. Why? Interestingly, the areas are quite close: 19.6 vs. 20. I believe strongly that this is the type of higher-order question that not only reviews important concepts but promotes deeper thinking, or should I say, thinking more than one inch deep!

What are your thoughts? Would you give students the e√3 formula before a standardized test or ever?Are these videos helpful to you? If you respond both on this blog and on my YouTube Channel, MathNotationsVids, and also rate these videos, that gives me the guidance I need to improve them.


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"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860) You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific

If We're 'Packing", Are We Going Somewhere?

Fascinating article from today's New York Times. In 2-dimensions we talk about tessellating objects to fill the plane. Circles of course will always leave gaps. In 3-dimensions, equal spheres will also leave gaps when packed as closely as possible, but the question then becomes, "How do we arrange the spheres which would result in the densest packing. Turns out that the grocer's method of stacking oranges solves that problem! Equal cubes can be packed together without any gaps, so we can say that the densest packing for cubes is 100%, that is, identical cubes can be packed so that they use 100% of the available space.

But packing regular tetrahedrons (a pyramid whose 4 faces are congruent equilateral triangles) as tightly as possible has defied the best logical mathematical minds, including Aristotle's, for nearly 2000 years. Recently, significant strides have been made, not only by the best mathematical and scientific minds, but also by graduate students, like Ann Chen from U. of Mich.,, who have taken dozens of tetrahedral dice from Dungeons and Dragons games and are using a hands-on approach to build various configurations and then computing the density of the packing. For the past several months teams from different schools have published their latest and best attempts, but, as of this moment, Ann has found the densest packing at 85.63%. That's right, she's outdone the best theoretical mathematicians and scientists in this quest for the new Holy Grail of packing problems. Aristotle mistakenly asserted that there exists a perfect 100% packing for tetrahedrons, but this was shown to be false. Now it appears that the percent is far more than we thought. Professor Nash, we need you!

I strongly believe that there is a place for this kind of discussion in our math classes from the earliest years on. Let students know that solving mathematical problems often involves hands-on experimentation as in science! Besides, who's to say that there isn't some middle schooler out there right now who might sit in her room playing with these dice who will arrive at 86%!!



 

"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)

Demo For Building An Investigation In Geometry For All Levels

Note: Diagram has been modified from original.

 For Figures 1 and 2, the following is given:

AD + AC = BD + BC

Perimeter of triangle = 36

AC = 15

Show that the length of AD = 3.

In other words, demonstrate that the length of AD is independent of sides AB and BC.

Instead of imposing or suggesting my way of using this question to build an investigation, how would you do it?

If you're new to this blog, I have published dozens of examples of investigations which are intended to develop process, conceptual understanding, generalization  and a different view of what mathematics is for our students. An investigation allows students to explore particular cases before attempting to generalize and abstract. Some might call this scaffolding. I see it as creating an experimental environment in the classroom, encouraging our students to become mathematical researchers! I know every argument against this approach but, remember, I'm suggesting that this type of activity only be used perhaps once a month...

The question above can be given as is to some groups of students but may not be appropriate in its present form for many others. The question can be reworded or changed completely.

What would you do?

The Return of the WarmUp Challenges!

Just when you thought that MathNotations is on permanent hiatus or in hibernation, here are a couple of WarmUps/Problems of the Day/Test Prep/Challenges/// to consider for your students. 

Actually, I'm embarking on a new venture - an online tutoring website with live audio and video for OneOnOne math tutoring for Grades 6-14 (through Calculus II). In addition, I'm also working on setting up a small group (5-10 students) online SAT or ACT Course grouped by ability (a 600-800 SAT group, a 450-600 group, etc.).  If you're interested in getting more information about these before the official launch just contact me at dmarain at gmail dot com.


Update: Answers/comments are at the bottom...

1.   NOTE: ANGLE B IS A RIGHT ANGLE IN DIAGRAM BELOW - THANKS TO JONATHAN FOR CATCHING THAT OVERSIGHT!

2.   If 10^-1000 - 10^-997 is written as a decimal, answer the following:


(a) How many decimal places are there, i.e., how many digits to the right of the decimal point?
(b) One can show that the decimal digits end in a string of 9's. How many 9's?
(c) How many zeros are to the right of the decimal point and to the left of the string of 9's?

Notes:
(1) If we write the negative exponent expressions as rational numbers, this is perfectly appropriate for middle schoolers and, in fact, I think they need more of these experiences!
(2) The "Make It Simpler - Look for a Pattern" Strategy should be second nature to our youngsters, but when they see questions like these on the SATs, how many of our students really think of it!
(3) The fact that some calculators return a value of zero for the expression in the problem is a teachable moment - seize it!!
(4) See below for an algebraic approach.



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ANSWERS


1. 9√3


2. (a) 1000   (b) 3   (c) 997


An Algebraic Approach to #2:
First, students need to be familiar with the basic pattern:
10^-1 = 1/10 = .1 Note that there is one decimal digit.

10^-2 = 1/10² = 1/100 = .01  Note that there are two decimal places, etc.


10^-1000 - 10^-997 = 1/10¹⁰⁰⁰ - 1/10⁹⁹⁷
Using 10¹⁰⁰⁰ as the common denominator, we obtain
1/10¹⁰⁰⁰ - 10³/10¹⁰⁰⁰ =
-999/10¹⁰⁰⁰ from which the results follow (with some additional reasoning)...

Note: I could have worked directly with the exponent form by factoring out 10^-1000 but I chose rational form for the younger student.

Updates, ODDS AND EVENS and some Geometry Packing Problems

Enjoying your summer hiatus or as busy as ever? I know that feeling!

1. MathNotations' Third Online Math Contest is tentatively scheduled for the week of Oct 12-16, a 5-day window to administer the 45-min contest and email the results. As with the previous contest, it will be FREE, up to two teams from a school may register and the focus for now will be on Geometry, Algebra II and Precalculus. Several other ideas are running through my head but I need the time to bring them to fruition. If any public, charter, prep, parochial or homeschool (including international school) is interested, send me an email ASAP to receive registration materials: "dmarain 'at' geeeemail dot com."

2. CNNMoney.com Article - Something to tell your students in September!
Here is the link. The 2nd paragraph says it all:

The top 15 highest-earning college degrees all have one thing in common -- math skills.

3. Silly Instruments for Math Teachers to Play
I always told my students that I'm predominantly left-brained -- analytical, organized, detailed, process-oriented, algebraic -- as opposed to most of my children and my wife who are creative, spatial, mechanical, who see the forest more than the trees. One of my sons is a musician and another is a dancer so we are not always on the same wavelength! So I mentioned to my SAT students that I wish I had a more creative side and perhaps be able to play an instrument, but, in fact, the only thing I can "play" is my iPod! One of my students in the front row immediately responded, "I know an instrument you can play, Mr. M -- the triangle! I congratulated her for the cleverness and told her that maybe I will learn how to play the "cymbals." (the class actually applauded that lame attempt at word play!). In fact, I've read that many famous mathematicians were also musicians, so let us know: Do you play an instrument or are passionate about music or do you have a silly instrument for a mathematician to play?

4. Circle Packing Problems
Even though I am dominantly left-brained, I still enjoy challenging spatial geometry problems. I find these questions have improved my creativity and my spatial sense and they often involve multi-faceted thinking. Here are a couple of famous 'packing' problems which are accessible to geometry students. More important than solving these is to give our students a sense of the importance of packing problems and the ongoing research in this area. There are still unsolved problems here!

Although you can easily research packing problems on MathWorld and Wikipedia, the diagrams below come from an exceptional website I discovered. The author, Peter Szabo (missing accents), provides diagrams for packing 2-100 circles with accompanying data (radii, density, etc).


PROBLEM I


The two congruent circles at the left are actually enclosed in a unit square which is not shown.The circles are tangent to each other and to the sides of the square. If these circles have the maximum radius possible, determine the radius.
Note: The indicated square (assume it is a square) is helpful in solving the problem. Trig is not necessary here.

Answer (Yes, I'm providing this since the objective is to discuss the method):
[The following is the diameter, not the radius, of each circle. Thanks to watchmath for correcting this error].









PROBLEM II



Again, imagine that the three congruent circles at the left are enclosed in a unit square and are tangent to each other and to the sides of the square. If the circles have the maximum radius possible, determine this radius.
Notes: The indicated square again may be helpful to solve this problem. Trig can be used but clever use of special right triangles is preferred.

Answer:
[The diameter is given below, not the radius. Thanks to watchmatch for correcting this]