Saturday, August 30, 2014

A Classic Algebra/Geometry Inequality Proof Without Words




In case one forgets that math can be beautiful...


Diagram is far from perfect. Hopefully you can make sense of it.


If you want more details let me know via Blogger Contact Form.
Have you signed up for free solutions to my Twitter/Blog problems via Contact Form?
















Use new contact form at top of right sidebar to contact me directly! 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.

Friday, August 29, 2014

Explore Inscribed Square in Rt Triangle - CCSS Activity




Want Free Answer/Solution/Extension sent to your inbox?
Just request this via Blogger Contact Form.

Incude
(a) Your full name
(b) A link to your website/blog if you have one
(c) State & Name of school district in which you are teaching (or a student)
(d) Grade Level(s) or Math subjects currently teaching or indicate grade level/current math courses if a student. Also indicate if you're teaching LD/Reg/Honors/AP Levels
(e) Preferred math problem types (E.g/, Middle School, CCSS, Algebra, Geometry, SAT, Precalculus, Explorations, etc)
(f) Indicate if you already subscribe to my blog feed (& which feed you use) and if you already follow me on Twitter.

Thursday, August 28, 2014

A Crazy Ages Problem - Is there a place for this in CCSS

A variation on a classic for your algebra group!

J is as old as K was when J was half the age he'll be in 10 years. If K is y yrs old, express J's current age in terms of y.

Want answer and detailed solution sent to your inbox? By now you should know what you have to do. Join the other teachers who have figured it out...

Reflections...
So you're probably thinking this question is either too hard or irrelevant in the Common Core. I beg to differ!

Before having students attempt to solve this algebraically I would encourage my students to experiment with numerical values. For example suggest a value for y, say y=40 and have them guess some values for x, J's current age. I'm sure you'd agree that this is still a challenging problem but someone will probably guess the correct value, x=30. In fact just verifying that 30 is correct is formidable enough!

Is making a table still a good idea for organizing information? I'll let you decide...

         Present         10 yrs from now

J           x                          x+10

K           y                          y+10

So is there still a value to these "un-real" types of puzzle problems? Do you see the benefits that I do?

Tuesday, August 26, 2014

f(x)=(25--x^2)(16-x^2) Algebra 2 CCSS Challenge

f(x)=(25--x^2)(16-x^2)
P=product of x-int, Q=y-int of graph of f
|P| - |Q| =?
Generalize!

Want free ans/soln sent to your inbox? Just go to
http://mathnotations.blogspot.com/2014/08/free-ccsssatchallenge-math.html

Back to School Challenge SAT/CCSS Problem

Posted on Twitter this morning...

(100+99+98+...+51)-(1+2+3+...+50)=?

NO calculator/Noformulas!
Mental math - 30 seconds!

Are you getting your FREE answers and detailed solutions with strategies sent to your inbox?  For details go to http://mathnotations.blogspot.com/2014/08/free-ccsssatchallenge-math.html

PLS SHARE THIS WITH COLLEAGUES!
Think there's a catch here because there's no such thing as free? Why not ask the teachers who have been receiving these?

Sunday, August 24, 2014

Free CCSS/SAT/Challenge Math Problems/Solns sent to your inbox?

I'm considering a trial run depending on response. Many of these will be similar to the hundreds of Twitter Problems I've posted but more developed and with Answers/Solutions.

I will update my progress with this venture and let my readers here and on Twitter know if I will continue.

Since it's free at this juncture, I request that you

(1) Follow me on Twitter @dmarain

(2) Send me request via Blogger Contact Form with the following info:

(a) Your full name
(b) A link to your website/blog if you have one
(c) State & Name of school district in which you are teaching (or a student)
(d) Grade Level(s) or Math subjects currently teaching or indicate grade level/current math courses if a student. Also indicate if you're teaching LD/Reg/Honors/AP Levels
(e) Preferred math problem types (E.g/, Middle School, CCSS, Algebra, Geometry, SAT, Precalculus, Explorations, etc)
(f) Indicate if you already subscribe to my blog feed (& which feed you use) and if you already follow me on Twitter.

As you may know my problems are almost all conceptually based and require strong skill.

As always you are free to use these problems for personal use or teaching purposes but they cannot be reproduced for commercial use.

I have no idea what the response will be or whether I can maintain this on a daily or weekly basis. I may be able to customize it for your needs by category but no promises!

Friday, August 22, 2014

An Algebra Puzzle to Start The Year! Grades 6-10

As I posted on Twitter today (@dmarain)...

If I were to give you $50 we'd have the same amount. If you were to give me $50 I'd have 9 times as much as you. How much do we each have?

Submit your answer and solution via the Blogger Contact Form in right sidebar.
Other comments welcome.

Reflections...

1. This was not intended to be highly challenging. It might engage students early in the year and I designed it to be accessible to most.

2. The language is open to interpretation by design! We want students to feel some disequilibrium. But we don't have to resolve ambiguities. I let my students do that among themselves.

3. Before jumping into an algebraic solution, I would allow my students to experiment with numbers - call it "plug in" or Guess-Test-Revise. After all on standardized tests this is what many will do in spite of all the algebra we teach!

Friday, August 15, 2014

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!