A brief respite...

I will not be posting for the next few days as my family and I observe the one year passing of my wife. Thank you for your understanding.

An SAT quiz to sharpen your brain for March 9

Click on the image to enlarge. Good luck trying to read my scrawl!
This is one I wrote from the 20th century! Feel free to use with your students but observe the copyright please.

No answers yet but you can share your thoughts...



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.

So is 75 the avg of the pos integers from 50 to 100 inclusive?

This very common type of question appears so straightforward. So why do variations of it recur so often on SATs and other standardized tests and math contests?

Why not test it out with your students and ask them to explain their reasoning. I am still surprised by the creativity of our students when given the opportunity to display it!

Again, my boring disclaimer...
This is not a conundrum for the math problem-solvers out there. It is intended as a discussion point for helping students develop some important ideas in mathematics.






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.

A RADICAL DEPARTURE - AN ALGEBRA 2 /CCSSM/MATH 2/MATH CLUB CHALLENGE

A Radical Departure...

(Inspired by Ramanujan and an excellent Wikipedia article on Nested Radicals)

Suggestion: Assign as a 2-day team or individual project after demonstrating a similar but simpler example such as the square root of 3+2√2 = 1+√2.

NOTE: The method below DOES NOT show a detailed algebraic solution, using substitutions and solution of resulting quadratic equations.  Rather, I suggested some reasonable educated guessing,  aka number sense. I would recommend both approaches. 

There is considerable more theory than is suggested by this example, e.g., justification of uniqueness of roots, conditions for roots to be of the form suggested in the solution, etc.  Encourage students to investigate further! 

PROBLEM: Demonstrate the following identity by simplification of the left-hand side only. No calculators permitted for derivation although numerical (decimal) verification that the left side equals the right is recommended prior to starting the 'proof'.

(SOLUTION GIVEN BELOW STATEMENT OF IDENTITY)

NOTE: Illegibility of next to last line of 3rd image!  Should be (Square root of 3 + Square of 2) not 'Square root of 4'.


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.

SAT/CCSSM: How many 3-digit positive integers satisfy...

Disclaimer/Reminder
Don't forget to comply with the Creative Commons License in sidebar. Thanks!

An acronym I just thought of for improving your students' performance on SATs or other standardized tests:


S: SPEED
A:ACCURACY
T:TERMINOLOGY

(For training purposes only)
TIME LIMIT: 45 sec
NO CALCULATOR

HOW MANY 3-DIGIT POSITIVE INTEGERS SATISFY BOTH OF THE FOLLOWING CONDITIONS?

• THE PRODUCT OF THE DIGITS IS 72
• THE 3-DIGIT INTEGER IS A PALINDROME (an integer that is the same when its digits are reversed)

Let me know how many of your students can do this within the time limit and no calculator.  And for those who could not? Guess that means they need more of these to practice! Why not ask each student to write a similar problem for hw! They learn more from writing their own and we give up control --- perfect!!


Answer: Read below shameless ad for my book...


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.

Answer (send yours if you believe I erred!):
2 (namely 383,626)

The Quintessential SAT Problem: If h hens eat p pounds of feed a day...

If one was to categorize every SAT question from the very first SAT ever published, I believe we would find the following type of algebraic ratio problem one of the most common type. Even with all the exposure students now have to SAT problems, my direct experience is that many students still struggle with these types of questions.
WHY?

More importantly, are these types of problems important enough in the CCSSM to justify the time investment to introduce them in middle school and reinforce in secondary algebra classes? IMO, ABSOLUTELY!

If h hens consume a total of p pounds of feed per day, then, at this rate, how many pounds of feed would c hens consume in x days?

Not only was a similar question the recent SAT Question of the Day on the College Board web  site, the statistics were also published:
35620 responded (up to the time I checked)
31% correct
So, about 7 out of 10 students attempting this question online got it wrong.

Note: The actual question was followed by 5 choices, allowing students to plug in numbers and test each choice, but I chose to focus on the question here rather than test-taking strategies.

IMO, the College Board hires highly competent math people who write succinct, accurate and helpful online solutions but this only scratches the surface. It only suggests one particular approach and has little to do with Instructional Strategies and the various ways children develop these important ideas.

REFLECTIONS...

1.  Where are ratio concepts introduced for the first time in the CCSSM? K? 1st? 4th 5th?

2. By your own estimate,  how many of these kinds of questions appear as sample problems or homework exercises in your elementary/prealgebra/algebra texts?

3.  Do you believe ALL your students receive adequate exposure to and review of these?

4. Would you be willing to share some of your favorite methods of laying the groundwork for and developing the skills and concepts needed for your students to be successful with ratio problems and ultimately algebraic types? If I take a risk, would you?

Putting myself out there...

The simplest and most instinctive approach usually makes the most sense, doesn't it? We know how we learn best and the same is true of all students.  Do you accept the following as a truism, an essential tenet of teaching and learning mathematics?

EVERYONE LEARNS BETTER WHEN PRESENTED WITH CONCRETE NUMERICAL RELATIONSHIPS BEFORE TACKLING ABSTRACTIONS. FURTHER, THE COMPLEXITY OF LANGUAGE SHOULD BE GRADUALLY INCREASED, STARTING WITH THE MOST ACCESSIBLE INFORMAL PHRASES.

For example,

If 6 hens eat a total of 12 pounds of feed each day, how many pounds of feed would one hen eat in one day?

When first introduced, should our focus be on which operation to perform? In my view, our goal should be to develop number sense, in this case, ratio sense. 

We all know that a powerful construct for developing ratio/proportion sense is the idea of first reducing the information to a UNIT.
Many of us were taught this way and most children tend to think like this at first.

Scaffolding...
If 6 hens eat a total of 12 pounds of feed each day, how many pounds of feed would nine hens eat in one day?

Working from one hen consumes 2 pounds per day, the child can usually move on to 9 hens eat 9x2 or 18 pounds per day.

Two points here...

First, I believe it is important to routinely use a variety of equivalent phrases:
"in one day" vs. "each day" vs. "per day."

Secondly, I would encourage students who can reason proportionally to share this with the group:

"Well, if 6 hens eat 12 pounds, then 3 hens will eat half as much or 6 pounds, so 9 hens will eat 12+6 or 18 pounds."

Teaching conceptually means NOT SETTING UP A PROPORTION initially. Procedures and algorithms turn off the child's sense-making and stifle intuition and number sense. You can fight me on this all you want, folks, but you will not win here on my blog!


So when do we introduce proportion problems involving variables and what are some good ways to solve the original problem??  I'll allow my readers to figure that out for themselves...




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.

What is the smallest positive odd integer which has exactly 10 factors?

PLS NOTE CORRECTION TO THE ANSWER TO THE PROBLEM. MY ERROR WAS CAUGHT BY NOVOTNY!!

Occasionally I like to respond to the topics in the Google searches which bring my readers to MathNotations.

Today's problem in the title of this post
(Humorous speech-to-text aside: "this post" was interpreted" as "disposed")
is a classic math challenge question, difficult SAT- or CCSSM-type question which is appropriate for grades 5-11.


What is the smallest positive odd integer which has exactly 10 factors?
Explain your method.


Answer at bottom of post after shameless promotion...
(Aside: Wouldn't Shameless be a a cool title for a premium cable TV show about a deadbeat dad mathematician!)

REFLECTIONS for my colleagues...
(Wouldn't it be awesome if someone actually read these!)

1.  How many of these types of questions have you seen in textbooks, math contests, SAT's, standardized tests or on other blogs?

2. How different would this question be if the word "odd" were removed? An easier or harder question in your opinion?
(Humorous speech-to-text aside: "the word odd" was interpreted as "The Word of God")

3. Would you like to share some math strategies you have used for this type of problem? Are there instructional strategies you prefer for this? Do you see these 2 questions as equivalent?

4.  Do we have to be the ones to devise variations on this?
My feeling is that all learners, including us, become more proficient at problem-solving and develop deeper understanding when we are asked to pose our own problems!

Do you think your students would, in groups or alone, arrive at variations like "use even in place of odd or drop the word completely? There's only one way to find out!  Perhaps you can share your experiences here...



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 first 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.

Answer to today's problem: 3^4 x 5 = 405  [Correction thanks to Novotny!!]

Do Parabolas Have Centers? Another PairoDucks?

Well, your chance to win a free copy of my Math Challenge book has come and gone but I thought I would post an original problem/investigation about graphs of quadratic functions.

In Alg2/Precalculus students learn about parabolas:
Vertex, axis, symmetry, intercepts then, perhaps, further into other defining properties involving focus & directrix. It's also fun to touch on other important and fascinating applications such as the reflecting properties of 3-dim parabolic surfaces. OK, enough overview.

So, is the focus of a parabola the closest analog to a "center"? It's my blog so I say no!

Consider the following sloppily drawn sketch...

Can you make sense out of this graph of y = (1/12)x^2?

Without referring to the focus-directrix form of a parabola (e.g., x^2=4py), determine the values of p and q.

Reflections...

1.  Is the "p" in this problem the same as the parameter p which defines the focus?

2.  Does the diagram suggest another way to define the focus? Explain.

3.  Of course, we do not refer to the point Q as the center. I just felt  like calling it that. Can you guess how the Circle Paradox posts led me to this? Hey, I may not be as creative as some of you but I am persistent!

4. OK, fellow (gender-free) colleagues. How might you extend this investigation? We need to share ideas, right?

5.  So what does "4p" represent geometrically? Refer to the diagram.

As always, feel free to share this but don't forget proper attribution.





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.

Making a Challenge Math Problem the Springboard for Concept Development

My problem-posing creativity peaks in the hours between 6-9 am so I'd better publish this fast!

Here's a sample of an SAT-type problem although it's verging on a math contest offering...

Disclaimer: Remember, the problem or its solution is never the objective of this blog. It's merely a framework for helping our students learn to think mathematically while developing concepts in a collaborative setting that builds self-esteem. Wow, where did that rhetoric come from? Uh, me...

If the sum of 2 positive integers is 2^16 and their greatest possible product is 2^k, then k = ?

Answer: 30

Reflections...

1.  What are some strategies you want your students to use for these types of questions? And what type is this!

2.  What do you see are the "big" mathematical ideas embedded in this problem? Are "exponents" a big idea?

3.  What are the prerequisite skills needed for success with this type of question? Would you review these first?

4.  How would you utilize this problem with middle schoolers? High schoolers? Algebra 2 vs precalculus?

5.  Would you begin with an easier problem first, then build up to this or let them struggle with it as is?

6.  How would you assess that students grasped the ideas here?  Make up 10 similar questions for homework? Give them another one to try in class? Include one of these on the next test?

7.  Would you ask students to generalize this problem? First demonstrate what "generalize" means?
[One possibility: If the sum is 2^n, then the greatest possible product is 2^(2n-2).]

8. How do you create an environment of making connections in and/or applying mathematics?
[One possibility: Relate this question to the problem of finding the rectangle of maximum area for a given perimeter!]

So, have I successfully killed off all potential comments!




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.

SAT QUADRATIC FUNCTION PARABOLA PROBLEM -- Level 4/5

This type of coordinate problem is occurring more frequently. Students need exposure to these...

The graph of the quadratic function f(x) = bx^2 + ax + c intersects the x-axis at 3 and 4 and the y-axis at 5.
b = ?

Answer: 5/12

Reflections...

1.  What in the question do you think  might cause students to struggle?
2. Do you use a standard approach to these types of coordinate problems, e.g., an x-y table?
3.  Do you usually discuss at least 2 methods for these? You know how I feel!

REMINDER
All the problems I post are original and are the property of MathNotations. Feel free to use them for classroom purposes according to the Creative Commons License in the sidebar.





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/@ 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.

WHAT IS MathNotations -- my annual rant...

The Circle Paradox Revisited seems to be provoking some interest however I have to remind my readers that the main purpose of this blog continues to be

HOW TO USE PROBLEM-SOLVING IN THE CLASSROOM TO DEVELOP CONCEPTUAL UNDERSTANDING AND STRENGTHEN STUDENT REASONING

I know that we all enjoy "solving the problem" but the Circle Paradox is not all that challenging and should prove straightforward for math teachers and mathphiles in general.

To clarify: My intent is to 

Generate dialog about how to effectively use non-routine problems in the classroom to enhance student thinking! 

WHY is it empowering for students when we encourage several approaches and not force feed our method or way of thinking?

IMO, the surest way to turn off students' minds is to "do it for them" or not allowing them the time to struggle.

Judicious guidance and applying "less is more" is, for me, the hallmark of the master teacher. I'm retired and I'm still learning how to do this with my grandkids!

For some reason, this intent has not taken hold in the six-year existence of this blog. When I try to shift the focus to how to use these problems with students, it is generally ignored. "Instructional strategies, Dave? Who cares!" 
Uh, I care...

Enough of my rant for now...

Oh, yes, hypocritically, I am reminding my readers that the deadline for submitting a solution is Fri 2-8-13, 12 noon EST. Of course, I'm hoping that your solutions are coming from your students!





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.

Circle Paradox Pt 2

"Proof" without words? There must be a dozen other ways...

No takers yet? My free offer will expire by Fri 2-8-13!






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.

The Circle Paradox Revisited

WHAT'S WRONG WITH THIS PICTURE OF A CIRCLE?

CHALLENGE YOUR STUDENTS TO EXPLAIN THE PARADOX!

FIND AS MANY WAYS AS POSSIBLE!

BEST EXPLANATIONS WIN A COPY OF MY CHALLENGE MATH WORKBOOK! OPEN TO ANYONE OVER THE AGE OF 4...




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.

The Super Bowl and the ratio 16 to 9

NOTE: The following classroom scenario requires a reimagining of how we normally present a math lesson and I already have heard all the negatives: not enough time for this, too much material to cover, this won't be on the test so why bother...
I hope you will be open-minded.


The 16 to 9 ratio of course refers to the aspect ratio of  high definition LCD or LED TV screens today.

Let's say you just purchased a 55 inch LED HDTV. We all know the 55 inches refers to the diagonal of the screen and, in fact it's slightly less than 55 inches. So what would the width of the screen actually be?

How does our method of presentation and the questions posed affect concept development? 

We know how to solve this, no nonsense. Just apply the Pythagorean Theorem with some algebra and voila. "Tradition" as the song title goes from Fiddler on the Roof.

A Non-Traditional Classroom Scenario

Let's try some estimates, boys and girls...
54" 52" 50" 48"?? Hmm, most 'guesstimated' 50?

Can anyone guess why there are sheets of paper, scissors and rulers on the table? Right, we will first "construct" a solution! Oh, so a 16" width is too big for standard 8.5×11 paper. Any ideas? Oh, it's a ratio so we don't have to use 16" and 9". Ok, we'll use 8"×4.5". OK, go to it...

Each member of your team should measure the diagonal to the nearest 1/8". Oh, that's right we could have used cm instead to make measurements more precise...

Alright, so most of you got around 9 1/8" for the diagonal.
(Aside: This is definitely an imaginary scenario!).
Guess, metric measurement would have been better.  Let's verify this using the Pythagorean Thm. Ok, 9.18" to nearest hundredth. So how will we apply this to a 55" diagonal? Oh, make a proportion, and we obtain 55×8/9.18 ≈ 47.9"!

In your groups, solve the ratio problem algebraically and compare results...

So a 55" screen is less than 48" in width. Wonder why they use diagonal measurements in the ads...

Yes, Alex. You found another way to estimate this mentally?

The closest Pythagorean triple to 16 and 9 is 15 and 8. The hypotenuse would be 17 and 17×3 is 51, sorta' close to 55. So we can triple the dimensions to get 16×3=48" for the width.  Hmm...



What non-traditional approaches for this kind of problem have my colleagues used? Share!

This imaginary lesson would consume the entire period, yes? Do you think it's worth it?




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.