MENTAL MATH TRICKS OR SOMETHING DEEPER

Hey folks, I do appreciate the increased readership and renewed interest in MathNotations.  I'm definitely a "binge" blogger, prolific for a few weeks, then burning out like a SuperNova, 'retweeting'  into oblivion again. I miss doing video solutions of some problems but I don't know if that will happen. Since I get lots of views but zero comments, I really don't know if I should continue...



EXPLAIN USING MENTAL MATH IN 10 SEC OR LESS...

72÷25=2.88
72 × 25 = 1800

Just curiosities to grab students and then fade away OR do you see these as worthwhile for developing deeper understanding of number operations?

For example, dividing by  25 is equivalent to multiplying by 4, then ÷ by 100.

How would you develop or extend this investigation?




If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175  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 multiplechoice, case I/II/III type and constructed response items. Price is $9.95. Secured pdf will be emailed when purchase is verified.

A CCSSM GEOMETRY ACTIVITY -- IS IT NECESSARY OR SUFFICIENT

IF THE ALTITUDE ON THE HYPOTENUSE OF A RIGHT TRIANGLE DIVIDES THE HYP INTO A 1:3 RATIO, PROVE THAT THE TRIANGLE IS 30-60-90. 

ASK YOUR STUDENTS TO

FIRST VERIFY THE CONCLUSION BY CONSTRUCTING A TRIANGLE WITH GIVEN CONDITIONS
  (1)  by construction with mechanical tools
  (2) using electronic tools (e.g., Geogebra or Geom SketchPad)

THEN HAVE STUDENTS
(A) FiND AT LEAST TWO SYNTHETIC (DEDUCTIVE) METHODS WHICH DO NOT INVOLVE TRIG.


Reflections...
(1) So are given conditions both necessary and sufficient? Of course in geometry we usually write "if and only if" or "iff".
(2) I know most educators are annoyed that I ask lots of questions but don't ever answer them. Like, "who has time for this!" My children, grandkids and my students have always thought I was annoying too. I do pride myself on consistency!
(3) So who's going to call my bluff and ask me to show at least TWO methods of proof?
[Alt on hyp theorems same as similar triangle methods?]


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 FUNCTIONS APPLICATION

C is a point on the portion of the graph of f(x) = 6x-x^2 in the 1st quadrant. If points A and B are the points of intersection of the graph of f with the x-axis, what is the greatest possible area of ΔACB?


Answer: 27



Questions:
 1) Do you think the College Board would provide the graph? Would you provide it or would you expect this from the student?

 2) Can you find examples like this in an Alg or Precalc text? Do students need more exposure to applied problems like these?

3) How would you rate the difficulty level of this question --- 3 is medium, 5 is hard.

4) EXTENSION
Change to f(x) = kx-x^2

Ans: k^3/8
Note: College Board is moving toward use of parameters. Can you guess why?



If interested in purchasing my NEW 2012 Math Challenge ç/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 TRIANGULAR PRISM PROBLEM

E

Animated gif showing one way to "draw" the prism above

gif make


SAT Geom questions often assess 3-D spatial visualization. Do you think your students will find the following trivial?

I'm still playing with my Handraw app so the diagrams will continue to be sloppy!

FIND THE VOLUME OF THE RIGHT ISOS TRIANGULAR RIGHT PRISM ABOVE.

Note: Markings are intended to indicate that the height of the prism equals the equal legs of the bases.

[NOTE: Error in labeling choices has been corrected.]

(A) 108 (B) 108√2  (C) 54√2 (D) 27√2 (E) (27√2)/2


Ans: D [Corrected]

How would you explain this?
What methods do you think students who solve this will use?
Apart from the poorly drawn diagram, how many students do you think would quickly recognize that the prism is half of a cube.





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 Variation on Using Powers of 2 to Eliminate Debt

Are you tired of the powers of 2 problems you've seen? Well, most students are not familiar with them, so why not impress another generation while developing their understanding of exponential growth and sharpening their mental math and exponent skills.

The title suggests there is a practical application for this activity, either for our national debt or our personal credit card or mortgage obligations. How you bring this in is all about your personal preference.

We'll start the usual way:

You receive a penny on January 1, 2013; 2 pennies on January 2nd, 4 on January 3rd, 8 on January fourth and so on.
How many dollars will you have accumulated by the end of the month?

To make this a little different why not have your students or children at home use 1 significant digit estimates and do this without pencil and paper!

For example we know 2^5 = 32, therefore 2^ 10 ≈ 1000 or 10^3, so 2^20 ≈ 1,000,000 or 10^6. Continuing, 2^30 ≈ (10^6)•(10^3) = 10^9.
But this is in pennies, so you'd have around 10^7 dollars or $10M!

So is this a fair estimate of how much would have accumulated by the end of Jan? If not, correct my error(s)!

As a teaching strategy, you could demonstrate one of these procedures first, then ask students to devise another path to 2^30 or
2^31.

So what is our approximate national debt today? First one to find it on your smartphone receives a penny!






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 GEOM: A special regular square pyramid

Consider the hand-drawn sketch of a regular square pyramid in which all 8 edges, both base and lateral, are congruent:

(1) Explain why angle PVR is a right angle.

(2) If edges each have length e, explain why the height PT has length e/√2.

(3) Any other interesting observations?


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.

Soln to SAT Quadr Problem 1-19-13

Sketch of solution to SAT quadratic problem

Note: I'm experimenting with a new Draw app on my Nexus 7 so pls

let me know if you can make any sense of 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.

Quadratic Function SAT Practice (Grid-in)

Student Constructed Response ("grid-in") type


If the graph of the quadratic function f(x) = -x^2 + kx - 25 intersects the graph of y = 0 in exactly one point, what is the value of k?

Answer: 10

NOTE: This is similar to but a notch above the College Board Problem of the Day of 1-19-13






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.

Tuneup for Jan 2013 SAT MATH -- TEST YOURSELF

The following questions reflect the medium to more difficult questions on a typical SAT math section. Both multiple choice and "grid-ins" are included.

ALLOW THREE MINUTES.


1.  The base of a parallelogram is 3 times the base of a triangle; the height of the parallelogram is twice the height of the triangle. What is the ratio of the area of the parallelogram to the area of the triangle?
(A) 24 (B) 12 (C) 9 (D) 6 (E) 3

2.  If (8x+8)^2 = 64 and x≠0, then x^2 = ?

3.  What is a value of x for which 1/(10x-20) > 10?

4.  The first 2 terms of a sequence are 4 and 6 and each term after that is the average of all the preceding terms. What is the sum of the first 22 terms?



Answers (formatting error corrected, hopefully! Thanks, Sue)
1.  B.
2.  4
3.  2.01>x>2
4.  110



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 WHY is 10 raised to the zero equal to 1 again?

This is a topic that recurs annually as we move traditional skills/concepts to lower grades according to the Common Core standards. I believe there will be a temptation to give rules without motivating them or explaining them. 

So perhaps raising a number to the power of 0 could be a fourth grade topic these days. I did publish a post several years ago on developing exponent concepts in the middle grades but I guess it can't hurt to revisit it now for even younger students.

One approach I have found useful is to have students construct a table in which the left column is the exponent and the right column is the corresponding power of 10.

I will start the table, suggest some questions but leave the rest for you to bring your own ideas and expertise to the  "table"!


EXPONENT  POWER OF TEN
          3           10^3 = 10x10x10 = 1000
          2           10^2 = 10x10 = 100
          1           10^1 = 10


Teacher: So what do you notice?
Remember --- a mathematician must first be very observant like a good detective!

You and your partner have 1 min to write as many observations as possible...

I HAVEN'T ADDRESSED THE TITLE OF THIS POST YET!
PLS COMMENT IF YOU WANT ME TO PROVIDE MORE DETAILS OR MAKE A SHORT VIDEO.






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.

How many positive integers less than 101 are not div by 2 and not div by 3?

Instead of posting this innocent little combinatorial problem to Twitter like I've been doing for the past 2 years, I decided to post it here so my readers will know MathNotations is not extinct!

Comments about this problem:

1) What would be an easier variation you might first pose? Oct 2009 SAT, Prob 17, "grid-in"!

2) To those who  have been teaching these types for awhile, SHARE your strategies!

3) A Black Box Soln:
100-(50+33-16)=33



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.

Math Teachers at Play 58 Up and Running -- DON'T MISS IT

Head on over to Denise's latest masterpiece collection of outstanding recent math blog posts. Something for everyone!




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.

RAISING AN EDUCATED CONSUMER - A CCSSM MIDDLE SCHOOL ACTIVITY

Ever wonder about the extra 9 in the cost of a gallon of gasoline at the pump? Here's a real-world activity for middle schoolers which might lead to a deeper understanding of decimals and fractions not to mention producing more educated consumers!

So what does the "raised" 9 mean at the end of $2.99? Some retailers display the extra 9 by writing it as 9/10. Nine-tenths of what? Hey, the cost is still less than $3.00 so that 9 is pretty insignificant, right, kids?

We would hope our youngsters would guess the extra 9 represents nine-tenths of a penny or nine thousandths of a dollar. We would hope...

First I would ask the students or my child or grandchild at home what she thinks the 9 means.  In the classroom, we ask the questions but we are not always obligated to confirm or reject students' replies! Research shows that the consensus of the group is usually correct so trust them and help them to develop more confidence!

So how would you develop this lesson to reinforce decimal/fraction skills and develop deeper understanding of decimals in a practical consumer setting? What are some questions you would ask?

Hey, it's my blog and I don't have to show my way first!






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.

Challenge all our children beyond what we think they can do!

So today I told my daughter to again challenge my 5 year old grandson with math problems and don't be afraid to ask him questions that appear to be beyond his knowledge, ability or developmental level. And don't clarify the wording if he hesitates.

I suggested she ask him "What is the biggest even number less than 100?" She expressed that he might not be able to understand the question and had learned only a little bit about odds and evens in class. I told her to ask anyway and if he makes an error just say, "Are you sure about that?"

She called me back a few minutes later and says "Dad, he immediately replied 99 so I said "Are you sure?" and he replied "No, no I meant 98." So of course my daughter followed up with "what is the biggest odd number" but before she could finish the question, he replied infinity plus 1. She looked at him incredulously and didn't have any idea how to respond to that so she said, "I was going to ask you what is the biggest number less than 100 and he replied,  "I already answered that ----  99."

Obviously I'm a proud grandparent like all grandparents are but the real  message is that whether at home or in school we need to stretch our children's minds so that their reach will exceed their grasp. Perhaps we are too concerned about frustrating our children. My kids and my students were never too concerned about frustrating me! It's only fair!





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.
Suitable for SAT I, Math I/II Subject Tests, 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. DON'T FORGET TO SEND ME AN EMAIL (dmarain "at gmail dot com") FIRST SO THAT I CAN SEND THE ATTACHMENT!

Another Inconvenient Truth and a New Year's Resolution

I needed a cause to start blogging again...


Let's state the obvious.
There are many problems in our educational system.

In suburban districts, both middle class and affluent, there is complacency. There is little impetus for meaningful change in curriculum or instruction. Perhaps the new core standards will have some impact but don't count on it. Why are even our top-performing students still not competing well with students from other countries, particularly Asian?

No matter how much money we dump into urban schools, we will not see significant improvement until we acknowledge that the root cause isn't the schools, the administrators, the teachers, the students or the parents. It is the pernicious effect of poverty and deprivation.

WHAT WE CAN DO - A NEW YEAR'S RESOLUTION

I know from direct experience that early intervention programs can make a difference.

When children of poverty enter our schools we need to make sure that their brains are nourished in the literal and figurative sense.

POUR THE MONEY INTO EXTENDING THE SCHOOL DAY

These children need more time in the classroom each day.
Extend the school day. No child should leave the building without having completed their daily work and homework.

Compensate teachers for the additional time. Utilize the many other available resources: retired teachers & professors, students who would be more than happy to work with these youngsters, college students on break, etc...

If I start a petition on change.org to reallocate funding for extending the school day and year for our neediest, will you support it?





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 and DETAILED SOLUTIONS/STRATEGIES for the first 8 quizzes. Suitable for CCSSM, SAT I, Math I/II Subject Tests, 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 by PayPal.