All posts by mathconfidence

July 2018 Brain Puzzler Solution

Q: For how many integer values of x is the value of 4000(2/5)^x  an integer?
A: 9
This can be done with guess and check either by hand or with a calculator.
If x = 0, then 4000(2/5)^0 = 4000 so 0 works.
If x = 1, then 4000(2/5)^1 = 1600   (4000 x 0.4) so 1 works.
If x = 2, then 4000(2/5)^2 = 640     (1600 x 0.4) so 2 works.
If x = 3, then 4000(2/5)^3 = 256     (640 x 0.4) so 3 works.
If x = 4, then 4000(2/5)^4 = 102.4 so 4 does not work.
We need to also try negative integers so here goes:
If x = -1, then 4000(2/5)^-1 = 10000 (4000 x 2.5) so -1 works.
If x = -2, then 4000(2/5)^-2 = 25000 (10000 x 2.5) so -2 works.
If x = -3, then 4000(2/5)^-3=  62500 (25000 x 2.5) so -3 works.
If x = -4, then 4000(2/5)^-4 = 156250 (62500 x 2.5) so -4 works.
If x = -5, then 4000(2/5)^-5 = 390625 (156250 x 2.5) so -5 works.
We have been ok all along because we keep getting even numbers (until x = -4) but now…
If x = -6, then 4000(2/5)^-6 = 976562.5 (390625 x 2.5) so -6 does not work.
There are 9 integers that work:
-5, -4, -3, -2, -1, 0, 1, 2, 3

June 2018 Brain Puzzler Solution

Q: Sabina’s car has 50% better fuel efficiency (miles per gallon) than her old car.  The new car uses diesel which is 20% more costly than the gas in her old car.  On a percent basis, how much will she save on a long trip?

A:  20%
For this problem, we made up numbers to make the Math easier 🙂
We can use 10 miles per gallon for Sabina’s old car which would make the new one 15 mpg.
We can pick an easy number that plays nice with 10 and 15 like 30.
If she goes 30 miles on a trip, she will need 3 gallons with the old car but only 2 gallons with the new car.
Now we need a price for gas so we assumed $4.
If gas is $4/gallon but diesel is 20% more at $4.80 per gallon, she would have spent 3 x $4 = $12 with the old car but now spends only $9.60 for the new car (2 x $4.80).
She saved $2.40 on $12 which is a 20% savings.

Equity and Access: Empowering Students for the Common Core Alg I Regents

Shout out to New York the Empire State aka the Regents State!  The Regents is coming!
CC Algebra I is the most important Regents as s’s need at least 1 Math Regents to graduate and Alg I is so much more accessible than Geo or Alg II.
The best prep: Do Regents questions!!

There are certain questions that all and I mean ALL students should be able to answer correctly and get 2 points!  Too often, students have not seen all the content and/or are not adept at the TI-83/4 calculator.

Make sure that you and your student(s), teacher(s) feel empowered to take the exam and learn Math in the process 🙂

Here is a link to the Bible of Algebra I (thanks to the jmap folks!)
If you prefer, you can call it an OED of Algebra I or the Primer of Algebra I.
http://www.jmap.org/JMAPArchives/CurrentVersion/JMAPAI_REGENTS_BOOK_BY_PI_TOPIC.pdf