Q: I was 29 the day before yesterday and next year I will be 32. This is true only one day in a year. What day is my birthday?
A: December 31st is the birthday so on December 30: age = 29
Dec 31: turns 30
Jan 1 : the day it is ‘now’ in the question
will turn 31 at the end of the current year (it is far away as it is Jan 1 and the birthday is Dec 31)
therefore the person will be 32 at the end of the next calendar year
At least 5 times a week, someone will share that their son or daughter is a poor test taker. We all know people who ace the test with no effort but much can be gained by focused studying and commitment and I do not just mean test scores!
Tests may not always show knowledge and skills — that 7 in Physics was tough freshman year (yes out of 100!) but I learned so much about attention to detail and focus. How many times do we see 2^3 (2 to the 3rd) = 6 rather than 8? I call these “fuzzy errors”. People can improve their test scores and their attention to detail by answering the exact question and noticing the finer points of what they are being asked.
How did I learn to do this? The hard way — btw, my next score in Physics was exciting because it was double digit — 11! (because class avg was so low, I still managed a C in the class).
It used to be that Math was lots of number crunching but Elementary Math has changed –my theory is the advent of the calculator and Google. Homework can no longer be a page of straight division problems as they can be Googled for the answer and even for all the steps involved!!!
I came across this working on homework with a 4th grader. Is it 9 divided by 2? or 9 divided by 4? Both have a remainder of 1.
See the diagram below for these two ways of interpreting division (albeit division without remainder).
Here is another example:
13 can also be grouped by quotition division see above with 3 groups of 4 with remainder 1. This method will yield the same Math results EXCEPT for problems like 12 divided by 5 which is 2 remainder 2. But divide 12 by 2 and we get 6 not 5 remainder 2 b/c the remainder goes in one more time (same goes for divide 21 by 6 is 3 R3 but 21 divided by 3 is exactly 7 not 6 R3.). So be aware that division these days is about interpretation and not computation.
Q: Tree A and tree B both have some birds. The tree A birds tell the tree B birds “if one of you comes to our tree, our population will be the double of yours.” The tree B birds tell the tree A birds “if one of you comes here, then our population will be equal to that of yours.” How many birds in each tree?
A: Tree A has 7 birds and tree B has 5 birds
This can be done with guess and check or algebra
This is for statement 1) A + 1 = 2(B – 1)
This is for statement 2) A – 1 = B + 1
This is A in terms of B from 2) A = B + 2
Then sub into statement 1 and solve for B
B + 2 + 1 = 2B – 2
B + 3 = 2B – 2
B = 5
then can solve for A using either equation A – 1 = 5 + 1 therefore A = 7
Yikes!!! She saw that superscripted exponent as just a plain vanilla x. How often do students not answer the exact question but something super wicked close? Scores do not always reflect student knowledge and skills due to these “fuzzy errors”.
People can improve their Math skills and scores by paying attention to the details to reduce fuzzy errors. I learned this the hard way in engineering school: a little m means milli or thousandth .001 buta big M means mega or million so if you write a M when you mean m your answer is off by a billionfold — that’s a BIG fuzzy error!
For lesson planning, multiple choice with “good wrong answers” can be used for instruction so that students see common errors and misunderstandings and get less fuzzy. Quantitative Comparison questions are amazing for this — they were sadly deleted from the SAT in 2005 but still alive on the GRE QC questions.
During exams, I walk around with a crayon or highlighter and make visual comments on student papers to point out details such as a missed negative sign (I obviously missed the above during the College Algebra final!!). This helps students to improve their metacognition and attention to detail which strengthens student skills, scores, confidence and enjoyment of Math.