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metacognition, non-calculator, Teaching Philosophy

Why Learn to do Math by Hand?

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            Division Jluy 2019       There’s a connection between pencil, paper and brain.

My chairman who is 20 years my junior has lit upon an old idea: help students understand Math without the crutch of a calculator.  This summer, I will teach arithmetic along with algebraic and geometric skills to rising college freshmen to boost their analytical skills, critical thinking and mathematical breadth.  One of the goals is to help students see how much understanding the times tables contributes to the knowledge and skills for high school and college Math.

After teaching Math for Elementary Educators for over 10 years, I can appreciate how the classroom can differ without the use of technology.    This is like going back home as I am from the last millenium aka BC Before Calculators.

Some of the topics we will cover include multiplication and division in various formats, adding and multiplying fractions, finding equivalent fractions for repeating decimals as well as using the “educated guess and check” method for computing square roots.  I may also dust off my knowledge of square roots by hand and share this division-related topic.

So how would you do 1472 divided by 5?  If you answered “With a calculator.”, you may find that most efficient but doing it by hand can help build estimation and computational skills while promoting Math facts and self-reliance.  In a tech-filled world, it can be wonderful to think independently without a calculator or Siri or Alexa or Google Assistant etc.

So sharpen your pencils and try my 3 favorite division problems on paper — yes please write these down and then close your laptop and/or power down your phone til it’s just you and the Math:

70 divided by 5
365 divided by 7
1000 divided by 8.

 

brain puzzler

May 2019 Brain Puzzler Solution

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Q: Let n = 2^2008 + 2008^2. What is the units (ones) digit of 2^n + n^2?

A:   This is related to the powers of two and also the last digits of squared numbers.

We first have to find n
Since the last digit of 2^(a power) has a pattern we can figure this out!

2^2008

2 to a power has a last digit pattern
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
2^9 = 512
2^10 = 1024
2^11 = 2048
2^21 = 2097152
2^31 =

Since it is a cycle of 4, every power divisible by 4 will end with a 6…therefore 2^2008 ends with a 6…but we actually need to know the last two digits as 2^1 and 2^11 do not share the same last digit but 2^1 and 2^21 do.  So when we think about very large exponents for powers of 2, we need to know the last two digits…
2^4 ends with 16
2^8 ends with 56
2^12 ends with 96
2^16 ends with 36
2^20 ends with 76
2^24 ends with 16
2^28 ends with 56 and so on
every multiple of 20 ends with 76
so 2^2000 would end with 76
2^2004 ends with 16
2^2008 ends with 56

2008^2 =4032064
Adding up 64 + 56 = 120 so the last two digits of n would be 20

so in evaluating 2^n + n^2
First 2^n: 2^20  or 2^120 or 2^ 220 always ends in a “6”

then n^2, anything with a 0 in the ones digit squared would have a ones or units digit of zero.

Therefore, the last digit would be 6.

 

 

 

 

 

brain puzzler, brain teaser

March 2019 Brain Puzzler Solution

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Brain Teaser March 2019

 

on the right side there are 3 pink and 3 blue
on the left side, there are 2 pink and 3 blue plus 5 green
Therefore the 5 green are equivalent to 1 pink.

On the bottom left, we can see that 1 pink = 2 green + 1 blue
so since 1 pink is 5 green and also 2 green + 1 blue,

5 green = 2 green + 1 blue
so 3 green = 1 blue or 1 blue = 3 green

Looking at the right hand side, we have 3 blue + 3 pink = 1/2(144)
If we replace blue and pnk in terms of green,
3 (3 green) + 3 (5 green) = 72
9 green + 15 green = 72
24 green = 72
1 green = 3
1 blue = 3 green = 3(3) = 9
1 pink = 5 green = 5(3) = 15

Thanks to https://solveme.edc.org/mobiles/