A wooden cube that is 20 cm on each side is composed of 1 cm x 1 cm x 1 cm cubes.
Q1: If you paint all 6 sides of the outside of the cube blue, how many cubes will have no blue paint?
With a 3 by 3 by 3 cube, there is just one cube in the middle that would have no paint.
With a 4 by 4 by 4 cube, there are 8 cubes in the center of the cube (2 x 2 x 2) with no paint.
With a 5 by 5 by 5 cube, there are 27 cubes in the center of the cube (3 x 3 x 3) with no paint.
Answer: The pattern continues so a 20 by 20 by 20 cube, there would be 18 x 18 x 18 (5832) cubes in the center of the cube with no paint.
Q2: What if the 20 x 20 x 20 cm cube is composed of 2 cm x 2 cm x 2 cm cubes?
Answer: There would now be 1000 cubes (10 x 10 x 10), there would be 8 x 8 x 8 (512) cubes in the center of the cue with no paint.
The pressure’s on!
Thanks to @jaz_math for the Day 1 Silent Conversation idea. The PreFreshman Seminar students had amazing ideas on how to be a great Math teacher including content knowledge, attitude and leadership.
Students appreciate teachers who care, who know the material and can impart that knowledge. Hope I can live up to some if not most of these suggestions.
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.
Q: Using the numbers 2 3 4 5 and just the symbols + and =, make a true equation.
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 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
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
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.