Mathematical Thinking 3

SEAMEO RECSAM

19 April 2017

18-Apr-17 Assoc. Prof. Dr Kor Liew Kee 2

What is Mathematical Thinking?

• http://study.com/academy/lesson/critical-thinking-math-problems-examples-and-activities.html

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4 fundamental processes on how to think mathematically

• Specializing – trying special cases, looking at examples

• Generalizing - looking for patterns and relationships

• Conjecturing – predicting relationships and results

• Convincing – finding and communicating reasons why something is true.

Task 1

Exponents

Exponents

https://aiminghigh.aimssec.ac.za/grades-8-10-powerful-thinking-2/

Questions

• What pattern can you see in the last digits of the powers of 2?

• What about the last digits of the powers of 3?

• What patterns can you see in the last digits for the numbers in the picture?

• Can you explain why all these numbers are multiples of 5?

Further questions

• What about the 99th powers 299+399?

• What can you find out about 199+299+399+499?

• What other patterns with powers can you find that have special properties?

Possible solutions

Explanation

• The table shows the last digits in the powers of 1, 2, 3, …9.

• The last digits repeat in cycles. • For example the last digits in the powers of 2 are

2,4,8,6,2,4,8,6,2… and • the last digits in the powers of 3 are 3,9,7,1,3,9,7,1,3…

both cycles of length 4. • Adding the sums of the last digits of the numbers with

odd powers: 21 + 31, 23 + 33, … are always 5 for all odd powers.

• So these numbers are divisible by 5.

Exploring 199+299+399+499

• The last digits in the powers of 4 repeat in a 2-cycle 4, 6, 4, 6,… with the last digit for odd powers always equal to 4.

• Also 1 raised to any power equals 1 so the last digit of 199 + 299 + 399 + 499 ends in 0.

• Hence, this number is divisible by 10 (also by 5).

What about the 99th powers 299+399?

• As 99 is an odd number we know that 299 + 399 is divisible by 5 (is a multiple of 5) because it has 5 as the last digit.

Key Questions

• What do you notice?

• Can you describe a pattern that you see?

• What is special about the powers 1, 3, 5, 7 and 9 in this question?

• What do you notice about the last digits?

• Can you explain how you know that?

• How do you know when a number is divisible by 5?

• How do you know when a number is a multiple of 5?

• Can make up your own pattern of numbers involving sums of powers that share the same property?

Possible support

Task 2

Exponents with nth power

https://aiminghigh.aimssec.ac.za/grades-8-10-powerful-thinking-2/

Questions

• Will the first number 2n+3n be a power of 10 for any value of n?

• What can you say about the values of n that make the second number 3n+7n a power of 10?

• Are there any other pairs of integers between 1 and 10 which have similar properties?

Specializing

Specializing and Generalizing

• The table shows the last digits in the powers of 1, 2, 3, …9. • The last digits repeat in cycles. • The last digits of 2n are 2, 4, 8, 6, 2, 4, 8, 6, … repeating the

cycle 2, 4, 8, 6 over and over again • The last digits of 3n are 3, 9, 7, 1, 3, 9, 7, 1 … repeating the

cycle 3, 9, 7, 1 over and over again … • So adding 2n + 3n gives the last digit 5, 3, 5, 7, 5, 3, 5, 7 …

repeating the cycle 5, 3, 5, 7 over and over again. • The numbers 2n + 3n are divisible by 5 for all odd values of

n. • The last digit is never zero so 2n + 3n is never a power of 10

for any value of n.

Conjecturing and Convincing

• The last digits in 31 + 71 , 33 + 73, 35 + 7 5, 37 + 77, … and so on for all odd powers all end in 0.

• So these numbers 3n + 7n are divisible by 10 (multiples of 10) for all odd values of n.

• Similarly the last digits in 13 + 93, 15 + 95, 17 + 97, … and so on for all odd powers all end in 0

• So these numbers 1n + 9n are divisible by 10 (multiples of 10) for all odd values of n.

• Similarly the last digits in 23 + 83, 25 + 85, 27 + 87, … and so on for all odd powers all end in 0.

• So these numbers 2n + 8n are divisible by 10 (multiples of 10) for all odd values of n.

• Similarly the last digits in 43 + 63, 45 + 65, 47 + 67, … and so on for all odd powers all end in 0.

• So these numbers 4n + 6n are divisible by 10 (multiples of 10) for all odd values of n.

Key Questions

• What do you notice?

• What do you notice about the last digits?

• Can you explain how you know that?

• Can you describe a pattern that you see?

• Can you use algebra to make what you are saying into a general statement?

Possible support

Task 3

https://aiminghigh.aimssec.ac.za/wp-content/uploads/2016/09/Teacher-Notes-POWERFUL-THINKING3.pdf

Questions

• Imagine the first number 34 × 45 × 56 is written out in full. How many zeros would there be at the end?

• Why?

• Can you find the answer without doing the whole calculation?

Possible solution

• 34 × 45 × 56

• = 34 x 210 x 56

• = (3 x 2)4 x 26 x 56

• = (3 x 2)4 x 106

• So there will be 6 zeros at the end.

Questions

• Work out a few values of the number 1n + 2n + 3n .

• What can you say about the sum for all values of n?

• For what values of n will 1n +2n +3n + 4n be even?

Possible solution

• 1 + 2 + 3 =6, • 12 + 22 + 32 = 14, • 13 + 23 + 33 = 36, • 14 + 24 + 34 = 98 … All these are even. • For all values of n, 1n = 1 is odd, 2n is even and 3n

is always odd. • The sum of two odd numbers is always even and

the sum of two odds and an even number is even.

• So 1n + 2n + 3n is always even for all values of n.

Questions

• What about 1n +2n +3n +4n +5n ?

• What can you say about the sums of powers of the counting numbers 1, 2, 3, 4, 5, 6, 7, …p?

Possible solution

• As 4n is even for all values of n it follows that 1n +2n +3n +4n is even for all values of n.

• As 5n is odd for all values of n it follows that 1n +2n +3n +4n +5n is odd for all values of n.

• All nth powers of odd numbers are odd and all nth powers of even numbers are even.

• So 1n + 2n + 3n + … + pn is even if p is even and odd if p is odd.

Key Questions

• Is 4 a prime number? How could you write it as a power of a prime number?

• What do you notice? • What do you notice about the last digits? • Can you explain how you know that? • Can you describe a pattern that you see? • Is the sum of 2 odd numbers odd or even? How do you

know? • Is the sum of 2 even numbers odd or even? How do you

know? • Is the sum of an odd number and an even number odd or

even? How do you know?

Possible extension

Further Exploration

• Is this number 3444+4333 divisible by 5?

• Investigate other big powers.

• Make up some similar numerical expressions involving powers that have interesting properties.

• What about 2666 + 6222 ,7888 + 8777?

Key Questions

• What do you notice? • What do you notice about the last digits? • Can you explain how you know that? • Can you describe a pattern that you see? • Can you see a repeating pattern? How many

digits are repeated? • What is the length of the cycle of days of the

week? (or months of the year?) • How many cycles of length 4 would you complete

before you get to the 666th power (or the 444th)?

Solution

• https://aiminghigh.aimssec.ac.za/wp-content/uploads/2016/09/Teacher-Notes-POWERFUL-THINKING4.pdf

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