Powers of Investigations

Prior knowledge

explain the meaning of 2ⁿ

Background:

In this activity students explore patterns of powers.

Further information to back up what they will meet:

• FIO Books. Refer to these for extra activities.
• National Archive of Virtual Manipulatives.

Comments on the Exercises

Exercise 1: The 64th Chess Square
Asks students to solve a puzzle involving powers of 2. This ancient problem has reappeared in many forms. The problem should be clearly explained to students who are then left to investigate patterns and ways of generating an answer. The answer should be manipulated into a sensible form which has meaning to the student. This may be kgs of wheat or buckets of wheat. Having a kg of wheat handy would be a good idea. The answer is enormous. An internet search reveals quite quickly that the annual harvest of the whole Earth is about 1/1000th of the the answer. Of course the Prince could never give the man the amount of wheat he asks for and the donkey of course could never carry it away! Presentation to an audience of a well investigated answer is a good focus for a small group of students.
2⁶⁴ = 1.845 x 10¹⁹ grains of wheat.

Exercise 2: Powerful Leap Frog
Asks students to explore the pattern of powers in a well known leap frog puzzle. This, like Exercise 1, is a popular problem. Carefully working out what is a “walk” and what is “leap” is important. The moves for 1 and 2 frogs on each side are straightforward. Three is the major hurdle. Master three and you master them all. No more help! Use matchsticks if you have no counters.

The number of walks is the even numbers. N frogs, 2n walks or in “kid speak” “just double the number of frogs”. This is good algebraic thinking. The number of jumps is the square numbers giving for n frogs we need n² jumps or similar. The Total then is just (n+1) ² -1 seen from looking at the patterns in the table. This is not exactly easy but is a great group problem.

Some algebraic work here is “Can you show that 2n + n² = (n+1) ² -1 . Use CAS calculator or model it with blocks.

Exercise 3: Regions in a Circle - All is not what it seems!
Asks students to generate a rule that predicts the next answer in a pattern. Beware of assuming you know the answer! The table and careful drawing and counting of regions reveals the pattern 2, 4, 8, 16 for the points 2,3,4,5 respectively. We are lured into thinking the 6 point circle will have 32 regions. Hunt for all the time you have and you will never find the 32nd region. Students tend to spend a long time trying to verify their guess! Seek the 7 point solution and 8 point solution to gather more evidence.
Rule for n<6 is Number of Regions = 2 ^(n-1)
This problem is included because it does not behave perfectly.

Exercise 4: Paper Folding
Asks students to explore a pattern and generate a rule.  Follow the instructions carefully and work through 2 and 3 folds before exploring on. It is important the paper has an up side to refer to all the time. Because of the folding in half it is almost certainly a double 2 series problem. It is challenging to guess the answer before the investigation provided some experience with powers has been developed.
The rule for the number of valleys is 2^F
The rule for the number of hills is 2^F -1
The rule for the total number of valleys and hills is 2F+1 -1.
Ask students to show algebraically that 2^F + 2^F -1 = is 2^F+1 -1. This can be done using a CAS calculator. Why is there always one less hill than the number of valleys? The answer is a bit like the answer to “Why is there always an even number between each odd number?”

Exercise 5: The Powers Charts
Asks students to complete a table using the power button on a calculator.  This exercise uses the x^y button extensively. The charts are related in that one is a horizontal version of the other.
The numbers repeat from time to time which should invoke the question of “Why?”
Other curiousities include:
Any pair of 2^x multiplied by 5^x is a power of 10. “Why?”
The powers of 5 always end in 25
Zeros do not appear in the first chart.
Digits in the powers of 6 add to 9. Likewise powers of 3.
Be aware that some calculators generate 0⁰ = 1. A graph of y = x^x is interesting and suggests the upper limit as x tends to 0 is in fact 1. There is no negative side to this graph. What meaning does 0⁰ have? All good questions.

Exercise 6: Martian Maths
Asks students to explore base 2. This is another application using the powers of two and an excellent opportunity to review place value understanding. What is the largest number you can count to using base 2 with one hand. Answer is a 16, an 8, a 4, a 2 and a 1. One more is 2⁶ so answer is 2⁶ -1 = 63. With 2 hands the biggest number is 2¹¹-1 = 2047. With all 10 fingers and toes the largest number is 2,097,151. This gets even bigger if you include using the head, an arm or two and the legs, a pair of ears, a nose and a tongue! This counting system is very powerful effective.

The obvious move from here is towards using this counting system and developing addition and multiplication basic facts and into computer complementary arithmetic with the understanding that computers can only add and then to digitial electronic circuits.

Lastly, draw the Martian! Play the students the CD of “War of the Worlds”.

Exercise 7: The Magic Powers of 3
Asks students to explore powers of 3. Using the powers of 2 and the binary operation called addition we can generate all other numbers. Using the powers of 3 and the binary operations addition and subtraction we can do the same.
This property does not work for powers greater than 3. Why is this? A clue is to look at powers of 10 where we need multiples of each power to construct all the numbers we use every day.
Answers
12L = R9 + 3, 13L =R9 + 3 + 1, 14 + 1 + 3 + 9L = R27, 25 + 3L=R 27 +1,
47 + 1 + 9 + 27L = R81 + 3, 74 + 1 + 9L=R81 +3
190 can be weighed in parts. Eg A sensible solution might be 6x30kg bags and a 10kg bag depending on what you can lift. Justify your solution!
One half is a tricky idea. One way is to measure with a 1kg and divide the pile of wheat being weighed into two equal parts. Anothere is to share the 1kg of wheat just weighed onto each side of the scale until it balances. The learning is the thinking and the communication of ideas.

Exercise 8: It All Adds Up!
Asks students to explore patterns and write a generalisation. A sequence is any succession of numbers and can be finite or infinite and may or may not have an obvious and predictable pattern. The sequence of prime numbers is not predictable but it is infinite. A series is made by adding any sequence and may be finite or infinite. Quite a lot is known about arithmetic and geometric sequences and series. See any Year 12 text.

A very important sequence is the counting numbers and the series formed by adding the counting numbers forming what we also know as the triangular numbers. Modelling these to students would be an excellent lead in to this exercise. Triangular numbers are fun to explore and will reveal the secret formula for adding n counting numbers. This is n(n + 1)/2 or the middle number x the number of numbers or T _n + T _n+1 divided by 2 where T _n is the nth triangular number. The even numbers is double this formula, the odd numbers is the even numbers subtract n lots of 1 which reduces to n2 of course. The multiples of 3 is 3 times n(n + 1)/2 and likewise 7 and y.

The secret formula for adding n powers of 2 is the (n + 1)th power of 2 minus 1 or 2 ⁽ⁿ⁺¹⁾ 1 which is easily seen from the table. The powers of 3, 4 and beyond is much more difficult but is for powers of y ¹ (y-1) (y⁽ⁿ⁺¹⁾ -1) and it is a useful exercise to show this is true.
For the powers of 3 the formula is 1/2( 3ⁿ⁺¹ -1), the denominator always being one less.

Exercise 9: Adding Fractions Step by Step
Asks students to explore a pattern of adding a series of related fractions. This series converges. Using the sum to infinity formula for a geometric series with first term = 1 and multiplier = 0.5 gives the answer 2 as expected. This answer is not at all obvious to a student so a first guess is important. This pushes the creative thinking buttons. Justifying an answer is part of this thinking.
The practical wall part is always an astonishing event for every student. It becomes obvious quite quickly that the answer must be 2. This is an unavoidable conclusion and will open up a miriad of ideas they may well wish to explore. For example what happens if I stand 3m from the wall? Does it work for thirding the distance?

Exercise 10: Alias Smith and Number
Asks students to use the power button (x^y) on the calculator. Students may need some simple lead in exercises like 2³ and other problems they can easily verify mentally.
Students with very little knowledge of decimals and fractions will almost certainly have trouble with this BUT it could well be an interesting event for them. For example if they guess 8^0.5 as the estimate to get to 2 and then see the answer is actually 2.828… do they “increase or decrease” the power and “what is a smaller number or a bigger number than 0.5?” accordingly.
The “Believe It or Not” is a really BIG idea and may well be a little too infinite to fit inside a Year 9 students head.