Powers of Numbers

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Purpose

These exercises, activities and games are designed for students to use independently or in small groups to practise number properties. Some involve investigation and may become longer and more involved tasks with consequent recording/reporting. Typically an exercise is a 10 to 15 minute activity.

Achievement Objectives
NA5-2: Use prime numbers, common factors and multiples, and powers (including square roots).
Specific Learning Outcomes
  • Explore powers of 2.
  • Explore powers of 3.
  • Add with two or more powers of 2.
  • Explore a pattern in an exponents series.
Description of Mathematics

Multiplication and Division, AP (Stage 8).

Required Resource Materials
  • 100s board, chart and flipblock board.
  • Multilink blocks.
  • Practice excercises with answers (PDF or Word).
  • Counters.
  • CAS calculator for factorising large numbers.
Activity

Prior knowledge

  • explain 1 x 2 = 2
  • double any two digit number
  • use of CAS calculator

Background

This activity explores pattern with powers of 2 and 3.

Further information to back up what students will meet:
FIO books. See Number Book exercises for more investigations.
Learning Objects.
National Archive of Virtual Manipulatives (Use Google)

Comments on the Exercises

Exercise 1
Asks students to list the powers of 2 up to 216. This exercise follows up teacher modelling of the powers linking the index notation, repeated multiplication and the value. From the approximation of 250 = 1x1015 it is deduced that there are 16 digits. There is always a pattern in the values that allows identification of the end digit which is 4 in this case. Students should be encouraged to see the x2 as the doubling operator.

Exercise 2
Asks students to explore the cube pattern. Any power with (23)n will generate a cube. Another way of looking at this is when the index is a multiple of 3. It must be 3 because the three dimensions are all the same length in a cube. This exercise would follow instruction on creating the powers 2x2, 2x2x2, 2x2x2x2, etc and observing some can be modelled into a cube.

Exercise 3
Asks students to explore the size of powers using physical material.  This exercise would follow instruction on creating the powers 2x2, 2x2x2, 2x2x2x2, etc and observing some can be modelled into a long tower. Allowing a small group to explore the huge numbers resulting from doubling to say 220 using the multilink blocks (2cm approx) would result in this power being over 20 km long. Is the playground big enough! The distance from the Earth to the moon is about 384000km or 234 cubes would nearly reach and 235 is about twice as far. Writing and reading big numbers could be revised here.

Exercise 4
Asks students to to complete a doubling chart. The chart patterns result from doubling and are obvious.

Exercise 5
Asks students to complete a chart of the powers of 3.  This exercise follows up teacher modelling of the powers linking the indice notation, repeated multiplication and the value. From the approximation of 350 = 7x1023 it is deduced that there are 24 digits. There is always a pattern in the values that allows identification of the end digit which is a 9 in this number. Students should be encouraged to see the x3 as tripling.

Exercise 6
Asks students to express numbers using powers. This exercise leads nicely to binary arithmetic and a revision of base 10. It is interesting to note any number can be represented by adding powers of 2 or by adding and subtracting powers of 3 and this can not be done with any other base power. Why is this? 219+218+217+216+214+29+26 is 1 million. The last digit is always 0 (zero) because it is like x10 (x2x5). Factorising a million to 10x10?10?10?10?10 allows 26 (64) x 56 (15625) to be identified as the two zero-less numbers. Investigate with all powers of 10.

Exercise 7
Asks students to play a Four in a Row game with powers. Four in a row is a common numeracy style game and students should be familiar with it. The odds and evens are separated for no particular reason. A better board arrangement may be found.

Exercise 8
Asks students to complete a table involving powers and primes.  This formula generated many but not all primes. There are probably an infinite number of similar formulae each of which would generate an infinitely large number of primes. 3123 – 2 for example is likely to be a prime but the CAS calculator could take 17 billion years to confirm this.

Exercise 9
Asks students to complete a table with powers of numbers of their own choosing. This is similar in style to exercise 1 but student should choose different numbers to explore.

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Level Five