Square Numbers

Task 111 ... Years 4 - 10

Summary

Limited numbers of coloured cubes are used to help students discover that each square number is formed from the previous one by adding an odd number of cubes. This relationship allows prediction of the nth square number, which means that the value of any square number can now be found in at least two ways.

This cameo has a From The Classroom section which includes an extension for Year 7 and a look at proof by mathematical induction for senior students.

Square Numbers also appears on the Picture Puzzles Pattern & Algebra A menu where the problem is presented using one screen, two learners, concrete materials and a challenge. The extra challenge in this puzzle investigates the difference between two squares.

 

Materials

  • Coloured cubes: 1 red, 3 blue, 5 yellow, 7 green, 9 white, 11 orange
  • Recording sheet

Content

  • square numbers
  • odd numbers
  • square numbers as the sum of odd numbers
  • properties of odd and even numbers
  • number patterns
  • algebraic generalisation
Square Numbers

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
   

The card provides guidance for finding answers that are recorded on the sheet. Answers can vary. For example, the openness of the question giving the answer as 36 and asking for the corresponding question might yield:

  • 36 = 25 + 11 (previous square plus an odd number)
  • 36 = 25 + (2 x 6 - 1) (previous square plus a line of blocks down the right side and along the bottom, which counts the bottom right corner square twice)
  • 36 = 16 + 9 + 11 (square previous to the previous square plus two odd numbers)
  • 36 = ...
  • 36 = 1 + 3 + 5 + 7 + 9 + 11
and students may suggest others, including 9 x 4 which uses only square numbers.

The 20th square number could also be answered in several ways:

  • S20 = 20 x 20 = 400
  • S20 = S19 + (20 x 2 - 1) = 261 + 39 = 400
  • S20 = 20 x 1 + 2(1 + 2 + 3 + ... + 18 +19) = 20 + 2 x 380/2 = 20 + 380 = 400
  • ...and, no doubt, others
Each of these has a visual explanation if thought of in terms of the blocks:
  • The first is square of twenty rows of twenty blocks, hence 20 x 20.
  • The second builds on the known result for the 19th square by adding a row of twenty down the right and another along the bottom. But this produces two blocks at the same place in the bottom right of the 20th square, so one has to be removed, hence 20 x 2 - 1.
  • The third builds on the properties that every odd number is one more than an even number and every even number can be shown as blocks lined up in twos. Also the fact that the 20th square number will be the sum of the first 20 odd numbers. Each of these 20 numbers will contribute a 1, hence 20 x 1. The remaining sum is 2 + 4 + 6 + 8 + ... + 36 + 38, Hence 2(1 + 2 + 3 + ... + 18 +19). One way of summing the natural numbers (as in this series to 19) is to make a staircase, then copy it, then place the two together to make a rectangle. This is twice the number of blocks required, hence 380/2.
Since each of these has a visual/concrete/tactile base, any one of them might be suggested by students even though we might not think they are 'simple'. Another value of their existence is to validate once again the mathematician's question:
  • Can I check this another way?
Follow this question through a little further in the From The Classroom section below, which includes an extension of the problem into Year 12.

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
   

With a good collection of wooden blocks, or linking cubes, and students working in pairs, the outline above provides the content for a rich lesson. Teachers with Poly Plug will be easily able to convert the investigation to this material and if the school is a Calculating Changes member, the activity Squares & Square Roots will enrich the lesson further.

With either material, you might like to begin the lesson at a floor space using cards about 20cm x 20cm to create the atmosphere of a learning community and to clarify the problem before the concrete materials are used to deepen the investigation.

At this stage, Square Numbers does not have a matching lesson on Maths300, however, Lesson 12, Gauss Beats The Teacher, explores several hands-on ways to sum number series.

Visit Square Numbers on Poly Plug & Tasks.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
   

The Square Numbers task is an integral part of:

  • MWA Pattern & Algebra Years 3 & 4
  • MWA Pattern & Algebra Years 7 & 8

From The Classroom

MacKillop College
Swan Hill

Damian Howison
Year 7


I read your recent cameo for Square Numbers. It came at a good time because one young lady took to the task but we weren't sure at the end of it where to go. Then I read your cameo the very next day and we came up with something nice.

Damian offers this interesting interpretation of the problem here for all to share.

Doug Williams
Consultant
From a senior school perspective, this problem could be revisited from the point of view of proof. None of the explanations above is a proof. They are demonstrations based on examples, but no matter how many examples may be successful, this is not a proof that all examples will be successful. None-the-less such demonstrations can point the way to an hypothesis, in this case, that the sum of the first n odd numbers is n2.

So we have to prove that:

1 + 3 + 5 + ... + (2n - 1) = n2
and Mathematical Induction is one way to go about this:

First test the hypothesis for n = 1.
L.H.S = 1
R.H.S = 12 = 1 = L.H.S

Now assume the hypothesis is true for k, k less than n, and prove it true for (k + 1).

So assume:
1 + 3 + 5 + ... + (2k - 1) = k2 ... Equation A
is true and, as a consequence, prove that:
1 + 3 + 5 + ... + (2k - 1) + (2[k + 1] - 1) = (k + 1)2

L.H.S
= 1 + 3 + 5 + ... + (2k - 1) + (2[k + 1] - 1)
= 1 + 3 + 5 + ... + (2k - 1) + (2k + 2 - 1)
= 1 + 3 + 5 + ... + (2k - 1) + (2k +1)
= k2 + 2k +1 ... by substituting from equation A
= (k + 1) (k + 1)
= (k+ 1)2
= R.H.S as required.

Think of this procedure as like climbing an infinite ladder. To guarantee reaching the top you must: This process works whether the ladder has one step, two steps or an infinite number of steps.

Green Line
Follow this link to Task Centre Home page.