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 extended lesson plan from a Year 5 class which could be modified to use at any level, an extension for Year 7 as an Investigation Guide 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. It is available as one of two sample Picture Puzzles and supported by Teaching Notes. 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 on the sheet giving the answer as 36 and asking for the corresponding question might yield:

  • 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 = 25 + 11 (previous square plus an odd number)
  • 36 = 16 + 9 + 11 (square previous to the previous square plus two odd numbers...)
  • 36 = 1 + 3 + 5 + 7 + 9 + 11 (...and when that reasoning is continued back the sum of odd numbers results because 1 is both odd and square
  • 36 = ...
...and students may suggest others, including 9 x 4 which uses only square numbers as requested in one of the questions on the card.

However, with the help of the blocks we can see several ways of using square numbers to equal 36.

-----------------------------------------------------------------------------------------

  • 3 rows with 3 objects in each row = 32
  • Each object represents 2 rows with 2 blocks in each row
    = 22
  • Total of blocks
    = 32 lots of 22
    = 32 x 22
Look again.
  • Can you see 3 x 2 rows?
  • Can you see 3 x 2 in each row?
  • Total of blocks
    = (3 x 2) rows with (3 x 2) blocks in each
    = (3 x 2)2
-----------------------------------------------------------------------------------------

  • 2 rows with 2 objects in each row = 22
  • Each object represents 3 rows with 3 blocks in each row
    = 32
  • Total of blocks
    = 22 lots of 32
    = 22 x 32
Look again.
  • Can you see 2 x 3 rows?
  • Can you see 2 x 3 in each row?
  • Total of blocks
    = (2 x 3) rows with (2 x 3) blocks in each
    = (2 x 3)2
-----------------------------------------------------------------------------------------

  • 6 rows with 6 objects in each row = 62
  • Each object represents 1 row with 1 block in each row
    = 12
  • Total of blocks
    = 62 lots of 12
    = 62 x 12
Look again.
  • Can you see 6 x 1 row?
  • Can you see 6 x 1 blocks in each row?
  • Total of blocks
    = (6 x 1) rows with (6 x 1) blocks in each
    = (6 x 1)2
-----------------------------------------------------------------------------------------

  • 1 row with 1 object in each row = 12
  • Each object represents 6 rows with 6 blocks in each row
    = 62
  • Total of blocks
    = 12 lots of 62
    = 12 x 62
Look again.
  • Can you see 1 x 6 rows?
  • Can you see 1 x 6 blocks in each row?
  • Total of blocks
    = (1 x 6) rows with (1 x 6) blocks in each
    = (1 x 6)2
-----------------------------------------------------------------------------------------

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.
Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.

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. Using cards which are a different colour on each side helps to reveal patterns. (If two colours isn't possible, mark one side with cross using the diagonals.)

A Year 5 lesson at Ashburton Primary School which began this way is described below. The lesson is also extended using the Square Numbers Picture Puzzle which is available in the Free Tour of that part of Mathematics Centre. (See link below in the lesson.)

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

Square Numbers Investigation

Ashburton Primary School
Year 5

Background

This lesson plan has grown from initial experiences at Ashburton Primary School. As described it fitted comfortably into a 50 minute time slot. It is presented both as 'a good idea for tomorrow' and as a self-directed professional development unit for a group of teachers.

You are encouraged to read this lesson plan through with a partner in a teachers' meeting and imagine yourselves as observers in the classroom. At the end you will find support for debriefing the lesson as a team.

Setting the Context

As students are entering and settling write on the board:
School mathematics is Learning to Work Like a Mathematician.
Draw attention to the statement and if the class is experienced in this approach go straight to...
And how does a mathematician's work start?
...expecting the choral response:
With an interesting problem.
However, although this class had been experiencing Working Mathematically, it was the first time this statement had been made explicit. So the conversation following the board statement began with the question:
What do you think a mathematician's work is about?
Briefly accept and record responses and work towards pointing out that the starting point of the work is an interesting problem.
So today I have found a problem for us that I hope you will be interested in.

Introducing the Problem

Arrange that all students are handed a card and ask them to move into a 'circle' around a floor space. Invite one student to place their card on the floor. (See above for explanation of the 20cm x 20cm cards.)
What shape is this?
Explain that the problem today is going to be about making squares.
So we can make a square with one card. That's why 1 is the first square number. Now I want people to make the next square number by putting more cards with this one.
If necessary, encourage using two colours to show which cards were placed first and second.
Now I can't tell which cards were placed second. Can anyone find a way to make it clear?
Ask how many cards are needed in total to make the second square number.

Move on to asking people to place cards to turn this into the third square number.

One student in this class changed the cards into a 'doughnut square'. What to do?
Now that's a great idea X. A square inside a square and another square could go around that and so on. I think you have created a new problem. Let's look at that idea another day.
Mental note to self: Use Squares Around Squares in the future and call it X's Squares.
  • Encourage again that colours are used to show the cards placed to make the third square number.
  • Ask the number of cards in total to make the third square number.
Now before I ask you to make the next square number, I want you to whisper to the person next to you and tell them the number of cards that need to be added to the model to make the next square number.
Ask students to make it and ask whose prediction was correct.
If your prediction wasn't correct, ask someone else how they figured it out.
Allow a minute or two for this and you will hear mathematical conversation and see a bit of pointing to the floor and gesturing. Again ask for the number of cards in total to this stage.

Now whisper to the person next to you and tell them the number of cards that need to be put down to make the fifth square number.
Ask whose prediction was correct - it will almost certainly be everyone - and ask for the total number of cards to this stage.
So we know it takes 25 cards to make the fifth square number. Your problem for today is to find out how many cards it takes to make the twentieth square number. I would like you to work in pairs and you can use these Poly Plug or this Square Paper or anything else in the room that might help you.
(You might have cubes or square tiles available in the classroom.)

Investigating - collecting & organising data, making and testing hypotheses, recording

Probably the first thing you will notice at this stage is that everyone knows exactly what to do and how they are going to go about it. Certainly that was the case in this class, although that doesn't mean that weren't children who needed assistance. Allow about 20 minutes for this exploration phase and move around from group to group.

As they work many students will find the need for recording, so when you notice that compliment them on working like a mathematician.

Making notes and drawings like that is just what a mathematician would do. They can't be certain they will solve the problem today, so they need to know where they were up to when they come back to it.
For this reason all the Year 4 & 5 classes at Ashburton have a mathematics journal as well as a mathematics exercise book.
Certainly some students will want to give you their answer before the 20 minutes is up. They will also expect you to tell them it is right or wrong. It will help to develop independent learners if you avoid that temptation and instead, after asking them to explain their answer, point out that a mathematician can't ask anyone if their answer is correct. They are working on a problem and it's a problem because it has been solved yet, so no one else knows the answer.
What a mathematician asks now is Can I check this another way?. See if you can find another way to work out the 20th square number. If you get the same answer as this one, you are very likely to be correct.
Use a calculator might be alternative approach which is suggested. It is certainly consistent with the fact that technology is used in much mathematical investigation these days.
And is there another way?

Bringing it Together

With 10 minutes left in the lesson pack up the investigation and bring students back to their tables or the carpet.
Okay who can tell me what our problem was today? ... Don't tell me the answer, just put up your hand if think you have worked it out? ... All right, I'll count to three and you all say the answer. One, two, three...
That gets the answer out of the way. Now take another couple of minutes to review ...how we have worked like a mathematician. Inexperienced classes can make suggests which are noted on the board. More experienced classes will have the Working Mathematically page available, either as a wall chart, or as a page in their journal, or both.

Looking Ahead

Congratulate the students on their attitude and activity in this lesson and hand out the Recording Sheet as homework.
Great effort today. You are really starting to work like mathematicians. I want you to fill this sheet in for homework and bring it back tomorrow to paste in your journal. You will soon see it is very similar to what we have just been doing.
Close the lesson in your usual way.

Debriefing as a Teaching Team

Hopefully this lesson plan forms a unit of self-directed professional development and not just a 'good idea for tomorrow'. If the words and images have helped you 'see' your way into this classroom and encouraged you to think about this investigation in terms of your own class:
  1. Discuss as a group what you have 'seen' supported by this Debrief Guide.
  2. Organise to teach the lesson and meet again to compare and contrast the experience.
  3. Plan the next lesson(s) based on the notes below.
  4. Repeat from 2 as appropriate.

Next Lesson(s)

The most common outcome of the first lesson is that students discover (or have reinforced) that it's easy to find the value of a square number simply by multiplying the number by itself. However, it's not always clear that the value of a square number can also be calculated by adding a sequence of odd numbers from 1. It seems that the pattern of odd numbers in the diagrams above comes to be seen as a step to discovering the multiplication calculation and the calculation by addition is left behind along with the L shape pattern that represents it. The L shape will be revisited and extended in this lesson.

Elements of this lesson could include:

  • Pasting homework into journals and discussing what was recorded, especially for the open question 36 = .....
    (See iceberg notes above for some possible responses.)
  • Connecting the various approaches with the mathematician's question Can I check it another way?.
  • Downloading and introducing the Free Tour Picture Puzzle Square Numbers.
    • For students this is intended to refresh and extend the initial investigation.
    • For teachers it is an opportunity to explore the teaching craft of one screen, two learners, concrete materials and a challenge.
      (The puzzle is well supported by the teaching notes linked below the download above and the teacher story linked below that.)
    • How will you distribute this PDF slide show to the students' devices?
    • What equipment will you use?
      1cm linking cubes, available in many schools, work well for this form of the investigation but it pays to have enough sorted into sticks of 15 in the same colour before the lesson starts.
  • Working through the first few slides together with the students then giving them time to explore the next slides themselves.
  • Creating a plenary discussion around this slide. The discussion to include a question such as:
    So we know we could calculate the value of, say, the 50th square number by adding a series of odd numbers starting from 1. But what would be the end number of that series and how do you know?.
  • Exploring the 'More' section of the slide show to discover how the L shape can be used to do a special type of calculation.
  • Using Damian Howison's Investigation Guide (see below) as the next homework sheet, or as a mini-assessment project.

 

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 a 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.