Task 111 ... Years 4 - 10
SummaryLimited 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.
IcebergA 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:
However, with the help of the blocks we can see several ways of using square numbers to equal 36.
The 20th square number could also be answered in several ways:
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.
- Visit the Home Page for more Background.
- For this specific activity click the Learners link and on that page use Ctrl F (Cmd F on Mac) to search the task name.
Whole Class InvestigationTasks 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:
From The Classroom
Square Numbers InvestigationAshburton Primary School
BackgroundThis 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.
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.
|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.
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.)
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?
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.
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.
Elements of this lesson could include:
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?.
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.
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) = n2and Mathematical Induction is one way to go about this:
Now assume the hypothesis is true for k, k less than n, and prove it true for (k + 1).
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
= 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: