4×4 Square with extra column of 4 results in two equal shaded areas- triangular numbers!
Algebra gets a ‘bum rap’. Then again, it has a lousy public relations manager. Whoever came up with the whole ‘letters and symbols’ campaign should be sacked. Yes, opening up to Exercise 7D and solving 50 variations of 2x + y = -7 is n0t anyone’s idea of fun. But as I said, Algebra needs a new PR campaign.
DISCLAIMER: I’m just a Primary/Elementary teacher without any official qualifications in High Level Mathematics – No Masters, no Ph.D, just an A+ Average in High School/College Maths and 25+ years teaching kids to enjoy,not stress about, Maths. I may be completely off base with the great mathematical minds out there in what I’m about to describe regarding Algebra but I make no apologies. my students get it this way – including the Year 7-11 students I’ve tutored at home to relieve the confusion caused at their schools. (WARNING: Bear with me, I’ll take a while to get to the point of this post’s title – skip ahead if you want to ignore my Algebra rant!)
Now we have that out of the way, back to my message for today. I have a certain belief about Algebra. I define it as a systematic way of organising, recording and explaining your mathematical thinking using numbers and symbols/letters instead of words and pictures. Where we seem to get lost is that we go straight to the symbol without developing the thinking through the words and pictures/objects. We provide no context or purpose; just a meaningless string of equations with Xs and Ys that need to be solved. I see Algebra as problem solving support, not equation solving.
Last week, I was called in to take a Grade 6 class to release a teacher for planning ( the usual release teachers were unavailable). Maths was on the agenda for the day and I had worked with some of the other Grade 6 students on a similar lesson earlier in the week as a support for some of the high achievers. This time, though, I was on my own and in control so I applied my full tech+Maths kit to the group of students I had for that session.
The lesson/task that preceded this actually had fractions as its focus. One of the teachers had introduced a task involving a a building pattern for shading in grids to make fractions.
The lesson was differentiated to allow for a range of responses. Some needed to build the patterns with counters to discover anything. And then there was “Sheldon” ( not the boy’s real name) whom I walked in on to find him showing his mate the formula for the relationship between square and triangular numbers! When I confronted “Sheldon” to explain his formula and why it worked, he didn’t know how. So began my challenge and the rationale behind the lesson I’m about to recount. In the end, Sheldon actually discovered the key to this lesson I led in the class I took later in the week.
SO…this fraction lesson turned into a pattern and algebra exploration. All the children were able to discover the growing patterns in both number sequences and could describe the change. Square number differences increased by +2, the triangular number differences increased by +1. But that additive thinking was as far as they got. They needed more support to think multiplicatively, to think ‘Algebra’.
Enter (finally we get to the title of this blogpost!) the iPad and AirServer. Yes, I could have done all of this without the technology. I had done so earlier in the week with my small group of advanced students. But the engagement and ease of use was no comparison between the ‘sheets of paper and coloured marker’ group and the iPad and AirServer. If you are unaware of AirServer, I explained its significance in a recent post. Basically it projects multiple iPad screens onto a computer connected to a projector/iWB.
We started with creating the fraction grids using the iPad App Hands On Maths Color Tiles ( I reviewed this and others in the Hands On Maths collection last year ). Again, we could have hand drawn grids or made them with counters but I had the students more engaged by getting them to make 1 grid each using Color Tiles and getting multiple students to project their grid onto the whiteboard using AirServer. This took 1 minute instead of 10 and allowed us to move straight into discussion with all the visuals needed on the screen – created by students, not me.
We then discussed the three properties visible in these tiles – side length, square size and the shaded (red here) area ( they hadn’t recognised them as triangles yet). I introduced the problem solving strategy of ‘Make a table’ – a strategy that should be embedded in their thinking by now, but it wasn’t. I created the table on my iPad and projected it on the screen. The students then created their own tables, using Numbers, on their iPads and filled in the side lengths, square sizes and shaded areas. Once they had the numbers in tables, they could start looking for relationships in numbers across the properties, rather than just look at the isolated number sequences. It was at this point that some students were able to recgognise that the shaded area numbers increased by adding on the next side length.
From that discovery, some children then saw that by adding the side length e.g. 4 to the square number 16 ( by this time we had recognised these as square numbers, not just square size), 20 the shaded area was half the size – 10. Here we talked about the importance of proving our theory by testing with other numbers. EVERY child in the class then tested this out with the other numbers, using Explain Everything as a whiteboard to quickly write out equations and project them on the screen to show their proof. Again, this could have been done on paper but by spotlighting everyone through the AirServer iPad mirroring it engaged those children who more often than not pretend to do the work and then let the teacher pleasers to put their hands up and call out the answers. This process really had everyone involved at all times. Some of the less than stellar mathematicians were excited about this discovery. But we were not finished.
I wanted them to see what type of numbers they were creating with the shaded areas – most still didn’t realise. This time I went back to old school methods -
counters. AirServer and my iPad still played a role. I asked the group to use the counters to create the sequence of numbers in the shaded area column in rows. As they began, some weren’t sure what to do. Instead of telling them what to do, I used my iPad’s camera to spotlight pairs who were building triangles onto the screen, thus giving support to others who needed a hint. Every group then wanted their triangles on the screen as well! This idea of spotlighting using iPad and AirServer can work in many ways to maintain engagement – kids like to be on show and recognised .
Once this was done, the students realised they were creating square and triangular numbers and that there was a relationship between them. Children started to recall the rule we had discovered – square the side plus the side then half it gave us the triangular number. But I posed one final challenge – why does this work and how can we show it with our tiles to explain the relationship? Back to Color Tiles we went. We recreated our two coloured square tile pattern. Then we added an extra column/side length. Bingo! The students recognised that this created two equal halfs, a red and yellow half- two triangular numbers!
4×4 Square with extra column of 4 results in two equal shaded areas- triangular numbers!
The final step in the process now was to put all of these theories into one explanation and come up with a formula – finally Algebra was coming into play. The important thing here is that they were thinking algebraically all along – I just didn’t tell them because Algebra is such a dirty word. Now they were quite excited that they were doing algebra.
I asked them to take screenshots of the tiles and the table and import them into Explain Everything. Then we looked at the table again. I explained that the only difference between what we had been doing and algebra was that we needed to replace our words and ideas with letters and symbols. What was the starting point? The side lengths. What will we call them – we decided on s ( could have been x,y, l etc). What is the square number? s x s or s^2. What did we do next? +s. Finally we halved the total ÷2 . With all these symbolic represenations students were able to create a formula for finding a triangular number: (s^2 +s)/2
Now thinking they were expert mathematicians, the students were able to record their understandings in Explain Everything AND find any square and triangular number without creating a long sequence. And they got it because we started with the thinking and investigating, not the formula that “Sheldon’ told us about. By the way, he worked this out independently and actually helped out my thinking with the idea of adding the extra side to the square grid – that’s the first time I had visualised the two triangular halves. This shows that our high achieving students can support the learning in the class – they just need a biy of guidance in their thinking, He was happy with knowing the formula. Now he UNDERSTANDS the formula and why it works. His discovery helped the less able students to also understand the thinking behind it all. And the iPad, the apps and AirServer kept them engaged long enough to get there.
Oh, one more thing. I mentioned earlier context and purpose. I put this whole task in the context of a tile designing company. I talked about how the construction of Federation Square ( a modern structure in the City of Melbourne laden with geometric designs ) was not a random design. It was very mathematical. I put to them the scenario of customers wanting a design like the one we investigated created at a size of their own choosing. As employees of the company, we needed a method for quickly calculating how many of each tile we would need – the formula we discovered would get the job done.
Algebra need not be hard. It’s just logical thinking written down in an organised, symbolic way. Taking students through the right process can demystify it all. And it doesn’t hurt to use a bit of tech like my good friends the iPad and AirServer to help them along the way.