xiPad + yApps + zAirServer = Engaging Algebra

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.

3 iPad apps for volume and surface area investigations

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This week, Grade 5 began a unit on Volume, Capacity and Surface Area. On a weekly basis, I take combined groups from the 4 grades consisting of the higher achievers, while the classroom teachers concentrate on the mainstream group and students needing more individual instruction to achieve success. I made a conscious decision this week to focus on using iPads with my group to explore both volume/capacity as well as surface area.

I chose 3 apps to assist me in this learning experience – Think 3D ( free version) and Skitch, which are both free apps and Numbers ($9.99- $4.50 through the Volume Purchasing Program if 20 or more bought). Note: you could substitute the currently free CloudOn app, which is basically a server based Office app, or Google Spreadsheets, a free component of Google Docs/Google Apps for Education.

In the past I would have run this lesson using a limited number of connecting blocks and would have asked the students to record their observations in their exercise books. In using the iPads and the selected apps, I wanted to trial how this type of investigation could be enhanced and improved upon by using technology rather than traditional tools.

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The lesson began with the following premise. Each pair of students ( didn’t have enough iPads for 1:1; would probably work in pairs regardless to encourage collaboration and discussion) was to create a cuboid or rectangular prism with a volume of 72 cubes using Think 3D. In the past, students would have used a limited supply of blocks and would only have had enough to make one model. Using the iPad app, they were able to explore multiple ways of making a 72 cube prism with a limitless supply of cubes with a simple touch of the screen adding or deleting  a cube to the prism each time.

Another advantage is that, while there are many benefits in physically seeing and touching a real 3D object rather than a 2D representation of one on a screen, the ability to rotate the prisms on the iPad to view the different surfaces with a simple swipe made for easy investigation and no chance of the object falling apart and needing to rebuild, thus saving time for more analysis.

Using Reflection on my Macbook ( also available for PCs), the children were able to mirror their iPad screens on our interactive whiteboard and share all of the possible prisms and cuboids. This allowed for easy comparison and discussion without having to move our models around as we would have in the past.

The next step was to save the models as images in the Photo library on the iPad so that we could import them into Skitch, (an annotation app linked to Evernote.) As you can see from the image below, the students were able to clearly label the dimensions of their prisms and record surface area measurements as well. The use of this app enables easy collection of data for assessment rather than the rather difficult alternative of taking photos with a camera and writing notes about each photo. It also makes it easy for the children themselves to keep records of their work and thinking, an improvement on the lesson for both teacher and student. They were also able to swipe back to Think 3D to manipulate the prism to investigate the dimensions closely during the annotation stage.

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We then opened up Numbers to systematically record and calculate the measurements using spreadsheet formulas. Being capable students, they already knew how to use the L X W for area and L X W X H for volume formulas. I wanted to skill them up in using spreadsheet formulas to make quick calculations so that more time could be used for analysing the measurement data and the 3D models.

The spreadsheet was laid out so all possible dimension combinations discovered by the students were recorded. We then inputted a volume formula to verify each prism had a volume of 72 cubes. We then used formulas of our own creation to calculate the surface area of each prism. Once one formula was created, we were able to copy and paste that formula for each prism to calculate each prism’s surface area. Once we had all of the volumes and surface areas, combined with the 3D models, students were then able to make informed conjectures, observations and proofs about why certain prisms of the  same volume had varying surface areas.

While I am not saying I haven’t taught this lesson successfully in the past, using these apps and the iPad allowed for more direct and focused engagement from all students. Previously, the recording of data would have been a whole class event, which I always feel has the potential for disengagement as children watch others do the work. Having limited resources in terms of blocks, early problem solvers are left waiting for others. With the use of Think 3D, they were able to continue on with their own investigations rather than waiting for another pair to make an alternative model.

With today’s lesson, the children were actively involved in all aspects. They had opportunities to explore as many options as they had time for, they inputted all mesurement data, they annotated all of their images, which enabled them to consolidate and record their thinking more efficiently. The technology used also enabled them to save a permanent record of all the work they did today, whereas in the past, it was lost once the cubes were packed up. I  think this is a good example of how technology, and the iPad in particular, can be used for greater engagement and deeper thinking in Mathematics. Yes, all of the steps in the lessons could have been done without tech or iPad specifically, but I don’t think it is as effective.

Hands On Maths with the iPad

Earlier in the year, I wrote a couple of posts on the iPad and Maths Apps. I questioned whether there were apps out there that went beyond number facts drills and calculation games. One of my readers of those posts, Melissa,  let me know about a group of apps called Hands on Maths. This set of apps provide a range of digital versions of hands on manipulative tools that are needed to develop important Mathematics concepts and skills. I am in no way suggesting that they replace the physical tools entirely but they do provide always available, easy to manipulate tools that are linked to independent investigations generated by the app itself.

These apps include digital versions of geoboards, counting charts, Base 10 blocks, attribute blocks, fraction strips, grids, coloured tiles, abacuses and other maniuatives that support the development of basic number and spatial concepts.They would be particularly useful in supporting individual and small group learning plans for students who need visual aids and teacher aide intervention. Each app is customisable and allows for different skill levels and different types of tasks within the same app through a simple user interface. The settings are changed through the “cog” icon, the activities are accessed via the arrow icon and there is a home button to return to the beginning. There is also a tutorial included to explain the use of each app.

What follows is a brief overview of some of the Hands On MAths apps available on the iPad used on how I have used them. For a more expensive look at the apps before purchasing them ( each app is $1.99 AU or the equivalent in your country) the company Ventura Educational Systems has an excellent website providing detailed information about all their apps, including downloadable PDF instruction guides. I wish other app creators would provide this much information about their apps so that you could make informed decisions about purchasing.

BASE 10 BLOCKS

 Hands On Maths:Base 10 Blocks is a virtual mamipuative app that allows you to explore both whole number and decimal place value using the familiar base ten blocks, known in some countries as MAB. It also allows for addition and subtraction of numbers with and without regrouping. It is limited to 3 digit numbers from 100s through to hundredths. It works through simple dragging and dropping of block into a work space and the values are automatically generated as you build the numbers. A useful feature is built in that allows for groups of smaller values to automatically transfer into the higher value accompanied by an arrow that shows where the values transfer to. ( e.g when you make 12 tens in the tens place, it will change 10 tens into a hundred and leave the remaining 2 tens intact). this works in the decimal format as well. As I said in the introduction, I’m not suggesting we do away with the physical block usage as many younger mathematicians in training need to manipulate physical models. Where digital virtual manipulative excel is in instant feedback, quick turnaround of use, instant access and reuse and unlimited resources ( we often run short of blocks in whole class settings). Together with discussion with a teacher on a one to one or small group basis while manipulating the virtual blocks, I see this as a good tool for working with at risk students. I like that the app allows for the use of decimal place value as well, even though here is a school of thought that we should use different models for decimal place value. Me personally, I like to maintain the link between the base 10 system across whole and decimal numbers to show the consistent relationship.

INTERACTIVE HUNDREDS  CHART

The Hands On Maths Interactive Hundreds Chart is a counting board which you can set up starting from 0 or 1 and use to investigate, explore and discover number patterns and sequences. Users can mark out multiple counting sequences using different tools including crosses, ticks, circles and squares( transparent, opaque and solid) of different colours. Using these tools, students can discover patterns, common factors and multiples, predict the next few numbers in the sequence by studying the pattern show so far. They can create their own or follow sequences given by the teacher or other students. Used effectively, much discussion can be generated about number sequences as a precursor  to Algebraic patterns through visual representation. Again the advantage of the digital tool is the quick turnaround in exploring patterns and the instant reuse of the board.

INTERACTIVE COLOUR TILES

Hands On Maths Color Tiles has a huge range of options for developing important Mathematical concepts. The tiles can be used to create arrays for exploring multiplication and division. Addition and subtraction can be explored by adding or subtracting tiles by dragging on or off the workspace. These operations mentioned are supported by a built in pad that supports the calculations being done with the tiles. This pad can be customised to show fractional. decimal and percentage proportions of tiles on the workspace as well. There are also built in grids that can be used to support calculations or be used as graphs or co-ordinates. Symmetry can also be explored through symmetrical grids that create duplicate reflections vertically, horizontally or both as you place tiles on the grid. By exploring this app you will find more and more applications for the range of tools it provides. Read the PDF guide that is available on the website listed above. It gives further ideas. The moe I explore it the better opinion I form on this app. Check it out.

OTHER NUMBER BASED APPS AVAILABLE IN THE RANGE

There are a number of other apps in the Hands On Maths Range that address number concepts. I’ll provide the links here and direct you again to the company’s website so you can check out for yourself what these apps offer.

Number Sense provides ways for exploring whole numbers, fractions and decimals

Number Balance can support the introduction and development of equality, equations and algebraic thinking by providing a balance tool that enables you to crate equations that equal different value combinations on either side.

Tangle Tables and Multiplication Toolkit both give many opportunities to explore basic multiplication concepts in a hands on, concrete way.

Hands On Maths also has a number of apps that support the teaching of geometry and other spatial concepts. I’ll discuss them in a later post.

When I first discovered these apps, I thought they were nice little activities for the juniors to explore. As I explore them deeper, though I can see their applications in higher grades as well, used creatively and in context. In tutoring middle/high school children on the side, I get frustrated by the lack of hands on explorations of concepts by teachers in these schools. I can see a place for some of these apps in the right  context.  I recommend that certainly elementary/primary school teachers give these apps a go. Even if you don’t buy them, check out the company’s website ( I have absolutely no affiliation with them – I just discovered the site today file researching for this post). You might find some great applications for using the real versions of these virtual manipulatives that you can use to improve your maths teaching.