Patterns and Prediction

“This was really useful for our last lab in science”.

That was a win. In mathematics this year I wanted to address a problem I saw when proofing reports last year. In a nutshell I was proofing middle school reports and there was a glaring area requiring assistance. Processing data needed attention. I delved further as mathematical assistance would probably help.

IB DP Question: Obvious Pattern. Dubious Data. Getting kids to observe and analyse

Bivariate data analysis. Kids weren’t choosing scatter graphs and were asking technology to find trend lines without much work at how that happens.

I think teaching this is in a scientific context is the best way. Teaching in context just makes sense. In our curriculum the mathematics came at the end of the grade 8 year, so this year it is combined with algebra. We’ve called it Patterns and Prediction. Science likes it so much that we are planning to kick off next year with this unit.

One of my goals for my students is for them to be able to converse with anyone. I have a few things I do for this, but talking about the weather, it’s handy. To prepare students for their Interdisciplinary Unit (IDU – MYP term) we used data banks of Latitudes and Temperatures. Each class had to form a hypothesis and then we used Google docs to collaborate when collecting data.

Using the science departments guidelines we took grade 8 students on a journey that included:

  • Data collection – Google Docs
  • Data storage – Google Docs, Excel
  • Data Manipulation – Excel and Autograph
  • Scatter Graphs – Excel and Autograph
  • The Language of Analysis – using the wonderful 100 People website (see page 23 of PDF)
  • Trend Lines (Lines of best fit)
  • By eye using the shape of the data and the centroid. Find the y-intercept via extrapolation, then find the equation in slope intercept form
  • Using technology – Excel and Autograph.
  • Presentation of a report – fonts, layout, hard page, table headers, table formatting and labeling

They are now forming their own hypotheses and hunting for data (we have set up a Diigo group to get them started) so they run their own lab. Students will be using all of their mathematics and science classes this week and will be assessed in both classes.

Have a look at the graph above that I included for practice. Can you see the problem? It’s not with pattern spotting. It’s what the pattern predicts. Glad I am not  a bunny.

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Don’t Eat the Beans!

Rearrangement proof for Pythagoras' theorem. P...
Image via Wikipedia

If you read this blog from time to time and don’t like too much maths in your eyes, read on, it will be okay. I won’t go into proofs, this is more about how the kids would be introduced to the revolutionary blasphemous mathematics that got Pythagoras in all sorts of bother. For the record there are many stories about how he may have died. For the class I have ignored the less interesting ones.  But was it his theorem?

One of the things that didn’t work so well in #AoD, mostly with pacing, was letting kids do a summative MYP assessment task when they were ready. They were all working on different things, so were ready at different times. Not that I’m saying they should when they aren’t ready, but it would have been better to get my classes to do the Gradient Task at the same time, perhaps even take them to the water park to photograph more interesting surrounds.

They still got some enjoyable and valid mathematical experiences out of it, but as with all new things tried it won’t necessarily be right first off. I am already looking forward to fine tuning the program next year. Three weeks after the winter break we had a week off for Tet (Vietnamese New Year – chuc mung nam moi!). We also churned out set two of the four sets of reports for the year, so my tippy tapping blogging fingers were a bit distracted. It didn’t mean I wasn’t thinking about mathematics and blogging. I found time to read some blogs and comment on blogs. Must try harder. I’ve also been very distracted by Africa.

Puzzling over the not so perfect fit

Our latest adventure is all about Pythagoras and his famous theorem. I looked at what worked and what didn’t last year. This included how much and what students had learnt, but also what was ineffective or dull or both. Pythagoras’ Theorem features in mathematics curriculum all over the world, but I never did anything interesting with it when I was kid and I never take the time to apply it to see just how much distance or time I could save by cutting through rather than follow a path. Amazingly I can see it is shorter and that is enough. What I think is so cool about the theorem are the stories around it, the history and the impact it had on mathematics and the beliefs of the people at the time. And the murders. 

It was time to throw out the construction of squares on the sides of right angle triangles to help kids ‘discover’ the relationship all by themselves, using calculations and analysing data. We, maths type folk, are very into nutting out patterns, it’s in our nature, but that has been done to death with Pythagoras. It’s rumoured that he was killed, along with his apostles, for unearthing irrational numbers. Why beat him to death again?

The angle taken (you see what I did there?) is with Criterion D: Reflection and C: Communication and getting the students to show that his theorem does work, rather than find it. Not all students are ready for algebraic proofs so we are using geometry and data. If a student feels the urge to present an algebraic proof, they will be most welcome to do so.

Today, The Geometric Proof It was so much fun.

Kids like “make and do” (so do I) and even though I LOVE teaching with 1:1 tablets, it’s nice to touch things again – blocks, scissors, tape measures, paper. Finding the right balance is so important.

Each pair had some coloured paper and some white paper. They had to make squares. Two coloured with sides of 15cm and 20cm and the white one with sides of 25cm. Fluoro paper everywhere, short stubby ruler owners were challenged  moreso than those who had remembered their 30cm rulers. I gave them 5 minutes to get their squares ready, then asked them to clear their tables of everything except for their beautiful squares, and to sit back and just admire them.

Then I asked them to imagine they represented something wonderful, like gold or chocolate or…

My favourite commodity were the “squares of love”. That student had been one the singing telegrams for Valentine’s Day not so long before.

Once chosen I then said they had to choose between them. One member got the two coloured squares and the other member of the group got the white square. It was important not to use any adjectives meaning big or small. There was some friendly banter and in under a minute happy and resigned faces placed their squares in front of them.

“Who got the white square?” lots of whooping and shouting of victory about scoring the big square.

I praised those who decided to forego the big white square for the smaller colourful ones, then told them they were equal. The confusion about the lack of victory was amusing.

The task of showing me they were equal was then assigned. Confused faces. “Show me, any way you’d like to, that the area of the white square is the same as the coloured squares”. One voice pipes up: “Can we use scissors?”. And they were off. No mention of the right angle triangle at this point, though, some students were talking about it.

The girls prove it. Step one in getting closer to the deadly numbers

 One pair found it didn’t work, but their 25cm sides were actually 30cm. Problem fixed, mostly.

Time to reveal the right angled triangle and where the squares fit. Story telling time begins too.

We then discussed the famous rule, which many had seen before. We looked at the diagram of the squares  sitting on their right triangle and then in their pairs they discussed the accuracy of their geometric proofs

It was a fun hands on, no tablet lesson. My next post will show the groovy technology we use to show this theorem at work.

Here is a preview

Mathsnet has a great section on Pythagoras and his theorem. Proofs 2 and 11 are accessible to any student, and proof 11 is pretty groovy in its simplicity. One of my kids then found this online version of number 11..

Braining Camp is new to me, so it has been added to this task too, for practice.

I haven’t told the kids why Pythagoras and his followers did not eat beans and I won’t tell you yet. We have one lesson for the history and the numbers that led to his death. Would you die for mathematics?