The proofs for these rules are relatively simple, but getting a class of teenagers to engage with it is a different matter! These worksheets give you the proofs, step by step, but all jumbled up. Students must rearrange the stages in order to create a proof. It worked brilliantly!
I have recently been teaching lower ability Year 9 students how to calculate the mean from grouped and ungrouped data tables. I didn’t want to teach them a method to learn by rote, so I used a more investigative approach.
The third table had minimal information. Each group used their own method to find the missing values. Some chose the largest value in the range, some guessed what the results could have been in each group and one group decided to calculate two means – one using the largest value and one using the smallest.
We collected our results together on the board and discussed their accuracy. The class decided to use the middle of each range to calculate the estimated mean. They had gone from no understanding of estimated mean to formulating their own method.
What mathematical questions could you set from this picture?
Here are a few to start you off:
1. Sequences – do the increasing number of chocolates in each layer form a sequence (in 2D, in 3D)? If so, what is the general term? Is it geometric or arithmetic?
2. Series – if it is an arithmetic sequence, can you find the sum of a finite number of layers? Which layer would have the 1000th chocolate?
3. Geometry – what shape must the layers be in order to form this structure? Is there a pattern to the layers? Could you stack these in a different way to form an equally stable structure?
4. Money – if a standard box holds 12 chocolates, how many boxes would a 2D or 3D version of this require? What is the cost? What if they came in a larger box? Could you save money?
5. Health – how many calories are there in the tower? How far would you have to run to burn off the calories? How many ‘average’ meals is it equivalent to? How many fastfood burgers? How sick would you feel after all that chocolate?!
Instead of setting a question, why not ask your students or even your trainee teacher what questions they can come up with?
Are you fed up of explaining the difference between a histogram and a bar graph/chart?
Cheer up! Help is at hand…
I teach a class of bright students with very little self-belief in their abilities and total fear of leaving their comfort zone. Instead of telling them what to do and set page X of textbook Y, I let them tell me what was going on and let them take small steps. After all, you wouldn’t take a beginner climber up the North face of the Eiger, would you?
First I gave the students individual time to write down what they observed. They then compared their answers in pairs/threes. Finally, I collected their observations together on the board (where I had projected up the comparison worksheet).
This hands on approach allowed the students to understand how a histogram is constructed. There were fewer students thinking that histograms are just bar-charts where the bars touch.
This worksheet allows students to get the feel for calculating frequency densities without stress. The instructions are gradually removed, until students are just working from a data source. Then students practise drawing histograms.
It is also a handy revision resource – my students referred back to this worksheet when they were stuck in subsequent lessons, rather than ask me!
It’s amazing what maths you see when you go for a walk along a canal on a beautiful afternoon. After helping a canal boat through a lock, the following problem occurred to me: how many times must you turn the handle to raise the sluice gate?
Fact: The sluice is controlled by a series of cogs. The handle turns a ratcheted cog with eight teeth.
Fact: The handle turns a small cog with thirteen teeth.
Question: The next cog has ten teeth on a quarter of it’s circumference. How many is this in total?
Fact: This large cog is attached to a small cog with ten teeth, which lifts the vertical post. Question: From the picture can you estimate how many teeth are on the vertical post?
Question: Given all this information how many turns does the handle need? Extension: Look at this picture. What is the angle between the foot supports?