# 246. ChrisMaths Cheer

Hey … it’s that time of year again! Baubles and cheesy jumpers are creeping into the most mundane of places. How about a more mathematical festive season?

Image credit: http://technabob.com/blog/

Here is a round up of the Sandpit’s Christmas resources:

Twelve Days of ChrisMaths

# 245. Fair share

I spotted this ‘Expert Tip’ whilst flicking through a supermarket magazine:

Image credit: tesco.com/foodandliving

Question
If this cake has a diameter of 18cm (7in), is this a fair way to split it between guests? Can you prove your result in general terms?

Of course, this assumes that the icing on the side doesn’t count in the diameter or guest preference.

# 244. Resource of the Week

I recently came across this splendid resource for introducing Sine and Cosine rule to students.

Proving the Sine and Cosine rule

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!

Thank you @mrslack_maths

# 243. Messy Means

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.

Image Credit: http://www.thisismykea.com/designs/mr-messy

Grouped Frequency tables discussion

I started with a table with all the working shown, but some information blacked out. Each group had an A3 version and they filled in what was missing.

The second table had more information covered up. After a discussion the groups decided there wasn’t enough information and they would have to guess what the missing numbers were.

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.

We followed this up a Splitting the Steps estimated mean worksheet that I wrote after seeing Bruno Reddy’s presentation after #MathsConf2014 (Mr Reddy’s blog).

# 242. Edible Inspiration

Calling all creative thinkers!

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?!

# 241. Histogram Hysteria

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?

Let us begin:

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.

(Alternatively you can download the worksheet with RAG123 self-assessment at the end: Histogram calculations step by step RAG123 )

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!

# 240. Cogged up

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?