Category Archives: Problem Solving

158. May I take your order, sir?

Imagine getting your class to think about a number topic in a real-life context and subsequently having students leave the lesson feeling happy they could use this skill.

About as real as the square root of minus one? Not if you relate it to breakfast*

Image credit: ifood.tv

I wanted to make estimation more relevant for my class, a low ability Year 10. Outside my classroom I put a breakfast menu and my associate teacher took their orders** as they entered the classroom. I had put mini whiteboards on tables and I instructed the class to work out an estimate and the accurate total for their menu choice(s). The lesson had barely begun and the class were already talking about what they were doing (rather than Halloween antics the night before)!

Once everyone had arrived and settled down, I asked if anyone had underestimated and what this would mean – not enough money and doing the washing up!

I then asked each table how much their group order would cost. Would their overestimates cancel out their underestimates? Would the waiter get a tip? Meanwhile the associate teacher had added up the orders, so we could quickly check their calculations.

What if everyone paid £10? Would you have enough? How much tip would you be leaving? Would it cover a 10% service charge?

We followed up this task with some standard estimating questions.

Image credit: www.fudds.ca

The menus I used for the lesson are from a restaurant chain in California. The useful thing is there are no units of currency, so this works for different countries. It will work equally well with KS2 and KS3 pupils.

Download resource: Breakfast estimation (pdf)

BTW The students decided if the waiter wanted a tip, he should actually feed them first!

*Strongly suggest you use this before students have break or lunch time, or else they’ll be drooling in their next lesson.

**Unless you are providing food, please add the disclaimer that you are not feeding them.

157. Receipt for learning

Do you ever really look at the bottom of your supermarket receipts?

This caught my eye today:

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The bill is broken down into three bands of tax, the relevant amounts, VAT and total amount payable.

It would be quite an easy task to enlarge a receipt on a photocopier and blank out quantities. Don’t forget to blank out financial transaction details. You could work out percentages of amounts, reverse percentages or find missing percentages. It could be extended to percentage increase using multipliers. It also links to decimal calculations, rounding and money.

All this learning from a bit of paper from the supermarket*.

*Not all supermarkets do this, but there is usually some kind of tax reference.

154. When will I need to work out the area of a circle?

Answer:
If this is your crop on your farm!

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This is center pivot (or central pivot or circle) irrigation. The area which is watered is pretty obvious. There are plenty of images on the internet and sites full of statistics.

I could imagine this making a good research homework. Apart from simply working out the fertile area, you could look at volume of water used – you could even work out the optimum size and number of circles for maximum coverage!

153. Sequences Starter 2

So, you’ve got term to term sequences sussed. Time to tackle Nth term!

This idea just sort of appeared in my sequences lesson.

Equipment
Giant playing cards (or numbers on two different colours of A5 card)
Numbered headbands (I made crowns out of corrugated border card)

Set Up
1.Lay out one coloured set of cards on a table or the floor – these are the ones we needed in class. We started with all the cards in the suit.

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2. Issue headbands to four pupils.

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3. Pupils stand in number order.
4. Give each pupil a different. coloured card from a sequence to hold facing them.

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Task
1. Explain that each person represents a term in a sequence, given by the headband.
2. Pupil 1 turns around their card – Red 3.
Question: What is the next number?
Answer: Don’t know
3. Pupil 2 turns around their card – Red 5.
Question: What is the next number?
Answer: Might predict 7
4. Pupil 3 turns around their card – Red 7.
Question: What is the next number? Why?
Answer: 9, add 2.
5. Reveal the last number – Red 9.
6. What is the pattern? Add 2 Which multiplication table has the same pattern? Twos
7. Give each pupil in the sequence the appropriate number from the two times table.
Question: How do you turn the two times table into the sequence?
Answer: Add 1

8. How do you get from the headband to the sequence?
Headband x 2 + 1 = sequence

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9. What about a headband with 10 on it? Or 100? Or a mystery number?
10. Try this with other sequences and develop the idea of Nth term.

Outcome
I used this as a plenary for a term to term sequences lesson with a shared class. In the following lesson my colleague, D, used this idea to develop the concept of Nth term with another class. He wanted to make something for the pupils to have in their book to remember this. This is what he came up with: Handout for sequences intro (pptx) or How to for sequences(docx). I’m currently trying out hosting my own resources, rather than using TES resources – so we’ll see how effective this is.

152: Sequence Starter 1

So many people have the preconcieved notion that there is only one right answer to a maths question. This is such a silly idea – they just haven’t had the right question!

Here is a simple starter for introducing term to term sequences.

Equipment
Classroom whiteboard or large sheets of paper
Imagination

Task
Write down the next three terms in the sequence 1, 2 … and the rule used.
Eg: 1, 2, 3, 4, 5, …      Add 1
Note: Rules should be one short sentence.

Outcome
My Year 7 were frustrated that I’d given them the obvious answer and was asking for more. After a few minutes adjusting their expectations, they went for it. Some methodically wrote down rules, some abbreviated rules to symbols, some wrote rules and didn’t check them. Some didn’t write rules at all.

I randomly picked pupils to share their ideas on the board. I did the writing as I wanted to control the wording of rules and half of them can’t reach the top of the board.

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Their sequences were brilliant and very creative. The stumbling block was the rules. They didn’t always work for every term of the sequence. This gave other pupils the opportunity to develop their ideas by improving or adjusting the rules to fit the sequences. We also discussed how many terms you need to make a unique sequence. By the end of the discussion we only had one sequence without a rule. I was really impressed by their numerical skills!

In the subsequent classwork, their solutions were precise and well explained.

We finished with this brain teaser:
1,2,5,10,20,50,100 …

It’s UK currency:
1p,2p,5p,10p … etc

151: TMNW 1 – Puzzle maker

As promised, I’ve been trying out ideas from TeachMeet North West at Calderstones School. Here’s the first post:

My colleague J had mentioned Discovery Puzzlemaker last term, but I’d not had time to try it out. Then Fiona Bate @fibate used it as part of her presentation on ‘Profound thinking in the classroom’.

How I used Puzzlemaker
I decided to test this out on my Year 9 students – they are a bright bunch and there are a lot of them. I put out tile puzzles on sheets of A5 and the class settled to the starter task, after they’d got their books out. There were lots of different strategies and eventually everyone cracked the code – the formula for the area of a circle.

Blank puzzle

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Different strategies

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The funniest part was later in the lesson. A student put his hand up and said he couldn’t remember the rule for the area of a circle. More than one of his peers pointed out he’d just spent ten minutes cracking a code where the rule was given and it was still on his desk in front of him!

Lesson Objectives
Luke O’Hanlon @funkwalkee did a presentation on ‘Ways to engage with Learning Objectives’. This linked nicely with using Puzzlemaker to discover the aim of the lesson, as well as encourage independent learning and problem solving. Once the class had cracked the code they knew what they’d be doing that day.

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Puzzlemaker

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As you can see, Discovery Puzzlemaker is a really useful tool. I’m going to use some of the larger puzzles as homework tasks for my lower ability classes as I can tailor them to their specific needs. I’m starting with the ‘Hidden Message’ task to reinforce circle vocabulary.

Thank you to J, Fiona and Luke for sharing this site/their ideas.

149. Cold Questioning

Cold calling is the infuriating practice of randomly contacting people in order to sell something that they don’t want and didn’t ask for. Don’t ask me for my opinion on this business idea!

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Cold questioning is the practice of putting a random question on the board and asking pupils to solve it.

Example: GCSE revision
I have a class of critical C/D students who are sitting their GCSE in November. We have been revising prime factorisation, indices, simplifying, standard form and HCF/LCM. Today I put this question on the board:

A pair of trainers are reduced by 30%. The sale cost is £75, how much were they originally, to the nearest £1?

I was pleased to see them talk about the problem and have a go.

Students were randomly selected to share their answers and only two answers occurred – both of them wrong, but with a hint of understanding. Answer A resulted from calculating 30% of £75 and adding it on. Answer B resulted from subtracting the 30% from £75.

Now this isn’t as depressing as you may think, because as we discussed this it became evident that the class were confident calculating simple percentages – they just struggled to apply this knowledge. One student said they had used a multiplier. This opened up the task as we developed the link between reverse percentages and multipliers. Some of the class weren’t convinced, so we had a quick ‘converting percentages to multipliers, including inc/dec’ quiz. We even extended it to fraction to decimal conversion.

Finally, we looked at rounding as it was specified in the original question.

Review
I could have prepared a step-by-step revision lesson, gently taking them through these topics. I think ‘ambushing’ the class with a topic they haven’t studied for a while was more effective as they used a variety of skills to solve the problem, rather than repeat given procedures.

Ambushing your class
*Pick a topic you haven’t looked at for a while
*Avoid easy/obvious questions
*Try to include more than one skill
*Allow sufficient thinking/discussing time
*Finish off with another question, which requires similar skills
(Eg my second question was a reverse percentage involving an increase)