Tag Archives: problem solving

361. Routes, Reindeer & Reasoning

Well, we are nearly at the end of a very crazy year. Congratulations on surviving it!

So, it’s been a while since the last blog post. Apologies for that. At the moment I am involved in Mixed Attainment teaching with Year 8. To finish off the term, I thought we deserved a bit of fun. We have a week of lessons left so I’m going for a mini project each lesson.

Lesson 1: Santa’s Route
I found this fab task on the Maths Drill website. There is a real chance for extension in this task, which is great for the mixed attainment classroom.

Lesson 2: Reindeer Ratios (Updated 13th Dec)
We have been following the White Rose Maths scheme for Year 8, which covers a lot of proportion and reasoning through ratio, multiplicative change and fractions. This task tries to cover some of these skills. The answers will be uploaded soon.

Lesson 3: Elf Box Packing Problem (Updated 14th Dec) Elf Box Packing Problem Solutions
This task involves using multiplicative change and fractional multiplication and division, with a dash of unit conversion. There is some work on shapes, but formulae are given where necessary. The first four pages print nicely into a folded A4 (A5) booklet. There is a help sheet for the box packing problem; this would be better printed on A4.

359. Proportional steps

Just a quick resource upload today.

Image credit: https://www.heyn.co.uk/

I’ve written a step by step resource on how to construct algebraic direct proportion relationships, including the answers.

Small steps in Direct Proportion (docx)

Small steps in Direct Proportion (PDF)

I used this with a Year 11 class who aren’t very confident with algebra. They were surprised by how straight forward the work was and were happy to now attempt problem solving with algebra.

328. Slice of genius

So, I was doing my usual Human Piechart  activity when an interesting point occurred. I had the class split into a group of 20 and a group of 18 (I had a student in charge of each group so that the circles would be factors of 360). I asked the class how we could combine the groups to make a whole class pie chart. One student suggested we add together the angles and divide by two. Several other students agreed that it was a good idea.

There goes my lesson plan.

I put this prediction on the board and asked them to prove it or disprove it using hard facts. I was very impressed by the different techniques they used. Most students started by adding the angles and dividing by two, then:

  • They went to the raw data and calculated the actual answers, disproving the prediction.
  • Some looked at the angles on the ‘combined’ pie chart and worked out the number of degrees per person for each angle. They used the irregularities in angles to disprove the prediction.
  • Looking at how many degrees there were per person and using logical deduction that you cannot add the angles.
  • Others noticed that categories with the same number of people had different sized angles.

All this before they’d answered a single pie chart question! The moral of this story is: don’t ignore the wrong suggestions, embrace them and use student knowledge to dispel the myths.

325. Mrs D’s Delightful Display

Anyone else been purging their classrooms of ragged wall displays ready for a fresh start in September? But then you end up rushing displays ready for the Autumn Y6 open evenings? And you need to get to know your new classes too!

Mrs D had a splendid plan to address all these issues. The first step was to introduce the problem: step through a piece of paper. It’s a classic problem involving maximising perimeter – I remember seeing it in a children’s Annual as ‘The journey through a postcard’.

This isn’t the easiest of tasks and takes a fair bit of determination and patience. Teamwork skills are also helpful. You can really get to know your class with this activity.

Once they’ve figured out how to do this you can reflect on how they overcame obstacles. All of this can be pulled together to make an amazing wall display on problem solving.

Thank you to the excellent Mrs D for allowing me to share her idea!

301. How much is my sandwich?

A visual discussion starter for you:


These three pots of sandwich filling cost £1 each. The flavours are egg mayo, chicken & bacon and cheese & onion.
How much would the 182g chicken filling cost if it weighed the same as the others?
The large pots contain 5 servings and the small pot contains 3 servings – are they the same size serving?

If you zoom in on the picture you could generate your own questions based on the nutritional information eg calories per serving.

You could extend this to the snacks in students’ bags. Are they as healthy as they think?

296. Jellybean Trees

How on earth can you create a maths lesson using these items?


Well, first sort them into colours, then put twenty jelly beans into each cup. Make sure there are only two colours in each cup, write the contents on a sticky label and use that to seal the cup. Each cup should have slightly different numbers or colours – it prevents copying.

Note: Eat all the orange jelly beans – you’ll be doing your dignity a favour!


Have you figured it out yet? No? We’re doing probability tree diagrams without replacement. Now I know you could do this with one experiment at the front of the class, but getting everyone involved means it’s more hands-on and memorable.

The Experiment
I did a demonstration of this on the board first, before handing out the cups and worksheets. I told the class what was in my cup and picked out a jellybean. It was orange. I drew the first stage of the worksheet (see below) on the board: What was the experiment? How many of each colour do we have? What is the probability of each colour? Then we filled in the first stage of the tree diagram.

I ate the jellybean.

But you can’t do that – it messes up the experiment! I asked what would be the probabilities for a second jellybean now. They figured out the slight change to the probabilities. Then we went back and thought about what would have happened if my first jellybean had been lemon.

I always encourage students to work out all the possible outcomes before they even look at the rest of the questions. And this is why you need to eat all the orange – the list on the board was:

  • P(LL) =
  • P(LO) =
  • P(OL) =

Do I really need to put the last one?

After much giggling, the class were let loose with their own cups. They did the experiment once with their standard cups and then had their work checked. They could then alter (eat) the contents of their cup so that a minimum of five beans of two colours remained. You can see an example of a student’s work here:


I summarised the lesson by looking at different types of probability problem where items are not replaced. I now have a nice ‘hook’ to refer to when discussing probability tree diagrams without replacement.

Download the worksheet here:
Tree diagram without replacement (pdf)
I printed out two per page as it fitted nicely in their books. The descriptions are deliberately vague to allow it to be used in different experiments.

(The usual warning regarding food allergies and beliefs stands. Some jellybeans have animal derivative gelatine – please check, you don’t want to accidentally upset a student)

293. Boxing Bounds

I thought this would make a nice little starter – address a few different topics, bit of problem solving, all over in 15 minutes. How wrong I was!

The Question: A company packs toys into boxes which measure 12cm by 8cm by 10cm (to the nearest centimetre). The boxes are packed into crates which measure 1m by 0.75m by 0.8m (to the nearest centimetre).
(a) Basic question – How many boxes fit into the crate?
(b) What is the maximum volume of a toy box?
(c) What is the minimum volume of the crate?
(d) Look at your answers to (b) and (c) – do they affect your answer to (a)?

It was a simple question about fitting toy boxes into a shipping crate. It extended to looking at upper and lower bounds, then recalculating given this extra information. Simple? No chance!

Problem One
Not changing to the same units

Problem Two
Working out the two volumes and dividing to find the number of toys. When challenged on this, it took a while to get through to the basics of how many toys actually fit – mangled toys and split up boxes don’t sell well.

Problem Three
Maximising the arrangement of boxes – remainders mean empty space

Problem Four
Using the information from Problem Three to find the total number of toys

Problem Five
Working out the dimensions and volume of the empty space in the box

Problem Six
Trying to convert centimetres cubed into metres cubed. I don’t even know why they wanted too!

Problem Seven/Eight
What’s an upper/lower bound?

Problem Nine
What do you mean that the original answer changes when the box size alters?

Problem Ten
All those who weren’t paying attention when you went over Problem Two and don’t ‘get’ why the answer isn’t 625!