# How do we create investigations that challenge, but remain accessible for all students?

Explaning, generalising, exploring, noticing links and patterns, discussing; These are the things I want my students to be doing in their maths lessons. Well-planned investigation activities can allow for all of this, whilst promoting deep mathematical thinking. But you knew that already.

The best students in my class are very able pattern spotters. They have the confidence to tackle unseen problems and the resilience to redirect their efforts after a mistake. They also have a deep understanding of the mathematics they met in primary education and in KS3.

What about the students who don’t have these skills? How do I help them access the task in a way that allows for all of the things listed in the first line of this post?

This is the question that has been at the forefront of my mind this week, after a lazily planned investigation lesson went horribly awry. Of course, the answer lies in adequate and appropriate scaffolding.

The task is shown here.

__Mirrors Investigation__

The Problem:

**Sasha uses tiles to make borders for square mirrors. The picture shows her design for a 5x5 mirror with a border of 1x1 tiles.**

**What is the total number of 1x1 tiles needed for a 5x5 mirror? What about a 7x7 mirror?**

**Investigate the total number of 1x1 tiles needed for different sized square mirrors. Is there a rule connecting the number of 1x1 tiles needed for a square mirror with a particular length for the side, n?**

**Extend your investigation to look at other related problems (you might want to look at rectangular mirrors, or having a border two tiles thick, or having different shaped or sized tiles).**

**Extend your investigation to look at other related problems (you might want to look at rectangular mirrors, or having a border two tiles thick, or having different shaped or sized tiles).**

Many students had a whole host of problems accessing this task. I will try to summarise the main issues here.

Firstly, the wordy explanation of the task, along with an over-simplified drawing of the mirror meant many students had no real clue about how to start the task. The clumsy description of the “5x5 mirror with 1x1 tiles” led to a problems as some students didn't know what this meant.

One student was adamant that the mirror was in fact 1x1 since it was made up of just one square. I can see the point he was making.

The task invites students to find the number of tiles for a 5x5 mirror, and then a 7x7. This is the introduction to the problem and as such should be easily accessible to all. Unfortunately, the lazy explanation of the problem made it difficult for students to understand what was been asked of them, a trivial task of counting the squares around the perimeter of the given mirror.

Some students were able to find the Nth term in this sequence of 4n+4. The students who managed this were exclusively the ones who had set out there work in a logical manner. A completed table of results, or another method of logically recording answers, helps students investigate the underlying pattern. There should be adequate scaffolding here for students who find it hard to organise their ideas.

Lastly, for the (very few!) students who managed to get to the extension activity, they found it very challenging. They were paralyzed by the choice. A better extension would be have chosen ONE variation and pushed students in that direction.

Thinking about the above issues, I sat down and redesigned the task. You can see it below.

__Mirrors Investigation__

**Sasha uses tiles to make borders for square mirrors. **

**The picture shows her design for a mirror (in blue) and a border made of grey tiles.**

**How many grey tiles does Sasha need to create this mirror design?**

**Sasha calls this design the ‘5-mirror’, because 5 grey tiles fit perfectly along one edge of the mirror.**

**What do you think the 3-mirror looks like? Can you draw it using the square paper in your book?****How many grey tiles does Sasha need to make the 3-mirror?**

**Sasha’s mirror shop sells designs up to and including the 8-mirror.**

**Can you draw all the designs that Sasha sells?****How many tiles are needed to make each design?**

** (Record your answers so they are easy to read!)**

**Sasha thinks there might be a rule connecting the number of grey tiles needed to create each design, but she isn’t sure.**

**Can you find the rule and explain it to her?****How many tiles would you need for the n-mirror design?**

__Extension__

**One of Sasha’s customers wants a specific mirror. **

**The mirror MUST be rectangular mirror, and it MUST have an area of 30cm ^{2}.**

**How many different designs can you find that fit the criteria?**

**Which one should Sasha make for the client? Explain why you have chosen that design.**

The instructions are still quite wordy, but the text has been broken up. The colours help differentiate between information and instructions, and so will help them extract the information they need.

The fact remains that a number of my students do not have the investigative skills needed to complete an investigation on a separate, blank sheet of paper. Further scaffolding could be implemented by giving lower attainment students a partially completed worksheet with the 1-mirror, 2-mirror and 3-mirror partially drawn. A table of results could also be constructed for the student to fill in, a much easier proposition than completing one from scratch.

I envisage the task to be completed in pairs, with a shared piece of A3 paper in order to record results. Less able students can be given prompts (such as the table) already drawn on their paper. Key words can be placed around the paper, along with guidance on the structure. The questions in the task can be best utilised when paired discussion is encouraged. Choosing these pairs carefully can also provide support for less able stundets.

I have found the above process very interesting, and I don’t think the task is perfect. But I will leave it there and look forward to delivering it next year!

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