In several of my previous posts I discussed the importance of promoting critical thinking in mathematics teaching and learning. I’ve also discussed at length various ways to contextualise mathematics to provide opportunities for students to apply prior learning, build on concepts, and recognise the relevance of mathematics in our world. In addition, investigations provide excellent assessment material – usually when we assess in mathematics we ask for specific answers. In investigations, students can show us a range of mathematics, often beyond our expectations. They are also a great way to integrate other subjects areas such as literacy and science.

In this blog post I am going to share some ideas for open ended and inquiry-based mathematical tasks based on two items that most students would be familiar with – beach towels and pencil cases!

### Pencil Cases

Let’s start with **pencil cases**. It’s the start of the 2018 school year next week and many children begin each school year with brand new stationery, in brand new pencil cases. Even if they’re not brand new, most children have a pencil case. I came across an interesting article relating to pencil cases a few days ago, and I think this could be used to spark interest and curiosity. The article can be found here:

https://honey.nine.com.au/2018/01/19/14/35/pencil-case-missing-letter

#### Short activities:

- Who has the heaviest pencil case? Compare the mass of your pencil case with the pencil cases of your group members. Who has the lightest? Estimate the mass, then use scales to test your estimations. How close were the estimations?
- Estimate, then calculate the surface area of your pencil case.
*What units are the most appropriate to use? Explain how you measured the surface area.* - Faber Castell is a famous brand of pencils. Investigate the history of Faber Castell and illustrate this on a timeline.
- According to the Faber Castell website, it takes one ‘pinus caribaea’ tree 14 years to be ready to be used to manufacture pencils. Each tree can produce 2500 pencils. If one tree was allocated to each school, how many pencils do you think each child in your school might receive?
*How did you work this out?* - If each of the 2,500 pencils were sold for $1.50, how much do you think the entire tree be worth in pencil sales?

#### Investigations:

- At the beginning of each school year many children get brand new pens and pencils to take to school. Investigate how much it would cost to buy your stationary. Which shop offers the best value for money?
- Some pencil cases like the one in the photo and in the Missing Letter article have small clear plastic pockets to put your name in. If a pencil case has only eight pockets, is this enough for your name? Investigate the length of names in your class. What would be the average length name in your class? What else could you explore about names?
- The pencil case in the picture came with some pre-printed letters for the clear pockets. There are more of some letters than others. Investigate the most common letter occurring in students’ Christian names.
*Do you think it would be the same in all countries?* - Design and make a pencil case to suit your individual stationery needs. Write about the mathematics you use to do this.

#### Extension Activities:

- Design a new and improved pencil and explain the changes you have made.
- Design, justify, and create a marketing campaign for a new, ‘miracle’ pen.
- Research and discuss the following statement: “To save the environment, wooden pencils will no longer be manufactured”.

### Promoting Curiosity and Wonder

Mathematical investigations should promote curiosity and wonder. The pencil case questions and investigations are open, yet provide some structure and support. They give enough detail to communicate the type of mathematics required to complete the task or investigation. Students should eventually be able to feel confident enough to come up with their own questions and follow their own path in terms of the mathematics they access and apply, just like mathematicians do.

### Round Beach Towels?

In the last year or two a new beach towel has emerged onto the beach towel scene. It’s round. Now this idea immediately caused some concern for my mathematical brain. I had questions.

- Is there more fabric in a round beach towel than a regular, rectangular beach towel?
- Is there more fringe, and wouldn’t this make the towel more expensive?
- How does one fold a round beach towel?
- Could you wrap a round beach towel around you the way you wrap a rectangular beach towel?
- How much more area on the beach gets taken up by people spreading round beach towels?
- Does this mean less people get to lay on the sand?
- Could you design a round beach towel that has a tessellating pattern?

** All of the questions above can be explored using a range of mathematics…I wonder how many more questions your students could come up with?**