# Exploring Space

Jack challenges his sister to a race. Jack says, “I will swim across the middle of the pool and back. And you swim all the way round the edge”. Who do you think will win the race?

In Exploring Space (G2), pencils, paces, and sports fields are used to develop students’ ability to estimate actual distances—something school aged children find particularly challenging. Students explore rich embodied experiences, like standing in a lift and tiling a floor, which provide meaningful contexts for working on the relationship between cm² and m². Active classroom experiments, like covering the classroom in A4 sheets of paper, help learners to develop an implicit notion of scale factor.

The contexts of swimming around the edge of a circular swimming pool and turfing underneath a trampoline serve to develop visualisations that give learners a good sense of the relationships between the lengths and areas of circular shapes. Through these visualisations, they hypothesise and then formally examine relationships through which the formulae C= πd  and A = πr2 emerge. In the final section of Exploring Space (G2), students return to the counting strategies developed in Seeing It Differently (N2) to make sense of volume. The contexts begin in stacking shelves, where students generate flexible counting systems. This helps them to make sense of finding the volume and surface area of standard 3-D shapes in a variety of ways.

# Overview of Exploring Space

#### Section A: Dimensions

Lessons 1: Measures of length; Covering and Tiling
Students develop a real sense of what metric and imperial measurements represent. Visual measurement is encouraged and used to develop a notion of scale. Through an investigation of length and area through practical, everyday contexts, students revisit the strategies used in the Fitting It In (N1) module. Practical activities, including the use of A4 paper to cover the classroom floor, allow students to develop their ideas of how length relates to area. The context of tiling a floor provides a further extension of this thinking, helping students to engage implicitly with scale factors.

Lesson 2: The Lift
Through the context of a lift, students develop more strategies for dividing space into equal parts. They revisit ideas from For Every One (PR2), using the ‘For every one …’ strategy with measures of area. Students develop their own approaches for gauging the size of a lift and assessing how people may fit inside it, encouraging equal sharing and fitting in strategies.

#### Section B: Round and About

Lesson 3: A Swimming Competition; A Sewing Challenge
Students work on contexts designed to compare the diameter and circumference of circles. They use string to compare the distance of swimming around a circular swimming pool with swimming across its middle. They also use the time taken to sew a line of sequins across the middle of a circular badge to predict how long it will take to sew sequins around its edge. Gradually, students establish that about 3 diameters fit in to any circle’s circumference, according to a ratio of 1:3. They use this relationship to find the perimeter of circular shapes.

Lesson 4: Curly Creations
Tasked with finding the length of string required to cover some ‘curly creations’, students view arc-lengths as fractions of a circumference, or as parts which can be re-assembled to make a full circumference. Students view the length of a semi-circle in two different ways: (1) As half of the whole circumference, (2) Partitioned into one diameter + half a diameter. Students consider how the general rules C = π x diameter and C = 2πr compare with their informal strategies for finding the length of the circumference.

Lesson 5: Mirrors
Students work out the cost of wooden edging for a variety of mirrors, taking into account whether the edging is curved or straight. They both estimate (C ≈ d x 3) and calculate (using π=3.14) the edgings length and cost the mirrors accordingly. In addition, students identify and label distances on a netball court, including the circumference of the centre circle and the shooting semi-circles.

Lesson 6: Trampolines
The context of ‘turfing’ a bare circle of grass under a trampoline introduces students to an informal strategy for estimating the area of a circle. Students consider whether the area of 3 squares drawn from the radius of the trampoline circle matches the area of the circle. Like the tasks in Section A, students make meaning of a particular distance (5 ft) by marking out that distance on the floor. In Q1 of Activity Sheet 6 – Covering the Circle, they practice visualising the three squares drawn on the radius and use this to estimate the area of other circles.

Lesson 7: The Circle Using Formulas
Students compare the informal strategy using the squares drawn on the radius with the formal rule, A=π x radius2. They use both formal and informal strategies to find the area of a variety of circular shapes.

#### Section C: Layer Upon Layer

Lesson 8: Counting in 3D
Students begin a series of lessons focused on counting cubes to calculate volume (instead of using formulas). This lesson is essentially about counting strategies. It encourages students to use multiplication and subtraction strategies, as well as addition.

Lesson 9: Packing Cubes
Students work on developing strategies for counting in 3D and seeing cubes inside 3D shapes. Volume is introduced as the repeated layers of cubes within a shape.

Lesson 10: Cheese and Pineapple
The price of a block of cheese becomes a mediating quantity for comparing volumes. The context of a box of sugar cubes helps students to visualise cubes within a cuboid and to decide on the dimensions of a box given a certain number of cubes that must fit inside it.

Lesson 11: More Cheese
Students find the volume of various 3D shapes, both in real life contexts and in mathematical images.

Lesson 12: An Old Cabinet
In this lesson, students engage with the concept of surface area. They generalise their strategies for finding the number of ‘tiles within tiles’ to help them recognise what happens when you look at the number of ‘cubes within cubes’.