# Photo Angles

### Task 156 ... Years 2 - 10

#### Summary

Working from a photograph of objects on a grid students are asked where the camera was placed. For those who are up to the challenge, the next step is to work out where the light was placed. Of course, shadows give some clue. The task can be tackled in a variety of ways from recreating the grid and physically moving around to find a position that 'seems to be right', to dipping into the mathematical skill toolbox and choosing to use Pythagoras' Theorem, similarity and trigonometry.

#### Materials

• 16 square grid made of 4cm x 4cm squares
• assorted geometric shapes and objects
• photo of pieces arranged on the grid
• small measuring tape

#### Content

• 2D representation of 3D objects
• equations, simultaneous
• language of space, position and order
• measurement, angle
• measurement, area
• measurement, length
• measurement, perimeter
• position in space, 2D or 3D
• Pythagoras theorem
• ratio & proportion
• scale drawing
• similarity
• spatial perception, 2D or 3D
• transformation experiences
• transformations, enlargement/dilation
• trigonometry

#### Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

The great value of this task is that at all levels from K to 12 it generates mathematical discussion. Reconstructing the physical situation from the photo is the first level of talk. This involves measurement concepts based around the unit square on the grid. Previous experience with shadows encourages discussion of where the light might have been placed and concepts related to distance and angle come into the conversation. Students usually make their hands into a lens and look through it as they move around the grid to match the camera angle. Estimation and approximation are important skills for a mathematician.

This informal level of investigation is encouraged further by the inclusion of a tape measure. Frequently, students who have explored the problem at the first level then ask each other What do we use this tape for?. In the primary school one practical way to follow this up is to use lamps or torches to try to recreate the position of the light. Then its position can be measured. But measured from where? Students will need to enter a new phase of discussion to choose a reference point and then work out how to tell someone else this position in three dimensional space. That might be achieved in one of two ways:

• A direct line measurement from the reference to the light source which will also require specifying the angle of the string.
• A system of right angle left/right, forward/backward, up/down measurements making an ordered triple, in the same way as an ordered pair of numbers specifies a point in two dimensions.
Using the school digital camera, or the student's phone camera or other personal gadgetry, in a similar experimental way can help to determine the camera position when the photo was taken, and again, this point can be measured.

Older students could use similar triangles, trigonometry and Pythagoras in three dimensional space to determine the position of the light and the camera; or create the 3D equation of the lines and solve these as simultaneous equations. In this context such calculations are likely to be more relevant than the traditional spider crawling from one corner to a diagonally opposite corner of a room to introduce Pythagoras in three dimensions. Moreover, in attempting Photo Angles by calculation, the 'spider problems' may develop more meaning.

In whichever way students tackle this problem, this is one task that deserves knowing the answer. For the photograph in the task:

Camera
From the front right hand corner of the grid, the camera position was 12cm to the right, 50cm towards the front and 30cm vertically.

Light
From the front left corner of the grid, the light was 75cm to the left, 15cm towards the front and 50cm vertically.

How close did your students get?

#### Extension

There is a link between the mathematics in this task and, for example, the challenge of measuring the height of tree using a shadow stick. One useful description of this process is at the Utah State University Forestry site. Other descriptions can be found by web searching 'meaure tree height'.

#### Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

For this task to become a whole class investigation in the usual sense you would require multiple copies of all the bits of equipment, even if the photo could be scanned for public presentation. This is probably more effort than is appropriate, so consider creating Mathematician Teams of 3 or 4 students and selecting this as one of several tasks with a related theme.

Photo Angles includes many aspects of mathematical content so it could fit into multiple themes. Use the Task Cameo Content Finder and Ctrl F to search Photo Angles in a way that reveals other tasks related to it by content. For example, one theme might be 'Reasoning in 3D', so content such as:

• 2D representation of 3D objects
• language of space, position and order
• position in space, 2D or 3D
• spatial perception, 2D or 3D
would be good starting points for building an appropriate set of tasks. Equally, a set of say 15 such tasks would be a perfect basis for using the Menu Maths model.

At this stage, Photo Angles does not have a matching lesson on Maths300.

#### Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Photo Angles task is an integral part of:

• MWA Chance & Measurement Years 5 & 6
• MWA Chance & Measurement Years 9 & 10
This task is also included in the Primary Library Kit for Maths Around The Kitchen Table.