# Kids On Grids

### Task 217 ... Years 4 - 8

#### Summary

Created to help students experience location on a grid, the task goes much further by exploring number patterns in the ordered pairs that are formed. It is ordered pairs in a pattern that are the basis of the algebra associated with defining lines and curves on a plane. The task leads students towards exploring some linear examples and, given they have the opportunity to invent their own rules in the Challenge, they might stumble upon a curve. The thrust of task is from shape to number - from what you can see to how you can symbolise it - and the hunt for a pattern.

#### Materials

• 10 'children' & 1 object to mark the camping spot
• Large grid board (consider enlarging to A3)

#### Content

• algebra, generalisation in words & symbols
• algebra, linear
• decimals, number line
• equations, creating
• equations, simultaneous
• fractions, calculations
• fractions, whole & parts
• graphical representation
• language of space, position and order
• measurement, length
• mental arithmetic
• number line
• patterns, number
• patterns, visual
• position in space, 2D or 3D
• recording mathematics
• spatial perception, 2D or 3D

#### Iceberg

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

The task begins by developing the skills of locating objects in two dimensional space using an ordered pair of numbers. This involves spatial perception, measurement (counting of grid units) and accepting the convention of reading the first number in the pair as the horizontal, in this case easterly, direction. Another convention is that the starting point for counting is always (0, 0) 'because G never leaves camp'. These skills relate to mapping, reading street directories and creating and interpreting graphs. The hands-on aspect and the story context lift the task out of an equivalent text book page. If you have a grid painted on the playground, or a large plastic mat, your students can act out the task.

However, the task is more than measurement. Kids can be placed on the grid in a random manner, but they can also be placed on the grid in a pattern. Creating a visual pattern by placing in this way is a doorway to graphical algebra. If there is a visual pattern there will always be a number pattern, and that pattern can be discovered by asking:

• What is the same in all of the number pairs?
For example:

 If the kids were placed in this pattern: the number pairs would be... (1, 3), (3, 3), (5, 3), (7, 3), (9, 3), (11, 3) and what is the same is that the second number is always 3... ie: y = 3. What happens if we start with the children in a line above 3 on the horizontal scale?

 If the kids were placed in this pattern: the number pairs would be... (1, 2), (3, 3), (5, 4), (7, 5), (9, 6), (11, 7) and what is the same is that twice the second number minus the first is always 3... ie: 2y - x = 3. Leave these children on the grid and add more in a different colour at: (2, 1), (3, 3), (4, 5), (5, 7), (6, 9) Explain what you see. What number will be the partner of 1? (1, ??)

 If the kids were placed in this pattern: the number pairs would be... (1, 2), (3, 3), (5, 4), (7, 4), (9, 3), (11, 2) and the description this time might best be in two parts. For x less than or equal to 5, the rule is still twice the second number minus the first is always 3. What is the rule for x greater than or equal to 7?

#### Extensions

• Explore what happens to the ordered pairs and their patterns if the camp is moved and the children stay still? For example, it might be moved 2 East and 1 North from where it is. What about 2 West and 1 South??
• Place the children in a shape rather than a pattern. Can you discover anything about the ordered pairs?
• Imagine the children had a long rope that joined them all together to keep them safe (like mountain climbers, or a walking bus). Choose a point on the rope between two of the children and work out its ordered pair.
• Some children walked to these positions: (0, 0), (1, 1), (2, 4), (3, 9). They thought they made a pattern. Explain their pattern. Where would a child in this pattern stand to complete the pair (4, ??)?
• One child went out from camp. He had a safety rope around his waist. So he wouldn't get lost it was tied to a stake at the camp. He walked out to (5, 0) and the rope was taught. Then he kept walking. What shape did he make? Find some of the pairs on the shape. Can you discover the rule for this shape?
• What happens if we put the camp in the centre of a sheet of graph paper?

#### 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.

As mentioned above, this task begs to be expressed as a whole class investigation outside or in a multipurpose room with the children acting the story. So that the actors can also see 'the big picture' they can take witch's hats (or cones from the art room) to place in their spot then move off the grid. A long length of rope or coloured cord will be useful. It can be held on the ground to highlight any patterns between points. After exploring in this way, return to the classroom and set up a table top model using the printed grid above and blocks. The ideas above will be the source of several lessons.

At this stage, Kids On Grids does not have an exact match on Maths300, but it is closely related to Lesson 22, Algebra Walk and Lesson 75, Walking With Children.

#### 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 Kids On Grids task is an integral part of:

• MWA Chance & Measurement Years 5 & 6
• MWA Chance & Measurement Years 7 & 8