What it shows
Pins at grid points on a four-quadrant board let a string be stretched through any two points to show a straight line. The string through (0, 1) and (2, 5) rises 4 for every 2 across, slope 2, and crosses the y axis at 1, so its equation is y = 2x + 1.
The science
Every point on the plane is fixed by two numbers, its distance along x and its distance up y. A straight line has a constant slope: the rise divided by the run between any two of its points is always the same. The slope and the y intercept give the equation y = mx + c. Parallel lines have the same slope, and perpendicular lines have slopes that multiply to minus 1, which the board shows with a set square held against two strings.
Materials
Cost: under Rs 50. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A cork sheet or soft board 40 cm square
- Centimetre graph paper with axes from minus 10 to 10
- A box of drawing pins
- Coloured thread or elastic
- Labels and a marker
- A set square for perpendicular lines
Safety
For this project: Drawing pins are sharp; store them in a closed tin and keep the board out of reach of small children.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Glue the graph paper to the board and draw the x and y axes through the centre with the origin at (0, 0).
Push a pin into (0, 1) and another into (2, 5), stretch thread between them and extend it to the board edges.
Count the rise and run between the pins: up 4, across 2, so the slope is 2, and the line crosses the y axis at 1. Label the thread y = 2x + 1.
Check a third point: at x = 3 the thread passes y = 7, and 2 times 3 plus 1 is 7.
Add a second thread through (0, 4) and (2, 8), slope 2 again, and show it never meets the first: parallel lines.
Stretch a third thread through (0, 1) and (4, minus 1), slope minus one half, and check with the set square that it is perpendicular to the first; 2 times minus one half is minus 1.
Mark the midpoint of the segment from (2, 5) to (6, 1) with a coloured pin at (4, 3) and show the midpoint formula.
Measure the distance between (1, 1) and (4, 5) with a ruler, 5 cm, and verify it with the distance formula.
What you should see
The thread lines pass through every predicted point, parallel threads never cross, perpendicular threads meet at a right angle, and the ruler distance from (1, 1) to (4, 5) is 5 units.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1What is the slope of the line through (0, 1) and (2, 5)?
2, because the rise is 4 and the run is 2.
2What is the y intercept of y = 2x + 1?
1, where the line crosses the y axis.
3How do you know two lines are perpendicular from their slopes?
Their slopes multiply to minus 1, for example 2 and minus one half.
4What is the distance between (1, 1) and (4, 5)?
5, from the square root of 3 squared plus 4 squared.
Take it further
Pin the three vertices of a triangle and use the coordinates to find its area with the shoelace formula, then check by counting squares.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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