Bands
Investigation Questions
Use these alongside the tool above. Stretch bands between pegs, toggle measurement overlays from the Measure menu, switch board types, and use split view and multiple pages to explore.
1
On the default 5 × 5 square board, stretch a band between two pegs in the bottom-left corner and two pegs in the top-right corner to make a rectangle. Toggle Lengths on in the Measure menu. What are the side lengths? Now toggle Area on. What is the area? Does the area match what you would calculate from length × width?
2
Make a square using four pegs. Toggle Shape Name on. Does the tool identify it as a square? Now drag one vertex to a neighbouring peg so the shape is no longer a square. What does the tool call it now? What property changed?
3
Create a right-angled triangle on the square board. Toggle Angles on. Which angle is 90°? Now toggle Area on. The area should be a whole number or end in .5. Can you make a right-angled triangle whose area is exactly 2? What about exactly 3? Is it possible to make a right-angled triangle with area 2.5?
4
Use the Grid toggle to show grid lines on the square board. Make a band that follows four grid lines to form a 1 × 1 square. What is its area? Now make a 2 × 2 square. What is the area now? Continue with 3 × 3 and 4 × 4. Write down the sequence of areas. What kind of numbers are these?
5
Stretch a band to make a tilted square — one whose sides do not follow the grid lines. Toggle Lengths on. Are all four sides equal? Toggle Angles on. Are all four angles 90°? Toggle Shape Name on. Does the tool call it a square? What is its area? Can you explain why the area of a tilted square is not a whole number of grid squares?
6
On the square board, make as many different triangles as you can that all have the same base (the bottom row of pegs). Toggle Area on for each one. What do you notice about the areas of triangles with the same base and the same height? Does this match the formula you know?
7
Make a triangle and toggle Angles on. Add up the three angle values shown. What is the total? Now make three more triangles of different shapes and sizes. Is the total always the same? What is the mathematical rule?
8
Challenge: can you make an equilateral triangle on the square board? Try it. Toggle Lengths and Angles on. Are all three sides equal? Are all three angles 60°? Why is it impossible (or very difficult) to make a truly equilateral triangle on a square grid?
9
Make a triangle where all three angles are less than 90° (acute). Now make one where exactly one angle is greater than 90° (obtuse). Finally, make one with a right angle. Toggle Shape Name on for each. Does the tool correctly classify them as Acute, Obtuse and Right-angled?
10
Make an isosceles triangle on the square board — one with exactly two equal sides. Toggle Lengths on to verify. Now try to make a different isosceles triangle with a different pair of equal sides. How many different isosceles triangles can you make on a 5 × 5 board? Can you make a scalene triangle (no equal sides)?
11
Make a rectangle (not a square) on the board and toggle Shape Name on. Now carefully move one vertex to create a parallelogram. What does the Shape Name change to? Toggle Angles on. In a rectangle, all angles are 90°. What are the angles in your parallelogram? What pattern do you see?
12
Make a trapezium (a quadrilateral with exactly one pair of parallel sides). Toggle Shape Name on. Does the tool identify it correctly? Now toggle Lengths on. What is special about the two parallel sides compared to the two non-parallel sides?
13
Make a kite on the square board (a quadrilateral with two pairs of adjacent equal sides). Toggle Lengths on to check your side lengths. Now toggle Angles on. What do you notice about the angles? Are any of them equal?
14
Investigate: how many different types of quadrilateral can you make on the square board? Try to make each of these: square, rectangle, parallelogram, rhombus, trapezium, kite, and an irregular quadrilateral. Toggle Shape Name on for each. Which ones does the tool recognise? Which are hardest to make on a square grid?
15
Make any quadrilateral and toggle Angles on. Add all four angles together. What is the total? Try five more quadrilaterals of different shapes. Does the angle sum ever change? Now make a pentagon (five sides) and add its angles. What is the total? What is the pattern for the angle sum of polygons?
16
Make a simple rectangle on the square board (e.g., 2 pegs wide by 3 pegs tall). Toggle ⊙ Interior (I) and ○ Boundary (B) from the Measure menu. Count the dashed rings (interior pegs) and solid rings (boundary pegs). Now toggle Area on. Check: does A = I + B/2 − 1? This is Pick’s Theorem.
17
Make three different rectangles on different pages. For each one, record I (interior pegs), B (boundary pegs), and A (area). Verify Pick’s Theorem for each. Now make a non-rectangular shape — a triangle or an L-shape. Does the formula still work?
18
Toggle only ○ Boundary (B) on (leave Interior hidden). Make a 1 × 1 square. How many boundary pegs does it have? Now make a 2 × 2 square, then 3 × 3. Record B for each. Can you write a formula for the number of boundary pegs on an n × n square?
19
Can you find a shape on the board where I = 0 (no interior pegs)? What is the simplest such shape? Using Pick’s Theorem with I = 0, the formula becomes A = B/2 − 1. Make several shapes with no interior pegs. Does this simplified formula always work?
20
Challenge: on a 5 × 5 board, what is the largest area shape you can make? Toggle Area, Interior (I) and Boundary (B) on. Can you beat an area of 12? What about 14? What is the theoretical maximum? Use Pick’s Theorem to explain why.
21
Make two different shapes that have the same area but different values of I and B. For example, find two shapes both with area 4, but where one has more interior pegs and fewer boundary pegs than the other. What does this tell you about the relationship between I and B in Pick’s Theorem?
22
Switch to the Coordinate board type. With the default settings (Max 6, single quadrant), toggle Coords on in the Measure menu. Click any peg. What coordinates does it show? Where is the origin (0, 0)? Where is the point (6, 6)?
23
On the coordinate board, make a horizontal line segment from (1, 2) to (4, 2). Toggle Lengths on. What is the length? Now calculate: what is 4 − 1? Make a vertical line segment from (3, 1) to (3, 5). What is its length? How does subtracting coordinates give you the length of horizontal and vertical lines?
24
On the coordinate board, make a triangle with vertices at (0, 0), (4, 0) and (0, 3). Toggle Lengths on. Two sides are easy to calculate. What is the length of the slanted side? Toggle the length display — does it match what you get from Pythagoras’ Theorem? (Hint: 3² + 4² = ?)
25
Toggle 4 Quadrants on. The axes now extend from −6 to 6 in both directions. Make a shape with at least one vertex in each of the four quadrants. Toggle Coords on. Which vertices have negative x-values? Which have negative y-values? What is special about points in the third quadrant (bottom-left)?
26
In 4-quadrant mode, plot the points (2, 3) and (−2, −3). Toggle Lengths on and connect them. What is the distance? Now plot (2, 3) and (−2, 3). What is this distance? A student says ‘to find the distance between two points with the same y-coordinate, just subtract the x-coordinates.’ Test this with your examples. Does it work when one coordinate is negative? Be careful!
27
On the coordinate board, make a square with vertices at (1, 1), (4, 1), (4, 4), and (1, 4). Toggle Perimeter and Area on. Note both values. Now make a second square on a new page with vertices at (0, 0), (3, 0), (3, 3), and (0, 3). What are its perimeter and area? Both squares have the same side length — does position on the grid affect measurements?
28
In 4-quadrant mode, make a rectangle with vertices at (−3, −2), (3, −2), (3, 2), and (−3, 2). The rectangle is centred on the origin. Toggle Area on. What is the area? The width spans from x = −3 to x = 3. What is the total width? How do you calculate this when dealing with negative coordinates?
29
Switch to the Circle board. Set the peg count to 8. Connect every other peg to make a shape. Toggle Shape Name on. What shape have you made? Now try connecting every peg. What shape is this? What would happen with 6 pegs on the circle?
30
On the circle board with 12 pegs, make a triangle by connecting pegs that are equally spaced (every 4th peg). Toggle Angles on. What are the three angles? Is this an equilateral triangle? Why is it easier to make an equilateral triangle on a circle board than on a square board?
31
Switch to the Isometric board. Try making an equilateral triangle. Toggle Lengths and Angles on. Are all three sides equal? Are all three angles 60°? Why does the isometric grid make equilateral triangles easy to construct?
32
On the Triangle board, make a shape using the edge pegs only. Toggle Shape Name on. What types of shapes can you make? Can you make a regular hexagon? How is building shapes on a triangular board different from building on a square board?
33
Using the Rectangle board, set W to 8 and H to 3. Make a diagonal line from the bottom-left corner to the top-right corner. Toggle Lengths on. What is the length of the diagonal? Now change to W = 6 and H = 4 on a new page and do the same. Which diagonal is longer? Can you predict the diagonal length without the tool?
34
On the square board, make three different shapes that all have an area of exactly 6 square units. Toggle Perimeter on for each. Do they all have the same perimeter? Which shape has the smallest perimeter? What type of shape tends to minimise perimeter for a given area?
35
Make a shape with a perimeter of exactly 12 units. Toggle Area on. What is the area? Now make a different shape, also with perimeter 12. Is the area the same? How many different areas can you achieve with the same perimeter of 12?
36
On a 5 × 5 board, make the shape with the largest possible perimeter. (Hint: think about shapes with many edges that zigzag around pegs.) Toggle Perimeter on. What did you achieve? Now make the shape with the smallest possible perimeter for a closed band with at least 3 pegs. What is it?
37
Make a 3 × 2 rectangle. Toggle Area and Perimeter on. Record both. Now make a 6 × 1 rectangle. Record both. Both shapes have the same area. Which has the larger perimeter? Try a 4 × 4 square and a 2 × 8 rectangle (area 16 each). What pattern do you notice about the relationship between shape and perimeter when area is fixed?
38
Use the Split button to show two boards side by side. On the left board, make a right-angled triangle. On the right board, make a triangle with the same side lengths but reflected (a mirror image). Toggle Lengths on for both. Are the side lengths identical? Toggle Angles on. Are the angles identical? Are the two triangles congruent?
39
On one board, make a shape that has exactly one line of symmetry. Use the drawing pen to draw the line of symmetry. Now make a shape with exactly two lines of symmetry. What about four lines? What type of shape has the most lines of symmetry on the square board?
40
In split view, make a small triangle on the left board and a larger triangle with the same angles on the right board (you may need a bigger board size). Toggle Angles on for both. Are the angles the same? Toggle Lengths on. Are the sides in the same ratio? These triangles are similar — same shape but different size.
41
On a 5 × 5 board, how many different triangles can you make using the bottom-left corner peg as one vertex? Make at least ten. Use a different page for each group. What strategy did you use to find them systematically?
42
Make a 1 × 1 square and record its area, perimeter, I and B. Now make 2 × 2, 3 × 3, and 4 × 4 squares. Record all four values for each. Can you write a formula for each measurement in terms of n (the side length)? Test your formulas by predicting the values for a 5 × 5 square, then check.
43
A student claims: ‘If you double every side length of a shape, the area doubles.’ Test this by making a 1 × 2 rectangle (area 2) and then a 2 × 4 rectangle. What is the area of the larger one? Is the student correct? What actually happens to the area when you double the side lengths? Try with a triangle too.
44
On the square board, make a diagonal line from one corner to the opposite corner. This creates two triangles. Toggle Area on. Are the two triangles equal in area? Now draw both diagonals of a rectangle. How many triangles are formed? Are they all equal in area?
45
How many different quadrilaterals with an area of exactly 4 can you make on a 5 × 5 board? Use a different page for each one. Toggle Shape Name on — how many different types of quadrilateral appear in your collection? Which type appears most often?
46
On the coordinate board in 4-quadrant mode, investigate: if you reflect a triangle in the x-axis, what happens to the y-coordinates of each vertex? What if you reflect in the y-axis? Make both the original and reflected triangles and toggle Coords on to verify.
47
On a 5 × 5 square board, what shapes can you make where the perimeter is numerically equal to the area? For example, a 4 × 4 square has area 16 and perimeter 16. Can you find a non-square example? Can you find a triangle? Use algebra and Pick’s Theorem to help your search.
48
Using the circle board, investigate regular polygons. Set the peg count to 12. Can you make a regular triangle? A regular square? A regular hexagon? A regular dodecagon? For each one, what fraction of the 12 pegs do you connect? Which regular polygons are impossible with 12 pegs?
49
Switch between different board types and make the ‘same’ shape (e.g., an equilateral triangle) on each. Which board makes it easiest? Which makes it impossible? Write a guide for a younger student explaining when to use each board type.
50
Design your own geoboard investigation. Choose a board type, decide which measurements to toggle, and write a question that another student could explore. Test it yourself first across multiple pages, then exchange with a partner. Which of your questions led to the most interesting discoveries?