Question 20
Try and redraw each sequence in Table 3 in your notebook. Can you draw the next shape in each sequence? Why or why not? After each sequence, describe in your own words what is the rule or pattern for forming the shapes in the sequence.
We can draw the next shape in each sequence by following its special rule.
Step 1 — Regular Polygons
Yes, we can draw the next shape. Look at the first row of shapes. The first shape is a Triangle. It has 3 sides. The next is a Quadrilateral. It has 4 sides. Then a Pentagon with 5 sides. This pattern continues. Each shape has one more side than the one before it. The last shape shown is a Decagon. It has 10 sides. So, the next shape will have 11 sides. We call a regular polygon with 11 sides a Hendecagon. We can draw a regular shape with 11 equal sides.
The rule for Regular Polygons is: Each new shape has one more side than the shape before it.
Step 2 — Complete Graphs
Yes, we can draw the next shape. Look at the second row of shapes. The first graph, K2, has 2 points. They are connected by 1 line. K3 has 3 points. Each point is connected to every other point. K4 has 4 points. Each point is connected to every other point. The last graph shown is K6. It has 6 points. All points are connected to each other. To get the next graph, K7, we add one new point. Then we draw lines from this new point to all the 6 old points. We can easily draw this.
The rule for Complete Graphs is: Add one new point, then connect this new point to all the points that are already there.
Step 3 — Stacked Squares
Yes, we can draw the next shape. Look at the third row of shapes. The first shape is a single small square. It is a 1 by 1 grid. The next is a 2 by 2 grid of small squares. Then a 3 by 3 grid. This pattern continues. Each shape is a bigger square grid. The last shape shown is a 5 by 5 grid of small squares. So, the next shape will be a 6 by 6 grid of small squares. We can draw a large square and divide it into 6 rows and 6 columns.
The rule for Stacked Squares is: Make the square grid one unit bigger on each side.
Step 4 — Stacked Triangles
Yes, we can draw the next shape. Look at the fourth row of shapes. The first shape is a triangle made of 1 small triangle. It has 1 row. The next has 3 small triangles. It has 2 rows. The bottom row has 2 triangles. Then it has 6 small triangles. It has 3 rows. The bottom row has 3 triangles. This pattern continues. Each shape adds a new row of triangles at the bottom. The last shape shown has 5 rows of triangles. The bottom row has 5 triangles. So, the next shape will have 6 rows of triangles. The new bottom row will have 6 small triangles. We can draw this by adding a new row of 6 triangles below the last shape.
The rule for Stacked Triangles is: Add a new row of triangles at the bottom, with one more triangle than the previous bottom row.
Step 5 — Koch Snowflake
Yes, we can draw the next shape. Look at the last row of shapes. The first shape is a simple triangle. The next shapes become more bumpy. This is a special fractal shape. For each straight line part, we remove the middle third. Then we add two new lines to make a small triangle sticking out. This happens again and again. The last shape shown has many small bumps. To draw the next shape, we repeat this rule for every tiny straight line part on the last shape. This would be very hard to draw perfectly by hand. But the rule tells us exactly how to make it.
The rule for Koch Snowflake is: For every straight line segment, remove the middle third and add two new segments to form an outward-pointing triangle.
Answer
Yes, the next shapes can be drawn by following the geometric rules. Regular Polygons: Each new shape has one more side than the shape before it. Complete Graphs: Add one new point, then connect this new point to all the points that are already there. Stacked Squares: Make the square grid one unit bigger on each side. Stacked Triangles: Add a new row of triangles at the bottom, with one more triangle than the previous bottom row. Koch Snowflake: For every straight line segment, remove the middle third and add two new segments to form an outward-pointing triangle.
More questions in FIO
Can you think of other examples where mathematics helps us in our everyday lives?
How has mathematics helped propel humanity forward? (You might think of examples involving: carrying out scientific experiments; running our economy and democracy; building bridges, houses or other complex structures; making TVs, mobile phones, computers, bicycles, trains, cars, planes, calendars, clocks, etc.)
Can you recognise the pattern in each of the sequences in Table 1?
Rewrite each sequence of Table 1 in your notebook, along with the next three numbers in each sequence! After each sequence, write in your own words what is the rule for forming the numbers in the sequence.
Copy the pictorial representations of the number sequences in Table 2 in your notebook, and draw the next picture for each sequence!
Why are 1, 3, 6, 10, 15, ... called triangular numbers? Why are 1, 4, 9, 16, 25, ... called square numbers or squares? Why are 1, 8, 27, 64, 125, ... called cubes?
You will have noticed that 36 is both a triangular number and a square number! That is, 36 dots can be arranged perfectly both in a triangle and in a square. Make pictures in your notebook illustrating this!
This shows that the same number can be represented differently, and play different roles, depending on the context. Try representing some other numbers pictorially in different ways!
What would you call the following sequence of numbers?
That's right, they are called hexagonal numbers! Draw these in your notebook. What is the next number in the sequence?
Can you think of pictorial ways to visualise the sequence of Powers of 2? Powers of 3?
Can you find a similar pictorial explanation for why adding counting numbers up and down, i.e., 1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, ..., gives square numbers?
By imagining a large version of your picture, or drawing it partially, as needed, can you see what will be the value of 1 + 2 + 3 + ... + 99 + 100 + 99 + ... + 3 + 2 + 1?
Which sequence do you get when you start to add the All 1's sequence up? What sequence do you get when you add the All 1's sequence up and down?
Which sequence do you get when you start to add the counting numbers up? Can you give a smaller pictorial explanation?
What happens when you add up pairs of consecutive triangular numbers? That is, take 1 + 3, 3 + 6, 6 + 10, 10 + 15, ... Which sequence do you get? Why? Can you explain it with a picture?
What happens when you start to add up powers of 2 starting with 1, i.e., take 1, 1 + 2, 1 + 2 + 4, 1 + 2 + 4 + 8, ... ? Now add 1 to each of these numbers—what numbers do you get? Why does this happen?
What happens when you multiply the triangular numbers by 6 and add 1? Which sequence do you get? Can you explain it with a picture?
What happens when you start to add up hexagonal numbers, i.e., take 1, 1 + 7, 1 + 7 + 19, 1 + 7 + 19 + 37, ... ? Which sequence do you get? Can you explain it using a picture of a cube?
Find your own patterns or relations in and among the sequences in Table 1. Can you explain why they happen with a picture or otherwise?
Can you recognise the pattern in each of the sequences in Table 3?
Try and redraw each sequence in Table 3 in your notebook. Can you draw the next shape in each sequence? Why or why not? After each sequence, describe in your own words what is the rule or pattern for forming the shapes in the sequence.
Count the number of sides in each shape in the sequence of Regular Polygons. Which number sequence do you get? What about the number of corners in each shape in the sequence of Regular Polygons? Do you get the same number sequence? Can you explain why this happens?
Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?
How many little squares are there in each shape of the sequence of Stacked Squares? Which number sequence does this give? Can you explain why?
How many little triangles are there in each shape of the sequence of Stacked Triangles? Which number sequence does this give? Can you explain why? (Hint: In each shape in the sequence, how many triangles are there in each row?)
To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment '—' by a 'speed bump' _/_ . As one does this more and more times, the changes become tinier and tinier with very very small line segments. How many total line segments are there in each shape of the Koch Snowflake? What is the corresponding number sequence? (The answer is 3, 12, 48, ..., i.e., 3 times Powers of 4; this sequence is not shown in Table 1.)