Patterns in Mathematics

 

Figure it out (Page 5):

1. Copy the pictorial representations of the number sequences in Table 2 in your notebook, and draw the next picture for each sequence!

Answer:

The next picture in each sequence is shown below:

2. 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?

Answer:

1, 3, 6, 10, 15, … are called triangular numbers because they can be arranged as dots in the form of equilateral triangles.

1, 4, 9, 16, 25, … called square numbers or squares because they can be arranged as dots in the form of squares.

1, 8, 27, 64, 125, … called cubes because they can be arranged in the form of cubes.

Figure it out (Page 8, 9):

1. 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?

Answer:

3. 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?

Answer:

The All 1’s sequence is as follows: (1, 1, 1, 1, ….).

Adding the All 1’s sequence up:

1 = 1

1 + 1 = 2

1 + 1 + 1 = 3

1 + 1 + 1 + 1 = 4

Therefore, we get the Counting numbers sequence (1, 2, 3, 4, ….)

Adding the All 1’s sequence up and down:

1 = 1

1 + 2 + 1 = 4

1 + 2 + 3 + 2 + 1 = 9

1 + 2 + 3 + 4 + 3 + 2 + 1 = 16

Therefore, we get the Squares number sequence.

4. Which sequence do you get when you start to add the Counting numbers up? Can you give a smaller pictorial explanation?

Answer:

Counting numbers are as follows: (1, 2, 3, 4, 5, …..)

When we start to add the Counting numbers up we get:

1 = 1

1 + 2 = 3

1 + 2 + 3 = 6

1 + 2 + 3 + 4 = 10

We get Triangular numbers (1, 3, 6, 10, …)

A pictorial explanation is shown below:

5. 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?

Answer:

1 + 3 = 4

3 + 6 = 9

6 + 10 = 16

10 + 15 = 25

7. 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?

Answer:

Triangular numbers:

1, 3, 6, 10, 15, …

Multiplying by 6 and adding 1:

1 × 6 + 1, 3 × 6 + 1, 6 × 6 + 1, 10 × 6 + 1, 15 × 6 + 1, …

= 7, 19, 37, 61, 91

The sequence is Hexagonal numbers.

9. 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?

Answer:

The Counting number sequence is as follows: 1, 2, 3, 4, 5, ….

We can see by adding consecutive counting numbers that:

1 + 2 = 3

2 + 3 = 5

3 + 4 = 7

4 + 5 = 9

Now adding 1 to the beginning to the beginning of the series we get the Odd number sequence as follows: 1, 3, 5, 7, 9, …

This can be explained with the help of a picture as follows:

Figure it Out (Page 11):

1. 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?

Answer:

Counting the number of sides in each shape in the sequence of Regular Polygons, we get the following number sequence: 3, 4, 5, 6, 7, 8, 9, 10, ….

This is the Counting Numbers sequence starting from 3.

Counting the number of corners in each shape in the sequence of Regular Polygons, we get the following number sequence: 3, 4, 5, 6, 7, 8, 9, 10, …. This is also the Counting Numbers sequence starting from 3.

Therefore, we get the same number sequence because the number of sides = number of corners in a regular polygon.

2. Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?

Answer:

Counting the number of lines in each shape in the sequence of Complete Graphs, we get the following sequence: 1, 3, 6, 10, 15, …which is the Triangular number sequence.

3. 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?

Answer:

Number of squares in each shape:

First shape = 1

Second shape = 4

Third shape = 9

Fourth shape = 16

Fifth shape = 25

This gives us the Squares number sequence: 1, 4, 9, 16, 25, … The reason is that the smaller squares have been stacked to form larger perfect squares.