Colour Spots on a Number Line Years 3 - 7

Preparation

• You need two (2) different coloured markers or pencils.
• Print Investigation Guide A.
• Write the title of this challenge and today's date on a fresh page in your maths journal.
Mathematician's know that if they can SEE a pattern, then there will be a number pattern connected with it.
They can't always find it easily, and sometimes they can't find it at all, but they know it will be there.
In this activity you will start with spot patterns and discover number patterns.

Getting Started

• Work through Investigation Guide A to gather data.
When you finish the sheet, stick it in your journal and return here.

Have fun exploring Colour Spots on a Number Line.

• Answer these questions in your journal after you finish the sheet.
1. What do the dotted arrows tell you?
2. What did you SEE the first time you looked at the spots in Part 1.
3. Suppose someone else has the same picture without the spots.
Write a rule that tells them where to put the first spot and how to know where to put all the others.
• The sheet assumes that the marks on the long line are counting by ones.
A mathematician might ask What happens if they mean counting by twos or fives or tens or...?.
If you want to, you can explore this question now before continuing.
Or, you can write the question in your journal and come back to it in the future.
When you continue, the marks will still mean counting by ones.

You know the spots make a pattern.
You know you can say and write the rule for the pattern.
You know that the spot pattern was hiding numbers and you found them.
You know if you change the rule (like in Part 2) the numbers change too.
For Part 1 and Part 2, you made a list of the first 12 line numbers that go with the first 12 spots.

All this information is your data.
Let's see if there is something else we can learn from the data.
Part 1 Data

Spot
(S)
Number
(N)
----- -----
1 3
4 12
2 6
5 15
11 33
6 18
8 24
3 9
9 27
10 30
12 36
7 21

In Part 1

• Spot 1 was hiding Number 3.
• Spot 2 was hiding Number 6.
Write the table in your journal, but change it so the Spot numbers are in order.
Keep the blank row at the top.

• If I tell you any Spot can you tell me how to calculate its Number? Explain.
• S = 9, N = ?
• S = 25, N = ?
• S = 100, N = ?
• N = 39, S = ?
• N = 63, S = ?
• N = 3000, S = ?
Aleki wrote that the rule is N = 3 x S.
• He said that means the hidden Number is three times the Spot Number.
Is that true? Explain.

In your journal you have written the table in order with the blank row at the top.
Tamara said she knew what to put in the top row. What do you think?

• Look up the Number column from the bottom. What do think the blank should be?
• Look up the Spot column from the bottom. What do think the blank should be?
Tamara said it worked with Aleki's rule. Explain.

In Part 2
• You chose where to put Spot 1.
• Aleki chose 4 and Tamara chose 5.
• Your table will be one of these.
Write your table in your journal, but change it so the Spot numbers are in order.
Keep the blank row at the top.

Aleki

• If I tell you any Spot can you tell me how to calculate its Number? Explain.
• S = 9, N = ?
• S = 25, N = ?
• S = 100, N = ?
• N = 49, S= ?
• N = 73, S = ?
• N = 3001, S = ?
What rule did Aleki write this time? How did he explain it?

What will Aleki's numbers be in the blank row?
Does it work with your rule? Explain.

Part 2 Data

Aleki
Spot
(S)
Number
(N)
----- -----
1 4
4 13
2 7
11 34
6 19
3 10
9 28
10 31
8 25
12 37
7 22
5 16

Tamara
Spot
(S)
Number
(N)
----- -----
1 5
4 14
2 8
5 17
11 35
6 20
8 26
3 11
9 29
10 32
12 38
7 23
 Tamara If I tell you any Spot can you tell me how to calculate its Number? Explain. S = 9, N = ? S = 25, N = ? S = 100, N = ? N = 35, S= ? N = 83, S = ? N = 3002, S = ? What rule did Tamara write this time? How did she explain it? What will Tamara's numbers be in the blank row? Does it work with your rule? Explain.

Challenge 1: More Spot Patterns

Print Investigation Guide B.
• Use what you have learned to do Parts 3 and 4 of the guide.

Challenge 2: Number Pairs and Graphs

 In 1637 the mathematician René Descartes published a book. The last section of his book changed mathematics forever. Descartes had invented a way of turning tables of number pairs - like the ones in the data above - into pictures. The mathematics he invented is called Co-ordinate Geometry. We still use it today to draw graphs. As a boy René was very sick and spent a lot of time lying on his back in bed. Some stories say that the ceiling was made of 4 big squares that met in the middle making a left/right line and an up/down line across it. Perhaps he first thought of graphs when he was a sick boy. Descartes taught us that you need two pieces of information to find the position of any point on a flat surface. You start where the lines cross and count right (or left) until you are under (or above) it, then count up (or down) until you reach it. Print Investigation Guide C. Answer the questions in your journal and on the sheet. Stick the sheet into your journal.

Just Before You Finish

For this part you need your maths journal and your Working Like A Mathematician page.
• How did you work like a mathematician today? Record 2 ways.
• What do you know now that you didn't know when you started Colour Spots?