Supporting the learning needs of ‘high achieving students’.

Exploring mathematical patterns with cards, circles and grids

Mā te huruhuru ka rere te manu.

It is the feathers which enable the bird to fly.

What do we mean by 2x?

Does it mean x + x, 2 × x, (x + x)?

Work through the following:

1. What does 2x mean?

2. What is the value of 2x when x = 3 ?

3. What is the value of 12 × 2 × 3 ?

4. What is the value of 12 ÷ 2 × 3 ?

5. What is the value of 12 ÷ 2 × x when x = 3 ?6. What is the value of 12 ÷ 2x when x = 3 ?

Did you get these answers?

1. 2 × x 2. 6 3. 72

4. 18 (convention tells us to work from left to right) 5. 18

6. Most people give the answer 2, different to that of question 5. What does it tell us about 2x and 2 × x? Nothing is quite as simple as it first appears, perhaps. Perhaps the moral of the story here is to use brackets everywhere where there might possibly be ambiguity

Food for Thought?

"Choose any two digit number, add together both digits and then subtract the total from your original number. When you have the final number look it up on the chart and find the relevant symbol. "

mysticalball

You have a fish tank containing 200 fish and 99% of them are guppies. You will remove guppies until 98% of the remaining fish are guppies. How many will you remove?

The Fish Tank

Using the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 once and only once.Where can they be placed so that the horizontal sum equals the vertical sum.

1

9 8

7

6 5 4

32

Some foundation work

• Calculate a)5 × 7

b)19 × 21

c) 299 × 301

Do you see a pattern?What about:a)24 × 26b)119 × 121c) (n − 1 )(n + 1)

Let’s try 14 × 16

10 × 10

4 × 10

10 ×

6

4 × 6

(n − 1 )(n + 1) = n2 - 1

Take a numberSquare that original number

Cube that original numberAdd five times the original number and add this

to the cube of the original numberWhat do you notice?

Take a number

10 playing cards

Watch the demonstration…

How can you do order the pack so that you can replicate this?

What is the least consecutive …

• What is the least number of consecutive natural numbers so that adjacent numbers are:

a)2 units apartb)3 units apartc) 4 units apartd)n units apart.

There’s a grid over there!

How long does it take to paint the house?

If Sally can paint a house in 4 hours and John can paint the same house in 6 hours, how long will it take for both of them to paint the house together?

61 2

1 house

4

Answer• Sally can paint 1/4 of a house per hour.

John can paint 1/6 of a house per hour. Let's say the answer is t hours for the total job. Then (1/4)t + (1/6)t = 1 (1/4 + 1/6)t = 1 (6/24 + 4/24 )t = 1 (10/24)t = 1 t = 24/10 hours = 2.4 hours

• If Sally can paint a house in 4 hours, then she can paint 1/4 of a house in 1 hour. If John can paint a house in 6 hours, then he can paint 1/6 of a house in 1 hour. Let x be the number of hours it would take them to paint the house. Working together, they can paint 1/x of the house in one hour, so 1/4 + 1/6 = 1/x Multiply by LCD, so 12x (1/4 + 1/6 = 1/x) 3x + 2x = 12 5x = 12 x=2.4 hours.

Also, 40% of 60 minutes is 24 minutes. Final answer = 2 hours and 24 minutes.

A third painter…What if Wirimu can paint a house in 5 hours and he was able to assist Sally and John, how long will it take for the three of them to paint the house together?

61 2

1 house

4

Looking for patterns.

Algebraically -generalising

Graphically Numerically

Plotting points on a graph to display your examples

What type of graph is it?

Straight LineLinear?

Other?

Draw / display examples to give you some ideas

Making links

What is the rule?

Generating a numerical pattern / sequence

Questions?