### Plum Tree

Label this plum tree graph to make it totally magic!

### Magic W

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

### 2-digit Square

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

# Interactive Number Patterns

##### Stage: 4 Challenge Level:

This problem, and the Excel interactivities, have been created to deepen students' understanding of the 'difference method'.

The Hints section contains additional direction for students engaging with this task.

Students often possess the method but without any sense of why it's valid. Hopefully this activity will not only help them to explore this method's validity but will also encourage in them a questioning attitude and a desire to establish the validity of other common methods as they meet them.

It may be valuable to notice that the quadratic expression and the red values can be set on either sheet and will then automatically update on the other.

There are a large number of values displayed on the sheet called 'Patterns with differences' (see the tab at the bottom of the work area on Interactive Number Patterns 2). Students need to be given the time to understand all parts of the display.

The following may be one useful way to use this resource with a class.

Below the formula box are three rows. These display the values of the quadratic, linear and constant terms respectively. Change the terms in the formula box to help students grasp what each row shows, and verify that the three values added together match the blue values. In the two rows of differences above the formula box and verify that the data showing is the correct difference for the blue values, and use the slider to change the red values, allowing students to verify the new values displayed within the sheet.

Draw attention to the first difference which changes, and the second difference which does not. Why is that?

The constant term contributes nothing to the first difference, the linear term contributes to the first difference but not the second, and the coefficient, or multiple, of the quadratic term is the only coefficient which influences the second difference.