Reverse and Add
1
2
3
Write down a two digit number, e.g. 62
Reverse the digits to form another two digit number, e.g. 26
Add the two numbers, e.g.
62 +
26
88
4
5
6
Repeat for other two digit numbers.
What do you notice?
Can you explain why this happens?
Convince a friend
Reverse and Subtract
7
8
9
Write down a two digit number, e.g. 62
Reverse the digits to form another two digit number, e.g. 26
Subtract the smaller number from the larger, e.g.
62 26
36
10
11
12
Repeat for other two digit numbers.
What do you notice?
Can you explain why this happens?
Convince a friend
Dot-to-dot Triangles
A
B
C
D
E
Draw a set of five labelled dots – [Don’t copy the ones I have drawn]
Make sure that you don’t choose three dots in a row
Make several copies of your set of dots
On each set draw a triangle(s) by joining any three dots.
How many different triangles can you draw?
How can you be sure you have found all the possible triangles?
Convince a friend
Can You prove it?
Sue Waring
The Mathematical Association
Staircase Numbers
A staircase is formed by adding consecutive integers.
Examples:-
12 because 3 + 4 + 5 = 12
53 because 26 + 27 = 53
80 because 14 + 15 + 16 + 17 + 18 = 80
Investigate staircase numbers and try to answer the following questions.
1.
2.
3.
4.
5.
6.
7.
8.
5
(3–step)
(2-step)
(5-step)
4
3
What do you notice about
(i) 3-step numbers, (ii) 5-step,
(iii) 7-step numbers?
What about “odd-step” staircase numbers?
Can you explain this result?
Do “even-step” staircase numbers behave in the same way?
Explain.
Describe how to find a staircase number.
Is it possible to find more than one staircase?
Are there any numbers which are not staircase numbers? If so, which?
Write a report about your investigation to
Convince a penfriend
Triangle Numbers
1
3
6
10
The first four triangle numbers are shown above.
Use the following questions to guide your investigation and write a report.
1.
By drawing similar diagrams find the next three triangle numbers and continue this table.
Position
(n)
1
2
3
4
Can You prove it?
Triangle Number
(T)
1
3
..
Sue Waring
The Mathematical Association
2.
3.
4.
5.
6.
7.
8.
9.
Write down the next three triangle numbers after the last one you have drawn.
Explain how you could obtain the 20th triangle number.
Find the most efficient way of doing this.
Test it on some triangle numbers you have found.
Try to write your method as a formula for finding the nth triangle number T.
Test your formula for n = 2, 3 and 4.
Explain how and why your formula works for the fifth triangle number.
Will it always work? How can you be sure?
Convince a penfriend
Calendar Squares
In the diagram showing the calendar for the month of May, a group of four numbers in a square has
been highlighted.
Adding diagonally opposite numbers in this square gives:15 + 23 = 38 and
16 + 22 = 38
MAY
Repeat for other squares.
Sun Mon Tue Wed Thu
Repeat for other squares.
Describe what you notice about the totals.
Explain why this happens.
Convince a penfriend
Partitioning
Fri
Sat
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Three objects can be partitioned in three ways:-
2 sets
A set of 2 objects and a set of 1 object
A set of 1 object and a set of 2 objects
3 sets
each of 1 object
In how many ways can
and
and
and
and
a) four objects
b) five objects
be partitioned?
Convince a penfriend that you have examined all possibilities
Can You prove it?
Sue Waring
The Mathematical Association