Calendar patterns
In this section from a calendar, put a square box around the 1st,
2nd, 8th and 9th. Add all the pairs of numbers. What do you notice
about the answers?
Age
7 to 11
Challenge level
Problem
Below is a copy of a calendar for 2008 showing December.
Put a square box around the 1st , 2nd , 8th and 9th .
Choose two of the numbers and add them together.
Choose another pair and add them. Keep going until you have made all the pairs.
What do you notice about the answers?
Put a box around another set of four numbers and try this again.
Look for other patterns.
Try multiplying - perhaps you could use a calculator to help.
What happens in other months?
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Student Solutions
Anil (Irmak Primary School, Turkey) explored some adding patterns:
$1 + 8 = 9$
$2 + 9 = 11$
$3 + 10 = 13$
$4 + 11 = 15$
$5 + 12 = 17$
$6 + 13 = 19$
$7 + 14 = 21$
$8 + 15 = 23$
Alice from Perse School for Girls spotted a pattern:
There is a pattern, it's that the two numbers diagonal to each
other added together make the same number as the other pair of
diagonal numbers make.
Chris found another pattern using multiplication:
If you cross multiply the set of four numbers, their
difference is always $7$.
$1 \times 9 = 9$ and $2 \times 8 = 16$, difference $= 7$
$3 \times 11 = 33$ and $4 \times 10 = 40$, difference $=
7$
$15 \times 23 = 345$ and $16 \times 22 = 352$, difference $=
7$ etc...
If you discovered any other patterns, then do let us know. Please don't worry that your solution is not "complete" - we'd like to hear about anything you have tried.
Teachers' Resources
Why do this problem?
This activity is a great example of how patterns and numbers may be investigated in everyday contexts. If you are looking for opportunities that give your pupils chance to follow things up themselves, then this may be your answer!
Possible approach
This investigation would work well with the children in pairs, each pair with their own copy of the December calendar section. It might be helpful to supply them with a paper frame to isolate the set of four numbers. This can easily be slid around the calendar to find new sets of four.
The problem begins with the lowest set of numbers simply to make the addition tasks easy. Later in the investigation encourage the children to move to the largest numbers they can cope with. Depending on children's experiences, encourage them to try and explain any patterns that they find.
Key questions
Tell me about the numbers you've found.
What have you done to get these answers?
Possible extension
If appropriate, guide the children to try multiplying the numbers and looking for patterns. If the children understand the basic concept of multiplication but can't readily manage the calculations, using calculators would be appropriate. This investigation could be revisited several times, trying different approaches each time. Encourage the children to discuss discoveries and suggest new
things to try. For example, what happens if the square box is enlarged to include nine numbers, or a rectangular frame of six numbers? Test discoveries on other months. What would happen if we lived somewhere where a week consisted of 6, 5, or just 4 days?
Possible support
Those who struggle a little may need some help to focus on which numbers they are dealing with at each moment - an adult helper would be good.