Calendar Calculations
Problem
Find the sum of all the dates in a week (you could use the dates of this week).
Write down the total.
Multiply the number of the first day in the week by $7$.
Add $21$ to the product.
Write down the total.
What do you notice about the two results?
Can you explain what accounts for the second answer?
Choose two consecutive weeks on the calendar and find the sum of the dates.
Multiply the number of the first day in the first week by $14$.
Add $91$ to the product.
How does this compare with using just one week?
Why do you multiply by $14$ and why add $91$?
Student Solutions
These investigations required you to use what you know about the multiples of seven.
How is this for a full explanation from James?
If you start the week, for example, on the 7th and add all of the numbers together, the result is 70. If you multiply the first number of the week - in this case the 7th - by 7 (7 X 7 = 49) and then add 21, you also get 70. Equal numbers will be reached whichever number you start with. If you multiply the smallest number by 7 (the number of days in a week) then each day you will get an extra number which will go up by one each day (8=7+1, 9=7+2 up to 6). The total of these numbers, 1+2+3+4+5+6 = 21. This is the same wherever you start on the calendar.
Scott
also explains, "The 21 stands for the difference between the number
of the first day of the week and the remaining days'
numbers".
Christina agrees with
Scott and James, "You add 21 because 1+2+3+4+5+6=21 and these are
the differences of the numbers between the first day of the week
and the other days."
The Reedmans showed
their answer in a table:
Monday = 1
Tuesday = 2
Wednesday = 3
Thursday = 4
Friday = 5
Saturday = 6
Sunday = 7
1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.
Multiply the number of the first day of the week by 7. Add 21 to the product:
| Monday = 1 | 1 x 7 = 7 | 7 + 21 = 28. |
"The final numbers are the same!!!"
Teachers' Resources
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.
Further explorations can be made by changing various parts and carrying out more investigative work. Here we see what happens when we pretend to have a four-day week and take our block to be 4 by 4.
The multiplication is done as follows;
This forms the first block of answers in the following table. The table also includes some other pretend weeks. The answers to the 6 multiplications that can be done around the 4 by 4 square are then shown as well as the digital roots of these answers.
Some of the blocks of 4 by 4 would not work because of the short weeks - they are shown with hatching. The numbers 1 to 5 show the starting number for each 4 by 4 block. Having given you this starting point, pretend weeks of different lengths could be explored in a similar way.
I found it very worthwhile to explore the digital roots that resulted. You can then change the size of the blocks and continue with more investigational work.
When working with your pupils in opening out a challenge it's really good to get some first results from a change in the question and then ask the pupils, "what do you see?". Then it's a matter of being brave enough to follow some of those leads and not just push for what you have seen. You'll probably be very surprised about all the different things that come up which may lead to further investigations.