Calendar Challenge:
What number is associated with the current month, i.e. January - 1, February -2, etc.
For each month, an equation can be created corresponding to the number of days, i.e., Januray 1-31, February 1-28 (29), etc.
Using the month number six times, create an equation for each day number.
Consider order of operations, fractions, decimals, exponent, factorials....
Example:
September 5th:
(9+ 9 + 9 + 9 + 9) / 9 = 5
Complete instructions for teacher implementation:
In the typical school setting, this would be introduced the first week of September. It can be used in any classroom, whether heterogeneously or homogenously grouped. A whole group introduction is needed to ensure a solid foundation of understanding. Large chart paper is posted on the walls, usually three sheets, the first numbered from 1-10, second from 11-20, and third from 21-30. On the whiteboard (or chalkboard, or chart paper) I write six 9’s spaced out.
9 9 9 9 9 9
As a group, students are challenged to find a way to insert mathematical symbols to make the equation = 1. (Calculators can be used, but for most gifted students it is a mental math challenge [with paper and pencil].) In many situations, providing parentheses to encourage grouping can be helpful.
(9 9) (9 9) (9 9)
(It can be solved without the parentheses, using order of operations, but it jump starts most students’ thinking.)
There are multiple solutions in this format.
(9 ÷ 9) x (9 ÷ 9) x (9 ÷ 9) = 1 (9 + 9) – (9 + 9) + (9 / 9) = 1
(9 – 9) + (9 – 9) + (9 ÷ 9) = 1
Other formats can include using 99 , 9.9, 99 (all of which count as two 9’s).
99 ÷ 99 x 9 ÷ 9 = 1 9.9 - .9 – 9 + 9 ÷ 9 = 1 99 ÷ 99 + 9 – 9 = 1
Once students demonstrate their understanding of the task, they are challenged to find equations for 2 and then 3. These can easily be created by making minor operation changes in some of the above equations.
(9 ÷ 9) x (9 ÷ 9) x (9 ÷ 9) = 1 can be changed to (9 ÷ 9) + (9 ÷ 9) x (9 ÷ 9) = 2
At this point in the activity it is helpful to summarize some strategic ideas:
(9-9) = 0 (9 ÷ 9) = 1 99 99 ÷ 9 = 11
If necessary, order of operations should be reviewed. Students are then challenged to find some of the other equations that correspond to the calendar dates. For any (or all) students who are feeling frustrated, I guide them to 16, 17, 18, 19 and 20. They can all be solved by starting with (9 + 9) and the above strategic combinations.
During he first week of October the challenge is resurrected. Now it’s six 10’s for the numbers 1-31. Have students refer to their work from September. Does this give them any ideas? This comparison is the springboard to creating algebraic formulas for the different numbers. (It is not a necessary part of the challenge, but is usually satisfying to gifted students.)
(9 ÷ 9) x (9 ÷ 9) x (9 ÷ 9) =1 (10 ÷ 10) x (10 ÷ 10) x (10 ÷ 10) = 1
(a ÷ a) x (a ÷ a) x (a ÷ a) = 1
Wanting this to be a year long project suggests that only a few formulas need to be attempted each month. If there is a concept (exponents, decimals, etc.) that you wish to emphasize to correlate to your curriculum, this is an excellent way to keep the “excitement” going by challenging students to use an exponent, for example, in each equation.
Sample solutions for June:
| (6 ÷ 6) x ( 6 ÷ 6) x (6 ÷ 6) | = 1 |
| 66 ÷ 6 + 6 – (6 ÷ 6) | = 16 |
| (6 ÷ 6) + ( 6 ÷ 6) x (6 ÷ 6) | = 2 |
| 66 ÷ 6 + 6 – (6 - 6) | = 17 |
| (6 ÷ 6) + ( 6 ÷ 6) + (6 ÷ 6) | = 3 |
| 66 ÷ 6 + 6 + (6 ÷ 6) | = 18 |
| (6 + 6) ÷ 6 + (6 + 6) ÷ 6 | = 4 |
| 6 ÷ .6 + 6 + 6 x .6 - .6 | = 19 |
| (6 + 6 + 6 + 6 + 6) ÷ 6 | = 5 |
| 6! ÷ (6 x 6) + (6 – 6) x 6 | = 20 |
| (6 x 6 + 6) ÷ 6 – (6 ÷ 6) | = 6 |
| (6 x .6 - .6) x (6 + [6 ÷ 6]) | = 21 |
| (6 x 6 + 6) ÷ 6 – (6 – 6) | = 7 |
| 66 ÷ 6 + 66 ÷6 | = 22 |
| (6 x 6 + 6) ÷ 6 + (6 ÷ 6) | = 8 |
| 6 + 6 + 6 + 6 – (6 ÷ 6) | = 23 |
| 66 ÷ 6 – (6 + 6) ÷ 6 | = 9 |
| 6 + 6 + 6 + 6 + (6 - 6) | = 24 |
| (6 + 6) - (6 ÷ 6) – (6 ÷ 6) | = 10 |
| 6 + 6 + 6 + 6 + (6 ÷ 6) | = 25 |
| (6 + 6) – (6 ÷ 6) – (6 – 6) | = 11 |
| 6! ÷ (6 x 6) + 6 + (6 – 6) | = 26 |
| (6 + 6) – (6 – 6) – (6 – 6) | = 12 |
| 6! ÷ (6 x 6) + 6 + (6 ÷ 6) | = 27 |
| (6 + 6) + (6 ÷ 6) – (6 – 6) | = 13 |
| (6 x 6 – 6) – (6 + 6) ÷ 6 | = 28 |
| (6 + 6) + (6 ÷ 6) + (6 ÷6) | = 14 |
| 66 ÷ 6 + 6 + 6 + 6 | = 29 |
| (6 + 6) + (6 + 6 + 6) ÷ 6 | = 15 |
| 6 x 6 – 6 + (6 – 6) x 6 | = 30 |