Expressions using Letter-Numbers · Lesson 9 of 11
Patterns in a Calendar
“Use calendar offsets to explain why diagonal and cross-shaped sums follow general rules.”
• Express neighbouring calendar entries using horizontal and vertical offsets. • Complete a two-by-three block from a reference date. • Prove equal diagonal sums in any full two-by-two calendar block. • Explain why a five-entry cross totals five times its centre. • Distinguish checking examples from explaining a pattern for every valid block.
Read the Calendar as a Number Grid
Within a calendar, moving one cell to the right adds 1 to the date. Moving one full row down adds 7, because a week has seven days. These two simple changes let us describe a small shape of dates using only one reference date.
Use a full block of numbered cells: a blank cell before the beginning of a month is not a date. A real month has edges, so a requested shape must actually fit. To study the general pattern, we can imagine continuing the numbers in endless seven-column rows. The row spacing remains seven even beyond the end of the month.
Problem
A date is 15. Find its immediate left, right, above, and below neighbours in a full calendar grid.
- 1.Left and right neighbours differ by 1: 15 − 1 = 14 and 15 + 1 = 16.
- 2.Above and below differ by 7: 15 − 7 = 8 and 15 + 7 = 22.
- 3.The offsets come from the grid positions, not from memorising this one set of dates. The same instructions apply to another centre date.
A Two-by-Three Block from One Date
Suppose the bottom middle cell of a two-row, three-column block contains w. The other bottom cells are w − 1 and w + 1. To get the top row, subtract 7 from each bottom entry, because it is one week earlier.
| Top left | Top middle | Top right |
|---|---|---|
| w − 8 | w − 7 | w − 6 |
For the top-left cell, move left once and up once: subtract 1 and subtract 7, giving w − 8. For the top-right cell, add 1 and subtract 7, giving w − 6. Following the movements in either order works because the same two offsets are added.
Problem
The bottom middle date is 18. Fill all six entries.
- 1.Bottom row: 18 − 1, 18, 18 + 1, giving 17, 18, 19.
- 2.Subtract 7 from each bottom entry to obtain the top row: 10, 11, 12.
- 3.Check the layout: horizontal neighbours differ by 1, and each vertical pair differs by 7.
Why the Two Diagonal Sums Are Equal
A two-by-two block containing 12, 13 in its top row and 19, 20 in its bottom row has diagonal sums 12 + 20 = 32 and 13 + 19 = 32. This suggests a pattern. To explain why it works for every full block, give its top-left date the general name a.
The top-right entry is a + 1, the bottom-left is a + 7, and the bottom-right is a + 8. Each diagonal contains two copies of a. One diagonal adds the offset 8 all at once, while the other adds offsets 1 and 7. Since both offset totals are 8, the sums agree.
Problem
Show the two diagonal sums are equal and check a block beginning at a = 9.
- 1.First diagonal: a + (a + 8) = a + a + 8 = 2a + 8.
- 2.Second diagonal: (a + 1) + (a + 7) = a + a + 1 + 7 = 2a + 8.
- 3.For a = 9, the block is 9, 10 above 16, 17. Its diagonals total 9 + 17 = 26 and 10 + 16 = 26, agreeing with 2 × 9 + 8.
A numerical example checks one block. The letter argument explains every valid block at once because a was not assigned a special value. This is one of algebra’s most useful roles: it shows why a pattern persists instead of merely predicting another result.
The Five-Entry Cross
Now take a centre date a together with its immediate left, right, above, and below neighbours. Opposite neighbours have equal-sized offsets with opposite signs. Their offsets cancel when added, leaving only copies of the centre date.
Problem
Find the total for the cross centred at 15, then explain the general result.
- 1.The five entries are 8, 14, 15, 16, and 22. Their sum is 75.
- 2.In letters, the total is (a − 7) + (a − 1) + a + (a + 1) + (a + 7).
- 3.Collect five a terms; the offsets −7 −1 +1 +7 total zero. The sum is 5a, and 5 × 15 = 75 confirms the example.
Discover Another Shape and Explain It
You can look for other shapes whose totals are multiples of a reference date. A simple three-cell horizontal strip centred at a contains a − 1, a, a + 1. Its total is 3a because −1 and +1 cancel. A three-cell vertical strip similarly totals 3a because its offsets are −7 and +7.
Choose a full three-cell strip in a calendar and add its entries. Compare the sum with its middle date. Repeat with another strip, then replace the middle date by a and explain the offset cancellation. When suggesting a new shape, state exactly which cells it contains.
For any shape you investigate, first decide on a reference cell. Express each selected entry relative to it, count how many copies of the reference number appear, and add the offsets. A multiple of the reference alone results only when those offsets cancel. This gives a method for investigating, rather than a promise that every shape has the same kind of total.
Do not use a move of 1 for a vertical neighbour: the weekly spacing is 7. Also keep the reference cell consistent. In the six-cell block w is bottom middle; in the diagonal block a is top left; in the cross a is the centre.
Check Your Understanding
Use the ideas from this lesson to choose an answer. Explain your choice to yourself before opening the explanations below.
Quiz
In a standard calendar grid, the date directly below a is what?
A two-by-three block has bottom middle w. What is its top-left entry?
A two-by-two calendar block starts at top left a. What is either diagonal sum?
What is the sum of the five-cell calendar cross centred at a?
Why does the algebraic diagonal argument establish the pattern for every valid block?
One row down is one week later, an increase of 7.
Practice Problems
- Fill a two-by-three block whose bottom middle date is w = 24.
- For a two-by-two block with top-left 12, list the entries and both diagonal sums.
- For a two-by-two block with top-left a, write the bottom-right entry and explain its offset.
- Find the cross entries and sum when the centre is 20.
- Prove the sum of a three-cell vertical strip is three times its centre.
- The top-left date of a two-by-two block is 4. Use the general diagonal formula and verify directly.
- A student adds +1 for a cell directly above the reference. Correct the movement.
- Can a five-cell cross be selected at every date in a real monthly calendar? Explain.
Top row: 16, 17, 18. Bottom row: 23, 24, 25.
Key Takeaways
• A horizontal calendar step changes the date by 1. • A vertical calendar step changes the date by 7. • Choose a reference cell and express every other entry relative to it. • Both diagonals of a full two-by-two block total 2a + 8 from top-left a. • A five-entry cross totals five times its centre because offsets cancel. • A general letter argument explains a pattern across all valid cases.