Expressions using Letter-Numbers · Lesson 8 of 11
Algebraic Expressions to Describe Patterns
“Use occurrence numbers, cycle remainders, and grid coordinates to predict distant positions.”
• Distinguish an occurrence number from a position number. • Derive expressions for each design in a repeating three-design border. • Use division remainders to identify an item at a distant position. • Analyse the four-position traffic signal sequence. • Find a number’s row and column in a four-column grid and describe column patterns.
Occurrence Number and Position Number
A border repeats the designs A, B, C, then A, B, C again. A position counts every item from the start. An occurrence number counts only the appearances of one chosen design. For example, B’s second occurrence is at position 5, not position 2.
C appears at positions 3, 6, 9, and so on. Each complete cycle contributes three positions, so its nth occurrence is at position 3n. B’s nth occurrence is one position earlier than C’s nth occurrence, and A’s is two positions earlier. These offsets produce formulas that describe all appearances of each design.
Problem
Where do A, B, and C each appear for the tenth time?
- 1.Set the occurrence number n to 10.
- 2.C is at 3 × 10 = 30. B is one position before it, at 29.
- 3.A is two positions before it, at 28. Positions 28, 29, 30 form the tenth A-B-C cycle.
Use Remainders to Read Distant Positions
To identify the design at a supplied position, reverse the viewpoint. Divide the position by 3 and see how much is left after complete cycles. A remainder of 1 lands on A, a remainder of 2 on B, and a remainder of 0 on C, at the end of a full cycle.
| Remainder after division by 3 | Design | Example positions |
|---|---|---|
| 1 | A | 1, 4, 7, … |
| 2 | B | 2, 5, 8, … |
| 0 | C | 3, 6, 9, … |
Problem
Which designs occur at positions 99, 122, and 148?
- 1.99 = 3 × 33, so its remainder is 0 and the design is C.
- 2.122 = 3 × 40 + 2, so its remainder is 2 and the design is B.
- 3.148 = 3 × 49 + 1, so its remainder is 1 and the design is A. Division replaces the need to draw the entire border.
The remainder and the occurrence number are different. B at position 122 is B’s forty-first occurrence because 122 = 3 × 41 − 1. The quotient 40 from division by 3 counts the complete cycles before the two positions left over. It is not automatically the occurrence number of the current design.
A Traffic Sequence with Two Yellow Positions
The traffic sequence used here is red, yellow, green, yellow, then red again. Yellow appears twice in every full cycle, so this is a four-position cycle. Analyse the stated sequence rather than assuming every colour occurs once per cycle.
Red occupies positions 1, 5, 9, and so on; green occupies 3, 7, 11, and so on. Yellow occupies every even position, 2, 4, 6, 8, and so on. The nth yellow occurrence therefore uses a different count from the nth four-position cycle.
Problem
Find the colours at positions 90, 190, and 343.
- 1.90 = 4 × 22 + 2. Remainder 2 selects the first yellow position in a cycle.
- 2.190 = 4 × 47 + 2, so it is yellow too.
- 3.343 = 4 × 85 + 3. Remainder 3 selects green. Remainder 1 would select red, and remainder 0 would select the other yellow.
Numbers in an Endless Four-Column Grid
A grid fills its rows with 1, 2, 3, 4, then 5, 6, 7, 8, and continues. Its row number and column number locate an entry. Unlike the repeating colour strip, the numbers keep growing, but the four-column arrangement creates predictable position rules.
Let r be the row number, starting at 1, and c the column number, one of 1, 2, 3, or 4. There are r − 1 full rows before the chosen row, containing 4(r − 1) numbers. Within the chosen row, count c more numbers. The entry is therefore 4(r − 1) + c.
| Column | Entries | Expression in row r |
|---|---|---|
| 1 | 1, 5, 9, … | 4r − 3 |
| 2 | 2, 6, 10, … | 4r − 2 |
| 3 | 3, 7, 11, … | 4r − 1 |
| 4 | 4, 8, 12, … | 4r |
Problem
Find the entry at row 7, column 3, and locate 124, 147, and 201.
- 1.Row 7 has six full rows before it. The entry in column 3 is 4 × 6 + 3 = 27.
- 2.124 = 4 × 31. It is at the end of row 31, in column 4.
- 3.147 = 4 × 36 + 3. After 36 full rows it occupies row 37, column 3.
- 4.201 = 4 × 50 + 1. After 50 full rows it occupies row 51, column 1.
For a number divisible by 4, the division remainder is 0, but there is no column 0. Such a number is the last entry of the row given by the quotient. With a nonzero remainder, the column is the remainder and the row is one more than the quotient. This distinction avoids moving every multiple of 4 into the next row.
Patterns Inside the Grid
Rows and columns also reveal properties of the numbers. Odd numbers occupy columns 1 and 3, while even numbers occupy columns 2 and 4. In every column, going down one row adds 4, matching the row-width count.
The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, and so on. Their columns are 3, 2, 1, 4, then 3, 2, 1, 4 again. Adding 3 moves through three positions of the four-column arrangement each time, causing that repeating column order. The multiples themselves increase while their column locations repeat.
Keep the meaning of n, r, and c explicit. An occurrence number counts copies of one design, a row number counts rows, and a column number counts positions within a row. A remainder of zero means the final position of a complete cycle or row, not a position numbered zero.
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, B, C, A, B, C, …, where is B’s nth occurrence?
What design is at position 148 in the three-design border?
In red, yellow, green, yellow, …, what colour is at position 343?
Where is 124 in the four-column grid?
What entry is at row r, column c in the four-column grid?
B is one position before the nth C at 3n.
Practice Problems
- Find the positions of the fifth appearances of A, B, and C in the border.
- At which occurrence of B does position 122 lie?
- State the traffic colours at positions 90, 190, and 344.
- Find positions of the tenth red, tenth green, and tenth yellow occurrences.
- Find the number in row 12, column 2 of the four-column grid.
- Locate 147 and 201 in the four-column grid.
- Write expressions for all four columns using the row number r.
- List the columns containing the first eight multiples of 3 and describe the repeating order.
- Which columns contain odd numbers? Explain using a row’s four consecutive entries.
3 × 5 − 2 = 13; 3 × 5 − 1 = 14; 3 × 5 = 15.
Key Takeaways
• Separate position numbers from occurrence numbers. • Cycle length determines the coefficient in an occurrence formula. • Offsets locate each design within its cycle. • Remainders identify positions in a repeating cycle. • Count completed rows before adding the column position. • A remainder of zero identifies the end of a complete cycle or row.