Finding the Unknown · Lesson 6 of 13
Equations Hidden in Patterns
“Count one shape in two ways and use its rule to find a position.”
• Derive the tile rule for a growing T shape. • Explain why two different counts describe the same total. • Use an equation to recover a position from a tile total. • Check whole-number restrictions before accepting a pattern answer.
Find what grows and what stays
Consider a T built from square tiles. At position 1 it has a horizontal row of three tiles and one tile below the centre. At position 2 the row has five tiles and the stem below the centre has two. Each step adds one tile to each horizontal arm and one below the stem, so the totals are 4, 7, 10, 13.
Let k be the position number. Each of the three arms extending from the centre contains k tiles. The centre itself is one more tile. This gives 3k + 1. Seeing the three growing arms explains the multiplier 3; identifying the fixed centre explains the extra 1. Neither part is an arbitrary rule guessed from a list.
Problem
Find the number of tiles at position 8 by counting three arms and the centre.
- 1.Each arm has 8 tiles. Three arms contain 3 × 8 = 24 tiles.
- 2.Add the one centre tile: T = 24 + 1 = 25.
- 3.The centre is counted once. Including it in all three arms would count it repeatedly.
The same shape, another count
You can instead separate the T into its complete horizontal row and the vertical tiles below that row. The row contains k tiles to the left, one centre, and k to the right, so it has 2k + 1. The stem below contains k. Adding gives (2k + 1) + k = 3k + 1. Both methods count every tile once, so their expressions must agree.
Problem
Find the number of tiles at position 12 in two ways.
- 1.The row contains 2 × 12 + 1 = 25 tiles. The lower stem contains 12 tiles.
- 2.The total is 25 + 12 = 37 tiles.
- 3.The arm rule gives 3 × 12 + 1 = 37, confirming that the two descriptions agree.
Work backwards from the total
A rule becomes an equation when a total is specified. If the T has 100 tiles, replace T by 100 and solve 3k + 1 = 100. The solution must be a positive whole number because it counts a position. Substituting verifies the count, while the whole-number check verifies that the answer represents an actual shape.
Problem
At which position does the T contain 100 tiles?
- 1.Write 3k + 1 = 100. Subtract 1 from both sides: 3k = 99.
- 2.Divide by 3: k = 33. This is a positive whole number.
- 3.Check: 3 × 33 + 1 = 100. Its horizontal row has 67 tiles and its lower stem has 33, also totalling 100.
Problem
Can one of these T shapes have exactly 99 tiles?
- 1.Write 3k + 1 = 99. Subtract 1 to get 3k = 98.
- 2.Divide by 3: k = 98/3 = 32 2/3. This satisfies the equation but is not a whole position.
- 3.There is no complete 99-tile T in this sequence. Nearby positions 32 and 33 have 97 and 100 tiles.
Do not count the centre once in the row and again in the lower stem. Also, a fractional algebraic answer can be valid as a number while failing to represent a tile-pattern position.
Quiz
What stays fixed in the three-arm description of the T?
How many tiles are in the complete horizontal row at position k?
How many tiles does position 9 contain?
Which equation finds the position of a T with 46 tiles?
Which total is possible in this T sequence?
Practice Problems
- Draw positions 1, 2, and 3. Colour the centre and the three arms separately.
- Explain in words why the expressions 3k + 1 and k + (2k + 1) count the same shape.
- Find the tile count at positions 11 and 20 using both descriptions.
- Find the position of a T containing 76 tiles. Check the row and stem counts separately.
- Can the sequence contain T shapes with 50 tiles or 52 tiles? Solve and interpret each equation.
- A learner counts 3(k + 1) tiles. Explain exactly which tile is counted too many times.
Key Takeaways
• Observe which parts grow before writing a rule. • The T has three arms of k tiles and one centre. • The same total is also a row of 2k + 1 plus a lower stem of k. • An equation lets you recover a position from its tile count. • A valid pattern position must be a positive whole number.