Finding the Unknown · Lesson 2 of 13
From Matchstick Patterns to Equations
“Turn growing patterns and balance puzzles into mathematical sentences.”
• Derive the matchstick rule for a chain of joined triangles. • Explain an equation, an unknown, its two sides, and a solution. • Write equations from balanced groups. • Decide whether a calculated position is possible in a pattern.
Count a growing pattern
Place triangles side by side so that each new triangle shares one edge with the previous triangle. The first triangle uses three sticks. Each extra triangle needs only two new sticks because the shared edge is already present. The counts therefore grow as 3, 5, 7, 9, rather than 3, 6, 9, 12.
Let n mean the position number, which is also the number of triangles. There is an initial group of 3 sticks and then n − 1 additions of 2 sticks. The count is 3 + 2(n − 1) = 2n + 1. Another way to see it is two sticks per triangle, plus one end stick. The letter n lets a single rule describe every position.
Problem
How many sticks make position 12?
- 1.Position 12 means n = 12, so S = 2 × 12 + 1.
- 2.Calculate S = 24 + 1 = 25 sticks.
- 3.The same result comes from 3 sticks for the first triangle and 11 additions of 2: 3 + 22 = 25.
An equation asks for equality
Now suppose you know the total number of sticks and want the position. With 99 sticks, the rule becomes 2n + 1 = 99. This is a mathematical sentence saying that two expressions have equal values. The equal sign is a claim to check; it does not simply mean “write the next calculation”.
A combination of numbers, letters, and operations that represents a value; for example, 2n + 1. An expression has no equal sign by itself.
A mathematical statement that two expressions are equal, joined by an equal sign.
A letter representing a number that is not yet known. Here n represents the position.
The expression before the equal sign is the left-hand side (LHS). The expression after it is the right-hand side (RHS).
A value of the unknown that makes the two sides of the equation equal when substituted.
Problem
Which position uses 99 sticks?
- 1.Write 2n + 1 = 99. The left side gives the pattern count; the right side gives the required total.
- 2.Remove the extra 1: 2n = 98. Divide by 2: n = 49.
- 3.Check by substitution: 2 × 49 + 1 = 99. Position 49 is a valid positive whole number.
Problem
Can this triangle chain use exactly 200 sticks?
- 1.Write 2n + 1 = 200. Removing 1 gives 2n = 199, so n = 99.5.
- 2.This number makes the equation true: 2 × 99.5 + 1 = 200. But half of a position is not an arrangement in this sequence.
- 3.Therefore no complete arrangement uses exactly 200 sticks. Every count 2n + 1 is odd when n is a whole number.
Translate a balance into an equation
Choose a letter for the weight of one unknown object. A group of two matching objects then weighs twice that letter. Known weights are added as ordinary numbers. Put the two branch weights on opposite sides of an equal sign. The equation keeps all the information needed to find the weight, even without the picture.
| Balance information | Equation | What the letter means |
|---|---|---|
| Two eggs balance three breads of weight 2 | 2e = 6 | e: one egg’s weight |
| One 4-unit cross and two rings balance 16 | 4 + 2r = 16 | r: one ring’s weight |
| Two bananas and a 4-unit orange balance 10 | 2b + 4 = 10 | b: one banana’s weight |
| Five sacks balance two sacks and 21 kg | 5s = 2s + 21 | s: one sack’s weight in kg |
Problem
For 3x + 4 = 7, identify the sides and test x = 1.
- 1.The LHS is 3x + 4 and the RHS is 7. The unknown is x.
- 2.Substitute 1 for x: LHS = 3 × 1 + 4 = 7. RHS = 7.
- 3.Because the two values agree, x = 1 is a solution. The expression 3x + 4 by itself is not an equation: it needs an equality statement.
Equations can have the unknown on either side, as in 20 = y − 3, or inside a quotient, as in a/3 = 50. They can also have the unknown on both sides, as in 2z + 4 = 5z − 14. The definition of a solution stays the same in every case: substitute and compare both sides.
Quiz
Which is an equation?
In 20 = y − 3, what is the right-hand side?
Which value solves a/3 = 50?
A chain has 37 sticks with rule 2n + 1. Which is its position?
Why is n = 99.5 unsuitable as a triangle-chain position?
Practice Problems
- Write equations for the sack balances: s + 2 = 10 + 2; two sacks against one sack plus 14 kg; and 90 sacks plus 50 kg against 60 sacks plus 500 kg. Define your letter.
- Find the LHS, RHS, and unknown in 2z + 4 = 5z − 14. Test z = 6.
- Use S = 2n + 1 to find the counts at positions 7 and 25.
- Can 75 sticks and 76 sticks each form a complete triangle-chain arrangement? Show your reasoning.
- Create five different equations whose solution is x = 4. Test each one.
- Explain why counting every triangle as three new sticks overcounts the pattern.
Key Takeaways
• An equation states equality between two expressions. • An unknown letter has a meaning chosen for the situation. • A solution makes the LHS and RHS equal. • Sharing an edge changes the triangle count to 2n + 1. • A pattern position must satisfy the equation and the whole-number condition.