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Home›Algebra›Expressions

Expressions as rules: patterns and formulas

One square table seats 4. Push n of them into a row and 2n + 2 people fit, so 20 tables seat 42.

Tables in a row

A square table seats one person on each side, 4 in all. Push two together and the sides that touch lose their seats, leaving 6. Three tables seat 8, four seat 10. The table count goes up by 1 and the seats go up by 2 each time. That “add 2” is easy to see, but it is a slow way to answer a question like “how many seats for 20 tables?”, because you would have to walk through every step. An expression in n, the number of tables, answers it in one go. The figure below shows the first three arrangements.

Top: 1, 2 and 3 square tables in a row seat 4, 6 and 8 people (orange dots): one above and one below each table, plus one at each end, 2n + 2. Bottom: 1, 2 and 3 squares in a row need 4, 7 and 10 matchsticks: one stick to start, then 3 more per square, 3n + 1.

Look at the drawing of n tables. Every table has one seat above it and one below, which gives 2n, and the two ends add one more each, so the total is 2n + 2. Check it against the counts: n = 1 gives 4, n = 3 gives 8. Then answer the question: 20 tables seat 2 · 20 + 2 = 42 people. The expression also runs backwards. To seat 30 people you need 2n + 2 = 30, so 2n = 28 and n = 14 tables. A rule written as an expression is a machine that takes any n and gives the answer for that n.

Different ways of seeing give equivalent expressions

Someone else might see the tables differently. The first table seats 4, and every extra table adds 2, so with n tables the count is 4 + 2(n − 1). That looks nothing like 2n + 2, yet it is the same expression: expand, 4 + 2n − 2 = 2n + 2. The same thing happens with the matchsticks in the figure. One person says “one stick to start, then 3 per square”, which is 3n + 1. Another says “4 for the first square, 3 for each one after”, which is 4 + 3(n − 1), and that also simplifies to 3n + 1. When two people find different rules, simplify both and compare. If they match, both of you saw the pattern correctly.

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Column A counts the tables. Write =2*A2+2 in B2 and fill it down to get 4, 6, 8, 10, 12 and 14. Then try =4+2*(A2-1) instead: the column does not change.

Put the pairs from the table on a grid, number of tables across and seats up, and the points (1, 4), (2, 6), (3, 8) lie on one straight line. The graph lab draws the line y = 2x + 2. Only the points where x is a whole number of tables, 1 or more, mean anything for the cafeteria, but the line shows the pattern at a glance: it climbs 2 for every step of 1, which is the “add 2” you saw at the start, and it would cross the vertical axis at 2, the two end seats. Rules like this are the doorway to functions, the subject of the next course.

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The line y = 2x + 2. Read the small table under the graph: at x = 1 it gives 4 and at x = 2 it gives 6, the seat counts. Move the slope to 3 and the intercept to 1 to draw the matchstick rule 3n + 1 instead.

Formulas are rules someone already found

A formula is the same idea with the work already done. The perimeter of a rectangle is P = 2L + 2W: two lengths and two widths. With L = 8 cm and W = 5 cm, P = 16 + 10 = 26 cm. By the distributive property it is also P = 2(L + W), and 2 · 13 = 26 gives the same answer. Distance, rate and time are linked by d = rt. A train running at 60 km per hour for 2.5 hours covers 60 · 2.5 = 150 km. Evaluating a formula is evaluating an expression: replace each letter with its value and follow the order of operations. Units travel with the numbers, so check that they match before you multiply.

  1. Using 2n + 2, how many people can sit at 20 tables in a row?
  2. How many tables in a row are needed to seat exactly 30 people?
  3. How many matchsticks make 10 squares in a row?
  4. Write the matchstick rule for n squares in a row.
  5. Which expression is equal to 4 + 2(n − 1)?
    • 2n + 2
    • 2n + 3
    • 6n − 2
    • 2n − 1
  6. The pattern 5, 8, 11, 14, … has which rule for the nth number?
    • 3n + 2
    • n + 3
    • 5n
    • 3n + 5
  7. What is the 100th number in the pattern 3n + 2?
  8. Write the rule for the nth number of 7, 11, 15, 19, …
  9. A rectangle has length 8 cm and width 5 cm. Find P = 2L + 2W.
  10. A train travels at 60 km per hour for 2.5 hours. Use d = rt.
  11. Evaluate the rule 5n − 2 at n = 6.
Common mistakes
  • “Each table seats 4, so n tables seat 4n.” Joined tables lose the seats where they touch. At n = 2 the rule 4n gives 8, but only 6 people fit. Count a drawing before you trust a rule.
  • “The rule for 5, 8, 11, 14 is n + 3, because each number is 3 more than the last.” “3 more than the previous number” is a step rule, and n + 3 gives 4, 5, 6, 7. The nth-number rule needs 3n, the growth per step, plus a fixed part: 3n + 2.
  • “The rule works for n = 1, so it is correct.” Many wrong rules fit one case: 4n and 2n + 2 both give 4 at n = 1. Check at least two more values, such as n = 2 and n = 3.
Teacher's notePatterns run through the whole 2022 Korean national curriculum. Primary grades 1 to 2 find rules in arrangements of objects and numbers. Grades 3 to 4 explain changing rules and write them as numbers or expressions, with a note to avoid problems that need mixed calculations or a general-term expression. Grades 5 to 6 write relations from a table with boxes and triangles, and the note limits these to a single addition, subtraction, multiplication or division. So a rule like 2n + 2 belongs to the middle school band, where letters, the value of an expression and later the function value come in. A strong classroom move is to collect the different rules learners find for the same pattern and simplify them together.
Key points
  • A rule written in n answers any case at once: 2n + 2 seats for n tables.
  • Check a rule against at least three counted cases, not one.
  • Different ways of seeing a pattern give equivalent expressions: 4 + 2(n − 1) = 2n + 2.
  • A formula is a ready-made expression: P = 2L + 2W, d = rt.
Sources for this course (10)
  • Curriculum: 교육부 고시 제2022-33호 [별책 8] 수학과 교육과정 (ncic.re.kr/inv/org/download.do?year=2022&seq=10003559&orgType=ogi4, PDF read 2026-10-19: 초등학교 3~4학년군 [4수02-01] 다양한 변화 규칙을 찾아 설명하고 수나 식으로 나타내기(규칙을 식으로 나타낼 때 혼합 계산식, 일반항을 나타낸 식 등을 이용해야 하는 복잡한 문제는 다루지 않음); 초등학교 5~6학년군 [6수01-01] 혼합 계산에서 계산하는 순서(계산 순서에 중점, 지나치게 복잡한 혼합 계산은 다루지 않음), [6수01-04] 약수·공약수·최대공약수, [6수02-01] 대응 관계를 □, △ 등을 사용하여 식으로(덧셈식, 뺄셈식, 곱셈식, 나눗셈식 중 하나로 표현되는 간단한 경우만); 중학교 1~3학년군 [9수01-02] 소인수분해로 최대공약수, 수와 연산 용어 거듭제곱·지수·밑·분배법칙, [9수02-01] 다양한 상황을 문자를 사용한 식으로 나타내어 유용성을 인식하고 식의 값 구하기, [9수02-02] 일차식의 덧셈과 뺄셈, [9수02-09] 다항식의 덧셈과 뺄셈, [9수02-10] (단항식)×(다항식), [9수02-14] 함숫값, [9수02-19] 다항식의 곱셈과 인수분해(역관계 이해); 변화와 관계 용어 대입, 다항식, 항, 단항식, 상수항, 계수, 차수, 일차식, 동류항, 등식, 전개, 변수, 인수, 인수분해; '식의 값', '전개식'은 교수·학습 상황에서 사용 가능; 문자와 수, 문자와 일상 언어의 공통점과 차이점 탐색)
  • Textbooks: OpenStax Prealgebra 2e, 2.1 Use the Language of Algebra (Greg 20 and Alex 23, g + 3; variable and constant; × not used in algebra; expression like a phrase, equation like a sentence; 4 + 3·7 gives 25, not 49; PEMDAS and "Please Excuse My Dear Aunt Sally"; multiplication and division left to right, 18 ÷ 9·2; 18 ÷ 6 + 4(5 − 2); 5 + 2³ + 3[6 − 3(4 − 2)]; 2³ + 3⁴ ÷ 3 − 5²; work down the page: openstax.org/books/prealgebra-2e/pages/2-1-use-the-language-of-algebra)
  • 2.2 Evaluate, Simplify, and Translate Expressions (evaluate by substituting; 9x − 2 at x = 5 and x = 1; term, coefficient, coefficient of x is 1; 9b + 15x² + a + 6; like terms 5x, 7, n², 4, 3x, 9n²; 3x + 7 + 4x + 5; 7x² + 8x − x² − 4x; eight more than y, seven less than 9z; window height 6 inches less than width; dimes 2 less than 5 times the quarters: .../pages/2-2-evaluate-simplify-and-translate-expressions)
  • 7.3 Distributive Property (three friends at the movies, 3(9.25) = 3(9 + 0.25) = 27.75; a(b + c) = ab + ac, a(b − c), (b + c)a; 3(x + 4); 2(x − 3); 6(5y + 1); 100(0.3 + 0.25q): .../pages/7-3-distributive-property)
  • 9.4 (perimeter of a rectangle P = 2L + 2W: .../pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids)
  • OpenStax Elementary Algebra 2e, 7.1 Greatest Common Factor and Factor by Grouping (GCF of 54 and 36 is 18; GCF of 27x³ and 18x⁴ is 9x³; four steps, distributive property in reverse, check by multiplying; 4x + 12, 5a + 5, 12x − 60, 4y² + 24y + 28, 5x³ − 25x², 21x³ − 9x² + 15x = 3x(7x² − 3x + 5), −8y − 24 = −8(y + 3), 5q(q + 7) − 6(q + 7): openstax.org/books/elementary-algebra-2e/pages/7-1-greatest-common-factor-and-factor-by-grouping)
  • 2.6 (distance, rate and time formula d = rt: .../pages/2-6-solve-a-formula-for-a-specific-variable)
  • Microsoft Support, "Calculation operators and precedence in Excel" (negation before %, ^, then * and /, then + and −; =5+2*3: support.microsoft.com/en-us/office/calculation-operators-and-precedence-in-excel-48be406d-4975-4d31-b2b8-7af9e0e2878a)
  • History for the story block only: MacTutor History of Mathematics, University of St Andrews, "François Viète" (1540 to 1603; studied law, son of a lawyer; decoded letters to Philip II of Spain for Henry IV; In artem analyticam isagoge, Tours 1591, first systematic algebraic notation, vowels for unknowns and consonants for knowns; convention of early letters for knowns and late letters for unknowns introduced later by Descartes in La Géométrie: mathshistory.st-andrews.ac.uk/Biographies/Viete/)
  • "René Descartes" (1596 to 1650; La Géométrie an appendix to the Discours de la méthode, published at Leiden in 1637: mathshistory.st-andrews.ac.uk/Biographies/Descartes/).

Checked: 2026-10-19