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Home›Algebra

Expressions

Six units on algebraic expressions: letters for numbers and how to turn words into expressions, evaluating with the order of operations, terms and like terms, the distributive property and expanding, factoring out a common factor, and expressions as rules for patterns and formulas. Spreadsheet labs let you test whether two expressions really are equal, and a graph lab shows what a wrong expansion looks like.

Progress

About this course

English-speaking classrooms remember the order of operations as PEMDAS, from the sentence OpenStax quotes, "Please Excuse My Dear Aunt Sally". The pairing matters more than the order of letters. "My Dear" sits together because multiplication and division have equal priority, and you do whichever comes first from the left. So 18 ÷ 9 · 2 is 2 · 2 = 4, not 18 ÷ 18 = 1.

The six units are letters for numbers and writing expressions from words, evaluating expressions and the order of operations, terms, coefficients and like terms, the distributive property and expanding brackets, factoring out a common factor, and expressions as rules for patterns and formulas. To simplify, you add the coefficients of like terms: OpenStax simplifies 7x² + 8x − x² − 4x to 6x² + 4x and stops there, because 6x² and 4x are not like terms. Negative multipliers carry their sign to every term, so −2(x − 5) = −2x + 10. Factoring runs the other way. With x boxes of 12 pens and 18 loose notebooks the total is 12x + 18 = 6(2x + 3), shown as a rectangle of area 12x + 18 that is 6 tall and 2x + 3 wide.

Spreadsheet labs let you test whether two expressions really are equal, and a graph lab shows what a wrong expansion looks like. In the last unit the line y = 2x + 2 gives the seat counts in a table, 4 at x = 1 and 6 at x = 2. The unit also notes that in the 2022 Korean national curriculum the order of mixed calculations with the four operations is a primary grades 5 to 6 standard.

Before you start

Whole-number arithmetic and the order of operations help when letters replace numbers. The lessons use substitution to check an expression, then compare forms by expanding and factoring. In the spreadsheet labs, try more than one value for the letter: a match for one input does not explain the general rule.

When you finish

You can turn a verbal rule into an expression, substitute a value, and calculate in the agreed order of operations. You can identify coefficients and like terms, expand brackets using distribution, and take out a common factor. You can also write a seating rule such as 2n + 2 and test equivalent expressions in the spreadsheet.

  1. 1Letters for numbers: writing expressions from wordsIn 1591 François Viète printed a book in which letters stood for numbers, vowels for the unknown ones and consonants for the known ones.
  2. 2Evaluating expressions and the order of operationsWork out 4 + 3 · 7 from left to right and you get 49; multiply first and you get 25. Only one of them can be the value, so the order is agreed in advance.
  3. 3Terms, coefficients and like termsPut any number in for x and 3x + 5x comes out equal to 8x. At x = 7 that is 21 + 35 = 56, and 8 · 7 = 56.
  4. 4The distributive property and expanding brackets7 · 98 is awkward in your head until you write it as 7 · (100 − 2) = 700 − 14 = 686.
  5. 5Factoring out a common factor12x + 18 and 6(2x + 3) are the same expression. Only the second shows at a glance that, for any whole number x, the value is a multiple of 6.
  6. 6Expressions as rules: patterns and formulasOne square table seats 4. Push n of them into a row and 2n + 2 people fit, so 20 tables seat 42.

Related courses

  • Linear equations
    Eight units on equations such as 3x + 5 = 20: what a solution is, the four properties of equality, equations with the variable on both sides, the general strategy, fractions and decimals, and formulas. A graph lab draws each side of an equation as a line, so you can see the solution where the lines cross.
  • Graphs of lines
    Six units on the coordinate plane and straight lines: plotting ordered pairs, graphing an equation from a table or its intercepts, slope as rise over run, y = mx + b, finding the equation of a line, and solving two equations by seeing where their lines cross. Each unit has a graph lab to drag and test.
  • Quadratics
    Six units on quadratic functions and equations: the parabola, vertex form, factoring, the quadratic formula, the discriminant and a thrown ball. A function lab draws y = ax² + bx + c as you move a, b and c, and marks the vertex, the axis of symmetry and the x-intercepts.
  • Functions
    Six units on functions: what makes a rule a function, domain and range, reading a graph, shifting and stretching graphs, putting one function inside another, and running a function backwards. Each unit has a function lab, from the vertical line test to a mirror in the line y = x.
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