The lesson defines variables as "letters used to represent an unknown number" and warns about notation (e.g., using the dot operator to avoid confusion with the variable x). Several parts use letters to represent numbers in properties (e.g., commutative: a × b = b × a; associative: (a × b) × c = a × (b × c); distributive: a(b + c) = ab + ac). A student task (#9) explicitly asks students to write the multiplication problem "4 times n" in four ways (4 × n; 4 ∙ n; 4(n); 4n), requiring students to write expressions with a variable.

Students use an Exponent Problem Sheet that includes boxes where they may fill in numbers or variables to form and evaluate exponential expressions. Students write and evaluate expressions using exponential notation as stated in the Parent Plan and practice writing expressions such as 12(5 + 3) to show totals using the distributive property. Students complete cost tables and compute totals, which produces numerical expressions for purchases and uses multiplication of grouped quantities.

Students are asked to consider a generic point written as (a, b) and to determine the coordinates of its reflections across the x-axis, y-axis, and both axes; the answer key explicitly states formulas (a, -b), (-a, b), and (-a, -b). The answer key also states that "a and b stand for any pair of rational numbers," which frames a and b as variables that can represent numbers from a specified set. Multiple activities require students to write ordered pairs and transform them, reinforcing the use of symbols to represent numbers in coordinate expressions.

The answer key for the Sam and Lindy bottles problem displays and solves the equation 4x + 3x = 91, using x to represent the size of one ‘box' of the ratio and finding values for each person. Several problems ask students to find missing values (for example, complete tables, find how many lemons for 4 pitchers, or determine quantities given a total), which require solving for an unknown quantity in context.

Students translate word sentences into number sentences and solve equations that include an unknown symbol (for example, activity prompts and answer keys use n: "n = 90/100 × 300", "80 = n/100 × 200", "75 = 15/100 × n"). An image and answer key explicitly label the unknown with the symbol n and describe n as "the unknown that needs to be found." The parent/teacher notes state that the idea of a variable was briefly introduced and that students may begin using variables for unknowns.

Students are asked to "Translate the word sentence into a number sentence, and then solve," and the answer key shows equations using a variable n (for example, n = 28% × 200; 17 = 85% × n; 37 = n% × 50). Several other answer keys use n for percent/part/whole problems (e.g., n = 40% × 25). The practice problems and answers demonstrate students setting up and solving equations where a variable (n) represents an unknown quantity.

The Answer Key for the "Ratios All Around" activity shows a worked example that uses a variable: it writes the math sentence n = 10/100 (x) $200 and solves n = 20 for coupon savings. Activity pages ask students to set up ratios and write ratio sentences (e.g., price per unit, recipe ratios) where students compute unknown totals by solving ratio equations. The unit price and conversion activities require students to set up and solve division or proportion expressions (price ÷ quantity, unit conversions) which can be written as algebraic expressions.

Students are asked to write expressions from word problems (e.g., Carlee has 12 board games → 12 + n; packs of pencils → 5n; 36 cookies in n boxes → 36n) and to translate verbal phrases into algebraic expressions on multiple activity pages. Students practice using variables as unknowns and as changing quantities with real-world contexts (e.g., pay for car washes: 5n; Jackson's home runs: a + 3 = 8; Jackson's batting time 20n). Activity 2 explicitly teaches standards about variable usage (variables can be unknown or changing, same vs different values in expressions, variables changing between expressions) and prompts students to write and evaluate corresponding expressions and simple equations.

Students are given a definition of variable as "a value that is unknown or that can change" and shown examples where a variable stands for an unknown (the square-with-n example and the discussion that a lone variable has coefficient 1). In Activity 3 and the associated student pages, students write expressions from given mathematical clues (e.g., write 6n + 3, 4x, n^2 - 16, 7n + 3m + 18). Multiple activities have students identify coefficients, variables, constants, and count terms, so students practice forming and reading algebraic expressions.

Students translate real-world word problems into algebraic expressions (Activity 2 and the Student Activity Page: e.g., "Milo has 6..." → 6 + n; multiple workbook problems require writing expressions such as 12 + n, 3n, 24/n). Students evaluate expressions by substituting numbers for variables (Day 2 Activity 3 and its student page: evaluate 5 + 4y when y = 6, evaluate (n−3)^2 + 4 for n = 4,7,12, and the table evaluating expressions for n = 4,9,12). Students are asked to choose reasonable values for variables (pizza slices example) and to substitute several different values to see how the expression's value changes, showing that variables can take different numerical values.

Students represent quantities with variables in multiple places: the 4n flowchart and box models show 4n = n + n + n + n and 3x + 5 from a visual model. Students write expressions from a real-world scenario in the candy activity (e.g., 12b + 9b + 5p + 3p + 8g + 10g → 21b + 8p + 18g). Students also evaluate and use variables as placeholders in tasks such as "Evaluate n − 14 if n = 25" and in problems that require substituting or combining like terms.

Students translate real-world situations into algebraic expressions in multiple activities (e.g., Tessa: 2n + 3n -> 5n; Raoul: 15p - 18) and complete word-problem pages asking them to write and simplify expressions. Students use variables in expressions and evaluate those expressions by substituting chosen real-number values (several examples show substitution and evaluation, including proving equivalence by evaluating 5(x+3)+2x-12 and 7x+3 at x=5). The lesson explicitly describes a variable as an unknown or changing value and repeatedly asks students to choose real-number values to substitute, reinforcing that variables can represent numbers.

Students translate verbal phrases into algebraic expressions (e.g., "a number decreased by 7", "twice a number plus four") and write expressions for real-world contexts such as Tasha's shells (n + 7), Kelsey's cookies (n - 12), Zane's pay (12n + 5), Jeremiah's earnings (5x + 3), and Alana's bead total (12(x + 6)). Students choose variable symbols and evaluate expressions for given values (e.g., evaluate expressions if x = 5 or x = 3) and rewrite expressions using properties (distributive and commutative) to show equivalence. Numerous problems require students to represent quantities with variables, form expressions from contexts, and compute those expressions with substituted values.

Students are asked to write and label expressions (Show and Tell Vocabulary Poster) and to create terms with variables and coefficients (Create a GoFish! Game). Students simplify expressions and then evaluate both the original and simplified forms by substituting numerical values for the variable (Prove It! Equivalencies). The Parent Plan and skills list explicitly instruct students to write, read, and evaluate expressions in which letters stand for numbers and to write expressions that record operations with numbers and letters.

Students are asked to translate word sentences into algebraic equations (e.g., "Five plus a number is twelve" → 5 + n = 12; "Six is four less than twice a number" → 6 = x - 4). Students use variables to represent unknowns in real-world contexts and write equations for those situations (Penelope's stuffed animals, Jaxon's marbles, Brian's muffins). Students practice substituting candidate numbers for variables and using guess-and-check to determine which values make the equations true (2x + 12 = 18, 36 - 4n = 12, multiple activity pages requiring substitution).

Students represent unknown quantities with variables in tape diagrams and hanger diagrams (examples: n + 60 = 100, 48 = 21 + x, p + 3 = 8). Students write equations from word problems (e.g., Everett: 16 + n = 72) and use variables in activity pages and answer keys to solve one-step equations. The Parent Plan and activities explicitly instruct students to use variables to represent quantities in real-world and mathematical problems.

Students repeatedly use variables (n, x, p, y, etc.) to represent unknown quantities and write equations from word problems (e.g., 5n = 40, Emma's plates, plant growth) and algebraic forms (2n = 8, n/2 = 5). Student activity pages require students to write equations for scenarios, draw tape/hanger diagrams that correspond to those variable expressions, and solve for the variable. The lesson also has students substitute values back into equations to check whether a chosen number makes the equation true.

Students are asked to identify unknowns and represent them with variables in multiple real-world word problems (e.g., Abby's snowfall -> 3n + 4 = 19; granola bars 3n + 2 = 14; various activity pages requiring writing equations from scenarios). The Independent and Dependent Variables section has students identify dependent and independent variables, write equations such as d = 65t, create tables/graphs, and analyze relationships. Student activity pages explicitly require students to write equations using a variable, solve them, and check solutions by substitution.

Students represent unknown quantities with variables in real-world contexts (for example, Casper's neighbor age is written as n > 11 and José's purchase is written as x > 5). Students write and evaluate algebraic expressions involving variables (the example 2x + 6 > 10 is formed from a problem statement and then evaluated by substituting values for x). Students practice translating between word statements, inequalities, and number-line graphs through matching, writing-inequality, and graphing activities and by using substitution/guess-and-check to determine which values from a set make an inequality true.

Students are asked to identify unknowns and represent them with variables (for example, using n for the number of striped shirts or chickens) and then write inequalities such as n + 5 < 12 and 3n - 5 ≥ 22. Activity pages and word-problem tasks require students to write an inequality from a real-world situation, solve it, graph the solution set, and list reasonable values from that set. The lesson explicitly instructs students to check solutions by substituting values from the solution set into the original inequality.

Students are shown multiple real-world examples that use variables to represent numbers and to write expressions or equations (d = 45t for distance/time, y = 15x for cookies/eggs, y = x - 2 for ages). Students translate word problems into equations (Kevin earning $10/hr → 10x = y; 3D printer and age problems) and are asked to write equations from situations. Students practice treating a variable as an unknown and as a varying input by substituting values, making input/output tables, and graphing coordinate pairs; the text explicitly states, "The independent variable is x. This value can be any number -- positive, negative, or zero."

Students write and solve equations from real-world contexts (e.g., Titus: n - 5 = 12; Heather: 3n + 2 = 23; Jenna: n + 4 = 7; Dominic: 3n - 5 = 115). Students represent relationships with variables and identify independent/dependent variables (Ms. Crisp bulbs; Brandi and Tessa ages) and rewrite equations into function form and create tables/graphs (x - y = 8, y = 2x + 1). Students solve and graph inequalities and list reasonable solution sets or restrict solutions to specified kinds of numbers (Bryan windows n - 3 < 6 with reasonable set {4,5,6,7,8}; Leah n/4 ≥ 25; Naomi uses whole-dollar values). The Parent Plan skills and multiple activity prompts require students to use variables to represent unknowns and quantities and to interpret solution sets.

Students are asked to brainstorm numeric facts and then create at least 10 problems, including 5–7 equations and 4–6 inequalities that use variables (examples use n, x, y). One required problem must include two variables with one independent and one dependent variable and an accompanying table and description of the relationship. Students must write inequalities of the form x &#<003e; c or x < c, graph solution sets on number lines, and check solutions by plugging answers back into equations/inequalities. The materials include prompts asking what a variable represents and how equations/inequalities represent unknown values.

Students represent unknown angle measures with variables and write equations such as 126 + n = 180, n + 2n = 90, and 5n = n + 92 to model geometric relationships. Students write equations from diagrams (vertical, complementary, supplementary) on the Geometric Equations and Solving Geometric Equations pages and solve them using inverse operations. Students substitute the found variable values back into expressions to calculate actual angle measures (for example, finding n = 30 then computing 2n = 60). The lesson's "Things to Know" and activities explicitly state that variables can represent unknown measures and may appear as part of expressions.

Students write and solve equations using variables, for example the quiz shows the equation 104 + n = 180 with solution n = 76°. The lesson uses general variables a, b, and c to state the triangle inequality a + b > c and asks students to apply that inequality to select possible side lengths. Students complete blanks like ∠1 + ∠2 + ∠3 = 180 and use symbolic expressions (a + b > c) to reason about which numerical values are possible.

Students are shown using variables to represent unknowns and writing equations to solve word problems (for example, the paver problem uses 24n = 4,320 and solves for n = 180). The Omar parallelogram problem has students set up 15 × n = 75 and solve for n, and the triangle problem sets up 1/2 × b × 5 = 20 and solves for b. Several answer keys explicitly demonstrate using a letter as a variable to represent the missing measurement and solving the resulting equation.

Students are introduced to and use variables such as C, d, r, and A in formulas (C = πd, C = 2πr, d = 2r, A = πr^2). Activity 2 shows algebraic manipulation of the equation π = C/d to solve for C using inverse operations and symmetry, and Activity 4 shows substitution of C = 2πr into A = (1/2)Cr to derive A = πr^2. Numerous problems ask students to compute circumference and area by substituting numeric values for the variables.

Students set up and solve proportions using variables in real-world contexts (e.g., 1 inch/5 feet = 6 inches/n feet to find n = 30). The materials include explicit proportion equations with variables such as 1/3 = x/18 and 1/3 = y/12 and show use of x and y to find scaled lengths. A review item and worked example use an algebraic equation with a variable (8 + 3n = 44) and direct text states, "You can use variables to represent the unknown value that you are trying to find."

Students repeatedly use variables (for example n, expressions such as 2n, 3n + 12, 3n + 35, 4n) to represent unknown angle measures and are asked to write and solve equations (e.g., n + 2n + (3n + 12) = 180; 3n + 35 = 4n). Practice and test items require students to set up equations from geometric relationships (complementary, supplementary, vertical angles) and solve for the variable, then substitute to find angle measures. The answer keys explicitly show the algebraic steps students are expected to perform to find n and then compute numerical measures.

The lesson includes the prompt "What could a variable represent in a geometry problem?" in the 'Ideas to Think About' section. Students plan and write problem cards (Stations Planning) and create 'Real-Life Area Problem' and 'Solve the Missing Angle Problem' activities where they design problems and solutions that could involve unknown values.

Students are shown Euler's Formula and asked to write an equation using a letter to represent an unknown (F + 5 - 8 = 2). The lesson walks students through algebraic steps: combining like terms and adding the inverse to isolate the variable, resulting in F = 5. Problem pages ask students to use the formula to find unknown counts (faces or vertices) given edges and vertices, applying the variable in context.

Students use formulas with variables such as SA = 2LW + 2LH + 2WH and SA = 6s^2 to set up and evaluate surface-area expressions for rectangular prisms and cubes. The Getting Started section explicitly asks students to evaluate formulas like V = s^3 and A = 6s^2 at a specific value (s = 1/2). Student tasks ask them to substitute given dimensions into variable expressions (plugging L, W, H, or s into the SA formulas) and to solve for unknowns in Basic Skills Review problems (e.g., 4x + 12 = 24 and evaluating n^2 − (−5) when n = 3).

The lesson repeatedly presents and uses symbolic formulas such as V = l × w × h, V = s^3, and V = B × h and asks students to substitute measured or given numeric values (including fractions and decimals) for l, w, h, s, B, and V. Students practice plugging fractional and mixed-number dimensions into these variable-based formulas to compute volumes (e.g., converting 1 1/2 to 3/2 and computing V = 3/2 × 3/4 × 4). Activity prompts and problems require students to use the variables in the formula form to calculate answers for real-world contexts like packing cubes and filling aquariums.

Students are shown how to use a variable to represent an unknown length in the volume example: 54 = L × 4 1/2 × 2, which is solved step-by-step to find L = 6. Students set up and solve an equation for a cube's side using s in 864 = 6s^2 and solve for s. Student problems and answer keys include writing and solving real-world equations with variables (e.g., 5n + 10 = 130 for Lucas's mowing fee) and instructions that students should "write an algebraic equation using the geometric formula for the solid and a variable for the missing dimension."

Students use symbolic formulas with letters (A = l × w, V = l × w × h, V = B × h, SA = 2LW + 2LH + 2WH) to compute area, surface area, and volume. Students solve for unknowns using those symbols, for example using Euler's formula F + V − E = 2 to find V and solving 182 = 6.5 × h to find h = 28. Several problems require substituting numeric measures for variable symbols and carrying out the algebra to find missing dimensions.

Students are asked to evaluate the expression 4n + 12 when n = 5 (Basic Skills Review Problem 1), showing use of a variable to represent a number and to write/evaluate an expression. A real-world word problem has students write the equation n + 26 = 102 to represent Rico and his brother's game scores (Basic Skills Review Problem 3). Students are also asked to solve and graph the inequality n + 3 < 5 (Problem 7), which has students interpret a variable as representing all numbers in a specified solution set.

Students write and evaluate algebraic expressions and equations in the Basic Skills Review: they evaluate 6p + 12 given p = 4 and solve 4n - 3 = 17. In a real-world context, students model a situation with an inequality (4 + n ≤ 9) to represent how many fish Josie could have bought. The inequality solution n ≤ 5 shows students using a variable to represent any number in a specified set (all numbers less than or equal to 5).

Students write expressions from word phrases (Activity 1 Problem 2: "four times a number increased by five" → 4n + 5) and represent a quantity with a variable (Activity 1 Problem 5: Marie's earnings as 9h + 3 and evaluation at h = 7). Students set up and solve equations from real-world problems (Activity 2: Naomi 5n + 3 = 18; Pedro n/3 = 9) and solve and graph inequalities that represent all numbers in a solution set (Activity 2 Problem 7: n + 2 > 5 → n > 3 with number-line graph). The Jayne problem and its answer note that some solutions are not reasonable in context (restricting solutions to nonnegative amounts).

Students are asked to write and solve an equation for vertical angles given expressions 3n and 2n + 16, leading them to set up and solve 3n = 2n + 16. The lighthouse scaling problem shows students using a variable in a real-world proportion (100/800 = 2/n) and solving for n to find the new height. These items explicitly require students to use variables to represent unknown numbers and to write and solve equations or expressions.

The lesson asks students to 'Let x = 0.3̅' and then multiply both sides by 10 or 100 (e.g., 10x = 3.3, 100x = 17.17), subtract the original equation, and solve for x, explicitly using a variable to represent the repeating decimal. Activity 5 includes step-by-step student notes and practice problems titled "Changing Repeating Decimals to Fractions" where students set a variable equal to a decimal, form expressions (10x, 100x), and solve algebraically to find the fraction.

Students encounter and fill in general exponent rules that use letters as placeholders (for example: x^0 = __, x^{-n} = __, x^m · x^n = x^{m+n}, (x^m)^n = x^{m·n}, (x·y)^m = x^m·y^m). The lesson repeatedly presents formulas using variables a, m, n, x, and y in the 'Things to Know' section, activity pages, and the answer-key flowchart. Students record these variable-based rules in their 'Properties of Exponents Notes' and use them to rewrite and simplify expressions in symbolic form.

The Parent Plan Skills explicitly states students will "Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p," which introduces a variable (x) as the unknown in those equations. Real-world problems on the activity pages show students writing and evaluating numerical expressions (for example 3 × 2^4 for the bacteria problem), and students practice forming and evaluating exponent/root expressions when solving those problems.

Students work with the equation N = 2^t where t is the number of 20-minute intervals and are asked to calculate the population after 6 intervals and compare 2^t to 3^t after 4 intervals. Students also compute 2^10 for DNA replication, using an exponent expression with an implied variable (number of cycles). These tasks place variables in real-world contexts (bacteria growth and DNA replication) and require evaluating expressions that include variables.

Students use the area formula A = π r^2 in the drop-zone task, representing area and radius with the symbols A and r as they calculate radius and area. Students evaluate expressions given in symbolic form (for example distances given as square roots and the fuel-cell energy written as 7×10^2) when they determine supply distances and convert scientific notation. Students are told to "use the equations provided to calculate delivery costs and supply costs," which requires substituting measured values into formulas.

Students set up proportions using a variable (x or n) in multiple real-world problems (e.g., tacos/cost, beads, Lexi reading books) and write labeled fractions with the variable in the numerator or denominator. Students solve those equations by isolating the variable, multiplying or dividing both sides, and by cross-multiplication (examples and step-by-step pages show 3/9 = x/27, 3/2 = x/12, 2/3 = 24/n). Students practice these processes across worksheets and word problems where they represent unknown quantities with a variable and compute its value.

Students set up proportions using a variable (n) to represent an unknown in the recipe and word-problem examples (e.g., 2 cups/4 people = n cups/10 people) and perform cross-multiplication (2×10 = 4×n). Answer keys and activity solutions show students writing and solving simple algebraic equations and expressions such as 2n = 750, 6n = 30, and solving for n. Several problems guide students to use the variable to find an unknown quantity in real-world contexts (flyers, time, boxes, miles).

Students identify and name independent and dependent variables in real-world contexts (Activity 1) by writing x and y for quantities like time, number of items, and cost. In multiple activities (Activities 2, 3, 5, and 6) students write equations in the form y = kx, rearrange given equations to isolate y, and use variables to solve word problems (e.g., y = 60x then substitute x = 4). Students also pick points from graphs and tables and compute k = y/x, showing they use variables to represent varying quantities in charts, graphs, and equations.

Students are asked to identify independent and dependent variables (x and y) and to write equations of the form y = kx for real-world contexts (e.g., Jim earns $12/hour → y = 12x; multiple activities ask students to write y = 5x, y = 80x, etc.). Students build tables by choosing values for x, compute corresponding y values, and plot points from those tables to make graphs that reflect the equation. Several activities explicitly direct students to find the unit rate k from the point (1, k) and to translate between equation, table, and graph representations.

Students write and use variables and equations repeatedly (e.g., t = pn, t = 12n, t = 15n, a = rh, a = 15h, c = 8m, y = kx). Students substitute numbers for variables and solve for unknowns (for example 156 = 10m leading to m = 15.6). Multiple activities and quiz items ask students to write equations for given contexts (e.g., "If proportional, write an equation", "Write an equation where y is proportional to x with constant 5").

Students set up and solve equations using variables in multiple real-world contexts (e.g., the image that sets I for income: 0.20 × I = $9,000 and solves I = $45,000; the sales-tax example that lets x be the original price: 1.08x = $212.00; and answer-key problems like p × 1.06 = $374.40). The materials use formulaic expressions with symbols (Commission = Sales × Commission Rate; Gratuity = Tip Percentage × Original Bill) and prompt students to "let" a variable represent an unknown when working backward to find prices, incomes, or values.

Students are asked to solve work-backward problems where the answer key sets up unknowns using a variable (e.g., "Let x be the original price: x × 0.60 = 18 → x = 30", "x * 1.20 = 72 → x = 72 ÷ 1.20 = 60"). Several answer keys explicitly write equations with x to represent unknown prices or unknown markup percents (e.g., "50 × x = 20 → x = 0.40"). The review and backward-problem pages require students to find original prices or markups, which the materials model by using variable-based equations.

Students are given and label the formula I = Prt, filling blanks that identify I, P, r, and t as interest, principal, rate (decimal), and time (years). Students solve multiple problems that require rearranging I = Prt to find unknowns such as r (annual rate) and t (time), with step-by-step solved examples and answer keys. Activity pages ask students to compute interest and total balance using the variables in the formula across a variety of real-world contexts (deposits, loans).

Students are asked to let a variable represent an unknown price and solve 1.06p = 212 to find the pre-tax price, showing use of a variable to represent an unknown number. Multiple items require writing equations for proportional relationships (e.g., y = (3/5)x, y = (2/7)x) and evaluating whether forms like y = 3.2x are proportional. Several problems and answer keys interpret variables as quantities in context (x = number of items, y = cost) and ask students to explain points like (1, r) and (0, 0) on proportional graphs.

Students are asked to write equations in the form y = kx and given specific examples (y = 2x and y = 1.5x) to model the recipe scaling. Students set up proportions using a variable (e.g., 1 lemon/3 tbsp = x lemons/___ tbsp) to solve for unknown quantities. Students use variables in cost expressions and formulas (for example, (Number of lemons × unit price) + (Amount of sugar × price per pound) = Total cost and Selling price = unit price × 2, 2.5, or 3). The activity pages label x and y as independent and dependent variables and require students to create tables and graphs using those variables.

Students define and use variables to represent unknowns in multiple activities (e.g., "Let r = number of rides" and write Total Cost = 25 + 3r). Students set up and solve equations with a variable for original price (e.g., 31.50 = P(1 + 0.05) and 54 = P × 0.9) and practice rewriting formulas with named quantities (Selling Price = Wholesale Price × (1 + Markup Rate)). Student activity pages repeatedly ask students to set up expressions and equations using variables for prices, rates, and counts and to solve them.

Students set up and solve real-world equations using variables (e.g., letting l and w represent rectangle side lengths to solve P = 2(l + w) and letting p represent number of marker packs in 15 = 4p + 3). Students define symbols for quantities in word problems (t, c, p, s for total, cost per pack, number of packs, shipping) and write corresponding expressions and equations (t = cp + s). Students solve and compare equations of the forms ax + b = c and a(x + b) = c across multiple practice problems, showing use of variables to represent unknown numbers.

Students create tables of values and substitute numbers for x in equations such as y = 3x, y = 2x, and y = 1/2 x (Activity 3). Students write equations from graphs by picking a point and computing k = y/x, then record the equation in the form y = kx (Activity 4). Real-world scenarios (earnings $10 per hour, cost per pound, car traveling 120 miles in 3 hours) require students to represent quantities with x and y and express the relationship with an equation.

Students repeatedly write and use equations in the form y = mx (for example y = 4x, y = 2.5x, y = 3x) to represent real-world situations and are asked to "write the equation representing the proportional relationship." Activities label x and y with contextual meanings (x = time, x = number of books, y = distance, y = total height) and direct students to create tables and graphs from those equations. Several tasks require students to form tables of values from equations and to plot points such as (1,3), (2,6), showing that x and y represent numerical quantities that vary.

Students set y = 0 and solve for x and set x = 0 and solve for y in equations such as 2x + 4y = 8, 3x - 6y = 12, and y = 5x + 10 to find x- and y-intercepts. Students work with and are asked to derive and interpret linear expressions such as y = mx and y = mx + b and to manipulate those expressions to find numerical intercepts. Students record intercepts as ordered pairs (x, 0) and (0, y) and plot those points on coordinate grids.

The lesson displays the slope formula m = rise/run = (y2 − y1)/(x2 − x1) and uses the symbols m, x1, x2, y1, y2. Students are asked to calculate slope by filling in RISE and RUN boxes and to set up proportions comparing rise/run values for different triangles. Several activity prompts have students record numeric rises and runs and compare ratios (e.g., 2/2 = 4/4) and complete slope calculation boxes next to graphs.

Students write and use equations in slope-intercept form (y = mx + b) throughout the lesson (Things to Know, Activities 1–6). They practice substituting values and solving for unknowns (Activity 6: Solving for b) and translate tables and real-world scenarios into variable expressions and equations (Activity 7: Using Tables; Activity 8: Linear Equations in the Real World including y = 10x + 50). The lesson also asks students to identify which variable represents which quantity (e.g., x = hours, y = total earnings) and to convert given equations into y = mx + b form.

Students write and use variables in real-world contexts such as earnings and costs (e.g., write equations like y = 12x + 20 for wages with a bonus, y = 40x + 35 for membership fees). Students set up and solve equations for unknown quantities in word problems (e.g., 30 + 10x = 80 → x = 5 GB; 24 + 3x = 66 → x = 14 rides). Students convert situations into slope-intercept form (y = mx + b), identify slope and y-intercept, and write equations for lines given contexts or characteristics (e.g., write an equation with slope 3 and y-intercept -5).

Students are asked to write equations using variables in multiple places (Step 3: Write the Equations directs students to use y = mx and labels y = total distance, x = time, m = speed). In the Comparing Travel Time activity students set up and solve equations of the form 500 = 60x to find an unknown time, treating x as an unknown to solve for. In the cost activity students write and use linear expressions y = mx + b (e.g., y = 0.15x + 31.50) and fill tables and graphs for multiple x-values, using variables to represent distance and cost across a range of values.

Students use variables x and y as inputs and outputs in multiple places (for example, y = 2x + 6 and y = (x+3)/2) and evaluate those expressions with given x-values. In Activity 2, students are asked to write equations from verbal rules (e.g., "The sum of twice a number and three" → y = 2x + 3). Students choose x-values, fill input/output tables, and plot (x, y) pairs on graphs, repeatedly using variables to represent numbers across tables, graphs, and mappings.

Students are given equations (for example y = 2x + 4, y = x^2, y = x^3 - x, y = x + 1) and asked to plug in specified x-values to compute corresponding y-values and complete tables. Students choose x-values, calculate y, and plot the resulting points on coordinate planes in multiple graphing exercises. Student directions repeatedly ask learners to evaluate expressions by substituting numbers for the variable and to compute rates of change from those tables.

Students are introduced to variables x and y and told to think of x as the independent quantity (e.g., time) and y as the dependent quantity (e.g., money earned). Students label axes (time and height/distance) and plot ordered pairs for scenarios like Sylvia the Sloth and Timmy the Turtle, using given rates (e.g., 4 feet per hour) to calculate positions over time. Students connect points to produce graphs and interpret how changes in x affect y, practicing the relationship between two quantities.

Students set x = 0 or y = 0 and solve algebraic equations to find intercepts (examples: 2y + 3x = 4 and y = 2x + 3). Students practice substituting values for variables and solving for the other variable on the 'Intercepts from an Equation' activity pages. Students identify intercepts from tables and graphs, connecting variable coordinates to numeric values (e.g., finding rows where x = 0 or y = 0). Real-world contexts (movie tickets, water consumption, basketball) ask students to interpret the meaning of intercepts in a situation.

The lesson repeatedly uses variables in formulas and equations — for example the slope formula m = (y2 − y1) / (x2 − x1) and the slope-intercept form y = mx + b — and shows students how to identify m and b and substitute numeric coordinates into these expressions. Activities ask students to pick points (x, y) from graphs and tables and plug those numeric values into the variable expressions to compute slope. The lesson also includes worked examples that rearrange equations (e.g., 4x + 2y = −8 to y = −2x − 4) showing solving for y in terms of x.

Students write and use the general form y = mx + b across activities (Things to Know; multiple activity pages). Students model a real-world situation with variables when converting subway trip tables (x = minutes, y = miles) into equations and when interpreting m and b as slope and y-intercept (Table to Equation examples). Students treat variables as unknowns and solve for them by substituting numbers (Activity 5: given m and a point, students substitute x, y, m into y = mx + b and solve for b).

Students repeatedly choose variable names for quantities and write expressions that relate them (e.g., "Let c = number of chores" and "Let A = total amount" and then write A = 6c + 12). The lesson defines input and output as values represented by variables and has students form function rules in the form output = slope × input + starting value from stories, tables, and graphs. Multiple activity pages ask students to identify slope and y-intercept and then write equations using the chosen variable names (e.g., P = 15h, C = 3h + 5).

The lesson presents multiple real-world equations using variables (e.g., Jordan: y = -3x + 100 with x defined as weeks and y as amount of money; H = 4x + 10; d = 3t + 2; d = t + 4) and asks students to interpret those expressions. Students are instructed to identify the y-intercept as the starting value (value of y when x = 0) and to compute rate of change from the coefficient of x. Student activities ask learners to compare and interpret given algebraic expressions alongside graphs and tables, and answer questions about starting values and rates using the variables provided.

Students write equations from verbal descriptions (e.g., "The difference between twice a number and 3 is 5" → 2x−3=5) and create function rules for real-world contexts (e.g., E=12h, E=15h, y=15x+25, y=2x+17). Students are asked to define the variable in context (e.g., h is the number of hours worked) and to complete function tables for given x-values using rules like y=4x−1, showing x taking values from a specified set. Multiple items require translating scenarios into algebraic expressions or equations and comparing/using those expressions to find intercepts, slopes, or outputs.

Students create Blue and Yellow cards that ask them to write equations from real-world descriptions (e.g., "Emma earns $10 for each hour she babysits" and writing an equation, or choosing the equation that models starting amount plus rate). The Blue cards include items that require plugging in a value and solving and "Solve for x or y (one-step)" tasks, so students practice using variables as unknowns. Green cards ask students to "Write an equation for the table," connecting variables to numeric patterns. Several examples explicitly use variable-based equations such as y = -2x + 5 and choices of equations representing linear relationships.

Students are asked to represent unknown side lengths with a variable (x) in multiple problems (e.g., a larger shape with base 6 cm and height x, and tasks that state "find x"). Worked examples show students forming and solving equations such as scale factor = 6 ÷ 3 = 2 and x = 10 ÷ 2, and instructions direct students to use the scale factor to compute missing lengths (5 cm × scale factor = 10 cm). Several activity pages repeatedly require finding x using proportional relationships and writing math sentences (e.g., x = 5 × (scale factor)).

The lesson presents the algebraic translation rule Ta,b → (x + a, y + b) and asks students to fill in and use the formula on the "Translation Notes" page. Examples show students applying the rule to specific coordinates (e.g., M(6,-2) → M' = (1,1) by computing x+a and y+b) and Activity pages require students to compute new coordinates using given a and b or to determine a and b from paired original/image coordinates. The checkers activity asks students to describe real moves using a translation rule Ta,b, connecting the algebraic notation to a real-world context.

The lesson repeatedly uses variables x and y in coordinate notation and formulas (for example, rules shown as (x, y) → (x, −y), (x, y) → (−x, y), and (x, y) → (y, x)). It presents equations of lines such as y = x and y = −x and asks students to apply those formulas to given coordinate points (many exercises require computing A(4, −2) → A'(4, 2), etc.). Students practice applying the symbolic coordinate rules to find reflected points and fill tables of original and image coordinates.

The lesson presents algebraic rotation rules written with variables, e.g., (x, y) → (−y, x), (x, y) → (y, −x), and (x, y) → (−x, −y). Examples show students assigning x and y specific numerical values (e.g., M = (6, −2), x = 6, y = −2) and computing M′ = (2, 6) by substitution. Student activity pages require students to apply these variable-based rules to rotate points, lines, and triangles and to write the new coordinates.

Students set up and solve equations using variables in Activity 4 where they use the formula new length = original length × scale factor and solve for unknowns (example: x = 8 × 2.5). The lesson explicitly shows solving for a scale factor using a variable (e.g., Scale factor = A′B′ ÷ AB and k in the answer key: 2.5 ÷ 5 = k). Multiple student activity pages ask students to write equations, compute new lengths, and solve for unknown side lengths or scale factors using variables.

Students see and use the mapping rule (x, y) → (y, -x) for a 90° rotation, which uses variables x and y to represent coordinates and expresses the transformed coordinates as expressions in those variables. Examples show students multiplying coordinates by a scale factor (e.g., A' = (1×2, 2×2)) and adding translations (e.g., A'' = (2+3, 4+1)), so students apply arithmetic operations to coordinate components as expressions. Several activity prompts require applying these coordinate rules to any given points on the grid.

Students label angles with letters (A, B, C and sometimes x, y, z) and use the equation ∠A + ∠B + ∠C = 180° on the Triangle Sum Rule notes. The activity examples show students computing a missing angle by subtracting given measures (180° − 55° − 75° = 50°). The proof/video notes present a triangle with angles labeled x, y, and z and ask students to use those symbols to explain how the three angles fit together to form a straight line.

Students are given the letters a, b, and c to label triangle sides and repeatedly write and use the expression a^2 + b^2 = c^2. Multiple worked examples show students substituting numeric values for variables (e.g., 3^2 + 4^2 = c^2) and solving for an unknown variable (solving for c or for a by isolating the variable). Activity pages require students to set up the equation with variables and manipulate the expression to find missing side lengths in both abstract and real-world contexts (ladders, distance on a grid, 3D problems).

Students write and use algebraic expressions and formulas such as V = πr^2h, V = (1/3)πr^2h, and V = (4/3)πr^3 to model volumes. Students plug numerical values into these variable expressions, solve for unknowns (for example solving 314 = 3.14×25×h to find h = 4), and complete activity pages that require finding missing radius or height from a given volume. The design challenge asks students to choose dimensions, record formulas, and compute volumes using variables for radius, height, and volume.

Students are asked to solve for unknown side lengths using a variable x (for example, find x for a dilated square where answers indicate x = 15 and x = 7). The curriculum displays and asks students to apply transformation rules written with variables (e.g., rotation rules (x,y) → (−y,x), translation rules T_{a,b}) and to map coordinates algebraically when reflecting, rotating, translating, and dilating figures. Multiple problems require students to write and use coordinate expressions to produce images (e.g., use T_{5,-3} or apply the rotation/translation formulas to given (x,y) points).

Students repeatedly solve equations in which a letter (x) stands for an unknown number (e.g., x+6=14, x−9=5, 4x=32, x/5=7) across one-step, two-step, fraction, and decimal activities. Students substitute solutions back into original equations to check that both sides are equal, demonstrating use of a variable as an unknown. Contextual scenarios (repair lab, space station) present real-world situations where students are given equations to solve for a variable.

Students are repeatedly asked to choose and use variables to represent unknown quantities in real-world and mathematical problems (e.g., "Let c represent the number of pounds of chicken," "Write the following as an equation and solve for the mystery number," and multiple problem statements that begin with defining a variable). Several activities require students to write equations from word problems (e.g., 12+15+3c=45, 50+8.75g=312.50) and to translate sentences into algebraic expressions. Instructions and answer keys explicitly show students writing expressions and equations, then solving for the variable.

Students work with many symbolic equations involving variables (e.g., 3x+4=10, 5y+2=5y+2, 3x+4=3x+7) and simplify them step by step to determine solution types. Students create equations with infinite solutions by filling in missing coefficients or constants so both sides match (examples: 4x+7=4x+_____, _____x+5=2x+5). Students are explicitly shown and told that some equations are true for any number (e.g., the cookie analogy, y = y from 5y+2=5y+2), so a variable can represent any number in those cases.

Students are repeatedly prompted to "define a variable" (e.g., "Let x = the number of months the person has been a member") and to translate word problems into equations (25 + 15x = 130; 18x + 20 = 146). Numerous activity problems require students to set up an equation from a real-world scenario (van rental, phone plans, shared dinner bill) and solve for the variable. The lesson and parent plan explicitly ask students to recognize and give examples of equations with one solution, no solution, or infinitely many solutions, and the quiz/true-false items include statements about equations that simplify to identities like 4 = 4.

Students repeatedly work with variables x and y in equations such as y = mx + b and y = 2x + 1, graphing those equations and using them to find intersection points. Students substitute coordinate pairs into equations to verify solutions (e.g., substituting (1, 3) into y = 2x + 1 and y = -x + 4, and substituting (1, -2) into 2x - y = 4 and 4x - 2y = 8). Students convert equations into slope-intercept form (y = mx + b) to compare slopes and intercepts to determine one, none, or infinite solutions.

Students work with equations using variables x and y (for example, y = x + 1, 2x + y = 7, 3x - 2y = 11) and write expressions to substitute one variable for another (substituting x + 1 for y). Students isolate variables, rewrite equations in slope-intercept or standard form, and perform substitution and elimination steps that require writing and manipulating algebraic expressions. Multiple student activity pages ask students to solve systems by substituting expressions (e.g., y = 2x + 1 into x + y = 7) and to show steps and check solutions.

Students write equations from given points (instructions to find slope and write each line in y = mx + b) and solve systems using substitution and elimination (worked examples and practice problems). The lesson includes real-world word problems (delivery fee, babysitting) where students set up equations with variables and solve for an unknown (miles, hours). Students also classify systems as one solution, no solution, or infinite solutions and solve algebraically for x and y in multiple practice items.

Students are repeatedly prompted to define variables in context (e.g., Let S = Sophie's earnings; Let x = price per pound; Let x = number of movies; Let y = total cost). Students write equations and expressions from those definitions (e.g., S + J = 65, 2x + 4y = 23.40, y = 20x + 50, y = mx + b form and Total Cost = Rate × Quantity + Fixed Amount). Students complete activity pages that require writing systems of equations and solving them with substitution or elimination in real-world contexts (earnings, prices, break-even, tickets, etc.).

Students set up and solve real-world equations such as 12h + 50 = 122 and 25 + 5c = 60 (Problems 20–21), using variables to represent quantities like hours and classes. Students write expressions and equations from word problems (e.g., lemonade cups, ticket cost systems) and solve systems of two linear equations to find unknowns. Problems that ask students to determine the number of solutions (e.g., 2x + 4 = 2x + 4 and 3x + 11 = 3x + 5) require them to interpret when a variable can be any number (infinitely many solutions) or when no value satisfies the equation.

Students are prompted to "Let n represent the number of months" (housing) and write cost equations C=1200m and C=1500+1050m, graph them, and solve for the break-even month by setting the expressions equal. In transportation and streaming activities students are asked to let x or h represent miles or hours, write linear cost expressions (e.g., C=225+0.60x, C=1.25x, C=20, C=5+1.5h), and use substitution, elimination, and graphing to solve for unknowns. In the phone-plan task students write and simplify expressions (y=5x+20 and y=(10x+40)/2), recognize they simplify to the same expression, and interpret that as infinitely many solutions.

Students are asked to identify and label independent and dependent variables on graphs (e.g., hours studied on the x-axis and test score on the y-axis) and complete multiple activity pages that require naming the independent and dependent variable for given scenarios. Explanatory text defines the independent variable as the one you control (x-axis) and the dependent variable as the one that changes in response (y-axis), and examples and answer keys repeatedly reinforce this representation. Parent/Activity pages prompt students to choose which variable goes on which axis and to identify variables in real‑world contexts.

Students label axes using variable notation (e.g., Hours Studied (x) and Test Grade (y)) and repeatedly plot ordered pairs (x,y) from real-world data tables. Activities ask students to identify independent and dependent variables and to set up axes and scales for those variable quantities. Several pages instruct students to decide which quantity goes on the x- or y-axis, explicitly treating those quantities as variable measures.

Students regularly identify independent and dependent quantities and assign them to variables (for example, x = degrees above 60, y = cones sold). They write linear expressions/equations in y = mx + b form for lines of best fit, find m by picking two points, and write equations for multiple scatterplots. Students substitute numerical values for variables to solve prediction problems (for example, substituting x = 10 into y = 2x + 50 to predict sales).

Students are asked to choose and write linear equations that fit scatterplots (e.g., Question 13 gives equation choices y = 2x + 4 or y = 4x + 0; Question 14 asks students to write an equation such as y = 4x + 2). Multiple items require students to identify independent and dependent variables and label axes for graphs, tying real quantities to variables. The Parent Plan and activities ask students to use the equation of a linear model to solve problems in context and to interpret slope and intercept, which uses variables to represent measured quantities.

The Parent Plan skills section explicitly states students will "Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept," and the scatterplot activity asks students to fit a straight line informally and assess model fit. The numerical activity pages label data columns as "Variable A" and "Variable B" and require students to set up axes and plot numeric pairs, which can correspond to using symbols or labels for quantities. The scatterplot pages prompt students to analyze trends and optionally draw a best-fit line, implying use of a linear relation to describe numeric relationships.

Students solve for an unknown variable in algebraic equations in the Exponents and Roots activity (e.g., items: "If x^2 = 81, then x =" and "If x^3 = 125, then x ="). Several items ask students to "write an equation" to represent real-world contexts (e.g., gamer loses 45 points over 9 rounds; a debt of $72 split among 6 friends), and answer keys show students set up and solve those numeric equations. The activities routinely ask students to interpret the meaning of calculated results in context.

Students are asked to write and work with equations using variables (examples include tasks to match tables to equations such as y = 10x, y = 3x, and to write equations of proportionality like y = 7x). Activities ask students to graph and label equations (e.g., graph y = 6x and label points (0,0) and (1,6)) and to interpret points (0,0) and (1,r) in context. The Parent Plan explicitly gives an example of representing a proportional relationship with an equation (t = pn) and directs students to "write equations to solve for y."

Students write and use variables in multiple real-world contexts: they write equations such as y = 14x, y = 75x, y = 18x + 12, y = 15x + 30, and C = 9t + 6 to represent earnings, rental cost, delivery cost, phone plans, and ticket fees. Students solve for unknowns using variables (for example finding width w = 12 cm from a perimeter problem and solving cost problems given expressions like 40 + 5x). Students interpret what variables and parts of expressions represent (identifying slope as unit rate, y-intercept as starting fee, and what (0,0) means in context).

Students write and use variable expressions and equations in real-world contexts such as C = 11t + 4 for movie tickets (problem 33) and 54 = 2(12 + w) to find a rectangle width (problem 32). Students manipulate and solve algebraic expressions and equations with variables in Unit 3 problems (e.g., simplify 4x + 9 + 6x + 1; distribute 3(y+5) − y; solve 5(x+4)=45 and 4x+6=30). Students model relationships with variables in proportional and linear contexts (write an equation for k = 3/2, find y = 4x, graph y = 4x, and find constant of proportionality in tables).

Students write functions from real-world situations (e.g., "Write a function representing total earnings after x hours" with answer y = 18h in Activity 3 and Activity 4). Students translate a verbal expression into an equation and solve it (Activity 4 Section 5: "The sum of a number and three times the number is 28" → x + 3x = 28). Students write an explicit rule and generate outputs for multiple inputs (Function Rule & Table: "Multiply by 2, then subtract 1," write y = 2x - 1 and complete a table for x = 0,1,2,3).

Students write and use algebraic formulas and variables in several problems: Activity 3 Problem 8 asks students to let angle B = x and write the equation 2x + x = 90 to find angle measures. Activity 4 asks students to write the Pythagorean Theorem (a^2 + b^2 = c^2) and use it to solve for a hypotenuse, and another problem requires writing V = (4/3)πr^3 and solving that equation for r given a numeric volume. The tasks require setting up symbolic equations, substituting numeric values, and solving for unknown variables.

Students define variables and write equations to model and solve real-world situations in Activity 4 (e.g., "Let m = number of months," 25 + 15m = 100). Students set up and solve single-variable equations and manipulate expressions in Activity 1, practicing combining like terms, using the distributive property, and solving equations with fractions. Students analyze cases with one solution, no solution, or infinitely many solutions in Activity 2, including identifying when both sides are identical (infinitely many solutions).

Students analyze scatterplots and interpret lines of best fit in Activity 3, answering questions about what a line of best fit indicates and assessing linear association. The Parent Plan explicitly states students will "Use the equation of a linear model to solve problems… interpreting the slope and intercept," and a linked resource is titled "Write an Equation for a Line of Best Fit." Students also interpret relationships between two quantitative variables and reason about model fit.

Students write variables and expressions for real-world situations (Item 8: "You earn $10 per hour. Write a function…" with answer y = 10x) and translate verbal rules into equations (Item 10: "The difference between twice a number and 4 is 7," answered as 2x − 4 = 7). Students solve for unknowns in many equations (Items 23–27) and set up and solve a real-world linear equation (Item 35: tutor charging $18/hr plus $30 fee). Students also determine the number of solutions for equations (Items 28–31), with the answer key showing an equation with infinite solutions (6x + 4 = 6x + 4).