Seventh Grade - MATH
5: Math
Unit 1: Operations
Lesson 2
Multiplication Review
Students solve real-world multiplication problems that have the structure of proportional situations (e.g., a basketball uniform costs $24 and the coach buys 12 uniforms, magazines: 52 pages × 850 copies, Kayla earns $5.25 per hour × 100 hours). The lesson asks students to represent multiplication with a variable (e.g., "What are the four ways to represent the multiplication problem '4 times n'?" and shows notation such as 4n). Variables and letter notation are introduced in examples and in property formulas (e.g., a(b + c) = ab + ac).
Lesson 6
Greatest Common Factor
Students set up and solve multiplication and division equations that show how items distribute into equal groups (e.g., "12 × ? = 24" and "24 ÷ 12 = 2", "6 × 7 = 42"). Students write factored forms that express totals as a common factor times a sum (e.g., 3(5 + 7), 6(3 + 8), 4(7 + 10)) when dividing items into identical bags or rows. Students use the distributive property backwards to convert sums into a product of a GCF and a sum, representing the relationship between number of groups and items per group.
Final Project
Planning a Party
Students multiply the number of items per bag by 12 to find total items and multiply the number of packages by price per package to find total cost (example: 3 × $4.80 = $14.40). Students are instructed to compute a cost per goody bag by dividing the grand total cost by 12 (example: $42.84 ÷ 12 = $3.57). Students also use the distributive property to show totals (example: 12(5 + 3) = (12×5) + (12×3) = 60 + 36).
Unit 2: Integers and Rational Numbers
Lesson 2
Fraction Multiplication
Students translate word problems into multiplication number sentences (for example, "2/3 of 6 = 2/3 × 6") and set up and solve multiplicative equations in context (Carlotta's total miles: 3 1/2 × 2 2/3; calculating total distance when a route is run twice). Students are asked to write number sentences for word problems and to use formulas written as equations (for example, "area = length × width") when solving applied problems. Students also practice using multiplication to find a part of a whole in multiple contexts (recipes, poster board, garden area).
Unit 3: Ratios and Percentages
Lesson 1
Introduction to Ratios
Students practice writing ratios in three forms (words, with a colon, and as a fraction) and make equivalent ratios by multiplying or dividing both quantities (e.g., 4/1 = 8/2 shown for paint:water). Students solve scaling word problems (bread recipe scaled by 3, cleaner mixing where 50(x)4 = 200 and 5(x)4 = 20 are used) that require finding and applying a constant multiplier to both parts of a ratio. The activity pages ask students to produce equivalent ratios and to determine whether given ratios are equivalent, reinforcing the multiplicative relationship between quantities.
Lesson 3
Equivalent Ratios
Students set up and solve multiplicative equations to find parts of a ratio (e.g., answer key shows 4x + 3x = 91 for Sam and Lindy and uses expressions like 5x and 7x for Jason and Zeke). Students use tables of equivalent ratios and scale factors to compute missing values and highlight proportional points on graphs (e.g., tables with (1,6),(3,18),(5,30) and highlighted point (4,24); (5,2),(15,6),(30,12) and highlighted (40,16)). Several answer keys show arithmetic written as equations and multiplicative scaling (e.g., multiplying both parts by a factor) to produce equivalent ratios.
Lesson 4
Unit Rates
Students compute unit rates by dividing totals to find a per‑one value (Elena: 216 miles ÷ 4 hours = 54 miles/hour; Jalen: 156 calories ÷ 12 chips = 13 calories/chip; strawberries: $5.25 ÷ 3 = $1.75/quart; jellybeans: $6.27 ÷ 3 = $2.09/pound). Students then use that unit rate multiplicatively to find totals (13 cal/chip × 25 chips = 325 calories; $1.75 × 5 quarts = $8.75; $2.09 × 10 pounds = $20.90). The skills list explicitly mentions using ratio and rate reasoning "by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations."
Lesson 6
Percentage Problems
Students are given and asked to use the formula "part = percent × whole" and several problems are set up and solved using that equation form (e.g., n = 90/100 × 300; 75 = 15/100 × n). The materials show algebraic manipulation of proportional equations such as 32 = 0.64x and x = 32/0.64 and include answer keys using variables (n or x) to represent unknowns. Students translate percentage word sentences into number sentences and write equations to solve for missing values in proportion problems.
Lesson 7
Unit Conversions
Students set up and solve unit conversion problems by creating equivalent ratios and using double number line diagrams (e.g., 1 kg : 1000 g scaled to 8.5 kg : 8500 g, 1 in : 2.54 cm scaled to 10 in : 25.4 cm). The lesson shows proportions written in fraction form for conversions (for example, 1 ounce / 28.35 grams ≈ 5 ounces / ? grams) and students solve for the unknown quantity by scaling the ratio. The activity problems ask students to determine the unit conversion ratio and use it to find unknown values (cups to fluid ounces, kilograms to grams, inches to centimeters, etc.).
Lesson 8
Unit 3 Test
Students set up and solve equations in percent problems (e.g., n = 28% × 200; 17 = 85% × n; 37 = n% × 50) and use equalities to find unit rates and unit prices (e.g., 4 pounds/$5.16 = 1 pound/$1.29). Students complete tables of equivalent ratios and scale values in tables and graphs (apples and pies table and plotted points) to find missing values by multiplication. Several answers show students using algebraic expressions and multiplication to compute proportional quantities (e.g., 16 × 7 = 112 ounces, 2.54 × 8 = 20.32 cm).
Final Project
What's the Best Buy?
Students set up ratios and compute unit prices (e.g., dividing $3.38 by 13 oz to get $0.26/oz) when calculating unit price in Activity 3. The Answer Key shows an explicit algebraic sentence for a percent problem (n = 10/100 (x) $200) and uses proportional equations in currency conversion examples ($1/0.85 = $3.25/2.76). The project asks students to use equivalent ratios to double or triple recipes and to use dimensional-analysis style ratio equations to convert units (gallons → fl oz).
Unit 4: Algebraic Expressions
Lesson 1
Introduction to Algebra
Students read and write multiplicative expressions that relate two quantities, for example representing pay as 5n for $5 per car and writing 36n for cookies in n boxes. The materials use formula notation in contexts such as V = s^3 and A = s^2 and present expressions like 20n for total minutes of practice, with tasks asking students to evaluate these when n is given. Activities prompt students to translate verbal descriptions (e.g., "How many pencils in p packs?" → 5p) into algebraic form.
Lesson 3
Working With Expressions
Students translate word problems into algebraic expressions that use multiplication to represent multiplicative relationships (examples: Martin wants three times as many miles -> 3n; Yuji's cookies -> 2n + 10; Damien -> 12 + n; Jade -> n - 8). Students create and evaluate expressions that model rates or per-item amounts (Josie earns $3 per dog -> 3n; Questions to Discuss: Janie earns $10 for every hour -> 10n). Students also practice substituting numeric values and evaluating these expressions (Activity 3 evaluates 5 + 4y when y = 6 and the student activity evaluates 3n when n = 7).
Lesson 6
The Distributive Property
Students translate real-world situations into algebraic expressions that multiply a unit rate by a variable and simplify to a single-term expression in several places. For example, Tessa's purchase is written as 2n + 3n and simplified to 5n and then evaluated for n = 4; another problem has Cesar's fruit baskets represented as 18y. Student activity pages repeatedly ask students to write total amounts as a coefficient times a variable (e.g., 18y, 5n) and to evaluate those expressions.
Unit 5: Algebraic Equations
Lesson 1
Algebraic Equations
Students translate word problems into multiplicative and divisional equations such as 3x = 24 (cost of notebooks), 3x = 30 (cars sold), p/2 = 85 (pills/doses), and 72 = 3x (candies divided into equal groups). Activity pages ask students to write these equations from verbal descriptions and to choose or substitute values that make the equations true. Several problems explicitly connect a per-item number (like $3 per notebook or 3 times as many cars) to a total through an equation.
Lesson 3
Solving One-Step Equations, Part 2
Students write and solve equations in the form px = q and x/p = q throughout the lesson (examples: 5n = 40 for the plant problem; 12y = 96 for Alex washing cars; many activity problems 4x = 24, n/8 = 9, etc.). The activities and answer keys require students to write an equation from a word problem (e.g., "Five times what number is 40?" → 5n = 40) and then solve and check the equation by substitution. The Parent Plan and student pages explicitly list the skill of solving real-world problems by writing and solving equations of the form px = q.
Lesson 4
Solving Two-Step Equations
Students are asked to represent dependent and independent quantities and to "write the equation d = 65t to represent the relationship between distance and time" in the Independent and Dependent Variables section. Activities direct students to identify independent and dependent variables in scenarios, write equations, create tables based on equations, and analyze relationships (e.g., as x increases, y?). The lesson explicitly instructs students to write equations that express one quantity in terms of another and to relate those equations to graphs and tables.
Lesson 7
Independent and Dependent Variables
Students are shown multiple examples of proportional relationships written as equations, e.g., d = 45t with the text noting 45 miles per hour (45:1) and y = 15x for cookies per egg. Word-problem activities require students to translate unit-rate situations into equations (Kevin earning $10/hr → 10x = y; Ron biking 5 mph → 5x = y) and to create input/output tables from those equations. Student pages and practice problems ask learners to write an equation from a described scenario and to use substitution and tables to produce solution pairs for equations of the form y = kx.
Lesson 8
Unit 5 Test
Students write equations that express one quantity in terms of another (e.g., Ms. Crisp bulbs: x − 6 = y; Brandi and Tessa: x − 8 = y) and rewrite linear equations into y = form (y + 5 − 2x = 6 rewritten as y = 2x + 1). Students create tables of values and graph linear equations on coordinate grids and identify points that are or are not solutions. The parent plan and activities ask students to use variables to represent two quantities and to write equations relating dependent and independent variables.
Final Project
All About Me
Students are required to write at least one equation that uses two variables with one independent and one dependent variable and to create a corresponding table and description of the relationship (e.g., "as x increases/decreases, y _______"). The project explicitly gives and uses a proportional example 2x = y ("I practice piano (y) twice as long as I practice soccer (x)") and asks students to solve and interpret that equation. The checklist and steps require students to include multiplication/division equations, fraction/decimal equations, and to include diagrams and tables that accompany two-variable equations.
Unit 6: 2D Geometry
Lesson 3
Triangles
Students study similar triangles and corresponding parts (Activity 4) and compare triangles that have the same angles but different side lengths. Students draw triangles on coordinate grids (Activity 3) and compare side lengths of triangles placed in different quadrants to decide if triangles are exactly alike or only the same shape. The engineer problem (Activity 5) asks students to reason that triangles with identical angles can be different sizes and that adding a specified side length makes triangles unique, which highlights scaling between triangles.
Lesson 4
Area
Students set up and solve algebraic equations in area word problems (for example, the paver problem where they are told to represent the computation with 24n = 4320 and solve for n = 180). In the parallelogram example students write and solve 15 · n = 75 to find the missing height. The lesson also instructs students to "write and solve an algebra equation" to find a missing dimension in triangle and area contexts.
Lesson 5
Circles
Students measure circles and compute the quotient C/d in Activity 1 to discover the constant ratio pi, directly linking circumference and diameter. Activity 2 shows algebraic steps where students (or the text) use inverse operations on π = C/d to produce the equation C = πd and then use d = 2r to rewrite it as C = 2πr. Multiple student activity pages require students to use and apply the formulas C = πd and C = 2πr to compute circumference for given diameters or radii.
Lesson 6
Scale Drawings
Students set up and solve proportional equations using variables in multiple places (e.g., 1 inch/5 feet = 6 inches/n feet to find actual dimensions; 1/3 = x/18 and 1/2 = x/6 to compute scaled lengths). Students are asked to write scale factor ratios, simplify them, convert them to percents, and use those ratios in equations to find unknown measurements (e.g., find dimensions using equivalent ratios and solve for n or x). Several activities require students to express the relationship between drawing and original as a ratio and then use that ratio in an equation to determine corresponding measures.
Lesson 7
Unit 6 Test
Students are asked to find and use scale factors in multiple problems (e.g., Heath/photograph reduction where diameter 12 in maps to photo radius 2 in giving scale 1/3; Jeremy's kite enlargement 6 in to 30 in giving scale 5:1). Problems require computing scaled lengths (use a scale factor of 1/3 to draw a rectangle and determine the new base and height) and computing scale factors for perimeter and area (answer key shows perimeter ratio 42/14 = 3 and area ratio 108/12 = 9). The answer key and problems repeatedly use multiplicative relationships and ratios to find corresponding measures between original and scaled figures.
Unit 7: 3D Geometry
Lesson 2
Surface Area
Students compute total cost by multiplying unit price by quantity in the Basic Skills Review problem (Michael bought 2.5 lb of grapes at $1.58/lb and finds $3.95). Students find a unit price by setting up an equivalent-ratio equation for the orange juice problem (24 oz / $3.12 = 1 oz / n) and solving for n. The worksheet asks students to use ratio reasoning to find unit rates and total costs in word-problem contexts.
Lesson 3
Volume
Students are shown and use formulas written with variables (V = l × w × h, V = B × h, V = s^3) to compute volumes. Students compute total volume by multiplying the number of unit (or fractional) cubes by the volume of a single cube and carry out numerical examples (e.g., 112 × 1/8 = 14 in^3). Students convert mixed numbers to improper fractions and substitute them into the volume equations to verify that counting cubes and using the formulas produce the same result.
Lesson 5
Problem Solving With Solids
Students compute total quantities by multiplying a unit measure times a count (for example, they calculate V = 80 in.^3 × 8 = 640 in.^3 for the eight boxed mugs). Students use division to find counts from totals (for example, 45,000 ÷ 600 = 75 boxes of salt and 528 ÷ 275 = 1.92 rolls of paper, rounded to 2). Students set up and solve algebraic equations that express a product relationship with a variable (for example, 54 = L × 4.5 × 2 leading to L = 6 and 864 = 6s^2 leading to s = 12).
Lesson 6
Unit 7 Test
Students are given and use multiplicative formulas such as V = B × h and V = l × w × h, and they set up and solve equations from those formulas (for example 182 = 6.5 × h to find the flute case height). Several word problems require writing and manipulating equations that express a quantity proportional to another (e.g., 6300 = 42 × h for the soap prism length, and using SA = 6s^2 and dividing 3000 by 150 to find how many boxes a roll will cover). The Kareem paint problem and the paint-can computation also require treating cans as proportional to surface area and performing the corresponding arithmetic.
Unit 8: Statistics
Lesson 8
Making Inferences
Students write part-to-whole ratios and use equivalent ratios to compute percentages in the candy activity (example: a sample with three green candies is recorded as 3:10 and converted to 30%). Students use ratios to find unit rates in the Basic Skills Review (600 miles in 8 hours → 75 miles per hour). In the food-truck activity students scale a mean to predict total revenue (mean $25 per customer → $2,500 for 100 customers), which uses multiplicative scaling.
Unit 9: Skills Review
Lesson 1
Decimals, Factors, and Multiples
Students compute total quantity for multiple identical items in Activity 1 problem 7, where they multiply 2.1 ounces by 72 bottles to get 151.2 ounces; the answer key shows the multiplication 2.1 × 72 = 151.2. Several problems require multiplying a unit amount by a number of items (for example, finding total ounces in a case), so students practice using multiplication to scale a unit quantity.
Lesson 3
Expressions, Equations, and Percentages
Students write and solve equations that relate two quantities using multiplication and division (e.g., 9n = 63; m/4 = 5). In a word problem about toy cars, students set up n/3 = 9 and solve to get n = 27, which represents the total as 3 times the amount per bin. In the expressions activity, students write rate-based expressions such as 9n + 3 for dollars earned given hours worked, showing use of a unit rate in an equation or expression.
Lesson 4
Geometry
Students solve scale-drawing problems that require finding and applying scale factors (e.g., enlarging a triangle by a scale factor of 2/1 and finding new side lengths). Students find scale factors by writing and comparing corresponding parts of rectangles (4/12 = 1/3 and 2/6 = 1/3). In the lighthouse problem students set up and solve a proportion written as an equation (100/800 = 2/n) to find the unknown height.
3: Math
Unit 1: Numbers
Lesson 1
Positive and Negative Rational Numbers
Students set up and solve multiplication equations that express a rate times a quantity (for example, Activity 4 shows -3.5 × 4 = -14 for temperature change and Day 2 shows -0.2 × 6.5 = -1.3). Several activity pages ask students to write an equation to represent a scenario (e.g., write 20 + (-12) for earnings/spending, or write (-)15 + 15 for dive/ascent scenarios). Division problems ask students to compute unit values from totals (for example, -60 ÷ 5 = -12 to find debt per person), which finds the constant per unit in contextual situations.
Lesson 6
Scientific Notation
Students set up and compute products that reflect proportional relationships in multiple places (for example, the grains-of-rice example multiplies 1.13 × 10^8 grains per pound by 50 pounds, and the fire-hose example multiplies liters-per-minute by minutes). Activity problems (Day 3 and the maze) ask students to multiply rates or per-unit quantities by counts or times expressed in scientific notation (e.g., test tubes × cells per tube, flow rate × time, production rate × hours). Several word problems require students to compute totals from a constant rate or per-unit value, reinforcing multiplicative relationships between two quantities.
Lesson 8
Unit 1 Test
Students solve recipe-scaling problems that require multiplying a quantity by a constant (e.g., doubling 0.25 cups to get 0.5 cups and tripling 0.125 cups to get 3/8). Several problems ask students to scale initial amounts (e.g., doubling or tripling ingredients) and compute the resulting totals. These numeric scaling tasks show students applying a constant multiplier to quantities.
Final Project
Mars Station Test Mission
Students multiply hourly energy to get daily and yearly energy (e.g., E_day = 24 × E_hour, E_year = 365 × E_day). They scale device output by a constant (calculate energy for 10 solar panels and 10 wind turbines: E_total = 10 × E_single). Students compute costs and quantities with direct proportional formulas (cost = cost_per_kg × weight; fuel_cells_needed = energy_shortfall ÷ 7×10^2 kWh). The drop-zone and construction cost work uses area and unit-cost equations (A = πr^2 and total_cost = area × cost_per_unit).
Unit 2: Proportions
Lesson 1
Proportional Relationships
Students set up proportions with a variable in multiple activities (e.g., 3 tacos/9 dollars = 9 tacos/x dollars and 3/9 = x/27). The Multiplication/Division notes show students rewriting a proportion as 3/2 = x/12 and transforming it to 3×12 = x×2 to solve for x = 18. The Cross-Multiplication notes have students form and solve equations such as 2n = 3×24 and then divide to find n = 36. Real-world word problems require students to write labeled ratio equations (e.g., 120 miles/3 hours = x miles/5 hours) and solve for the unknown.
Lesson 2
Unit Rates
Students set up and solve proportions using variables (e.g., 2 cups/4 people = n/10 people and the cross-multiplication 2×10 = 4n) and solve for the unknown. The materials present formula-style expressions such as Folding Rate = (Total Sheets Folded) ÷ (Total Time in Seconds) and examples where students compute unit rates (e.g., $ ÷ quantity) and compare them. Several activity pages require students to write and solve equations in context (e.g., 3x = 48 for strawberries, 2×10 = 4n for recipe adjustment).
Lesson 3
Constant Rate
Students repeatedly write proportional relationships in the form y = kx and identify k from tables, graphs, and real-world contexts (Activities 2, 3, 6, and Build Your Own Problems). Activity 5 explicitly has students rearrange equations into y = kx (e.g., 4y = 8x → y = 2x) and find the constant of proportionality. Multiple student pages require setting up equations from word problems (printer, car speed, cost per item) and solving using the equation y = kx.
Lesson 4
Graphing Proportions
Students are asked repeatedly to write equations in the form y = kx: Activity 2 has students derive Jim's equation y = 12x from a table, Day 2 (Activity 3) explicitly teaches y = kx with examples (y = 4x, y = 5x, y = 8x) and has students build tables and graphs from given equations. Multiple student pages prompt learners to "Write the equation" from a scenario (apples y = 2x, ribbon y = 1/2 x, train y = 80x, etc.), and the answer keys give equations in y = kx form for many contexts.
Lesson 5
Proportional Relationship Equations
Students write equations in the form y = kx and t = p × n with explicit examples such as t = 12n and t = 15n. Students complete tables and derive equations from table data (movie tickets, apples, painting, theme-park tickets) and identify the constant of proportionality. Students substitute values and solve proportional equations (e.g., 156 = 10m solved to m = 15.6) and are asked to write equations on quizzes and activity pages.
Lesson 6
Taxes, Tips, and Commissions
Students write and use equations of the form Quantity = Base × Rate (e.g., Sales Tax = Price × Tax Rate; Gratuity = Tip Percentage × Original Bill; Commission = Sales Amount × Commission Rate). Students set up and solve variable equations in context, for example 0.012 × V = $2,400 to find property value, 0.20 × I = $9,000 to find income, and 1.08 × x = $212.00 to find a pre-tax price. Multiple activity pages and answer keys require students to represent proportional relationships with algebraic equations and solve forward and backward problems.
Lesson 7
Markups and Discounts
Students use and write multiplicative equations that express one quantity as a constant times another, for example: "Discount = Original Price × Discount Percentage" and "Markup = Cost Price × Markup Percentage." Students solve backward problems set up as equations such as x × 1.20 = 72, x × (1 - 0.40) = 18, and 50 × x = 20 in the answer key. Practice problems and answer keys require students to form and manipulate these multiplicative equations to find unknown original prices, selling prices, or percentages.
Lesson 8
Simple Interest and Percent Error
Students are given and use the equation I = Prt repeatedly (presented as the simple interest formula) and complete multiple problems that require substituting values into that equation to find interest and balance. Students rearrange the equation to solve for missing quantities (for example solving for r or t in problems where I, P, and one other variable are given). The Parent Plan and activity directions explicitly state that students will "use proportional relationships to solve... percent problems" and will "apply" the formula to real-world scenarios.
Lesson 9
Unit 2 Test
Students are asked to write equations for proportional relationships directly (e.g., tasks requiring y = (3/5)x and write an equation where k = 2/7). Several problems present equations in y = kx form and ask students to identify whether they are proportional (e.g., "Is y = 2.5x a proportional equation?") and to find the constant of proportionality (e.g., "What is the constant of proportionality in the equation y = 7x?"). The assessment items and answer key explicitly use and expect students to produce equations such as y = (3/5)x and y = (2/7)x and to graph and interpret equations like y = 4x.
Final Project
Lemonade Stand
Students are explicitly prompted in Part 2 to "write the equation modeling the relationship in the form y = kx," and the answer key gives y = 2x for lemon juice and y = 1.5x for sugar. In Part 3 students set up and use equations to compute total cost ((Number of lemons × Unit price) + (Amount of sugar × Price per pound) = Total Cost for 1 gallon) and then compute cost per cup by dividing total cost by 16. Part 4 gives formulaic representations for markup (Selling price = unit price × 2, ×2.5, ×3) so students practice writing and using multiplicative equations tied to real quantities.
Unit 3: Expressions
Lesson 2
Rewriting Expressions
Students write and use one-step multiplicative equations such as Total Price = Original Price × (1 + Sales Tax Rate) and apply them to specific numbers (e.g., Total Price = 40(1.05) ⇒ 42). Students represent discounted prices by equations Discounted Price = Original Price × (1 − Discount Rate) and use examples like 80 × 0.75 = 60. Students also express selling price with markup as Selling Price = Wholesale Price × (1 + Markup Rate) and set up and solve equations for unknowns (e.g., 31.50 = P(1 + 0.05) to find P).
Lesson 3
Algebraic Expressions
Students set up and solve equations that express one quantity in terms of another, for example writing t = cp + s for the markers problem and solving 15 = 4p + 3. They also write and use s = r(m + t) to represent total ride cost as a unit-rate times total rides and solve 36 = 2(6 + t) for t. The perimeter formula P = 2(l + w) is used repeatedly, with students substituting known values and solving P = 2(l + w) for a missing side.
Lesson 4
Graphing Proportions
Students are introduced to the algebraic form y = kx in the "Things to Know" section, including that k is found by dividing y by x and that the unit rate equals the constant of proportionality. In Activity 4 students pick a point (not the origin), compute k = y/x, and write the equation y = kx for multiple graphs with worked answers (e.g., y = 3x, y = 1/2 x, y = 2.5x). In Activity 3 students create tables from equations such as y = 3x and y = 2x, plot the points, and confirm proportionality, reinforcing translation between equations, tables, and graphs.
Lesson 5
More Graphing Proportions
The lesson repeatedly presents and uses the equation form y = mx and explicitly states that m is the unit rate and slope (Things to Know). Multiple activities require students to write equations from contexts (e.g., y = 4x, y = 3x, y = 60x) and to translate tables and graphs into equations (Activities 1–6 and several student pages). Several activity prompts explicitly instruct students to "write the equation representing the proportional relationship" and to graph those equations, showing representation across contexts.
Lesson 6
Intercepts
The Parent Plan Skills section explicitly states students will "Derive the equation y = mx for a line through the origin" and interpret y = mx + b as a linear function. Activity instructions teach students to find intercepts algebraically by setting y = 0 or x = 0 and include many practice problems with linear equations (e.g., y = 3x - 9, x + 5y = 10). Graphing tasks include a line with intercepts at (0,0), and students plot and connect intercept points to form lines.
Lesson 7
Rise Over Run
Students draw right triangles on coordinate grids and count rise and run to compute slope as a ratio (rise/run). Students set up and compare proportions for different triangles (for example, checking 2/2 = 4/4) to show triangles are similar and that the rise-to-run ratio is constant along a line. Activity pages prompt students to fill RISE and RUN boxes, simplify slope fractions (including negative slopes), and decide whether triangle ratios match.
Lesson 8
y = mx + b
The lesson begins by recalling y = kx and explicitly notes that the special case y = mx (b = 0) is a proportional relationship that passes through the origin. The Parent Plan states students will derive y = mx for a line through the origin and interpret y = mx + b as a linear function. Students practice writing equations from tables and scenarios (e.g., a student activity with x:1,2,3,4 and y:2,4,6,8 and a real-world "Selling Bracelets" scenario with no base charge) and graph lines that start at the origin (y = mx).
Lesson 9
Unit 3 Test
Students are given multiple tables of quantities (e.g., Babysitting Earnings: hours vs. earnings and Car Rental Charges: days vs. cost) and are asked to determine if the relationships are proportional and to "Write the equation." The Unit 3 Test problems (dog walker and lawn care) likewise ask students to identify proportionality, slope, y-intercept, write the equation, and graph the data. The answer keys explicitly provide proportional equations in the form y = kx (for example, y = 10x, y = 50x, y = 12x, y = 60x) and the Parent Plan lists the skill "Graph proportional relationships, interpreting the unit rate as the slope of the graph."
Final Project
Planes, Trains, and Automobiles
Students are asked to write equations for distance vs. time using the formula y = mx, with y = distance, x = time, and m equal to each mode's speed (e.g., y = 60x, y = 80x, y = 400x). Students fill tables of distance at various times, plot those lines on a graph, and answer questions that identify that each line passes through the origin and that the unit rate is the slope. The cost activity also has students write linear equations and compare slopes, reinforcing the representation of relationships with equations.
Unit 4: Probability
Lesson 2
Observing Probability
Students calculate experimental probability using the formula Experimental Probability = (Number of times it happened) / (Total number of spins). Students then use the equation Prediction = Experimental Probability × 600 to compute predicted counts for 600 spins (e.g., 0.59 × 600 = 354). The activity guides students to convert frequencies to a multiplier (the experimental probability) and apply it to a larger quantity, directly using a multiplicative equation that relates two quantities.
Lesson 3
Probability Models
Students repeatedly set up and compute expected counts by multiplying a probability by the number of trials (for example, 1/2 × 500 = 250 for coin flips and 1/6 × 120 = 20 for die rolls). Multiple activity pages show equations like P(2) = 1/6 × 120 = 20 and P(6) = 1/8 × 64 = 8 to predict outcomes. The lesson also has examples where students compute expected counts from non‑uniform models (e.g., 7/12 × 50 for the spinner) and use probability = group/total before multiplying by the number of trials.
Lesson 4
Compound Events
Students compute proportions and scale them to populations or repeated trials (for example, 14/60 = 0.233 then 0.233 × 1143 ≈ 266). The lesson shows students multiplying a probability or percent by a number of trials to predict expected counts (e.g., 16.7% × 120 = 20). Student activity pages repeatedly ask learners to build a probability model and then calculate how many times an outcome would occur if the situation were repeated, requiring proportional multiplication.
Final Project
Happy Tails Dog Shelter
Students use the explicit formula "Percentage = (Number of dogs in the group / 60) × 100" to compute probabilities for each size/color combination. They write all 15 size+color combinations in the Probability Model and calculate each percentage (examples shown: 3/60×100=5%, 9/60×100=15%). Students also compute aggregated probabilities (e.g., total medium dogs = 30/60×100=50%), demonstrating conversion of counts to probabilities by a constant scaling factor.
Unit 5: Functions
Lesson 1
What Is a Function?
Students practice writing equations from verbal rules and representing those rules with algebraic expressions (e.g., y = 2x + 6, y = (x + 3)/2, y = 3x + 4). Students use those equations to compute outputs for given inputs, complete input/output tables, and plot the resulting (x,y) pairs on coordinate grids (Activities 2, 3, and 4). Students match tables to graphs and identify linear versus non-linear equations by observing the shape produced when points are plotted.
Lesson 2
Linear and Nonlinear
Students work with and interpret linear equations in the form y = mx + b (Skills section) and complete tables and graphs from given equations such as y = 2x + 4, y = x + 1, and y = 2x. Students plug in x-values, compute y, fill tables, calculate rate of change, and plot points to decide whether a relationship is linear or nonlinear.
Lesson 6
Slope-Intercept Form
Students repeatedly find a slope (m) from two points or from a table and solve for the y-intercept (b), then substitute into y = mx + b to write an equation (e.g., Activity 1: points (0,2) and (3,8) lead to y = 2x + 2). Several tasks ask students to find the y-value when x = 0 in a table and then write the equation (Activity 4 examples explicitly show y-intercept = 0 and write equations like y = 5x). Students also compute unit rates from tables (the example converts a slope to miles per minute and then to miles per hour) and then express the relationship as an equation in slope-intercept form.
Lesson 7
Creating Functions
Students write equations in the form output = slope × input + starting value (y = mx + b) across multiple activities and rename variables to match contexts (e.g., A = 6c + 12). Several tasks produce functions with a zero starting value and are simplified to proportional forms (e.g., P = 15h → P = 15h, Graph 2 answer y = 0.25x, and the cooking activity shows F = 0.5s and notes the line goes through the origin). The cooking extension explicitly connects the recipe scaling to a direct proportional relationship and states that the graph is a straight line through the origin.
Lesson 8
Comparing Functions
Students calculate rates from verbal descriptions (e.g., Alex: 1.5 miles in 30 minutes → 0.05 miles/min) and compute slope from graphs using two points (Bella: slope from (0,0) and (30,2) → 0.067). Students read and interpret linear equations that include slope and y-intercept (e.g., y = -3x + 100, H = 4x + 10, d = 3t + 2) and extract rate of change and starting value. Multiple activities present situations as tables, graphs, equations, and descriptions for students to compare rates and starting points across representations.
Lesson 9
Unit 5 Test
Students are asked to write functions for situations that map a constant rate to total amount, for example Problem 13 and answer key show E = 12h for $12 per hour and other items show E = 15h for $15 per hour. Several tasks require writing equations of the form y = mx or y = mx + b (e.g., Liam: y = 2x + 17 and gym example y = 15x + 25), and many items ask students to determine slope (rate) from tables and graphs. Students also compare rates of change between functions, which reinforces forming equations that express a constant multiplier between quantities.
Lesson 10
Final Project
Students are asked to derive the equation y = mx + b for a line through the origin, which addresses writing equations for linear relationships that pass through (0,0). Yellow card examples explicitly ask students to write an equation from a real-world description (e.g., "Emma earns $10 for each hour she babysits"), requiring an equation of proportional form. Green and blue card tasks require students to "Write an equation for the table" and "Write the equation from the story" and to match equations to graphs or stories, giving repeated practice representing relationships with equations.
Unit 6: Geometry
Lesson 1
Congruence and Similarity
Students compute and use scale factors (for example, scale factor = 6 ÷ 3 = 2) and set up multiplicative relationships to find missing side lengths (for example, 5 cm × scale factor = 10 cm and x = 10 ÷ 2 = 5). The lesson gives numeric examples of proportional corresponding sides (e.g., AB = 4, DE = 8; BC = 5, EF = 10) and asks students to find x in many practice problems by applying multiplication or division. Activities repeatedly prompt students to identify scale factors and write equations or calculations that relate one side to another using that factor.
Lesson 6
Dilations
The lesson repeatedly gives and has students use the equation new length = original length × scale factor (Day 2 Activity 3 and Activity 4) and models solving for unknowns with equations such as x = 8 × 2.5. Activity 4 directs students to compute the scale factor via scale factor = new ÷ original and then apply it in equations to find missing side lengths. Student activity pages require students to set up and solve these equations for multiple problems (e.g., finding new lengths and solving for scale factor).
Lesson 7
Sequences of Transformations
Students perform dilations by applying a scale factor to coordinates (e.g., Example 1 and Example 2 show A' = (1×2, 2×2) and similar computations). The student activity pages require dilations with various scale factors (2, 1.5, 0.25, 2.5, 0.5) and the answer key lists the resulting coordinates after multiplying x and y by the scale factor. Students are asked to dilate around the origin and then apply a second transformation, so they repeatedly compute products to find image coordinates.
Lesson 10
Volume
Students write and use equations such as V = πr^2h, V = (1/3)πr^2h, and V = (4/3)πr^3 to represent how volume depends on radius and height. Students plug numeric values into these formulas, manipulate the equations algebraically, and solve for missing quantities (for example, solving V = πr^2h for h when V and r are given). The lesson describes a cylinder as a "stack of circles," linking the area-of-a-circle factor (πr^2) to a multiplicative relationship with height.
Lesson 11
Unit 6 Test
Students solve multiple dilation and scale-factor problems (e.g., square side scaled by factor 3 giving x = 15; triangle sides dilated by factor 2 producing A'B' = 10; dilations by 0.5 and 2 appear in several problems). Students identify scale factors, determine whether a transformation is a dilation, and compute new side lengths after dilation (several answer keys show multiplied or divided side lengths).
Final Project
Abstract Art Gallery
Students create shapes on a coordinate grid and are instructed to dilate a figure about the origin by a factor of 3, and to "calculate and plot the enlarged version." The student activity asks learners to identify two shapes that are similar and to create similar and congruent shapes for the painting. The parent notes and descriptions explicitly describe exploring "proportional growth and scale" and using scale factor 3 for dilation.
Unit 7: Linear Equations
Lesson 1
Linear Equations With One Variable
Students solve many one-step and two-step equations that have the multiplicative form kx = b or x/k = b (e.g., 7x = 42, 4x = 32, x/5 = 7, 2x = 6) across Activities 1 and 2. Students also solve equations involving fractions and decimals by multiplying by reciprocals or dividing (e.g., (3/4)x + 3 = 7, (2/5)x = 6, 0.4x + 2 = 6.8) and check their solutions by substitution. The tasks require students to isolate the variable using multiplication or division, reinforcing manipulation of equations that can represent proportional relationships in algebraic form.
Lesson 2
Multi-Step Equations
Students write and solve equations that include a variable multiplied by a constant in several real-world items (for example, the running track problem: 1.5d = 36). Students set up equations with per-unit rates in context (Mr. Patel: 12 + 15 + 3c = 45; raffle/ticket and contractor problems use expressions like 3t or 85h). Activity prompts ask students to define a variable and translate phrases like "per" or "each" into multiplication by a constant when writing equations from word problems.
Lesson 4
Multi-Step Word Problems
Students set up and solve equations that are purely multiplicative in several places: the cupcake problem leads to 6x = 90 (x = 15), the Review Quiz includes 4x = 28, and the comic example reduces 10x + 40 = 120 to 10x = 80. Students also model situations with rate times quantity (e.g., monthly cost 18x, per-box cupcake counts 6x) and solve for the unknown. Multiple problems require writing an equation from a real-world context and isolating a single-term product to find x.
Lesson 7
The Point of It All
Students write equations of lines in slope-intercept form (y = mx + b) from two given points in multiple activities (e.g., tasks where Line A: (0,0),(2,4) yields y = 2x). The review and practice pages include explicit examples y = 2x and y = 3x (graphing and solving those systems) and ask students to graph and solve systems that feature lines through the origin. Several problems require students to find equations from points and represent those linear relationships algebraically before solving the system.
Lesson 8
Linear Algebra In the Wild
Students write equations of the form y = rate × quantity in multiple places (e.g., Shiny Solutions: y = 30x; StreamMore: y = 5x). The lesson explicitly gives formulas like Distance = Rate × Time and Total Cost = Rate × Quantity and models problems with equations such as y = 1.50x and y = 6x on activity pages. Student tasks ask them to define variables and write these single-term linear equations before solving systems or doing break-even analysis.
Lesson 9
Unit 7 Test
Students set up and solve word problems that use price times quantity (e.g., the lemonade problem: $2 per cup and $38 total) requiring an equation that relates total revenue to number of items. Students are asked to find equations of lines from two points and to write linear equations (several answer keys show lines written as y = x or y = 2x + 1). Graphing tasks and examples include lines that pass through the origin (for example, a solution showing Line A: (0,0),(4,4) → y = x), which corresponds to proportional relationships of the form y = kx.
Final Project
Getting Ready for College
Students write and graph equations that have the form C = k·n in several activities (e.g., Dormitory: C = 1200m; Rideshare: C = 1.25x; Meal Plan: C = 80w). Students plot these lines on coordinate grids, label axes, and use the equations to compute and compare costs (including finding intersections). Students also interpret slope as a unit rate (cost per month/week/mile/hour) and read y-intercepts to understand initial costs.
Unit 8: Data
Lesson 5
Categorical Data
Students compute and use relative frequencies using the formula Relative Frequency = (frequency of an event) / (total number of events) and convert frequency tables into relative frequency tables (e.g., 18/90 = 0.20). The Parent Plan and activities explicitly tell students to "use proportional reasoning to describe patterns and possible associations" and have multiple tasks where students calculate proportions for rows or columns and compare percentages. Several activities require students to express cell values as decimals or percents and to compare those proportions to draw conclusions about groups.
Lesson 6
Unit 8 Test
Students are asked to choose which linear equation fits a scatterplot (e.g., deciding between y = 2x + 4 and y = 4x - 0) and to write an equation for a scatterplot in Questions 13–14. Several answer keys show students producing equations in slope-intercept form, including y = 4x + 0 and y = 4x + 2. Multiple tasks require students to interpret linear relationships and produce an algebraic equation that models the data.
Final Project
Collecting and Organizing Data
Students plot paired numerical data (e.g., jumping jacks vs. heart rate), label axes, choose scales, and create scatterplots. Students are prompted to informally fit a straight line or draw a line of best fit and to use the equation of a linear model to solve problems and interpret slope and intercept in context. Students write numerical summary statements that describe how one quantity increases with another, connecting the plotted line to a verbal relationship.
Unit 9: Semester Exams
Lesson 2
Proportions Review
The Parent Plan explicitly states "Represent proportional relationships by equations" and gives the example t = pn. Activity 2 asks students to "Write the equation of proportionality," matches tables to equations (e.g., y = 6x, y = 4x, y = 10x), and includes problems where students determine k and write y = kx. Activity 3 requires students to write the equation for a table (x:1,2,3; y:5,10,15 -> y = 5x), graph equations such as y = 6x, and create their own proportional equation, table, and graph.
Lesson 3
Expressions Review
Students are asked to determine if relationships are proportional from tables and then write equations (Activity 2: Babysitting Pay → y = 14x; Car Rental → y = 75x). Multiple prompts require writing equations from contexts (e.g., "A tutoring service charges $40 per hour. a) Write an equation. → y = 40x") and from graphs and tables (Activities 2 and 3). Activities also ask students to interpret the slope as the unit rate and to use whether a line passes through the origin to decide proportionality, connecting the equation form y = kx to real situations.
Lesson 4
Probability Review
The Game Probability item and its answer explicitly compute an expected value using multiplication: the answer key shows "expected number of wins is 20 x 1/4 = 5," linking number of trials to wins by a multiplicative relationship. The Probability Models section gives a formula as an equation: "Probability = number of favorable outcomes ÷ total number of outcomes," which represents a ratio relationship in equation form.
Lesson 5
Semester Exam
Students are asked to find the constant of proportionality and write an equation from a table in problem 17 (table gives x,y pairs and answer key: y = 5x). Problem 18 explicitly asks students to "Write an equation for a proportional relationship where k = 3/2." Problems 19, 20, 34, and 37 require students to find unit rates from points or graphs and write equations in the form y = kx (answers show unit rate 5, y = 4x, y = -2x, etc.).
Lesson 6
Functions Review
Students write equations that model real-world linear relationships (for example, a delivery driver earning $18 per hour is written as y = 18h) and they write total-cost equations for situations with and without a fixed fee (bike rental y = 12x + 10). Students practice constructing functions from contexts, identifying the slope (rate per unit) and the y-intercept (starting value) in multiple activities. Activities have tasks that ask students to write functions from situations, compare rates of change, and interpret the meaning of the slope in context.
Lesson 7
Geometry Review
Students solve triangle similarity and dilation problems that require finding scale factors from corresponding side lengths and using those factors to compute missing side lengths (for example, given sides 5, 7, 9 and a corresponding side 10, students find scale factor = 2 and compute 14 and 18). Students perform numerical proportional calculations in dilation tasks (for example, computing scale factor 15 ÷ 6 = 2.5 and then finding the other sides as 22.5 and 30). The materials repeatedly prompt students to "use ratios to reason" and ask them to identify scale factors and decide enlargement vs. reduction, reinforcing multiplicative relationships between quantities.
Lesson 9
Data Review
Activity 3 asks students to analyze scatterplots, interpret trends, and consider what lines of best fit show about data. The lesson includes a web link titled "Write an Equation for a Line of Best Fit" and the Parent Plan skills list states students will use the equation of a linear model and interpret slope and intercept. These items show students engage with linear models and equations for lines in the context of bivariate data.
Lesson 10
Semester Exam
Students are asked in problem 8 to write a function for "You earn $10 per hour," and the answer key explicitly gives y = 10x, which directly represents a proportional relationship. Several other problems ask students to find slopes, intercepts, and to graph linear equations (e.g., find slope through two points, graph y = 2x − 4), giving practice with linear equations and equation-writing. Problem 35 requires students to set up and solve an equation from a real-world cost situation (18h + 30 = 138), showing students translate contexts into equations.