Students perform division to find unit amounts and per-item values in multiple word problems (e.g., $77.40 ÷ 12 = $6.45, $82 ÷ 8 = $10.25, 630 ÷ 45 = 14 cups). Students practice long division with decimals and learn to convert decimal divisors to whole numbers (e.g., 1.75 ÷ 0.25 → 175 ÷ 25) so they can compute exact quotients. Students solve contextual division problems that determine numbers of groups/items (e.g., 3,750 ÷ 25, 54 ÷ 4.5), which practices finding unit rates and equal-share calculations.

Students multiply the number of items per goody bag by 12 to find total quantities and then determine how many packages to buy given package sizes (e.g., 5 bars × 12 = 60, then 22-bar packages → buy 3 packages = 66 bars). Students multiply the number of packages by the price per package and add item costs to find a grand total, then divide the grand total by 12 to find cost per goody bag. Students convert a given sales tax percentage to decimal form and multiply the grand total by that decimal to find the tax amount and then add tax to determine whether they are under or over the $50 budget.

Students practice representing ratios in three forms (words, with a colon, and as a fraction) and create visual models of given ratios. Students generate equivalent ratios by multiplying or dividing both quantities and apply scaling to solve problems (e.g., doubling a recipe from 3:1 to 6:2, computing flour and milk for 9 loaves). Students solve proportional word problems that require finding and applying a multiplier (e.g., the cleaning-solution mixture and the piano-to-soccer hours problem).

Students draw pictures and write ratios in words and numbers to represent part-to-part, part-to-whole, and whole-to-part relationships. Students compute and use equivalent ratios and scaling (for example, using a 2:5 ratio to find 15 girls when there are 6 boys). Students simplify and rewrite ratios (for example, reducing 12:21 to 4:7) and practice translating situations into ratio form on multiple activity pages.

Students use tape diagrams, double number lines, and tables/graphs to represent and solve proportional relationships; they set up and solve word problems that scale ratios (e.g., Jason and Zeke 5:7 with total 144, hamburgers vs. drinks, cost of sodas, lemonade pitchers and lemons). Students create tables of equivalent ratios, find missing values, plot ordered pairs on coordinate planes, and use multiplicative scaling to compute unknown quantities (e.g., using a unit or multiplier to go from given ratio parts to totals).

Students learn to find unit rates and unit prices by writing ratios and making equivalent ratios (e.g., divide 216 miles by 4 hours to get 54 miles per hour; $5.25/3 = $1.75 per quart). Students practice multiple solution methods (division, equivalent-ratio diagrams, double number line, tape diagrams) and then use the unit rate to solve follow-up problems by scaling (e.g., find unit price then multiply to get cost for 10 pounds). The quiz and practice problems require students to use tape diagrams and double number lines to solve proportional problems and to scale unit rates to new quantities.

The lesson directly has students convert among percents, fractions, and decimals through guided examples and practice exercises (Activities 1–3 and corresponding student pages). Students solve single-step percent tasks such as writing 3/10 as 30%, converting $0.83 to 83% of a dollar, and using a 10% coupon to determine the fraction paid (9/10). Students also practice converting percentages to decimals and decimals to percentages using both the standard fraction method and the decimal-point shortcut in multiple exercises.

Students practice representing percentages as part-to-whole ratios and creating double number line diagrams to make equivalent ratios (Activity 1). Students translate word sentences into number sentences and solve problems using the formula part = percent × whole and fraction/decimal methods (Activity 2 and answer key). Students solve contextual percent word problems such as discounts (15% coupon), sales tax (6% on a $150 purchase), and percent-of scenarios (Tim reading 25% of 600 pages, fans wearing team colors).

Students set up and solve unit conversions by forming equivalent ratios (e.g., 1 quart = 2 pints used with a double number line to find 3 quarts = 6 pints). The lesson has students use double number lines and equivalent-ratio setups for metric and customary conversions (e.g., 19 cm → 190 mm, 5 oz → 141.75 g) and includes an activity sheet with multiple conversion problems to solve. The answer key and activities show students multiplying both parts of a ratio by a scale factor to produce equivalent ratios and compute converted values.

Students convert between percentages, fractions, and decimals and solve percent-of problems (e.g., "What number is twenty-eight percent of two hundred?", "Seventeen is eighty-five percent of what number?"). Students compute unit rates and unit prices and then use them to find totals (e.g., find unit price per rose then cost for 5 roses; find pages per minute then pages in 15 minutes). Students set up and solve proportional situations with tape diagrams, double number lines, tables, and graphs (e.g., Marco and Stanley model-car ratio, Farmer Ned cows/chickens, apples-to-pies table/graph).

Students set up ratio problems to find unit prices by dividing price by quantity (e.g., $3.38 ÷ 13 oz = $0.26/oz) and convert units using chains of equivalent ratios (e.g., 1 gal → 128 fl oz via gallon→quart→pint→cup→fl oz). Students use equivalent-ratio reasoning to scale recipes (double/triple ingredient amounts) and to convert distances and times (1 mile/6 minutes → 6 miles/36 minutes). Students solve percent problems such as computing a 10% coupon savings on weekly grocery cost and use ratios for currency conversions (e.g., $1 = 0.85 euros) to find equivalent prices in other currencies.

Students write and evaluate expressions that represent proportional situations such as Josie earning $3 per dog (3n) and then evaluate it for a given number of dogs. The Basic Skills Review includes a ratio scaling problem (Max runs 4 miles for every 3 miles Gina runs; find Gina's miles when Max runs 24) and a percent computation (What is 45% of 80?), so students practice single-step proportional and percent calculations. Several word problems require translating multiplicative and divisive relationships into expressions (e.g., 3n, 24/n, unit conversions gallons → quarts).

The lesson explicitly has students translate unit-rate proportional situations into two-variable equations (e.g., d = 45t, y = 15x, 10x = y, 5x = y) and use those equations to compute missing values by substitution. Students create input/output tables from equations (e.g., 2x - 4 = y, 2x + 4 = y) and generate coordinate pairs to graph proportional relationships. Word-problem activities ask students to identify independent/dependent variables and solve for one variable given the other (hours ↔ earnings, miles ↔ time). The student worksheets include practice computing simple percentages once (e.g., "What is 70% of 40?") and various ratio/unit-rate problems (cookies per egg, miles per hour).

Students are required to write at least one two-variable equation relating an independent and dependent variable (e.g., 2x = y) and to create a corresponding table and description of the relationship. Students must create 5–7 equations (including one two-variable equation), 4–6 inequalities, and at least two problems that require two steps to solve (involving addition/subtraction and multiplication/division). Students are also directed to use tape/hanger diagrams and number-line representations for at least one equation and one inequality.

Students set up and use ratios as scale factors and convert those ratios to percentages (examples: trapezoid 15/5 = 3 → 300%; parallelogram 4/16 = 1/4 → 25%). Students use proportional relationships to find actual measurements from scale drawings (example: 1 in/5 ft → 6 in equals 30 ft for the fort; architect 1 cm/4 ft → 120 ft equals 30 cm). Students carry out multistep scaling and percent calculations (enlarge then reduce the butterfly for the stamp; photograph scaled 1/30 then reduced by 1/2 to 1/60) and apply percent scale factors to compute new perimeters and areas (Mrs. Yee: compute 200% of dimensions, compare perimeter and area and note area scales by square of factor).

Students calculate and apply scale factors in multiple problems (e.g., Heath's photograph: find the reduction scale factor 1/3 or 33 1/3%; Jeremy's kite: find the enlargement scale factor 5:1 or 500%). Students use a given scale factor to draw scaled figures and compute new side lengths (Question 16: use 1/3 to draw a scaled rectangle and find scaled base/height). Students compute and compare perimeter and area scale factors (questions that ask for the scale factor of the perimeter and the scale factor of the area, and answer key shows area factor = square of linear factor).

The materials include a "Create a Scale Drawing" activity that asks visitors to make a scale drawing based on a given scale factor and to sketch on a laminated grid, which requires converting between actual and scaled measurements. The Parent Plan explicitly lists "Reproduce a scale drawing at a different scale," indicating students will work with scale factors and proportional resizing. The Stations Planning pages and planning prompts ask students to specify scale factors and details for problem cards, supporting practice with proportional conversion for drawings.

Students solve ratio and unit-rate problems in the Basic Skills Review: Michael's grapes cost (2.5 lb at $1.58/lb) requires multiplying a unit price by a quantity, and Evie's 24-ounce bottle for $3.12 asks students to find the unit price per ounce. The Carlotta cube/tape question asks students to compute a surface area and compare it to an available amount of tape, which uses proportional area calculation and a quantitative comparison. The lesson also includes unit conversion (meters to centimeters) and other single-step multiplicative reasoning tasks.

Students use scale factors and proportional reasoning in Activity 3 to find cross-section dimensions and areas (for example, using a 1/4 scale factor and noting that area scales by (1/4)^2 = 1/16). Students solve a percent-style problem in the Basic Skills Review ("Eighteen is what percent of 50?") and carry out division and rounding to determine number of mailing-paper packages (528 ÷ 275 = 1.92 → 2). Several tasks require setting up and solving proportions or algebraic equations that rely on ratio reasoning (e.g., using V = B × h to find missing dimensions).

Students compute surface area and then use a unit rate to find quantity needed (Kareem's paint problem: find SA, then divide by 10 sq ft per can to get 4 cans). Students compute surface area of a cube and divide total coverage to find how many boxes a roll of decorative paper will cover (150 in2 per box, 3000 ÷ 150 = 20). Students solve for a missing linear dimension using volume and base area (Bella's flute case and the soap factory problems: 182 ÷ 6.5 = 28 and 6300 ÷ 42 = 150).

Students answer questions that ask for percentages of a data set, for example: Box Plot #1 asks "What percent of the total data is represented by the whisker from 50 to 200?" Box Plot #2 asks "What percent of data values are shown between 30 and 40?" The Parent Plan questions explicitly state that 50% of the data lie in the interquartile range and prompt students to reason about these percentages.

Students write part-to-whole ratios for candy colors and use equivalent ratios to calculate percentages in Activity 1 (example: 3:10 → 30%). Step Six and the optional challenge ask students to compute percentages for the whole population (e.g., 9/60 = 15%). The Basic Skills Review includes a percent conversion problem ("64 is what percent of 200?") that requires ratio-to-percent work.

Students represent ratios in three forms and identify ratio types (part-to-part, part-to-whole) on the "Working with Ratios" page. Students solve proportional word problems such as finding counts from a given ratio and total (dog park), scaling production using ratios (necklaces: 3 in 5 minutes to 15 in 25 minutes), computing unit rates (train speed: 310 miles in 5 hours), and finding unit price from a total cost (apples: $6.12 for 3 lb). The Parent Plan explicitly lists "Use ratio and rate reasoning to solve real-world and mathematical problems," which aligns with students practicing proportional reasoning in variety of contexts.

The Parent Plan explicitly lists percent skills: finding a percent of a quantity as a rate per 100, solving real-world and mathematical problems involving percents, and finding the whole given a part and the percent. The Wrapping Up section directs students to practice solving problems with percentages and unit conversions via an online quiz (link provided).

Students compute and apply scale factors (Activity: enlarge a triangle by a 2/1 scale factor) and find scale factors from corresponding parts of rectangles (4/12 = 1/3). Students convert a scale factor to a percentage (answer key: 1/3 = 33 1/3%) and solve a percent enlargement problem (make a 2 in. lighthouse 800% larger to get 16 in.).

Students set up and compute products that use rates (e.g., temperature drop −3.5 × 4 = −14; submarine descent −4/5 × 3 = −12/5) and solve contextual multiplication problems (Jamie owes $1.25 per day × 8 days = −$10). Students also compute unit rates and quotients to find per‑unit values (e.g., splitting debt −60 ÷ 5 = −12; average temperature change −48 ÷ 6 = −8). Several activity pages require students to write equations, draw number‑line representations, and interpret signed quotients in real contexts like debt, temperature, and distance.

Students multiply and scale quantities using proportional reasoning (for example, multiplying a rate by time: 1.2 × 10^3 L/min × 60 min to get liters per hour). Students multiply counts and factors to find totals (for example, grains per pound × number of pounds, and test tubes × cells per tube). Students compare magnitudes and determine how many times larger one quantity is than another (for example, comparing populations written as 3 × 10^8 and 7 × 10^9).

Students compute and compare measurements by writing cell lengths in scientific notation and determining which organism has larger cells and how many times larger the cod's cell is compared to the sponge's, which requires forming and evaluating a ratio. Students also compare cell densities given in scientific notation (1.2 x 10^8 vs 9.5 x 10^7 cells per ounce), asking them to compare counts in equal tissue samples, which involves dividing or forming a ratio of quantities. These tasks explicitly ask students to use division to compare sizes and counts of quantities.

Students solve multiplicative scaling problems such as doubling sugar from 0.25 to 0.5 cups and tripling 0.125 cups to 3/8, which require using proportional relationships. Students solve an exponential growth problem where a bacteria culture doubles each hour (population modeled as 2^(4+3)), demonstrating multiplicative reasoning over repeated steps. Several problems require converting between fractions and decimals and scaling quantities, supporting students' practice with ratio-style computations.

Students compute energy needs and scale those needs from per hour to per day and per year (Task 1), and they calculate energy production for multiple solar panels and wind turbines (Task 2), which requires multiplying by unit rates and scaling quantities. Students calculate supply costs by multiplying monthly supply weight by cost per kg and compute delivery fees based on distance, then add delivery, supply, and construction costs to find total supply cost (Part 2). Students convert scientific notation to a whole number to find how many 700 kWh fuel cells are needed and then compute space required by multiplying cell area by number of cells (Task 3).

Students set up and solve proportions using labeled fractions and methods (Eyeball Method, Multiplication/Division, and Cross‑Multiplication) on multiple activity pages. They practice finding unknowns in real‑world one‑step problems (recipes, travel distances, shopping costs, paint coverage, bead counts) and record steps in an Interactive Notebook. Students also compare ratios as fractions to decide whether relationships are proportional in dedicated 'Is it Proportional?' activities.

Students practice finding and using unit rates across many contexts (price per item, speed, hourly wage) in Activities 1–2 and Activity 4. They solve multi-step ratio problems using both the unit-rate method and proportions (Activity 6 shows recipe scaling and travel problems solved by unit rate and by setting up proportions). Students work with complex fractions and division of fractional quantities (Activity 3) and apply Keep-Change-Flip to find rates when quantities are fractional.

Students repeatedly identify the constant of proportionality k (k = y/x) in tables, graphs, and equations (Activities 2, 3, 5). Students set up and solve real-world word problems by identifying variables, computing k, writing y = kx, and solving for unknowns (Activity 6: printer, car, faucet, machine, recipes). Students perform unit-conversion style multi-step reasoning in at least one problem (runner: 5 miles/hour → k = 5/60 miles per minute → y = (1/12)x).

Students plot points from tables and equations, draw straight lines through the origin, and write equations in the form y = kx to represent proportional relationships. They identify independent and dependent variables, compute unit rates (k) from tables or the point (1,k), and compare proportional relationships by comparing unit rates and slopes. Multiple activities require students to convert between tables, graphs, and equations and to test tables for equivalent ratios to decide proportionality.

Students write and use equations of the form t = p × n and y = kx to model proportional situations (movie tickets, apples, pizza, theme park tickets) and complete tables by multiplying by the unit rate. Students compute unit rates and solve for unknowns in proportional equations (e.g., c = 8m, a = 15h, 156 = 10m → m = 15.6). Students work with percent-style discounts in problems and answer keys (a 25% discount applied as Final = 0.75 × Original, group-ticket pricing using t = 50n and t = 50n × 0.9 for a 10% discount). Students identify non-proportional situations caused by buy-one-get-one/free deals, tiered pricing, or changing rates and justify why the ratio is not constant.

Students convert percent rates to decimals and compute percent amounts (e.g., Sales Tax = Price × Tax Rate; Tip = Bill × Tip Percentage; Commission = Sales × Commission Rate) in multiple examples. Students solve forward problems (finding tax/tip/commission and adding to totals) and backward problems (dividing by 1 + rate or dividing tax/commission by rate to find original price, value, or income) in Activity 2, Activity 3, and Activity 4. Students work on multi-step scenarios that combine steps (e.g., calculate tip after tax, find total earnings = base salary + commission) and complete challenge problems that require chaining proportional calculations.

Students calculate discounts and final prices using Discount = Original Price × Discount Percentage and apply multiple discounts sequentially (stacked discounts) in worked examples and practice problems. Students compute markups with Markup = Cost Price × Markup Percentage, add markups to find selling price, and solve multi-step scenarios that combine markups, markdowns, and additional fees. Students use percent change formula and solve percent increase and decrease problems, and they work backward by dividing by multiplicative factors (e.g., x·1.20 = 72 → x = 72 ÷ 1.20) to find original prices. The review and activity pages require students to solve multi-step, real-world percent problems and show their work across a range of contexts.

Students use the simple interest formula I = Prt in multiple activity pages to compute interest earned, calculate total balance (principal + interest), and solve for missing values such as rate and time. Student activity pages include worked examples and problems that require converting percent to a decimal and performing the algebraic steps to find I, P, r, or t. Students also use the percent error formula (|Actual - Estimated| / Actual) × 100 in several contextual problems (weights, temperatures, prices, counts) with answer keys provided.

Students solve many percent-and-ratio problems: finding pre-tax price from a total with sales tax, computing markups/discounts (25%, 30%, 15%, 20%), calculating tips on meals, computing simple interest for given principal, rate, and time, and finding percent error. Students also practice unit rates, constants of proportionality, writing equations (y = kx), and deciding whether tables or graphs are proportional, which supports using proportional reasoning to set up and solve these percent problems.

Students calculate unit rates for lemons, sugar, and cups and record these in tables and graphs. They set up and solve proportions to scale the recipe (y = kx) to find tablespoons of lemon juice and sugar for up to 16 cups and then use proportions to convert tablespoons to lemons and pounds of sugar. Students compute total cost per gallon and cost per cup by combining ingredient quantities with unit prices, and they calculate selling prices using 100%, 150%, and 200% markups. In Part 5 students apply percentage computations to find discounted prices (30% discount), sales tax (student-found local rate), and a 15% gratuity, combining those percent steps with prior unit-rate and cost work.

Students set up and use one-step multiplicative formulas such as Total Price = Original Price × (1 + Sales Tax Rate) and Discounted Price = Original Price × (1 − Discount Rate) to compute final prices (Activities 1 and 2). Students rewrite selling-price expressions as Selling Price = Wholesale Price × (1 + Markup Rate) and solve for selling price or wholesale price (Activity 4). Students solve for original price or unknown percent by writing and manipulating equations (e.g., 31.50 = P(1.05); 48.75 = 65(1 − D)) and compute totals for multiple items before applying a percent (multi-item tax and discount problems). Students write and solve linear expressions for fixed plus variable costs and handle tiered calculations in challenge problems (Activity 5).

Students set up and solve equations for sales tax (e.g., 60 × 1.07 = total) and compute discounts and markups (e.g., Discounted Price = (1 - 0.25) × 100, Selling Price = (1 + 0.30) × 500). Word problems require writing and solving equations that include fixed fees and per-unit costs (e.g., t = cp + s, 100 = 5 × 4p + 20) and finding unknowns after substitution and algebraic manipulation. The review and activities explicitly ask students to calculate final prices after tax, final prices after discounts, and selling prices with markups, giving practice with percent calculations using multiplicative relationships.

Students practice identifying proportional relationships on graphs and finding unit rates by dividing y by x (e.g., earnings of $10 per hour, walking 6 miles in 3 hours → 2 mph, apples 3 for $9 → $3/pound). Students translate between equations and graphs by making tables of values for equations like y = 3x, y = 2x, and y = 1/2 x and plot points to confirm proportionality. Students derive equations from graphs by picking a non-origin point, computing k = y/x, and writing y = kx for several provided graphs.

Students practice representing proportional relationships in multiple ways: they write and graph equations of the form y = mx (e.g., y = 4x, y = 6x), compute unit rates from tables and equations (e.g., speeds, savings, candy prices, streaming costs), and calculate slope from two points using m = (y2 - y1)/(x2 - x1). Students compare rates by reading the y-value at x = 1, comparing slopes (steepness) on graphs, and rank people/services by unit rate in several applied contexts.

Students are explicitly taught that y = mx is a proportional relationship (e.g., "If the y-intercept is 0, like in the equation y = 2x, this is a special case called y = mx"). Students practice identifying and using a rate (slope m) and an initial amount (y-intercept b) in many activities: they graph y = mx + b, convert equations to slope-intercept form, find equations from tables, and model real-world linear scenarios such as hourly pay with a starting bonus, subscription fees plus per-item charges, and parking meters (Activity 8). Students also compute slopes from two points and use m = change in y / change in x to extend lines and write equations from given data.

Students solve multiple percent and fee problems: they calculate discounts (e.g., $80 with 25% off), compute sales tax (e.g., $120 with 8.25%, $150 with 6.5%), and find final prices in answer keys. Students set up and solve linear equations that combine fixed fees and per-unit rates (e.g., 30 + 10x = 80 → x = 5 GB; 24 + 3x = 66 → x = 14 rides) and write equations for situations with hourly rates plus bonuses. The unit also includes many proportional relationship tasks: students determine if tables/graphs are proportional, find unit rates/slopes, write equations like y = kx for proportional situations, and graph those relationships.

Students write and use proportional equations for rate problems (y = mx) when modeling distance = speed × time and graph those proportional relationships as slopes/unit rates. Students write and use linear cost equations in the form y = mx + b to represent cost per mile plus fixed fees, then graph and compare those equations. Students compute and combine percent operations (5% tax on tickets/food, 10% discount on a plane ticket, 15% tip on dinner) and use those results in multistep total-cost calculations for a 500-mile trip.

Students compute probabilities as ratios (probability = number of favorable outcomes / total outcomes) and convert those ratios to fractions, decimals, and percentages in the coin-toss and spinner activities. Students record experimental outcomes (e.g., 6 heads out of 10 → 6/10 → 0.6 → 60%) and complete tables that require fraction, decimal, and percent representations. Students also compare and place numeric probabilities on a 0–1 probability line and reason about likelihood using these numeric forms.

Students record counts from spinner trials and compute experimental probability using Number of times it happened / Total number of spins. Students convert those probabilities to decimals and percents (e.g., 59/100 -> 0.59 -> 59%). Students use a proportional calculation to predict counts for 600 spins by computing Prediction = Experimental Probability × 600 and rounding to whole numbers.

Students calculate probabilities as ratios and convert them to decimals or percents (e.g., 1/6 × 120 = 20; 1/2 × 500 = 250; 29/50 = 0.58 or 58%). Students set up probability models as fractions (number in group / total) and then multiply those fractions or decimals by a number of trials to predict expected counts (e.g., 75 × 1/2 = 37.5). Students also compute experimental probabilities from counts (relative frequency = count / total) and compare those percentages to theoretical values.

Students build sample spaces, count favorable outcomes, and compute probabilities as fractions, then convert those fractions to percents and multiply by a number of trials to predict expected counts (Activity 3 example: probability 1/6 = 16.7% then 16% × 120 = 20). The Making Inferences activity has students compute a sample proportion (14/60 = 23.3%) and apply that percentage to a population (0.233 × 1143 ≈ 266). Student pages repeatedly ask for probability (fraction), percent chance, and predicted frequency if repeated, requiring sequential use of proportional reasoning.

Students assign percentages to digits in the Music Playlist activity (Pop 40% → digits 0–3, Country 30% → 4–6, Rap 20% → 7–8, Instrumental 10% → 9) and then run simulations using that proportional mapping. In The Library Hunt activity, students record counts out of 20 trials and are explicitly instructed to write the fraction of trials meeting a condition and "turn it into a percent." Students are also asked to predict and compare outcomes when the percentage of a category changes (e.g., if instrumentals were 20% instead of 10%).

Students compute percentages for size/color combinations using the formula Percentage = (Number of dogs / 60) × 100 and complete example calculations for (Small, Brown) = 5% and (Medium, White) = 15%. Students add percentages to find compound probabilities (e.g., total probability of meeting a medium-sized dog = 30/60 × 100 = 50% and total white dogs ≈ 23.33%). Students use the probability model values to run simulations and compare observed frequencies to the model percentages.

Students read and match graphs that represent constant-rate situations (e.g., Graph B: "Hours Worked" vs. "Money Earned" showing earning the same amount per hour). In Activity 2, students compute positions over time using given rates (Sylvia climbs 4 ft/hour for 3 hours, naps, then climbs 5 ft/hour) and plot those multistep segments on a coordinate graph. Several student pages describe scenarios like "a student saves the same amount of money each week" and turtle/balloon stories where students must calculate distances or heights from constant rates across successive time intervals.

Students write linear function rules from stories, tables, and graphs (e.g., modeling Liam's chores as A = 6c + 12 and the bike-rental cost as C = 3h + 5). Students model percent-as-rate situations (Maya's phone battery: starting at 100% and using 5% per hour, answer key gives B = −5h + 100) and create directly proportional functions in scaling contexts (Cooking with Functions answer key shows F = 0.5s, Su = 0.5s, E = 1s, M = 0.33s, B = 0.5s; other items show y = 0.25x and tables with (0,0)). The activities have students find slope and y-intercept from tables and graphs and write function rules in the form output = slope × input + starting value.

Students calculate and interpret rates in multiple items (e.g., Problem 20 asks for the train's rate of change = 60 miles per hour; several problems ask students to write earnings functions E = 12h or E = 15h and to model money earned per hour). Students set up and interpret linear models with starting fees and per-unit rates (e.g., write an equation for a gym membership with a $25 sign-up fee and $15 monthly fee; streaming subscription cost problems ask students to identify starting fees and monthly increases). Students match piecewise distance–time stories to graphs (car driving at 40 mph then 60 mph) and use slope/intercept skills to analyze real-world linear relationships.

Students compute and use scale factors to find missing side lengths (e.g., Activity 4 shows scale factor = 6 ÷ 3 = 2 and uses it to find missing sides). The lesson includes an explicit similar-triangles example with side lengths (AB = 4, BC = 5, AC = 6 and DE = 8, EF = 10, DF = 12) showing corresponding sides are proportional. Multiple student activity pages require finding scale factors and solving for unknown side lengths and angles using proportional relationships.

Students calculate scale factors by comparing new and original coordinates (new x ÷ original x and new y ÷ original y) and decide whether the transformation is a dilation. Students multiply original side lengths by a scale factor to find new lengths (new length = original length × scale factor) and plot dilated points by multiplying coordinates by the scale factor. Students set up and solve equations for unknowns using the relationship new = original × scale factor and solve for the scale factor by division (scale factor = new ÷ original).

Students set up and solve percent-and-fee problems such as a 20% markdown on a sofa with a $50 delivery fee (p - 0.2p + 50 = 610) and a 15% discount plus $5 shipping (0.85x + 5 = 42.50). Students model and solve rate problems using per-unit language (e.g., $3 per pound of chicken, $8.75 per guest, $0.15 per text, $0.25 per mile) by defining a variable, writing an equation, and solving for the unknown. The materials instruct students to identify what they know, define a variable, translate the situation into an equation, and then solve using the routine Distribute → Move → Isolate.

Students set up and solve linear equations that model rate and fee situations, for example writing 25 + 15x = 130 for a sign-up fee plus monthly cost and 18x + 20 = 146 for a subscription activation fee plus monthly charge. Students solve per-unit rate problems such as a van rental 40 + 0.30x = 94 (miles driven) and compare unit-rate plans with equations like 25 + 7x = 10 + 8x (GB of data). Students also translate multiplicative relationships (length is 3 times width) into equations and solve multistep perimeter problems, and they set equal-earnings equations for hourly-rate comparisons.

Students define variables and write systems of linear equations to find unit prices (e.g., peach rings and cherry sours, apples and bananas) and to solve motion and mixture problems. Students set up and solve cost models of the form Total Cost = Rate × Quantity + Fixed Amount and use those equations to calculate break-even points (e.g., comparing subscription plans or cleaning services). Students solve multi-step word problems by substitution or elimination and check solutions by substituting back into the original equations.

Students write and solve linear cost equations that include per-unit rates (e.g., car: C = 225 + 0.60x; rideshare: C = 1.25x) and find break-even points by setting equations equal and solving. Students set up per-hour and per-gigabyte cost equations (StreamScape: C = 5 + 1.5h; BingeBox: C = 20; phone plans: y = 5x + 20) and graph them to compare options. Students perform multi-step computations such as converting annual fixed costs to monthly rates (dividing $2,700 by 12) and splitting one-time costs with a roommate to produce new per-person equations.

Students learn and practice computing relative frequencies using the formula Relative Frequency = frequency / total and convert those decimals to percents (e.g., 18/90 = 0.20 = 20%). Students construct two‑way frequency tables, create row- or column-based relative frequency tables, and use those proportions to compare groups and describe possible associations. Multiple activities ask students to fill in relative frequency tables and answer interpretation questions that require comparing percentages/decimals across categories.

Students are instructed to build a two-way relative frequency table and to "Calculate Percentages" by converting tally counts into percentages for each cell. The Parent Plan and activity pages explicitly direct students to use relative frequencies for rows or columns to describe associations and to fill out percent totals for rows and columns. Activity pages and instructions guide students to compute and record percentages from counts when analyzing categorical data.

Students solve a dedicated set of percent application word problems (Activity 4) that explicitly ask for sales tax, tips, markups and markdowns, simple interest, percent increase/decrease, and percent error. Problems 5 and 6 require computing simple interest and then the total amount after 4 years, providing multistep percent work. Earlier activities ask students to compute unit rates, identify constants of proportionality, and represent relationships with equations (e.g., y = kx), which students can use to set up and solve the percent problems.

Students solve percent problems such as calculating total cost with sales tax and finding a final price after a discount (Activity 1). Students write and interpret linear equations that include fixed fees plus per-item costs (movie ticket C = 9t + 6; delivery service y = 18x + 12) and translate percent increase language into multiplicative form (Parent Plan example a + 0.05a = 1.05a). Activities 2–4 have students identify proportional relationships from tables, write equations of the form y = kx, find unit rates (slopes), and graph and compare proportional situations.

Students convert probability values to fractions, decimals, and percents in multiple tasks (e.g., experimental probability 7/25 = 0.28, and the hamster probability 4/20 = 20%). Students are asked to "reason using fractions and percents" and to explain how a spinner with 10 equal sections could model a 40% probability. Students create probability models and express probabilities as percents in Activity 2 and write probabilities as decimals and fractions in Activity 1.

Students compute unit rates and identify proportional relationships in Unit 2 (problems 14–19), including writing equations of the form y = kx, finding constants of proportionality, and interpreting graphs that pass through (0,0). Students solve percent and rate problems in Unit 3 and Unit 2–4 items: they calculate sales tax (problem 21), tips (problem 22), markups (problem 23), and simple interest (problem 24), and they write linear cost equations with fees (problem 33). The answer key provides numerical solutions and shows students set up and evaluate proportional equations (e.g., y = 4x, C = 11t + 4, I = PRT).

Students write and interpret linear functions from real-world situations (e.g., y = 18h for hourly pay, y = 12x + 10 for a bike rental with a $10 fee) and answer questions about which intercept represents a starting cost. Students compute and compare slopes (e.g., find slope from two points, compare Function A: y = 6x - 2 and Function B with points) and interpret slope as a rate of change in context. Several activities ask students to identify when a function passes through (0,0) and to construct functions from situations involving rates and fixed fees.

Students set and use scale factors in multiple activities (e.g., given two similar triangles with sides 5, 7, 9 and a corresponding side 10, students find the scale factor and compute the other side lengths). Students perform dilations on coordinate-plane figures (e.g., dilate Triangle LMN by a scale factor of 3 and find L'(6,3), M'(12,3), N'(9,9)). Tasks explicitly instruct students to "use ratios to reason through each problem" and include problems finding scale factors less than 1 (reduction) and non-integer scale factors (2.5) to compute corresponding side lengths.

The lesson asks students to compute relative frequencies in a two-way table and convert those fractions to decimals and percentages (for example, 12/50 = 0.24 or 24%). Activity 4 has explicit calculations of fractions-to-decimal conversions for table entries (e.g., 10/30 = 0.33, 12/50 = 0.24) and asks students to interpret which category is most popular based on those percentages. These tasks require students to use proportional reasoning to compare parts of a whole within categorical data.

Students write a unit-rate function in problem 8 ("You earn $10 per hour. Write a function...") and set up/solve a cost equation in problem 35 ("A tutor charges $18 per hour plus a $30 fee. If the total cost was $138, how many hours were worked?"). Students also work with constant-rate contexts elsewhere (slope problems such as #5 and writing linear equations), which requires forming and using proportional relationships between quantities. The relative frequency table in problem 50 asks students to compute decimals from counts, which is a ratio computation.