Students compute quantities labeled as "Number of item per goody bag" and multiply those numbers by 12 to find total items (e.g., 5 candy bars per bag → 12 × 5 = 60). Students are asked to compute a cost per goody bag by dividing the grand total by 12 (unit-rate calculation). Students are prompted to show totals using the distributive property (example: 12(5 + 3) = (12×5) + (12×3) = 60 + 36 = 96 items), which frames quantities in a per‑bag context.

The lesson defines a ratio with concrete examples (the biscuit 3 cups flour to 1 cup milk example and the statement 'A ratio shows how one value compares to another value'). It explicitly teaches three forms for expressing ratios (in words, with a colon, and as a fraction) and includes student pages that require filling in "There are ___ to ___" and writing the same ratio in all three forms. Multiple activities and word problems (the blueberry pie 'Think About It', shirt/pants and cocoa examples, and equivalent-ratios problems) require students to describe relationships between two quantities using ratio language.

Students are given multiple concrete examples that define and label ratios (pizza 3:8, cookies 10:3, apples to bananas 5:9) and are shown part-to-part, part-to-whole, and whole-to-part relationships. Students are asked to write ratios in three forms (a to b, a:b, a/b), to describe ratios in words (e.g., toast and eggs "1 to 2" and bottles cracked to total), and to draw pictures that illustrate given ratio situations (bottles, pizza slices, puppies). Several activities require students to identify the type of ratio, express equivalent ratios, and use "for each/for every" language in contextual problems (toast per camper).

Students represent and solve ratio situations using tape diagrams, double number lines, and tables/graphs in multiple activities (e.g., Amanda:Malika 3:5, boys:girls 2:3, hamburgers:chicken 5:4). Students create tables and plot coordinate pairs to find equivalent ratios (Leo's pitchers/lemons, Kara's key chains) and use double number lines to scale rates (sodas, hamburgers vs. chicken sandwiches). Students complete a Basic Skills problem asking them to write a ratio in three forms (2:7, "2 to 7", 2/7), and many problems use "for every" phrasing in prompts (e.g., "for every 5 hamburgers for every 4 chicken sandwiches").

Students write and represent ratios in multiple forms (e.g., 5:3, 5/3, "5 to 3") on the quiz and in activities. Multiple problems use ratio language such as "for every" and "per" (e.g., Hondo's cookie recipe, "2 cups of sugar for every 5 cups of flour", and many "miles per hour" examples) and ask students to produce equivalent ratios and unit rates. Tasks require students to classify ratio types (part-to-part, part-to-whole, whole-to-part) and solve word problems using tape diagrams, double number lines, and division to find ratios and unit rates.

The lesson explicitly states that the word "per" signifies a ratio and that a percentage is a part-to-whole ratio with the whole equal to 100. Students convert between fractions, decimals, and percents (e.g., 93/100 = 93%) reinforcing the ratio-as-fraction idea. Student problems ask for ratio language/practice: Rory's home runs are written as a home-run-games to total-games ratio and percentage, and the Basic Skills Review asks "What is the ratio of won games to lost games?" using "for every" language.

Students are asked to represent percentages as part-to-whole ratios and to write the part as a fraction of the whole (e.g., part/whole) and the percent as a ratio over 100. Students create and use double number line diagrams to find equivalent ratios and solve percentage problems (e.g., finding the missing whole, part, or percent). Student activity pages require forming equivalent ratios and solving problems by setting up percent/100 = part/whole and computing unknowns.

Students are presented with a definition that a ratio shows the relative size of two quantities and explicit part-to-whole and whole-to-part examples such as "12 inches = 1 foot" and "1 quart = 2 pints." Students set up and interpret ratios in fraction form and a:b form and use double number lines and equivalent-ratio setups to solve word problems (e.g., converting 3 quarts to 6 pints, 19 cm to 190 mm, 5 ounces to 141.75 grams). Students are directed to create an Interactive Notebook page organizing common conversion ratios and complete practice problems that require describing and using those ratios to find unknown quantities.

Students are asked to define "ratio" and demonstrate different ways to show a ratio in the unit objectives and skills list. Multiple student tasks require writing ratios in three forms (e.g., 3 to 7, 3:7, 3/7) and labeling ratios as part-to-part, part-to-whole, or whole-to-part (soccer games and fishing problems). Students solve contextual ratio problems using tape diagrams, double number line diagrams, tables, and graphs (Marco & Stanley, Farmer Ned, apples/pies), and the skills list explicitly states students should "use ratio language to describe a ratio relationship between two quantities."

Students set up ratio problems and record a "Ratio" column and unit price for each product (Activity 2 and Day 2), and they calculate unit price by dividing price by quantity (Activity 3). The "Ratios All Around" extension asks students to write the ratio of an ingredient to number of servings and to use equivalent ratios to double/triple the recipe. Students classify ratios as part-to-part, part-to-whole, or whole-to-part and use conversion ratios for unit and currency conversions. The Parent Plan explicitly lists "Use ratio language to describe a ratio relationship between two quantities" and includes several student tasks that require forming and using ratios in context.

The lesson uses "for every" language in examples such as "If you get paid $5 for every car you wash... the expression for your pay is 5n," and presents rate-like situations (e.g., two friends paid a rate per job leading to the expression xy). Several problems ask students to write expressions that relate one quantity to another (e.g., 20 + n = 35, 36n for cookies in n boxes), showing multiplicative relationships between two quantities. The activities require students to translate verbal phrases into algebraic expressions that model 'per' or 'for every' relationships.

Students encounter a word problem that uses explicit ratio language: "Max runs 4 miles for every 3 miles that Gina runs," and they compute Gina's miles when Max runs 24 (answer key shows using a multiplicative comparison to get 18). Students write expressions that represent sharing or rates, for example translating "Piper has 24 cookies. She plans to divide them among some of her friends" into 24/n. The Basic Skills Review includes unit-rate and unit-conversion items (gallons to quarts) that require reasoning about quantities per unit.

Students solve a rate problem in the Basic Skills Review where they calculate Elias did 42 push-ups in 3 minutes, find the unit rate (14 push-ups per minute), and use equivalent ratios to determine how many push-ups in additional minutes (e.g., 70 push-ups in 5 minutes). The answer key shows the ratio written as "14 push-ups/1 minute = 70 push-ups/5 minutes," which represents the rate as a ratio and a unit rate.

Students translate real-world "for every" or "per" language into algebraic expressions in several problems: Jeremiah earns "$5 for every dog he walks" and select or write 5x + 3, Zane earns "$12 per hour" and write 12n + 5, and Alana's beads problem uses 12(x + 6) to represent beads per bag. These items require students to represent multiplicative relationships between two quantities using expressions and to evaluate those expressions for specific values.

Students see tape diagrams explicitly linked to ratios in the tape-diagram example that compares Kali (12 boxes) and Lina (16 boxes) by dividing each into groups of 4, and the lesson states, "In ratios, the parts of the tape diagram show the relationship between two values." The Basic Skills Review includes a problem that directs students to "Solve using a ratio" (calculating cost for four tickets by scaling a unit price). These passages show students encountering ratio representations and using a multiplicative ratio method in at least one practice problem.

The lesson explicitly defines and uses unit rates as ratios: Tonya's driving example states a unit rate of 45 miles to 1 hour (45:1) and shows d = 45t. The cookie example states the ratio of cookies to eggs is 15:1 and links that ratio to the dependent/independent relationship in the equation y = 15x. Several word-problem activities (Kevin earning $10/hr, Ron biking 5 mph) have students write equations from rate language (10x = y, 5x = y) and create tables/graphs from those rates.

The project requires students to create at least one equation with two variables and a corresponding table and description of the relationship (e.g., "The equation 2x = y shows that I practice piano (y) twice as long as I practice soccer (x)"). Students are asked to use variables to represent two quantities that change in relationship to one another and to describe how y changes as x increases or decreases. The examples and poster prompts ask students to model multiplicative relationships (e.g., "twice as long") and to include diagrams (tape/hangar) to represent relationships.

The Basic Skills Review includes Problem 5 asking students to use equivalent ratios to determine how many gumdrops a machine makes in 1 minute given 208 gumdrops in 4 minutes. The answer key shows the ratio setup 208:4 = n:1 and directs students to divide both parts by 4 to get 52:1. Students therefore practice converting a two-quantity relationship into a unit-rate ratio.

Students measure the circumference and diameter of three circles, record those values, and compute the quotient C/d rounded to two decimal places. The lesson explicitly presents the equation π = C/d and directs students to notice that the computed quotients are all about 3.14. The lesson also discusses ratios in number form (example 5/2 and 2.5/1) when contrasting rational and irrational numbers.

Students set up and interpret scale-factor ratios throughout the lesson (for example, Mia's garden: 8 cm/2 m simplified to 4 cm/1 m and described as "each group of 4 centimeters ... represents 1 meter"). The lesson defines scale factor as a ratio or percent and asks students to write scale factors as ratios and percents (several Student Activity pages ask for ratio and percent and whether the drawing is an enlargement or reduction). Students solve proportion problems using given ratios to find actual or drawing measurements (architect 1 cm/4 ft → find 120 ft as 30 cm; multiple worksheet problems require forming equivalent ratios and simplifying).

Students compute scale factors and ratios in several problems (e.g., find the scale factor for a photograph of a nest where the actual diameter is 12 in and the photo diameter is 4 in, answer key shows 4/12 = 1/3 and 33 1/3%). Students find enlargement scale factors (Jeremy's kite: 6 in to 30 in and 4 in to 20 in, answer key gives 5:1) and use colon and percent notation (5:1 or 500%). Students calculate scale-factor effects on perimeter and area (questions ask for perimeter scale factor = 3/1 and area scale factor = 9/1), showing use of ratios between two quantities (dimensions, perimeters, areas).

Students are asked to create a scale drawing of a shape based on a given scale factor and to reproduce a scale drawing at a different scale (Create a Scale Drawing activity and Parent Plan). Students plan and execute stations that require use of measurement tools and scale factors when laying out and resizing shapes on a laminated grid. The Stations Planning sheets and station creation steps require students to specify materials and text for problem cards that may include scale-factor instructions.

The Basic Skills Review includes a unit-rate problem: Evie bought a 24-ounce bottle for $3.12 and is asked for the unit price per ounce. The answer key explicitly tells students to "Write equivalent ratios: 24 oz/$3.12 = 1 oz/n" and to divide both parts to obtain "1 oz/$0.13." This shows students set up and solve an equivalent-ratio equation to produce a unit rate.

The lesson uses scale factors and proportions when finding cross sections: it gives a scale factor of 1/4 to find cross-section side lengths and shows that the area ratio is the square of the scale factor (1/16). Students set up equivalent ratios (for example, 1/4 = n/8 and 1/2 = 4/8) to solve for unknown dimensions of cross sections. Several problems ask students to use proportional relationships to find dimensions (e.g., using area ratios or corresponding side ratios in similar figures).

The Basic Skills Review problem about the candy factory (85 pieces per minute) asks students to compute production in 12 minutes and the answer key describes the rate as a ratio: "The rate is 85:1. Multiply both parts of the ratio by 12 to get 1,020:12." This shows students set up and scale a ratio to find an equivalent ratio for a different time interval.

Students are asked to write part-to-whole ratios for each candy color (Activity 1) and record them in a table labeled "Color, Ratio, Percentage," with the example explicitly showing "If your sample has three green candies, the ratio is 3:10." Students convert those ratios to percentages using equivalent ratios (example calculation showing 3/10 → 30%). In the Basic Skills Review students work with a rate expressed as a ratio (600:8) and compute a unit rate (75:1, interpreted as 75 miles per hour).

Students compute and interpret a quotient that compares two quantities: they calculate the difference between two means and divide by the larger mean absolute deviation (examples show 1.25 and 5). The Parent Plan and Step Five explicitly present this computation as a mathematical comparison (described as using a "ratio" or formula to compare the difference of means with spread). Student activity pages require students to compute means, mean absolute deviations, and perform the division to produce a single comparative number and interpret whether overlap is large or small.

Students are asked to write the ratio of carrots to peppers in three forms (e.g., 3 to 4, 3:4, 3/4). Students write the ratio of onions to carrots and identify the type of ratio (part-to-part) and write the ratio of peppers to vegetables and identify it as part-to-whole. Students solve contextual ratio and rate problems (large vs. small dogs, necklaces per minutes, train speed, unit price of apples) that require forming equivalent ratios and computing unit rates.

The Parent Plan lists skills that reference ratio ideas: "Find a percent of a quantity as a rate per 100" and "Use ratio reasoning to convert measurement units," and the wrapping up section directs students to an online quiz on percentages and unit conversions. In Activity 1 students work with rates in context (Marie makes $9 per hour), which uses "per" language that expresses a rate between two quantities. In Activity 2 the Pedro problem (each bin contained 9 cars) and similar contextual problems require students to compute quantities related by a constant multiplicative relationship.

Students work with scale factors and set up ratios in several problems: Problem 3 asks students to enlarge a triangle by a scale factor of 2/1 and compute the new side lengths. Problem 4 asks students to find the scale factor between two rectangles, and the answer key explicitly shows writing ratios 4/12 and 2/6 to get 1/3. Problem 5 uses percent scaling (800% larger) and the answer key shows using an equivalent-ratio equation (100/800 = 2/n) to find the unknown height.

Students work with contexts that use per-unit language (e.g., "2.5 degrees per hour," "descends 0.2 miles per hour," "$4.25 per week") and solve problems by multiplying rates by time or dividing totals by number of units to find a per-unit amount. Students compute and interpret quotients as unit rates in activities such as finding monthly deductions (−72 ÷ 6 = −12) and average temperature change (−48 ÷ 6 = −8). Several prompts ask students to explain the meaning of results in context and to create their own real-world problems using per-unit phrasing.

The Parent Plan Skills explicitly instruct students to "express how many times as much one is than the other" and gives the concrete example comparing U.S. population (3 × 10^8) and world population (7 × 10^9) to determine the world is more than 20 times larger. The lesson's comparing activities ask students to compare numbers in scientific notation (e.g., problems that have students compare 3.2 × 10^6 and 1.5 × 10^7 and a challenge comparing 3.2 × 10^5 and 2.9 × 10^5), which requires students to reason about relative size and factors between quantities. Several word problems ask students to compute totals and rates (e.g., multiplying grains per pound × pounds, liters per minute × minutes) that could be interpreted as multiplicative comparisons between quantities.

Phase 1 asks students to determine which organism has larger cells and to "calculate how many times larger the cod's cell is compared to the sponge's," which requires forming a multiplicative comparison. Phase 1 also asks students to "compare the number of cells in equal tissue samples" for sponge and cod, which involves comparing two quantities and could require forming a quotient.

Students set up and interpret ratios as labeled fractions in multiple contexts (e.g., "3 blue beads for every 2 red beads" written as 3/2 = x/12). Students write relationships as equivalent fractions and check proportionality (Activity 3 shows simplifying 6/10 and 9/15 to determine proportionality). Real-world word problems (recipes, tacos, travel, shopping) require students to describe quantities and set up proportions using language such as "for every" and labeled ratios.

The lesson defines a unit rate as "a ratio that tells us 'how much for one'" and repeatedly has students set up labeled fractions (e.g., $4.99/6 apples) and divide to find a per‑one value. Students calculate and interpret unit rates in many contexts (price per pencil, miles per hour, cups per batch, sheets per second) and solve complex fraction division problems that produce rates (e.g., (3/4) miles ÷ (1/2) hours = 1.5 miles per hour). Activities require students to label results with "per" language and compare rates to decide which option is the better deal.

Students compute k = y/x and identify unit rates in multiple activities (Things to Know, Activity 2 tables, Activity 5 equations). Students describe rates using per/for every language in real-world contexts (Activity 6: "for every 1 minute, 2 liters are added", examples like "$10 per hour", "3 cups flour for every 4 cups sugar"). The Parent Plan and skills list explicitly state understanding a unit rate a/b associated with a ratio a:b and using rate language in context, and the review quiz asks students to compare and simplify ratios (e.g., 7:14 and 12:24).

Students repeatedly work with unit rates and the language "per" or "for every" (e.g., Jim earns $12 per hour; movie tickets cost $12 for every ticket). The lesson requires students to test tables for equivalent ratios (Activity 6) and to interpret the unit rate as the y-value when x = 1 and as the constant k in y = kx. Classroom tasks ask students to describe relationships in words, write equations y = kx, and state unit rates (e.g., $8 per t-shirt, 50 miles per hour).

Students calculate and use unit rates in multiple activities (e.g., finding unit price per ounce, miles per hour, calories per minute, and interpreting d = 4.5t). Students work with explicit ratio statements and notation in problems such as a width-to-length ratio of 5:8, a recipe of 2/3 cup sugar for every 1/4 cup butter, and a blue-to-red marbles ratio of 5:3. Students write and use equations of the form y = kx or t = p×n to represent ‘per‑unit' relationships (e.g., t = 12n, t = 15n, c = 8m) and interpret points like (1,5) as the unit rate.

Students convert percent rates to decimals and compute parts of whole quantities by multiplying (e.g., 7% → 0.07 and 15 × 0.07 = 1.05) when calculating sales tax, tips, and commissions. Students set up and solve proportional equations to work forward and backward (for example, 0.012 × V = 2400 to find property value and 1.08 × x = 212 to find pre-tax price). Students practice applying proportional reasoning in multi-step problems involving tax, gratuity, and commission calculations.

Students are asked to compare ratios directly (e.g., Question 3 asks whether 4:5 and 8:10 are proportional and to explain), compute unit rates from paired quantities (multiple problems ask for miles per hour, price per item, and unit rates from fractional amounts), and interpret ratio relationships presented as scales or "per" language (map scale 1 inch = 5 miles; recipe 3/4 cup sugar per 1/3 cup flour). Several items ask students to interpret points on a proportional graph (e.g., "What does the point (1,4) represent?" and "What does (0,0) represent?"), requiring verbal description of the relationship between two quantities.

Students calculate unit rates for lemons, sugar, and cups by using the unit rate formula and by dividing total cost by quantity (e.g., price per lemon, price per pound, price per cup). Students set up and solve proportions in multiple places (e.g., 1 lemon = 3 tablespoons, 1 lb sugar = 36 tablespoons) to find how many lemons or pounds of sugar are needed for a gallon. Students create tables, graph number-of-cups versus tablespoons and write equations in the form y = kx to represent the proportional relationship and identify the unit rate (k). Students compute cost per cup by dividing the total cost for one gallon by 16 cups, explicitly using a ratio to find cost per single cup.

Students set up and solve word problems that use unit rates and multiplicative relationships, for example writing t = cp + s for the marker problem (where c = cost per pack = $4) and s = r(m + t) for the Ferris-wheel rides (where r = cost per ride = $2). Students plug values into these formulas, subtract fixed fees, and divide to find unknown quantities (e.g., 15 = 4p + 3 → p = 3; 36 = 2(6 + t) → t = 12). The lesson repeatedly has students interpret and manipulate expressions that involve 'each' or a per-unit cost and solve equations that reflect one quantity per another.

Students compute unit rates by dividing y by x in multiple activities (e.g., earnings: $10 per hour, walking 6 miles in 3 hours → 2 miles/hour, apples $9 for 3 pounds → $3/pound). Several tasks require students to identify whether a graph is proportional by checking that it is a straight line through the origin and then state the unit rate (constant of proportionality) in "per 1" language. The Walk the Graph activity directs students to move a specified amount "for every" one step forward (e.g., move 2 steps up for every 1 step forward), reinforcing ratio language in context.

Students repeatedly compute and interpret unit rates as "how much y changes for every 1 unit of x," including explicit statements like "the unit rate is the y-value when x = 1" and examples in equations (y = mx). Students calculate and compare rates in concrete contexts (miles per hour, marbles per second, price per candy, cups of flour per cup of milk) using tables, graphs, and equations, and they use language such as "per" and "for every" (e.g., "2 cups of flour per 1 cup of milk"). Activities ask students to state which relationship has the higher rate and to describe rates in words (e.g., which person is saving faster, which faucet fills faster).

Students draw right triangles between two lattice points, count the vertical change (rise) and horizontal change (run), and record rise/run to compute slope. The student pages require students to fill in rise and run (with +/− signs), simplify the fraction rise/run, and use that ratio as the slope. Activities also have students set up proportions (e.g., 2/2 = 4/4) to compare rise-to-run ratios of different triangles and decide triangle similarity.

Students compute slope using m = change in y / change in x and are instructed to read that as how much y changes for a given change in x (e.g., "Up 2 → Right 1"). The table activity explicitly has students note that "When x increases by 1, y increases by 2" and then uses that to write y = 2x + 1. Real-world examples use rate language (e.g., "Liam gets paid $10 per hour" leading to y = 10x + 50) and student tasks ask them to identify slopes as rates like dollars per hour or cost per mile.

Students work with multiple real-world tables (babysitting, car rental, dog walker, lawn care) asking whether the relationship is proportional and asking for slope, y-intercept, equation, and graphs. The materials instruct students to "interpret unit rates and slopes from tables and graphs" and include using the formula m = change in y/change in x to find unit rates. Answer keys show students compute constant rates (e.g., slope = 10, equation y = 10x) from context-based tables.

Students use speeds given in miles per hour and complete tables of distance versus time, showing the numeric relationship between miles and hours. Students write equations in the form y = mx for car, train, and plane and compute rise/run (unit rate) for the train (480/6 = 80), linking the rate to slope. Students graph distance vs. time, identify proportional relationships that pass through the origin, and interpret unit rates as slopes, and they compute cost per mile as slopes in cost equations.

Students are shown probability as a ratio when the coin example converts chances to "1 out of 2 → 1/2 → 0.5 → 50%." The spinner section gives the formula "Probability = Number of favorable outcomes / Total outcomes" and has students fill a table using counts (e.g., 6 out of 12) and convert those counts to fractions, decimals, and percents. Activity pages and answer keys repeatedly use "out of" and fractional forms (e.g., 3/15, 5/15) to describe relationships between counts of outcomes.

Students record counts for each color and compute experimental probability using the formula Number of times it happened / Total number of spins, expressing results as a fraction, decimal, and percent (e.g., 59/100, 0.59, 59%). The materials frame results as counts "out of" a total (out of 100, predicted out of 600) and define relative frequency as how often an outcome occurs compared to the total number of trials. Students use those ratios to make numeric predictions (Prediction = Experimental Probability × 600) and fill tables that show "Total" and "Predicted Spins out of 600."

Students compute probabilities as fractions and use those fractions to predict counts (e.g., P(roll a 4) = 1/6 and 1/6 × 120 = 20). Students form probability models from counts (e.g., 16 choir, 10 band, 8 orchestra → probabilities 16/34, 10/34, 8/34) and calculate relative frequencies (e.g., 29/50 = 0.58 for experimental probability). Students group outcomes and count favorable versus total outcomes (e.g., even numbers 3/6, outside chores 2/6) which requires comparing two quantities.

Students count and represent unequal quantities (e.g., a pencil box with 3 red pens and 1 blue pen) and repeat outcomes accordingly when building the sample space. Students compute fractions and percentages from sample counts (for example, 14 out of 60 → 0.233 → 23.3%) and apply that proportion to a larger population (0.233 × 1143 ≈ 266). Students also set up proportions in several inference problems (e.g., 18/50, 12/30) and use those ratios to predict counts in larger groups.

Students compute and use the rate-of-change formula (y2 - y1) / (x2 - x1) in multiple tables and are asked to determine whether the difference in y-values stays the same as x increases. The lesson repeatedly has students fill tables, calculate Δy/Δx, and state that "Each time x goes up by 1, y goes up by 2," and students complete activities that ask whether a relationship is linear based on that constant change. Several worked examples and answer keys show students calculating and interpreting the numeric change in y relative to the change in x for given data and equations.

Students are given several scenarios that state rates (e.g., "Sylvia climbed at a rate of 4 feet per hour for 3 hours," Timmy the Turtle crawls "2 meters per minute," Bella ascends "6 meters per minute"). Students plot positions over time using those rates, marking points each hour or minute and connecting them to show movement and pauses. Several activities require translating a verbal rate description into a graph (Graphing Real Life Functions) and interpreting steady increases or decreases that come from constant rates.

Students are asked to compute "slope = rise/run" and repeatedly find how much y changes compared to how much x changes (e.g., "Slope = 1 up / 2 right = 1/2"). Students read and write statements using "for every" language (e.g., "the line goes down 2 for every 1 step right" and "every time x increases by 1, y increases by 2"). Students calculate these comparisons from graphs, tables, ordered pairs, and equations using the formula m = (y2 - y1) / (x2 - x1).

Students compute slope using the formula m = (y2 - y1) / (x2 - x1) and apply it to pairs of points (e.g., m = (8 - 2) / (3 - 0) = 2). The materials instruct students to use "rise" and "run" (for slope = 2, rise 2 units, run 1 unit) and to interpret slope as a rate in context (e.g., 1/5 mile per minute = 0.2 miles per minute, converted to 12 miles per hour). Multiple activities ask students to find the slope from tables, graphs, or two points and then write equations, reinforcing the idea of change in one quantity relative to change in another.

Students repeatedly identify and describe rates as "per" relationships (e.g., "for every chore he completes, he earns $6" and "15 pages per hour"), and they calculate and label the rate of change with units (slope = 6 dollars per chore; slope = 15 pages per hour). Activities require students to name input and output quantities and to interpret the slope in context (e.g., C = 3h + 5 interpreted as $3 per hour plus a $5 flat fee). Multiple tasks ask students to write functions from stories, tables, and graphs using language like "for each" or "per," which frames relationships between two quantities in ratio language.

Students calculate rates by dividing change in quantity by change in time (e.g., Alex: 1.5 miles ÷ 30 minutes = 0.05 miles/min and Bella: slope 2 miles ÷ 30 minutes = 0.067 miles/min). Multiple activity items present situations in "every" or "per" language (e.g., "4 miles every 20 minutes," "3 gallons every 2 minutes," "5 miles per hour"), and students compute and compare those unit rates. Several tasks ask students to read tables, graphs, equations, and verbal descriptions to determine which situation has the greater rate (unit ratio) or larger starting value.

Students calculate and interpret rates from tables and contexts (e.g., determining rate of change for a train as 60 miles per hour and identifying earnings functions like E = 12h or E = 15h). Several problems ask which service's cost "increases the fastest per month" or note that a function "increases by 4 each time," using "per" language to describe change. Students also write linear functions that express total amount as a unit rate times a quantity (money earned per hour, cost per month).

Students write and solve real-world rate problems on Yellow Cards (examples include "Emma earns $10 for each hour she babysits," temperature dropping 3 degrees each hour, and a car driving 60 miles per hour), requiring them to identify rate of change and write equations. Students create Blue, Green, and Red cards that ask them to identify slope from equations, tables, and graphs, and to determine whether relationships are linear or functions, which has them compute change in output per change in input (rate). Students match tables, graphs, and stories to equations and identify slopes from tables, which engages them with "per-unit" relationships between two quantities.

Students calculate and use scale factors to compare side lengths (for example, finding scale factor = 6 ÷ 3 = 2 and using 1/2 for the reverse). Students set up and solve proportional relationships to find missing side lengths (for example, multiplying a side by the scale factor to get the corresponding side or dividing a larger side by the scale factor to get the smaller). Students reason about corresponding sides being proportional and practice matching corresponding parts with measurements on multiple activity pages.

Students calculate scale factors by dividing new coordinates or lengths by original values (e.g., "Scale factor = A'B' ÷ AB" and examples where x: 6 ÷ 3 = 2). Students multiply original side lengths by the scale factor to find new lengths (new length = original length × scale factor) and perform dilations by multiplying coordinates by the same scale factor. Activity pages require students to determine whether a transformation is a dilation by checking that all x- and y-values are multiplied by the same number and to solve for unknowns using new ÷ original = scale factor.

Students work with scale factors and dilations in several quiz items and answer keys (e.g., "Multiply each side by the scale factor: 3×2 = 6, 4×2 = 8, 5×2 = 10" and "6 ÷ 2 = 3 → Triangle B is 3 times larger than Triangle A"). The curriculum asks students to find scale factors between triangles and to decide whether pairs of triangles are similar (AA similarity), which requires reasoning about multiplicative relationships between corresponding side lengths.

Students are told explicitly that a cone holds "exactly one-third as much" as a cylinder with the same base and height and are shown the demonstration that filling a cone three times fills the cylinder. The lesson states that a sphere has "exactly 2/3 the volume of the cylinder" with the same radius and height and uses multiplicative language (1/3, 2/3) when deriving and applying the formulas. Students apply these fractional comparisons when deriving the sphere formula (showing V_sphere = 2/3 × V_cylinder → 4/3 πr^3) and when using the cone factor 1/3 in volume calculations and problems.

Students are asked to identify and use scale factors in multiple problems (e.g., determining x when a 5×5 square is scaled by factor 3, dilation questions with scale factors 2 and 0.5, and exercises asking if a pair of figures is a dilation and to find the scale factor). Several problems require computing side lengths after a dilation (e.g., if AB = 5 and the figure is dilated by factor 2, find A'B' = 10). The materials include tasks where students compare corresponding side lengths and decide whether a transformation is a dilation, implying multiplicative comparisons between two quantities.

Students set up equations from word problems that use "per" and "each" language (for example: 3c for $3 per pound of chicken in Mr. Patel's problem, 50 + 8.75g for $8.75 per guest in the catering problem, and 85h + 200 for $85 per hour). Several tasks require translating statements with rates into algebraic expressions (car rental $0.25 per mile, phone plan $0.15 per text, raffle tickets $3 per ticket). The cupcake and baker problem uses "each" to indicate equal quantities per person and students write a multiplicative expression 4(y+2)=16 to represent that relationship.

The lesson explicitly teaches slope as a rise/run and gives directions like "Go up 2 and over 1 every step," and shows points (0,3), (1,5), (2,7) to practice that movement. Activities require students to convert equations to slope-intercept form and compare slopes to decide intersections, which has students work with the numerical ratio of vertical change to horizontal change. Several graphing tasks ask students to use the slope to plot additional points and determine line behavior.

Students are asked to compute slope using the slope formula (y2 - y1) / (x2 - x1) in multiple examples and activities. Several teacher notes and example graphs describe slope as a "rise of 2 units for every 1 unit" horizontally, and students find equations from two points then solve for intersection. Practice problems require finding slopes from point pairs and writing lines in slope-intercept form.

Students define and use unit rates (e.g., price per pound, cost per hour, cost per movie) and write equations using the template Total Cost = Rate × Quantity + Fixed Amount. Students set up and solve systems like 2x + 4y = 23.40 and y = 30x to find unit prices or break-even points, and they compare per-unit costs to decide which option is cheaper. Several activity pages ask students to let a variable represent a unit price or a number of items and solve for those quantities.

Students compute slopes in several problems (e.g., find slope through (2,3) and (4,7); multiple slope exercises asking for m = (y2 - y1)/(x2 - x1)). Several word problems use rate language such as "$12 per hour," "$2 per cup," and "$25 per month plus $5 per class," which require students to set up equations using a per-unit rate. Graphing tasks ask students to interpret intersection points and slopes of lines, which involves using the ratio of rise to run when finding slope.

Students set up and work with many per-unit rate equations (e.g., transportation: C = 225 + 0.60x and rideshare: C = 1.25x; streaming: C = 20 and C = 5 + 1.5h; phone: y = 5x + 20). Students interpret slopes and units (Meal Plans: slope described as $80/week and $50/week) and compute per-person rates when sharing costs (apartment monthly cost divided by 2 to get per-person rates). Students solve for break-even points by equating these rate-based equations and graph the lines to compare costs.

Students plot paired quantities (e.g., hours studied vs. test grade, hours of art practice vs. completed sketches) and answer questions about how one quantity changes as the other changes. The lesson explicitly states that "each additional hour studied corresponds to approximately a 10% increase in test grade" and asks students to make predictions such as what to expect "for every" extra hour (e.g., predict grade for 8 hours or sketches at 9 hours). Multiple activities ask students to describe positive or negative relationships and rates of change between two quantities.

Students repeatedly interpret slope as a change "for every" one unit in the independent variable (e.g., "The slope (m = 2) means that for every 1°F increase above 60°F, 2 additional ice cream cones are sold"). Activity prompts ask students to identify independent and dependent variables and to "Explain the Relation in Words" and to interpret slope and y‑intercept in context (plant growth: "For every week that passes, the plant grows 3 centimeters"). Multiple problems and the bird migration activity require students to compute a rate of change from two points and then write a sentence describing how one quantity changes for every unit of another quantity.

Students compute relative frequencies by using the formula Relative Frequency = frequency / total and convert those fractions to decimals or percentages (for example 18/90 = 0.20 = 20%). Multiple activities have students build two‑way frequency tables, create row- or column-based relative frequency tables, and compare the resulting proportions to describe which group is more or less likely to have an outcome (e.g., bikers vs. drivers catching a cold). The lesson asks students to interpret and explain these proportions in context and to use them to describe possible associations between categorical variables.

Students calculate and compare relative frequencies as fractions, decimals, and percentages and complete a Relative Frequency Table (e.g., questions asking "What fraction of evening viewers chose soccer?" with answers like 0.33 or 1/3). Multiple activities require students to construct and interpret two-way frequency tables and use those tables to describe relationships between categories. Several review items and answer keys show students finding fractions and decimal relative frequencies from categorical data.

Students are asked to convert tally counts into percentages and fill out two-way relative frequency tables (e.g., "Each cell represents the percentage of all 20 people who fit into that activity–time combination"). Instructions explicitly include a step to "Calculate Percentages" and prompts ask students to identify which category combination had the highest percentage and which had the lowest. The Parent Plan references using relative frequencies calculated for rows or columns to describe possible association between two categorical variables.

Students compare and test pairs of ratios (e.g., 8:12 and 14:21) and are asked to explain how they know they are proportional. Students are prompted to write real-world proportional situations, find the unit rate, and "explain what it means," which requires expressing the relationship between two quantities. Activities ask students to interpret points such as (1,r) and (0,0) in context and to write the start of a ratio and an example in a real-world situation.

Students determine whether relationships are proportional, find the slope/unit rate, and write equations from tables and contexts (e.g., Babysitting Pay table: hours vs. earnings; Car Rental Costs; tutoring service charges $40 per hour). Activity prompts ask students what the slope (unit rate) is and what it represents in context, and the Skills list explicitly mentions "interpreting the unit rate as the slope" and writing relationships in the form y = kx.

Students compute probabilities as fractions in multiple tasks (e.g., marbles: 3/10; letters: 21/26; spinner: 4 out of 10 for 40%). Tasks ask students to write probabilities as fractions and decimals and to explain what the probability means, with answer-key language such as "3 out of every 10 selections would be blue." The curriculum also states the probability formula (favorable outcomes ÷ total outcomes) and has students build probability models that use part-to-whole relationships (e.g., Dogs: 10/20 = 1/2).

Students solve multiple problems that require working with ratios and rates: they find unit rates (problems 14, 15, 19, 34), determine whether pairs of ratios are proportional (problem 16), and find constants of proportionality from a table (problem 17) and write proportional equations (problems 18, 34). Students also interpret linear relationships as proportional (problem 20 and its answer explaining the slope as "for every 1 unit increase in x, y increases by 4"). The answer key explicitly shows simplified ratio statements ("both ratios simplify to 2:3") and unit-rate answers (e.g., 1.6 miles per hour).

Students calculate slope from two points and explain it as a unit rate (e.g., "slope = 2 means the value increases by 2 units for every increase of 1 unit in x"). Students write functions for real contexts (y = 18h) and describe the slope as money earned "per hour" or "for each hour worked." Several answer keys and prompts ask students to interpret slope/rate of change in words and in context, using language like "per" and "for every."

Students are prompted to "use ratios to reason through each problem" in Activity 1 and to find scale factors when given corresponding side lengths (e.g., a triangle with sides 5, 7, 9 and a similar triangle with the side corresponding to 5 equal to 10, where students find scale factor = 2 and compute other sides 14 and 18). Multiple tasks require computing scale factors from side-length ratios (e.g., shortest side 6 becomes 15 so scale factor = 15 ÷ 6 = 2.5) and classifying dilations as enlargements or reductions based on whether the scale factor is greater or less than 1. The answer keys and parent notes explicitly state proportional reasoning language such as "the side lengths are proportional (each side is doubled)."

Students are asked to find the slope of a line through two points, which requires computing a ratio of vertical change to horizontal change. Students are asked to write a function for earnings when told "You earn $10 per hour," which involves the unit rate (10 dollars per 1 hour). Students must compute relative frequencies from a contingency table by dividing category counts by totals, which requires forming and calculating ratios as decimals.