Students multiply the number of each item per goody bag by 12 to find total items, showing they use multiplicative scaling (e.g., 5 candy bars × 12 = 60). Students divide the grand total cost by 12 to compute cost per goody bag, demonstrating calculation of a unit rate. Students use the distributive property to show totals (for example 12(5+3) = (12×5)+(12×3)), reinforcing multiplicative relationships between quantities.

Students practice creating and identifying equivalent ratios in Activity 2, where they are instructed to write two equivalent ratios for given ratios and to determine whether provided ratios are equivalent and explain their reasoning. Multiple word problems (bread recipe, cleaning solution, Kendyll's hours) require students to scale quantities up or down and to check that the ratio remains the same (e.g., multiplying/dividing both terms). The 'Think About It' blueberry pie question asks students to reason whether doubling one quantity requires changing the other to keep the same ratio.

Students practice writing and simplifying ratios (for example, Ellen sells 12 red pens for every 21 blue pens and students find the smaller equivalent ratio 4:7). Students scale ratios to find equivalent pairs (for example, given a 2:5 boys-to-girls ratio, students compute that 6 boys correspond to 15 girls). Students represent ratios in multiple forms (1 to 2, 1:2, 1/2) and draw pictures to model ratio situations and compare part-to-part, part-to-whole, and whole-to-part relationships.

Students make and complete tables of equivalent ratios (Activity 3 and multiple student pages show tables such as (1,6), (3,18), (5,30) and (5,2),(10,4),(15,6)). Students plot the ordered pairs on coordinate grids and are instructed to draw a line through the points (several problems state "plot the points... and draw a line through the points"). Students also perform ratio/fraction scaling to show equivalence (answer key shows fraction multiplication like 1/6 × 4/4 = 4/24), demonstrating testing for equivalent ratios in table form.

Students practice forming equivalent ratios and finding unit rates by dividing the numerator by the denominator (e.g., Elena: 216 miles in 4 hours → 54 miles per hour; Jalen: 156 calories in 12 chips → 13 calories per chip). Students use visual tools such as double number line diagrams and tape diagrams to create equivalent ratios and solve unit rate problems (described in Activity 1 and practiced in Activities 2 and 3). Students are given a table of hours vs. calories and instructed to create a graph from that table and answer questions about values (Unit 3 Quiz 1, problem 7).

The lesson defines a percentage as a part-to-whole ratio with the whole equal to 100 and has students convert percents to fractions by writing the percent over 100 and making equivalent fractions (e.g., 3/10 = 30/100 = 30%). Students also perform ratio scaling in the Basic Skills Review (e.g., Jamie's team won 5 games for every 1 lost, then find wins if losses = 3). Several activities require forming and recognizing equivalent ratios when converting among fractions, decimals, and percents.

Students create double number line diagrams and use equivalent ratios to solve percentage problems (Activity 1 and multiple practice problems). Several tasks require writing part/whole ratios and making equivalent ratios to find missing values (e.g., "48 is 12% of what number?" and other student activity pages). The Parent Plan and Skills sections explicitly note using ratio and rate reasoning and finding equivalent ratios.

Students set up and solve unit conversion problems by creating equivalent ratios (Activity 2 examples: 1 quart = 2 pints used to find 3 quarts = 6 pints; 1 cm = 10 mm used to find 19 cm = 190 mm; 1 oz ≈ 28.35 g used to find 5 oz ≈ 141.75 g). The lesson directs students to use double number line diagrams and to multiply/divide both parts of a ratio to produce equivalent ratios (Student Activity Page and Answer Key show multiple worked problems). Students are asked to determine the unit conversion ratio and use it to solve each problem, which explicitly practices forming and using equivalent ratios.

Students are asked to make tables of equivalent ratios, find missing values, and use those tables to compare ratios (listed explicitly in the Skills and practiced in multiple activity pages). In Problem 10 (apples and pies) students complete a table of values (10 apples → 3 pies, 20 → 6, 40 → 12) and plot those pairs on a coordinate grid labeled Apples (x) and Pies (y). Several problems require students to complete tables and use them to scale quantities (e.g., find apples for 24 pies or pies from 30 apples) and to use graphing and diagram methods (tape diagrams, double number lines) to solve ratio problems.

Students set up and compute ratios to find unit prices (e.g., dividing total cost by number of ounces) and record those ratios in data tables. The materials instruct students to use equivalent ratios to convert units (gallons → quarts → pints → cups → fl oz) so that comparisons use the same unit. Students also use equivalent-ratio reasoning to scale recipes (doubling/tripling) and to convert currencies using dimensional analysis.

The lesson's "Independent and Dependent Variables" unit asks students to list and graph ordered pairs of distances and times and to write equations such as d = 65t, explicitly connecting tables, graphs, and equations. Activities ask students to create tables based on equations and to identify the dependent and independent variable and how y changes as x increases or decreases. Several activities reference using graphs and tables to analyze relationships between two quantities (motion/speed, money/work).

Students create input/output tables and graph equations of the form y = kx (examples: Kevin's earnings 10x = y and Ron's biking 5x = y) and record solution pairs such as (1,10), (3,30), (1,5), (2,10). Students plot those pairs on coordinate grids and see lines that begin at the origin for those examples. The lesson also has students make tables and graphs for many two-variable equations and notes that linear equations produce straight lines.

Students create tables of ordered pairs and graph linear equations on coordinate grids (e.g., rewrite y + 5 - 2x = 6 as y = 2x + 1, list points, and plot the line). Students also rewrite equations (x - y = 8 to x - 8 = y), complete tables of values, and identify points that are and are not solutions on the graphed line. Multiple activities require students to list and graph ordered pairs that show the relationship between independent and dependent variables.

Students are asked to create at least one independent/dependent variable equation with a corresponding table and to describe the relationship (e.g., 'as x increases/decreases, y ______'). The lesson includes the explicit example 2x = y (practice piano twice as long as soccer) and requires students to answer questions for two-variable equations and fill a table. The project checklist and parent plan explicitly list 'Use variables to represent two quantities in a real-world problem that change in relationship to one another' and require an equation with a corresponding table and description.

Students practice equivalent-ratio reasoning in the Basic Skills Review problem that asks how many gumdrops a machine can make in 1 minute given 208 gumdrops in 4 minutes, explicitly instructing them to use equivalent ratios. The review also includes percent/ratio reasoning in the pan-price problem, which requires using a fractional/percent relationship to compare costs.

Students measure three die-cut circles, record circumference and diameter in a table, and compute the quotient C/d for each circle (Activity 1). They compare the three C/d values, observe they are all approximately 3.14, and identify that the constant ratio (pi) shows a consistent proportional relationship between circumference and diameter. Activity 2 further has students manipulate the equation π = C/d algebraically to produce C = πd, reinforcing that C and d are related by a constant multiplier.

Students compute and simplify ratios to find scale factors for pairs of figures (e.g., trapezoid 15/5 = 3:1, Mia's garden 8 cm/2 m simplified to 4:1) and use equivalent ratios and double number lines to determine corresponding measures. Activity worksheets ask students to find ratios and convert them to percents for multiple shape pairs (problems comparing triangles, squares, rectangles, circles) and to use proportions to find unknown lengths (e.g., 1 in : 5 ft = 6 in : n ft). Students also draw original and scaled rectangles on a coordinate grid to produce and compare enlarged and reduced figures.

The lesson asks students to compute and use scale factors (e.g., find the scale factor for a photograph reduction, use a 1/3 scale to draw a rectangle, and compute scale factors for perimeter and area). Students solve similarity/scale problems to find corresponding side lengths (e.g., determine side MN from CD = 4 cm, identify corresponding sides in triangles and trapezoids, and use a 5:1 enlargement for a kite). Students also draw triangles and shapes on coordinate-style grids when creating scaled or classified figures.

Students plan and create scale drawings using a given scale factor (Create a Scale Drawing activity) and the Parent Plan explicitly states students will "Reproduce a scale drawing at a different scale." The activities ask students to use a laminated grid to sketch shapes to scale and to apply a scale factor when making drawings, which requires applying consistent multiplicative relationships to lengths.

Students set up and solve proportions to find missing dimensions (e.g., using 1/4 = n/8 to find a cross-section length and 1/2 = 4/8 in a scale-factor problem). Several activities require creating and solving equivalent-ratio equations to find unknown lengths (Activity 2 and cross-section problems). Students also use division to scale quantities (e.g., dividing total volume by single-item volume to find how many items fit).

Students write part-to-whole ratios for candy colors and use equivalent ratios to calculate percentages (e.g., 3:10 scaled to 30%). A Basic Skills problem has students take the ratio 600:8 and reduce it to a unit rate 75:1. Several activities ask students to record frequencies and percentages, reinforcing ratio and proportion computations.

The "Working with Ratios" page asks students to write ratios in multiple forms and to find equivalent ratios (e.g., turning 3:5 into 15:25) and to identify ratio types. Problems ask students to scale ratios to solve word problems (making 15 necklaces from a 3:5 rate) and to find unit rates by division (train speed and unit price of apples). The parent notes and answer key explicitly show students creating and using equivalent ratios and computing unit rates.

Students compute and apply scale factors in the Circles and Scale Drawings activity (Problem 3 asks for enlargement by a scale factor of 2/1). Students write and compare corresponding ratios to find a scale factor in Problem 4 (answer key shows 4/12 = 1/3 and 2/6 = 1/3). Students also use proportional scaling in Problem 5 by converting 800% to a multiplicative factor and applying it to a linear measure.

Students compute quantities that scale (e.g., heater energy per hour, then per day and per year), fill tables of energy produced per day and per year for solar and wind, and multiply unit rates (cost per kg × monthly supply weight) to find total supply costs. The activities require students to complete and compare data tables for multiple locations (energy and supply tables) and to use given unit-rate values (like cost per kg and fuel-cell energy 7×10^2 kWh) to calculate totals.

Students practice deciding whether two quantities are proportional by writing relationships as fractions and testing for equivalent ratios (Activity 3: Is it Proportional? includes examples, problems, and answer key showing simplification to check equality). Students set up labeled fractions and determine proportionality in real-world word problems (Activity 4 and multiple problem pages require writing ratios with labels and solving for unknowns). Students repeatedly use equivalent-fraction methods and cross-multiplication to confirm proportional relationships across several practice pages and review activities.

Students repeatedly compute unit rates (e.g., cost per item, miles per hour) in Activity 1 and Activity 2 and use division of quantities to find a rate. Activity 6 explicitly presents two solution methods including "Method 2: Proportions" and shows setting up equivalent fractions (e.g., 2/4 = n/10) and solving for unknowns. Complex fraction activities have students divide fractional quantities (e.g., (3/4) miles ÷ (1/2) hour) to produce rates, reinforcing the idea of a constant rate.

Students are asked to compute k = y/x for tables (Activity 2) and to mark "YES" or "NO" when every pair gives the same k, directly practicing the equivalent-ratio test for proportionality. Students examine graphs (Activity 3) and are instructed to determine whether a line passes through the origin and to pick points on the line to compute k, directly using the straight-line-through-origin criterion. Students also rewrite and analyze equations into y = kx (Activity 5) and classify relationships as proportional or not, which reinforces deciding proportionality across representations.

Students are asked to test tables for equivalent ratios (Activity 6: "Is it a Proportional Relationship?" asks students to check whether all ratios are equivalent) and to graph points on coordinate planes and judge whether the line is straight and passes through (0,0) (Activity 7: "Graphing to Test Proportionality" and multiple "Analyzing Graphs" tasks). Multiple student activity pages require plotting table values, drawing a line through the points, and answering "Does the graph show a proportional relationship?" with justification. Answer keys and parent notes explicitly direct students to use y = kx and the point (1,k) to find the unit rate and to use both equivalent-ratio checks and graph-through-origin tests.

Students complete numerous tables (movie tickets, pizza, concert tickets, coffee beans, bikes, apples, painting, calories burned, driving, factory output) and are instructed to test proportionality by checking for a constant ratio and filling missing values. Students write equations in the form y = kx or t = p·n (e.g., t = 12n, t = 15n, c = 8m, a = 15h) to represent relationships and use unit rates to decide proportionality. Students interpret graphs and are explicitly told that a proportional relationship makes a straight line through the origin; quiz items ask whether given tables and graphs are proportional and to identify unit rates from graphs.

Students set up and solve equations that use proportional reasoning (e.g., Sales Tax = Price × Tax Rate, Commission = Sales Amount × Commission Rate) and compute values by converting percentages to decimals and multiplying. Activities require students to work forward and backward with percent relationships (examples: 0.012 × V = 2400 to find property value; 1.08 × x = 212 to find pre-tax price). The Parent Plan and multiple activity pages instruct students to apply proportional reasoning to multistep percent problems such as tax, tips, and commissions.

Students practice and apply the formula I = Prt to calculate interest, solve for missing values (rate, time, principal), and compute balances in multiple word-problem contexts. The Parent Plan explicitly states students will "use proportional relationships to solve multistep ratio and percent problems" and that activities will "strengthen his understanding of proportional reasoning." Several activity pages walk students through solving for one quantity when the others are held constant, which relies on proportional reasoning.

Students are asked to test pairs of ratios for proportionality (e.g., questions asking whether 4:5 and 8:10 are proportional and whether 6:8 and 9:12 are proportional). Several problems require students to decide if values in a table are proportional and to identify the constant of proportionality from tables (multiple table problems and items asking students to circle "Yes, it's proportional" or "No"). Students graph equations (e.g., graph y = 4x, y = 5x) and examine given graphs with plotted points (including points through (0,0)) to determine whether the relationship is a straight line through the origin and thus proportional.

Students create tables of number-of-lemons versus total cost and fill in values for 5, 10, and 20 lemons, providing data to compare ratios. Students are instructed to graph number of lemons (x) against total cost (y), plot a line for each store, label axes, and answer questions about which line is steepest and which is least steep. In the recipe scaling activity students complete tables, graph the data, and explicitly write equations in the form y = kx; the answer key shows the graph as a straight line beginning at the origin and the equations y = 2x and y = 1.5x. The Part 3 worksheets ask students to set up proportions (ratio equations) to find amounts and use unit rates in cost calculations.

Students analyze multiple graphs and are prompted to decide whether each relationship is proportional by checking for a straight line that passes through the origin (Activity 1 and Student Activity Pages A–D). Students create tables of values from equations (Activity 3: Turning Equations into Graphs) and plot the points to determine proportionality. Students are taught to find the unit rate by dividing y by x from a point on the graph and to write y = kx from a graph (Activities 2 and 4), which provides a numerical test for proportionality.

Students calculate unit rates from tables by dividing y by x (Activity 2 table method and multiple student pages ask for y/x to find speed or price). Students graph equations written as y = mx and plot points including (0,0), then compare slopes to decide which rate is greater (Activities 1, 4, 5, and Day 3 notes instruct plotting (0,0) and labeling y = mx). Students are asked to determine unit rate from a graph by reading the y-value at x = 1 and to recognize that proportional relationships produce straight lines through the origin (multiple activity instructions and parent plan statements).

Students identify x- and y-intercepts from graphs and write them as ordered pairs, including examples where the y-intercept is 0 (the origin). Students set y = 0 to find x-intercepts and set x = 0 to find y-intercepts and solve given linear equations (e.g., 2x + 4y = 8) to compute intercepts. Students graph lines using the two intercepts and practice plotting lines that include examples of lines crossing the origin.

Students draw right triangles between pairs of lattice points on coordinate grids and count rise and run to calculate slope. They set up and compare ratios of rise/run for different triangles and use proportions (e.g., 2/2 = 4/4) to decide whether triangles are similar. Students graph lines and use these ratio comparisons to determine whether different point pairs lie on the same straight line.

The lesson explicitly identifies y = mx as the special case that is a proportional relationship and states that such a line "passes through the origin." Students graph many linear equations, convert tables to equations (Activity 7), and compute slope using change in y/change in x from tables and pairs of points (Activity 5 and Activity 7). Multiple activities ask students to plot lines and compare their position relative to the axes (e.g., noting y-intercepts and whether lines shift up/down).

Students are given multiple tables of paired quantities (hours vs. earnings, days vs. cost) and are asked explicitly "Is this relationship proportional?" and to graph the data. Students calculate slope and y-intercept for those tables and graphs and write equations such as y = 10x, y = 50x, and identify when the y-intercept = 0. Students also compare graphs and equations to determine which are proportional (e.g., selecting y = 3x + 7 as not proportional and noting that proportional lines pass through the origin).

Students complete a table of time (hours) and distance (miles) for car, train, and plane speeds, giving data they can use to test equivalent ratios. Students write equations in the form y = mx for each mode (car: y = 60x, train: y = 80x, plane: y = 400x). Students plot all three on a coordinate grid labeled with time on the x-axis and distance on the y-axis and answer questions that explicitly ask whether the lines represent proportional relationships and why (noting lines pass through the origin and slopes equal unit rates).

Students record counts of outcomes for 50- and 100-spin trials and compute experimental probabilities using the formula (number of times it happened / total number of spins), expressing results as fractions, decimals, and percents. The activity provides tables for spinner results (50 and 100 spins) where students fill tallies and totals, and a Student Activity Page guides them to calculate experimental probabilities from those tables. Students then use a scaling step (Prediction = Experimental Probability × 600) to predict counts for 600 spins, applying a constant ratio to scale results.

Students compute theoretical probability as a ratio (favorable outcomes / total) and use that ratio to predict expected counts for different numbers of trials (e.g., 1/6 × 30 = 5, 1/6 × 60 = 10, 1/6 × 120 = 20). Students calculate relative frequency as count/total and build probability models from observed data (e.g., 16/34 for choir ≈ 0.47). Students apply proportional scaling when making predictions across varying trial sizes and when converting probabilities to expected counts.

Students plot points from rate-based descriptions (e.g., Sylvia the Sloth starts at (0,0), climbs 4 ft/hr for 3 hours, naps, then climbs 5 ft/hr) and connect them to form line segments. Students match scenarios that describe constant rates (e.g., "earning the same amount of money every hour" or "saving the same amount each week") to straight, upward-sloping graphs. Students are asked to identify whether a graph is straight (linear) or curved (nonlinear) and to note when a line shows a constant rate of change.

Students compute slope from tables by choosing two rows and using the formula m = (y2 − y1) / (x2 − x1) (Activity 5). Students plot points and find slope from graphs including examples that start at the origin (e.g., (0,0),(2,1),(4,2)) and practice identifying consistent rise/run patterns. Students also identify the slope m in y = mx + b and practice rearranging equations into slope-intercept form (Activity 6). The parent/skills notes mention deriving y = mx + b for a line through the origin.

Students repeatedly compute slopes (rates) from tables and pairs of points (e.g., Table to Equation example finds m = 1/5 and identifies the rate as miles per minute) and identify y-intercepts from tables and graphs (several activities direct students to find b by checking x = 0). Multiple activities require students to graph lines from equations and to rewrite standard-form equations into y = mx + b so they can plot the line using slope and the y-intercept.

Students read and interpret tables (e.g., the Pages Read table) and compute a constant rate of change (15 pages per hour) and then infer the y-intercept of 0 to write P = 15h. They use the slope formula on graphs (e.g., the bike rental graph with points (0,5), (1,8), (2,11)) to find slope and y-intercept and write linear functions. The cooking activity explicitly states that ingredient amounts double with servings and describes this as a direct proportional relationship whose graph is a straight line through the origin. Multiple student tasks require writing functions from tables and graphs, including examples that result in equations of the form y = mx (origin) and y = mx + b.

Students compute rate of change (slope) from tables, graphs, and verbal descriptions (e.g., Alex: 1.5 miles/30 min = 0.05; Bella: slope from (0,0) and (30,2) = 0.067). Students identify the starting value (y-intercept) as the value when x = 0 and use it to compare who started ahead (Jordan's y-intercept = 100; Taylor's table gives starting value 275). Multiple activities ask students to compare pairs of functions presented as tables, graphs, equations, or descriptions and determine which has the greater rate of change or higher starting point.

Students compute rate of change from tables (e.g., a train table: time 1,2,3,4 with distances 60,120,180,240 and rate = 60) and write linear functions from contextual situations (e.g., E = 12h, E = 15h, y = mx + b). Students identify y-intercepts and x-intercepts from equations and graphs and determine whether a graph is linear or nonlinear by checking if it is a straight line. Several tasks ask students to compare rates of change between two functions (given algebraically and in a table) and to note starting values (y-intercepts).

Students create and analyze tables on the green cards (drawing tables with x- and y-values, identifying slopes, completing missing values, and writing equations from tables). Students create and analyze graphs on the red cards (drawing or interpreting small graphs, deciding if a graph is linear or nonlinear, and identifying slope). The yellow and blue card examples include story problems and equations that lead students to write relationships such as "Emma earns $10 for each hour" (which yields y = 10x) and to identify slopes and y-intercepts.

Students compute and use scale factors to compare side lengths (for example, finding scale factor 6 ÷ 3 = 2 and using it to find a missing side). Students identify that corresponding sides of similar shapes are proportional and solve problems by multiplying or dividing by the scale factor (several activity pages ask for scale factor and missing side x). Examples show pairs of triangles with side measurements (e.g., 4→8, 5→10, 6→12) and ask students to write corresponding side relationships and use them to find unknown lengths.

Students calculate new x ÷ original x and new y ÷ original y for corresponding points (step-by-step tables and examples show these divisions). Students use those ratios to decide whether all coordinates are multiplied by the same scale factor (multiple worked examples label a transformation as a dilation when all ratios match). Students plot original and transformed points on the coordinate plane and apply the rule "(x, y) → (scale factor × x, scale factor × y)" when performing dilations.

Students work with similarity and scale factors: the lesson asks them to decide whether pairs of triangles are similar using angle-angle and to find scale factors from one triangle to another (Quiz items: "Find the scale factor from Triangle Y to Triangle X", "Multiply each side by the scale factor"). Students perform coordinate dilation tasks (e.g., "dilate the triangle by a scale factor of 2" and find final coordinates) and are given explanations that dilations make side lengths proportional while preserving angles. The lesson answer key explicitly states that dilations keep angles the same and keep sides proportional.

The lesson explicitly states multiplicative relationships between volumes (e.g., "a cone holds exactly one-third as much" as a cylinder) and has students use the cone formula V = (1/3)πr^2h to compute and solve for missing measures. The lesson derives the sphere formula by showing a sphere has 2/3 the volume of a corresponding cylinder and has students compute sphere volumes and solve for radius from a given volume. Students apply these scalar relationships in worked examples, practice problems, and the design challenge when choosing dimensions and computing capacities.

Students are asked to identify whether pairs of figures are dilations and to determine scale factors (e.g., Exercise 9 asks "Is this a dilation?" and to find the scale factor; multiple problems give original and image vertices showing coordinates scaled by 2). Problems ask for new side lengths after dilation (e.g., "Triangle DEF is dilated by a factor of 2; DE = 8; find D'E'" and squares with side lengths and a given scale factor). Several coordinate tasks present corresponding point pairs (A,B,C and A'B'C') where students must identify the dilation and compute the scale factor from coordinates.

Students write linear equations in slope-intercept form (y = mx + b) and set up equations of the form y = mx in multiple examples (e.g., y = 30x, y = 5x, y = 1.50x). Students solve systems of linear equations and are sometimes prompted to create graphs of both lines to 'see a picture of the solution' when finding break-even/intersection points. Several student activity pages give equations and ask students to solve and compare total costs for different x values.

Students write linear cost equations for real situations (e.g., Dorm: C = 1200m; Rideshare: C = 1.25x; Meal Plan: C = 80w) and complete tables of values for those equations. Students graph those equations on coordinate grids, label axes, plot lines, identify intersection (break-even) points, and analyze y-intercepts (e.g., noting groceries start at $100 while the meal plan starts at $0). Students also complete a table of values for phone plans and simplify equations to compare lines algebraically (showing when two equations represent the same line).

Students practice identifying linear versus non‑linear relationships by labeling scatterplots and deciding whether points form a straight‑line pattern. They draw and choose best‑fit lines, judge whether about half the points fall above and below the line, and compare graphs for low or high variability. Students also identify independent and dependent variables and match scatterplots to real‑world scenarios to decide if two quantities move together (positive/negative/no relationship).

Students repeatedly create and analyze scatterplots, identify linear vs. nonlinear relationships, and determine positive/negative association (multiple exercises ask them to label axes, draw scatterplots, and decide if a relationship is linear). Several items ask students to select or write linear equations that model scatterplots (for example choosing between equations like y = 4x + 0 and y = 2x + 4, and writing equations from plotted points). The parent/skills section and some problems ask students to informally fit a straight line and interpret slope and intercept in context.

Students are guided to create scatterplots: they label axes, set scales, plot paired numerical data points, and answer analysis questions about whether the dots form a pattern or trend. The materials prompt students to draw an informal line that best fits the data and to describe positive, negative, or no correlation. The Parent Plan/Skills section also references fitting straight lines and interpreting linear models (slope and intercept).

Students test whether pairs of ratios are proportional by computing equivalent fractions (Activity 1 asks whether 8:12 and 14:21 are proportional and to explain why). Students decide if given relationships are proportional and identify the constant of proportionality from tables and options (Activity 1 Q4 and Activity 2 matching tasks ask 'Is this relationship proportional?' and to write proportional equations). Students graph proportional equations (e.g., graph y = 6x, label points, and circle the origin) and analyze graphs to determine whether they represent proportional relationships (Activity 3 asks students to identify which graphs are proportional and to explain that proportional graphs pass through the origin).

Students are asked directly whether relationships are proportional in table problems (e.g., Babysitting Pay and Car Rental Costs) and to compute the slope/unit rate and write equations (Activity 2). Students graph those relationships and are asked whether the graph passes through the origin and what that indicates (Activity 2, Activity 4). Students compare lines given as y = kx and y = mx + b (e.g., Plan A y=20x vs Plan B y=15x+30, Job A vs Job B) and identify which are proportional and which are not.

Students compute unit rates (problems 14, 15, 19, and 34) and write proportional equations (problem 18 and 34). Students are asked to decide if pairs of ratios are proportional and to explain (problem 16) and to determine whether a table represents a proportional relationship and find the constant of proportionality (problem 17). Students graph lines that pass through the origin and interpret proportionality from the graph (problems 19, 20, and 37).

Students work with tables of (x,y) pairs and are asked to interpret and use those values (for example, tables with points like (0,0), (1,5), (2,10) and other tabulated x-y values). Students graph linear equations, find and interpret y- and x-intercepts, and calculate slopes/rates of change from points and equations. Students also compare rates of change between two functions and decide whether relationships are linear or nonlinear from equations, tables, and graphs.

Students compute and apply scale factors to determine similarity (e.g., given triangles with sides 5, 7, 9 and a corresponding side 10, students find the scale factor = 2 and compute the other sides). Students use ratios to reason about dilations (sections ask whether a dilation with scale factor 1/2 is an enlargement or reduction and to justify using scale-factor reasoning). Students plot figures and perform dilations on a coordinate plane (e.g., draw Triangle LMN with vertices L(2,1), M(4,1), N(3,3) and dilate it by factor 3 about the origin to find L'(6,3), M'(12,3), N'(9,9)).

Students graph linear equations (e.g., "Graph the equation y = 2x − 4", solving systems by graphing y = x + 2 and y = −x + 6) and compute intercepts (Find the y-intercept of y = 3x + 4; find x- and y-intercepts in #9). Students work with a table of (x,y) values to find an x-intercept (question 4) and compute slope between two points (question 5). Problem 8 has students write a unit-rate function for earnings (answer key: y = 10x), which is an example of a proportional equation.