Students identify the constant of proportionality k = y/x in tables (Activity 2) and use k to decide if a table is proportional. Students read graphs, pick points on lines that pass through the origin, and compute k = y/x, with explicit instruction that the line's slope is the constant of proportionality (Activity 3). Students rearrange equations into y = kx to find k (Activity 5) and solve real-world distance/time and rate problems using y = kx (Activity 6).

The lesson repeatedly defines proportional relationships as y = kx and identifies the unit rate k as the value at the point (1, k). Students build tables from equations and plot points (Day 2 Graphing Equations and Activity 2), and they are asked to find k from graphs by reading (1, k). Activity 5 has multiple tasks where students compare proportional relationships using equations, tables, and graphs (pools, pizzas, cyclists) and decide which object has the greater rate by identifying the largest unit rate or steepest line.

Students write proportional equations in y = kx form (t = pn, a = rh) and complete tables to generate coordinate pairs (Activity 1, Activity 3). Students are asked to create and interpret graphs (Theme Park Tickets asks students to model cost and create a graph; Review Quiz includes graph interpretation questions and asks whether a graph represents a proportional relationship). The Review Quiz and its answer key require students to find unit rates from equations (d = 4.5t) and from graphs (Water Tanks Filling Over Time), and to compare which tank fills faster by noting the steeper line.

Students are asked to graph proportional equations (e.g., "Graph the equation y = 4x" and "Graph the equation y = 5x") and interpret graphs (multiple problems ask whether a graph is proportional and what points like (0,0) or (1,r) represent). Several problems require students to find unit rates from graphs or points (e.g., a line through (0,0) and (2,8), and "Hours Worked vs. Money Earned" with points given). Students also identify constants of proportionality from tables and write equations of the form y = kx from given k values.

Students calculate unit rates for lemons, sugar, and cups by dividing total cost by quantity and record those unit prices. They create and complete tables (number of lemons or cups vs total cost or tablespoons), write equations in the form y = kx for recipe scaling, and graph the data with axes labeled and lines drawn through the origin. Students are asked to compare multiple lines on the same graph (different stores/options), identify which line is steepest or flattest, and answer that steeper lines mean a higher price per lemon (interpreting slope as unit rate).

Students plot and identify proportional relationships by checking that a line is straight and passes through the origin (Activities 1 and 3). Students compute the unit rate by dividing y by x for points on a graph and use that value as the constant k in y = kx (Activities 2 and 4). Students translate between representations by making tables from equations, plotting equations (y = 3x, y = 2x, etc.), and writing equations from graphs (find a point, compute k = y/x, write y = kx).

Students identify the unit rate as the y-value when x = 1 and see that in equations of the form y = mx the coefficient m is the slope/unit rate. Tasks ask students to compare an equation y = 4x to a graphed walk where y = 3 at x = 1 and decide which is faster. Multiple activities require students to find slopes from tables (unit rate = y/x), from graphs using the two-point formula, and from equations, then compare two relationships (e.g., drivers, cyclists, cars with y = 50x and y = 60x).

Students draw right triangles between lattice points, count the rise and run, and compute slope using m = rise/run (Activities 1 and practice problems). Students draw multiple triangles on the same line and set up proportions comparing rise/run to show triangles are similar and that the slope is constant (Activity 2). Students complete problems that require calculating and comparing slopes on different graphs and consider real-world ramps to connect steepness to rate of change.

Students learn that y = mx is the special-case proportional form and that m represents slope/rate (e.g., the lesson states "If the y-intercept is 0, like y = 2x, this is a special case called y = mx" and several table problems (x:1,2,3,4; y:2,4,6,8) lead students to write y = 2x). Students practice interpreting slope as a unit rate in real-world contexts (hourly wage: m = 10 represents $10 per hour) and repeatedly graph equations, convert standard form to y = mx + b, and find slope from two points or a table. Many activities ask students to compare slopes (e.g., "Which line is steepest?", "Which increases fastest?", and Group tasks comparing equations and graphs).

Students are given multiple table-to-graph problems (babysitting, car rental, dog walker, lawn care) where they determine if the relationship is proportional, compute the slope/unit rate, write the equation (e.g., y = 10x, y = 50x), and graph the points. Students plot lines from given coordinates (Line A and Line B problems) and are asked to identify which line has the greater rate of change and which graph is proportional. Review prompts and answer keys explicitly have students interpret unit rates as slopes from tables and graphs and convert representations between tables, equations, and graphs (slope-intercept form tasks and equation-writing from contextual rates).

Students fill a distance-vs-time table for car, train, and plane using given speeds, write equations in the form y = mx for each mode (Car: y = 60x, Train: y = 80x, Plane: y = 400x), and plot all three lines on the same graph with time on the x-axis and distance on the y-axis. They compute rise/run for the train line and answer which mode is fastest or slowest by comparing slopes, and they are asked whether the lines are linear and proportional because they pass through the origin. The activities explicitly link the unit rate (m) to speed and use comparisons of graphs, equations, and tables to decide which option has greater speed.

Students create tables of values from linear equations (for example y = 2x + 1) by choosing x-values, computing y, plotting the (x,y) points, and connecting them to form a line. Students complete activities that match tables, ordered pairs, and graphs (e.g., y = 2x − 3, y = −2x + 3) so they represent the same relationship in multiple forms. Students are asked to observe that linear functions change at a constant rate (the examples state that when x increases by 1, y increases by a constant amount).

Students calculate rate of change using the formula (y2 − y1)/(x2 − x1) in tables and use that constant rate to identify linear relationships. Students fill tables and plot points from given equations (for example y = 2x and y = 2x + 4) and decide whether the plotted shape is a straight line or a curve. Student activities ask them to complete tables, compute ∆y/∆x, and label graphs as linear or nonlinear, and the answer key explicitly identifies a proportional-looking equation y = 2x in one graph.

Students are given several linear graphs and prompted to identify straight-line (constant) rates — for example, Graph B is described as a straight line meaning a constant rate like money earned per hour. In the Sylvia the Sloth and Turtle/Balloon activities, students are given unit rates (e.g., 4 ft/hour, 5 ft/hour; 2 m/min, 3 m/min, 6 m/min) and asked to compute positions over time, plot points starting at (0,0), and connect segments, which ties rates to the slope of each line segment. Part 3 includes scenarios that map a linear upward slope to "a student saves the same amount of money each week," reinforcing the idea that constant-rate straight lines represent proportional-like situations.

Students calculate slope from graphs, from two points using m = (y2 - y1)/(x2 - x1), from tables by treating rows as points, and from equations by identifying m in y = mx + b (Activities 4, 5, and 6). Students interpret slope as how much y changes when x changes and see unit-rate phrasing in the table example ("every time x increases by 1, y increases by 2"). Students practice identifying positive, negative, zero, and undefined slopes and computing rise/run on multiple graphing activities.

Students find slopes from tables of time and distance (Table to Equation) and explicitly interpret the slope as a rate (example: slope = 1/5 mile per minute → 0.2 miles/min → 12 mph). Students graph lines from equations and plot/provide equations for lines that pass through the origin (e.g., y = 5x) and convert among graph, table, and equation representations in multiple activities. Multiple tasks require identifying slope (m) and y-intercept (b) and using slope as the rate of change when x represents time and y represents distance.

Students identify and calculate slope (rate of change) from stories, tables, and graphs (e.g., Liam earning $6 per chore, the reading table with pages per hour, and the bike-rental graph where m = 3). Students write linear function rules in the form output = slope × input + starting value from tables and graphs (multiple activities require writing equations such as A = 6c + 12, P = 15h, and C = 3h + 5). The cooking activity and its answer key explicitly present proportional (directly proportional) relationships that pass through the origin and show students how ingredient amounts scale (e.g., F = 0.5s), connecting unit rate to slope in a proportional context.

Students calculate unit rates by dividing change in distance by change in time (Example 1: Alex 1.5 miles/30 min = 0.05 miles/min) and compute slope from two points on a graph (Bella using (0,0) and (30,2) to get m = 0.067). Multiple activities require students to compare rates and starting values across graphs, tables, equations, and verbal descriptions (e.g., comparing Jordan y = -3x + 100 to a table for Taylor). Several student tasks present distance-time contexts and ask which person is faster, directly interpreting slope as speed/unit rate from different representations.

Students calculate rate of change from a table of time and distance (e.g., Time: 1,2,3,4 and Distance: 60,120,180,240 with answer 60 miles per hour). Students match a distance–time story (car driving at 40 mph, stopping, then 60 mph) to one of three graphs and explain reasoning about slopes. Several items require comparing rates from different representations (e.g., comparing Function A given by an equation to Function B given in a table; streaming subscription lines asking which increases fastest). Students also write and interpret proportional equations for earnings (E = 12h, E = 15h) that represent rate per hour.

Students create and solve Red (graph) cards that ask them to identify slope, decide if a graph is linear, and match graphs to scenarios. Students create Blue (equation) cards that require identifying slope and y-intercept, matching equations to graphs or stories, and writing equations from situations. Students create Green (table) and Yellow (description) cards that ask them to identify slope from a table, write an equation from a story, and match descriptions to graphs; the parent plan explicitly directs students to compare two functions represented in different ways and determine which has the greater rate of change.

Students graph many linear equations in slope-intercept form (y = mx + b) and plot example lines such as y = 2x + 3 and y = -x + 3. The materials explicitly teach that m is the slope and b is the y-intercept and have students convert equations into slope-intercept form to compare slopes. Activities require students to compare slopes to determine whether two lines intersect, are parallel, or coincide (including tasks that decide the number of solutions without graphing by comparing slopes).

Students are asked to find slopes from two points and write equations in slope–intercept form (several activities require computing m from two given points and writing y = mx + b). The review quiz and activities include graphing and comparing lines such as y = 2x and y = 3x and plotting lines through the origin. Multiple practice pages require students to plot lines, estimate intersection points, and note slopes (e.g., Activity 2 and the Intersection Challenge ask students to derive equations from two points and graph them).

Students write and use equations in slope-intercept form (y = mx + b) when the lesson introduces Total Cost = Rate × Quantity + Fixed Amount and explicitly displays examples such as y = 20x + 50 and y = 30x. Students are asked to create or look at graphs (e.g., "Look at the graph on the following page of the price of apples" and "you can create a graph of both lines to see a picture of the solution") to see where two lines intersect. Students solve examples that compare rates (slopes) to determine which option is cheaper before and after the break-even point, as in the Sparkle Clean vs. Shiny Solutions and CinemaNow vs. StreamMore examples.

Students are asked to compute slopes from two points (e.g., find the slope through (2,3) and (4,7); multiple problems ask for slope from point pairs). Students graph linear equations (for example, graph y = 2x - 1 and several systems of linear equations) and find line equations from two points (several problems require deriving y = mx + b). Some problems and answer keys include proportional examples (e.g., y = x from points (0,0) and (4,4)) and many tasks require comparing slopes when determining parallel vs. intersecting lines.

Students write and graph linear cost equations that are proportional in several activities (e.g., Dormitory: C = 1200m; Rideshare: C = 1.25x; Meal Plan: C = 80w) and are instructed to label axes and plot both lines. Students calculate slopes as unit rates and interpret them as real-world rates (answer keys compute $80/week for meal plan, $0.60 vs $1.25 per mile for car vs rideshare, and $1.50/hour for StreamScape). Students compare representations by writing equations, completing tables, and matching those equations to graphs (e.g., phone plans table and graph, housing cost table and graph) to decide which option is cheaper or where lines intersect.

Students identify independent and dependent variables, label axes, and determine whether relationships are positive, negative, or none by analyzing scatterplots (several activity pages ask them to name the x- and y-variables and classify the relationship). Students draw and choose best-fit (trend) lines and use those lines to make predictions and compare strength of linear relationships (activities ask them to pick the best fit line, draw lines, and compare variability). One example graph pairs hours spent driving with miles driven and describes a direct, straight-line relationship, which students examine visually for linear patterns.

Students practice finding slopes and y-intercepts from lines of best fit and writing equations in y = mx + b (Activity 1 and multiple student pages). Students interpret slope as a rate in context (for example, "for each 1°F increase, 2 additional ice cream cones are sold") and use those rates to make numerical predictions. Students match scatterplots with best-fit lines to given equations and write equations from plotted trend lines, using slope as the unit change in y per one unit of x.

Students create and analyze scatterplots by labeling axes, plotting provided data, and answering questions about linearity and trend (multiple activities ask students to draw scatterplots and identify linear vs. nonlinear relationships). Students are asked to select or write linear equations that fit plotted data (e.g., choosing between y = 2x + 4 and y = 4x or writing y = 4x + 2), and students compare multiple lines on a graph to decide which line best represents a relationship (lines A, B, and C for sunglasses or sunglasses/ sunshine problems). Students also interpret slope and intercept in context in the parent-plan skills list and apply slope-related reasoning when matching graphs to scenarios and predicting values from scatterplots.

Students make scatterplots from numerical data, label axes, choose scales, plot points, and draw or informally fit a line of best fit (Part 4 scatterplot instructions and images). The Parent Plan skills list explicitly tells students to informally fit a straight line to scatter plots and to use the equation of a linear model to solve problems and interpret the slope and intercept (with an example interpreting slope in context). The Part 5 reflection and summary pages ask students to describe the correlation and explain what the trend (slope) means for their situation (e.g., jumping jacks vs heart rate).

Students graph proportional equations (e.g., graph y = 6x), label points including (0,0) and (1,r), and are asked explicitly what those points represent. Students identify constants of proportionality and the equivalence of the constant and slope in True/False and matching tasks (Activity 2 and Activity 3). Students compute and compare unit rates from real-world situations (Activity 1 runner and cyclist problems) and match tables, equations, and graphs to determine proportional relationships and their k-values.

Students work with tables, equations, and graphs that show proportional relationships (Activity 2 babysitting and car rental tables ask if relationship is proportional, find the slope/unit rate, write an equation, interpret (0,0), and graph). Students find slope and write equations in slope-intercept form from points, tables, and graphs (Activity 3 asks for slope from two points, y-intercepts, and writing y = mx + b). Students compare rates across different representations and determine which relationship has the greater rate of change (Activity 4 compares y = 3x and y = 5x, phone plans y = 20x vs y = 15x + 30, and job offers given as equations and graphs).

Students compute unit rates in multiple problems (e.g., #14 find unit rate for 4/5 mile in 1/2 hour; #15 runner travels 18 miles in 3 hours). Students identify constant of proportionality and write proportional equations (e.g., #17 asks if a table is proportional and to find k; #18 asks for an equation when k = 3/2). Students graph and interpret proportional lines (e.g., #20 graph y = 4x and describe the slope as 4; #19 and #37 ask for unit rate/slope from points through the origin; #35 has students graph two lines from points and determine which has the greater rate of change).

Students compute slope from two points (e.g., (2,1) and (6,9)) and from equations (e.g., y = -1.25x + 5), graph linear equations (e.g., y = 3x - 2), and find and interpret y- and x-intercepts. Students write and interpret linear functions from contexts (e.g., y = 18h for $18 per hour) and compare rates by finding slopes for functions given in different forms (e.g., y = 6x - 2 and points (0,1),(1,5),(2,10)). Several tasks ask students to explain the meaning of slope in real-world contexts and to determine which function has the greater rate of change without graphing.

Students analyze scatterplots and lines of best fit in Activity 3, answering questions about types of correlation, interpreting trends, and what it means when points are close to a line of best fit. The Parent Plan explicitly states students will "use the equation of a linear model to solve problems... interpreting the slope and intercept," and a web link in Activity 3 is titled "Write an Equation for a Line of Best Fit." These items show students practice fitting linear models and interpreting slope in context.

Students calculate slope directly (Problem 5 asks for the slope of the line through (2,4) and (16,12) and the answer key gives slope = 2). Students graph linear equations (Problem 6: graph y = 2x − 4 and several graphing tasks are provided, including a plotted line image). Students write a proportional equation from a unit rate (Problem 8: "You earn $10 per hour. Write a function" with answer y = 10x).