Students pick points on a line that passes through the origin and compute k = y/x to find the constant of proportionality. Students rewrite equations into the form y = kx (examples and practice problems show converting 4y = 8x to y = 2x and identifying k in several equations). Students identify that the slope (k) is the rate of change for straight lines through the origin and distinguish equations with an extra constant term (e.g., y = 2x + 3) as not proportional.

Students repeatedly build tables of values, plot points, and write equations of the form y = kx (e.g., Jim's earnings y = 12x, examples y = 4x and y = 5x). Students practice finding the unit rate as the y-value when x = 1 and connect graphs, tables, and equations by plotting (0,0), (1,k), etc. Activities ask students to draw straight lines through the origin, label axes, and write the equation y = kx from data and graphs.

Students set up and use equations in the form y = kx (and t = p × n) to model proportional situations and write equations for tables and word problems. Students compute unit rates, identify that a proportional graph is a straight line through the origin, and interpret points like (1,y) as the unit rate. Students complete activities and quiz items that require writing equations in y = kx and explaining why proportional relationships form straight lines through (0,0).

Students are asked to identify and use the constant of proportionality in tables and equations (e.g., problems that ask for the constant from tables and to write equations such as y = (3/5)x and y = (2/7)x). Students graph and describe proportional equations (e.g., tasks to graph y = 4x and y = 5x and to decide whether graphs are proportional by checking if they are straight lines through the origin). Students interpret key points on proportional graphs, including (0,0) and (1, r), and compute unit rates from graphs and point pairs (e.g., problems asking for unit rate from points like (0,0) and (2,8) or (3,12)).

Students collect cost data, create tables, and plot number of lemons (x) versus total cost (y) with labeled axes and separate lines for each store. They are asked to compare line steepness, interpret steepness as price per lemon, and identify proportional relationships by observing whether graphs are straight lines through the origin. Students write equations for proportional relationships in the form y = kx (e.g., y = 2x for tablespoons of lemon juice), connecting unit rate to the slope of a line.

Students repeatedly pick points on lines (often including (0,0) and another point) and compute the unit rate k by dividing y by x (Activity 2, Turning Graphs into Equations and Turning Graphs into Equations). Students make tables of values, plot points such as (0,0),(1,3),(2,6), connect them to form straight lines, and write equations explicitly in the form y = kx (examples: y = 3x, y = 2x, y = 1/2 x). Students also physically walk out points and compare how y changes when x increases, reinforcing that the rate of change is constant for lines through the origin.

Students repeatedly compute slope using the Two-Point Formula m = (y2 - y1) / (x2 - x1) and are instructed to "pick any two points" to find the same slope. Students write and graph proportional equations in the form y = mx (multiple activities ask them to create tables, plot (0,0), and label lines like y = 4x or y = 3x). Activities ask students to identify the unit rate as the y-value when x = 1 and to compare steepness of lines (unit rate = slope) across tables, graphs, and equations.

Students draw right triangles between pairs of lattice points, count the rise and run, and compute slopes using rise/run (Activity 1). In Activity 2 students draw different right triangles along the same line, set up proportions comparing rise/run, and determine triangle similarity to show the rise-to-run ratios match. The student pages and answer key provide examples and guided practice where students compare triangle ratios (e.g., 2/2 = 4/4) to conclude the slope is constant.

The lesson defines slope as m = (change in y)/(change in x) and has multiple activities where students compute slope from two points, extend lines using that slope, and graph lines in y = mx and y = mx + b form. Students convert equations into slope-intercept form (e.g., 3x + 2y = 8 → y = (-3/2)x + 4) and solve for b by substituting a point into y = mx + b, and they identify the special case y = mx when the y-intercept is 0. Multiple student activity pages and examples require students to write, graph, and interpret equations y = mx and y = mx + b from tables, point pairs, and real-world scenarios.

The Skills list in the Parent Plan explicitly includes the statement: "Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line...; derive the equation y = mx... and y = mx + b..." Students repeatedly compute slope using m = change in y/change in x, find y-intercepts, convert equations into slope-intercept form (y = mx + b), and write equations for lines that pass through the origin or have a nonzero intercept in multiple activity problems. Several activities ask students to graph lines, extend lines to identify y-intercepts, and write equations for lines given points (including lines through the origin).

Students plot distance vs. time for car, train, and plane and compute rise/run for a line (e.g., "What is the rise/run for line B (train)?" with calculation 480/6 = 80). Students write equations of the form y = mx for the speed activities (car: y = 60x, train: y = 80x, plane: y = 400x) and identify that these lines pass through the origin and represent proportional relationships. Later, students write and graph cost equations in the form y = mx + b (e.g., y = 0.15x + 31.50, y = 0.20x + 20.75, y = 0.50x + 63.25) and analyze slope (cost per mile) and y-intercept (fixed starting cost).

Students work with linear rules such as y = 2x + 1 and y = 2x − 3, compute outputs for given x-values, and plot the resulting points to see they lie on a straight line. The lesson explicitly states that linear functions change at a constant rate and gives examples showing that when x increases by 1 the corresponding y increases by a fixed amount (slope). Students also identify where a line crosses the y-axis (for example, ‘‘the line crosses the y-axis at 1''), linking the constant change to an equation of the form y = 2x + 1.

Students calculate rate of change using the formula (y2 - y1)/(x2 - x1) in multiple tables and use those calculations to determine whether relationships are linear or nonlinear. Students complete tables and plot lines for equations such as y = 2x + 4, y = 2x, y = x + 1, and y = x - 4, noting the constant change in y for a unit change in x and identifying the y-intercept (for example, (0,4) for y = 2x + 4). Students also label graphs as straight lines versus curves and identify slope values (e.g., rise of 2 for run of 1) from both tables and graphs.

Students repeatedly compute slope as rise/run and Δy/Δx using pairs of points, tables, and graphs (examples include computing m = (16−2)/(−3−4) = −2 and slope between (0,0),(2,1),(4,2) = 1/2). The material shows students rewriting equations into slope-intercept form y = mx + b and identifying m and b (examples include rearranging 4x+2y=−8 to y=−2x−4 and the special case y=−2x where b=0). Students practice picking any two rows of a table or any two plotted points to find the same slope, and they label slope and y-intercept from given equations.

Students repeatedly compute slope using the formula m = (y2 - y1)/(x2 - x1) from two points (Graph to Equation, Slope Intercept from Two Points) and identify the y-intercept where x = 0 (Graph to Equation, Table to Equation). Students substitute m and b into y = mx + b to write equations (examples: y = 2x + 2, y = 15x) and they isolate y to rewrite standard-form equations into slope-intercept form (Simplify & Graph). Students practice finding b by plugging a known point and slope into y = mx + b (Slope Intercept from Slope and Point) and then graphing from slope and intercept (Equation to Graph).

Students compute slope from two points using the slope formula (e.g., the bike-rental example uses (0,5) and (1,8) to get m=3). Students identify the y-intercept as the value at x=0 (examples: (0,5) gives b=5; table examples give b=0) and write equations in slope-intercept form (examples: C = 3h + 5, P = 15h). Students also produce equations for lines through the origin (P = 15h, cooking functions like F = 0.5s) and for lines with a nonzero vertical intercept (A = 6c + 12).

Students calculate slope by choosing two points and applying the slope formula (m = (y2 - y1) / (x2 - x1)), as in the Alex/Bella example where they use (0,0) and (30,2) and are told "you can use any two points on the graph to calculate slope." Students identify and use the form y = mx + b, interpret m as the rate of change and b as the starting value (y when x = 0), and apply these ideas across graphs, tables, equations, and verbal descriptions (e.g., Jordan: y = -3x + 100 and discussion that b = 100 is the starting amount).

Students repeatedly compute slope from two points and from graphs (several problems ask for slope and the answer key shows slope calculation from point pairs). Students identify and use y-intercepts and write equations in slope-intercept form (tasks ask for y-intercept, to write y=3x-1, y=mx+b, and model situations with E=12h or y=2x+17). The Parent Plan and review items explicitly instruct students to interpret the equation y = mx + b as defining a linear function and to determine rate of change and initial value from tables or graphs.

The Skills section explicitly states that students will "derive the equation y = mx + b for a line through the origin and the equation y = mx + b for a line intercepting the y-axis." Multiple activities require students to identify slope and y-intercept (blue cards), write equations from tables and descriptions (green and yellow cards), and match equations to graphs and stories. Gameplay and card construction require students to compute slopes from graphs and tables and to write corresponding linear equations as part of their answer keys.

Students identify and use the slope-intercept form y = mx + b throughout the activities, labeling m as the slope and b as the y-intercept and graphing lines by starting at b and using rise/run. Students convert equations (for example 4x − 2y = 8 to y = 2x − 4) into slope-intercept form to find m and b. Students compare slopes of two equations to decide whether lines intersect, are parallel, or coincide, and they practice plotting points such as (0,3), (1,5), (2,7) to show consistency of rise/run.

Students repeatedly compute slope from two given points using the slope formula (for example, (4-0)/(2-0)=2 and (2-6)/(3-1)=-2) and use that slope to write equations in slope-intercept form (y = mx + b). Several activities ask students to find b by plugging a point into y = mx + b (e.g., finding y = -2x + 8), and examples and problems include lines through the origin such as y = 2x. Practice problems and parent notes explicitly require students to write equations in the form y = mx + b and to graph lines given two points.

Students write linear equations in slope-intercept form and interpret m and b as rate and fixed fee (e.g., y = 20x + 50, y = 30x, y = 2x + 15). Students set those two expressions equal and solve systems (by substitution or elimination) to find break-even points and produce coordinate solutions like (5,150) or (5,25). The activity pages prompt students to define variables, write y = mx + b type equations from contexts, and use those equations to compare costs.

Students calculate slope from two points in multiple problems (for example, find the slope through (2,3) and (4,7); exercises asking for slope through (1,2) and (5,10)). Students graph equations in slope-intercept form (e.g., graph y = 2x - 1, identify slope 2 and y-intercept -1) and are asked to find line equations from two points (several problems require finding an equation given two points). Students also analyze pairs of equations to identify same slope / parallel lines (e.g., y = 3x + 1 and y = 3x - 5 labeled as parallel).

Students write linear equations in slope-intercept form across multiple contexts (e.g., housing C=1200m and C=1500+1050m; meal plan C=80w and groceries C=100+50w; streaming C=20 and C=5+1.5h; phone plans y=5x+20). Students compute slope from two points (meal plan slope computed from (0,0) to (2,160)), identify and interpret y-intercepts as fixed costs, and simplify or manipulate equations (e.g., simplifying Plan B to y=5x+20). Students graph lines, mark intersections, and solve systems to find break-even points, using these graphs to write and use equations of lines through the origin and with vertical intercepts.

Students are instructed to compute slope using the formula m = (y2 − y1)/(x2 − x1) and are explicitly told to "pick any two points on the line" to find m. Multiple activities require students to identify the y‑intercept b and write equations in slope‑intercept form y = mx + b, then interpret slope and intercept in context and use the equation to make predictions. Examples (e.g., the ice cream and bird migration tasks) show students substituting values into y = mx + b to solve problems and to write linear models from scatterplots.

Students collect numerical bivariate data, plot scatterplots, and are prompted to plot points and (optionally) draw a line that best fits the data. Students are asked to interpret a linear model using language about slope and intercept (Parent Plan: "Use the equation of a linear model... interpreting the slope and intercept") and to use the equation of a linear model to solve contextual problems. Students practice labeling axes, choosing scales, and describing positive/negative/linear associations, which supports understanding slope as a rate of change between paired points.

Students graph equations of the form y = kx (e.g., y = 6x) and are asked to label points and circle where the line crosses the origin (Activity 3). Students identify proportional graphs as those that pass through the origin, compute and compare unit rates (Activities 1–3), and answer true/false items including that the constant of proportionality is the same as the slope. Students also work with nonzero intercept equations (e.g., y = x + 2, y = x + 6) and are asked to interpret points such as (0, 6).

Students repeatedly compute slope from tables, graphs, and pairs of points (Activities 2 and 3 ask for the slope from a table, from two points, and from graphs). Students write equations in the forms y = mx and y = mx + b for proportional and non-proportional linear situations (Activity 2 answer keys show y = 14x, y = 75x; Activities 3 and 4 ask students to write y = mx + b and identify m and b). Students interpret the meaning of the origin and the y-intercept in context (questions ask what (0,0) represents and ask students to explain what the slope and y-intercept represent in real-world problems).

Students are asked to work with lines through the origin and proportional relationships (e.g., Unit 2 problem 19: line through (0,0) and (2,10); Unit 2 problem 18: write equation y=(3/5)x). Students graph and describe y = 4x (Problem 20) and are asked to identify slope and intercept for y = 3x + 8 (Problem 36). Students plot a line through (0,0) and (4,-8) and compute its slope and equation (Problem 37), and several answer keys state forms like y = 4x, y = 5x, and y = −2x, showing practice writing y = mx and y = mx + b in specific cases.

Students compute slope from two points and compare rates of change (tasks: find slope from (2,1) and (6,9); compare slopes of y=6x-2 and points (0,0),(1,5),(2,10)). Students identify and interpret y-intercepts from equations and graphs (tasks: find y-intercept of y=3x+4, set x=0 to find y-intercept from 2x+3y=12). Students write linear equations from contexts, including a line through the origin (y = 18h) and a line with a vertical intercept (y = 12x + 10).

Students practice finding slope from two given points (Activity 2, Part B asks for slopes between (3,2) and (7,10) and between (-2,4) and (-8,-8)) and graph lines given in slope-intercept form (graph y = x - 3 and y = -2x + 1). Students write and solve linear equations from real-world contexts that include a fixed fee plus a rate (Activity 4 constructs equations such as 25 + 15m = 100), which represent lines with slope m and y-intercept b. Students solve and manipulate equations in slope-intercept form across activities and solve systems where graphing and slope are used (Activity 3).

Students analyze scatterplots and are asked to interpret trends and lines of best fit (Activity 3). The Parent Plan explicitly says students use the equation of a linear model to solve problems and interpret the slope and intercept. The materials include a linked resource titled "Write an Equation for a Line of Best Fit," indicating students are prompted to form and use linear equations from data.

Students compute slope from two distinct points (question 5 asks for the slope through (2, 4) and (16, 12) with answer slope = 2) and graph linear equations (questions 6 and 32 ask students to graph y = 2x − 4 and to graph/solve systems including y = x + 2 and y = −x + 6). Students identify intercepts and write linear equations (question 3 asks for the y-intercept of y = 3x + 4, question 8 expects y = 10x for earnings, and question 9 asks students to identify x- and y-intercepts and state linearity). A coordinate-plane diagram shows a straight line through several points, supporting student practice with constant rate of change.