Students work with exponential functions given algebraically (N = 2^t and N = 3^t) and compute numerical populations after specified intervals (e.g., after 6 intervals and after 4 intervals). Students are explicitly asked to compare the growth of the two models and identify which bacterium grows faster. Students calculate and compare numerical outputs from the algebraic expressions to support their conclusions.

Students use numerical tables throughout the project (Energy Data Table and Supplies Data Table) to record and compare quantities for each location. Students apply algebraic formulas (for example A = πr^2 to compute drop zone area and converting 7 × 10^2 kWh for a fuel cell) to produce numeric outputs. Students compute and compare yearly energy production and energy shortfalls, and they calculate total supply costs for each location, producing numeric comparisons across sites.

Students calculate the constant of proportionality k from tables (Activity 2) by dividing y by x and decide whether a table is proportional. Students find k from graphs and pick points on lines, and they determine which of two lines (Line A and Line B) represents a proportional relationship and find its k (Activity 3). Students also compute and compare numeric rates in real-world contexts (e.g., Review Quiz problem comparing price per ounce for two cereal boxes).

Students compare multiple proportional functions using equations, tables, graphs, and words in Activity 5 (Comparing Proportional Relationships): e.g., Pool A/B/C are given as equations (y=500x, y=750x, y=600x), tables of values are built, graphs are drawn, and students determine which has the greatest rate. The pizza and cyclists examples similarly give tables, equations, and graphs that students use to calculate unit rates and compare steepness. Student activity pages ask learners to write equations, complete tables, plot graphs, and explain in words which function has the larger rate of change (e.g., printer, train, hiker problems).

Students calculate unit rates from equations (e.g., d = 4.5t, c = 8m) and identify constants of proportionality from tables in multiple activities. In the Review Quiz (Question 13) students compare two water tanks by reading their graphs to determine unit rates and decide which tank fills faster. Several activities require students to complete tables and then write equations in y = kx form from table data (e.g., apples, concert tickets, painting fence).

Students compute unit rates from numeric situations (e.g., find the unit rate for 3/4 mile in 1/2 hour and 5/6 mile in 2/3 hour) and from graphs (several "Hours Worked vs. Money Earned" graphs ask for unit rates). Students identify constants of proportionality from tables (multiple tables ask whether the relationship is proportional and to find the constant k) and write or recognize proportional equations (tasks like "y = 2.5x", "y = 4x", and "y = (2/7)x"). Students also graph linear proportional equations and interpret points such as (0,0) and (1,r).

Students collect price data and compute unit prices for lemons, sugar, and cups, filling tables of number of items versus total cost. Students plot those tables on a graph (one line per store/option) and are asked to identify which line is steepest or least steep and what steepness indicates about price per lemon. In Part 2 students create tables, graphs, and write equations in the form y = kx for proportional recipe relationships, connecting numeric, graphical, and algebraic representations.

Students create and use tables from equations (Activity 3) and plot points from tables and equations to make graphs (Activity 3 and Activity 4). Students find the unit rate/constant of proportionality by dividing y by x for points on a graph and write equations in the form y = kx from graphs (Activities 2 and 4). The Parent Plan and Skills sections explicitly state that students will compare two different proportional relationships represented in different ways and give the example of comparing a distance-time graph to a distance-time equation to determine which object has greater speed.

Students compare unit rates between an equation and a graph in Activity 1 (e.g., y = 4x vs. a graphed walk where y = 3 at x = 1) to determine which walk is faster. In Activity 2 and its student pages, students compute speeds from tables (e.g., miles and hours) and directly compare those unit rates to speeds given by equations (e.g., y = 70x), then rank drivers. Day 2 and Day 3 activities require students to find slope from tables, graphs, and equations (using m = (y2 - y1)/(x2 - x1) and identifying m in y = mx) and to graph one representation alongside another to compare steepness and rate of change. Multiple student tasks and challenge questions ask learners to create, plot, and compare functions presented algebraically, graphically, numerically, and verbally (word problems about recipes, faucets, cars).

Students pick pairs of lattice points on graphs, draw right triangles, count rise and run, and compute slope as rise/run (Activity 1 and problems 6–8). Students draw multiple triangles along the same line, set up proportions comparing rise/run, and decide whether triangles are similar by comparing slopes (Activity 2, Part B). The answer keys and activities ask students to compute slopes for several different graphs and to compare those slopes to determine similarity or difference.

Students graph and compare lines given by equations (Activity 1, Activity 4 Group A/B) and answer which is steepest or which increases/decreases fastest. Students translate tables into equations and then graph them (Activity 7), finding slope from a table and writing y = mx + b. Students match verbal descriptions to graphs and identify slope and y-intercept from those verbal scenarios (Activity 8 Part 2).

Students calculate slope and write equations from tables, graphs, and pairs of points in multiple problems (e.g., babysitting/car rental tables, plotted lines with slope identification, and tasks to write equations from coordinates). Several items ask students to determine which line has a greater rate of change (e.g., Line A vs. Line B problems with coordinates and answer key comparing slopes). The parent-facing skills list explicitly tells students to compare proportional relationships represented in different ways and gives the example of comparing a distance-time graph to a distance-time equation.

Students fill tables of time vs. distance, write distance equations in the form y = mx for car, train, and plane, and plot all three lines on the same graph. They answer questions that ask for rise/run (slope) for the train, identify which mode is fastest or slowest by comparing slopes on the graph, and justify proportionality by noting lines pass through the origin. In the cost activity students write cost equations in the form y = mx + b, use tables of values for specific distances (0, 300, 500), graph those equations, and find the break-even point by setting two equations equal.

Students identify and test functions shown as tables, ordered pairs, graphs, and mapping diagrams (Activity 1) and use the Vertical Line Test to judge graphical representations. Students translate verbal rules into equations and complete input/output tables (Activity 2) and then plot those (x,y) pairs to graph both linear and non-linear rules (Activities 3–4). Students also complete a matching exercise that links tables of values with their corresponding graphs, connecting numerical and graphical representations.

Students compute the rate of change from tables using the formula (y2 - y1)/(x2 - x1) and use those results to decide if a table represents a linear or nonlinear relationship. Students plug x-values into given equations (for example y = 2x + 4 and y = x^2), complete tables of values, compute successive y-differences, and classify the relationship. Students also plot points from equations and inspect graphs to identify straight lines versus curves on multiple student activity pages.

Students match graphs to written scenarios in multiple activities, directly comparing a graphical representation to a verbal description when they select which graph fits each story. Students sketch graphs from verbal descriptions (e.g., Sylvia the Sloth, Timmy the Turtle, Bella's Balloon) and plot points using given rates, translating numerical/word descriptions into graphical form. Students analyze graph shapes to identify increasing, decreasing, constant, linear versus nonlinear behavior and write descriptions of what each drawn graph represents.

Students learn that slope is the rate of change (Things to Know) and practice finding slope from graphs (Activity 1, graph exercises), from two points using m = (y2 - y1)/(x2 - x1) (Activity 4), from tables by treating rows as points (Activity 5), and from equations by identifying m in y = mx + b or rearranging to that form (Activity 6). Multiple student activity pages require calculating rise/run, computing slope from tables, plotting points and finding slopes on graphs, and identifying slope and y-intercept from equations.

Students repeatedly compute slope and y-intercept from graphs (Activity 1, Graph to Equation) and from two points (Activity 6), extract slope and intercept from algebraic equations (Activity 2 and Day 3 Simplify and Graph), and calculate slope from tables of values (Activity 4). Students practice converting among representations by writing equations from graphs, tables, points, and algebraic forms and by graphing equations given in y = mx + b form. The review and cumulative quiz present problems in mixed formats (tables, points, graphs, and equations) for students to convert and solve.

Students identify rate of change (slope) and starting value (y-intercept) from stories, tables, and graphs (e.g., Liam chores, reading table, bike-rental graph). Students compute slope using the slope formula from two plotted points and extract y-intercept from graphs or by reasoning about the 0 input, then write function rules in the form output = slope × input + starting value. Multiple student activity pages ask students to find m and b from tables and graphs and to write corresponding linear equations.

Students calculate and compare rates of change from different representations: Alex's verbal description (miles per minute) is converted to a rate and Bella's rate is found by computing the slope from a graph. In the Jordan vs. Taylor example students extract the y-intercept from an equation and compute the slope from a table to compare starting values and rates of change. Multiple student activity problems require comparing pairs of functions presented as tables, graphs, equations, or verbal descriptions and asking which changes faster or who started higher. Activities also include interpreting two-line graphs to compare slopes and intercepts visually.

Students are asked directly to compare functions in mixed representations: Problem 14 asks which function has the greater rate of change when Function A is given algebraically (e.g., y = 2x + 1 or y = 5x - 1) and Function B is given in a table. Problem 15 asks students to compare starting values where one function is algebraic and the other is shown graphically. Problem 18 and Problem 23 require matching verbal descriptions or contextual stories to graphs and identifying which graph or line has the faster increase, demonstrating comparisons between verbal and graphical representations.

Students design and solve four decks of cards (graphs, equations, tables, and real-world descriptions) that require identifying slope/y-intercept, testing whether a relation is a function, matching representations, and writing equations from tables or stories. Students create red cards that ask about slope and whether a graph is linear, blue cards that ask for slope/y-intercept and matching an equation to a graph or story, green cards that ask students to identify slope from a table or write an equation for table values, and yellow cards that ask students to write equations from descriptions and match descriptions to graphs or tables. During play, students draw cards from different decks and must interpret the representation on the card to answer questions and check their answers against an answer key.

Students are asked to rewrite equations in slope-intercept form and compare slopes (Activity 6: "Once you get your equations into slope-intercept form, the first thing you should look at is the slope"). Multiple activities require graphing pairs of lines and identifying slopes and y-intercepts (Day 2 Activity 5 and many student pages). One student page explicitly includes a table of values alongside graphs (Activity 1, Second Section), so students encounter algebraic, graphical, and numerical representations of linear relationships.

Students plot lines from two given points, find each line's slope and write equations in slope-intercept form (y = mx + b) in multiple activities (e.g., Activity 2 points and Activity 3 examples). Students use both graphs and algebraic equations to determine intersection, parallelism, or identical lines (examples: problems with y = 2x and y = 3x, parallel lines y = 4x - 2 and y = 4x + 3, and systems solved by substitution/elimination). The tasks require students to compute and use slope to decide relationships between two lines, which involves comparing a key property (rate of change) when lines are given graphically and algebraically.

Students are given pairs of linear situations expressed verbally and then written as algebraic equations (for example, Sparkle Clean: y = 20x + 50 and Shiny Solutions: y = 30x; CinemaNow: y = 2x + 15 and StreamMore: y = 5x). Students solve those paired equations to find break-even points and then interpret which option is cheaper before and after the intersection. The lesson explicitly tells students to look at or create graphs of both lines to "see a picture of the solution," linking algebraic and graphical representations.

Students calculate slope from two given points (several problems ask for the slope through two points) and graph linear equations given algebraically (e.g., graph y = 2x − 1, y = −3x + 2). Students also are asked to find equations from two points (e.g., Problems 24/25) and to determine whether two lines intersect, are parallel, or coincide (classifying solutions as one/none/infinite). Word problems require students to create algebraic equations from verbal descriptions and solve systems to find unknowns.

Students write algebraic expressions for each option (e.g., Dorm: C = 1200m and Apartment: C = 1500+1050m; Car: C = 225+0.60x and Rideshare: C = 1.25x; BingeBox: C = 20 and StreamScape: C = 5+1.5h; Meal Plan and Groceries equations derived from a graph; Phone Plans y = 5x+20 and y = (10x+40)/2). Students graph those equations on shared axes and complete numeric tables (housing table of costs for 9, 10, 12 months; phone plan table for x = 0..4) and use the graphs/tables/algebra to find break-even points and compare costs. The phone-plan activity has students simplify one algebraic expression to show it is the same line as the other, and the meal-plans activity has students read y-intercepts and slopes from a graph to write equations.

Students practice matching scatterplots with lines of best fit to their algebraic equations (multiple-choice matching and write-the-equation tasks). Students calculate slope from two points on a graph, identify y-intercepts, and interpret slope as a rate of change in context (ice cream example, plant growth, fuel, tickets, etc.). Students collect numeric data in a table for the bird migration activity, plot points, draw a best-fit line, and form an equation in y = mx + b from those plotted points.

Students match algebraic equations to scatterplots (e.g., Question 13 asks them to choose which equation fits a given scatterplot and Question 14 asks them to write an equation from a scatterplot). Students compare multiple lines on a graph and justify which line best represents a relationship (Problems with lines A, B, and C). Students also interpret and create scatterplots, read tables, and match verbal scenarios to graphical patterns in several exercises.

Students collect numerical data in a table (e.g., jumping jacks and heart rate) and then create a scatterplot by plotting each pair of values. The materials instruct students to informally fit a straight line to scatterplots and to analyze that line (the Parent Plan explicitly says to use the equation of a linear model and interpret slope and intercept). Students also record data numerically in tables and convert categorical tallies into two-way relative frequency tables, giving practice with multiple representations (numerical tables and graphical plots).

Students compute and compare unit rates (Activity 1, Q5 compares Runner A and Runner B to determine who is faster). Students match tables to equations and identify constants of proportionality (Activity 2 asks students to match tables with y = 10x, y = 3x, etc., and to write proportional equations). Students graph proportional equations, label points, identify that proportional graphs pass through the origin, and determine k from graphs (Activity 3 asks students to graph y = 6x, identify (0,0) and (1,6), and to choose which graphs are proportional and state k).

Students read and interpret tables of values and extract the unit rate/slope (Activity 2 babysitting and car rental problems). Students write equations from tables and graphs and identify slopes and y-intercepts (Activities 2 and 3 ask for equations from tables/graphs/points and identify slope = unit rate). Students compare rates of change from graphs and equations (Activity 2 comparing two lines asks which has the greater rate of change, and Activity 4 has students graph and compare Plan A y=20x and Plan B y=15x+30 and determine which plan has the greater rate).

Students are asked to compare rates of change for two lines given as sets of points (Problem 35 asks which line has a greater rate of change and the answer key computes slopes for Line A and Line B). Students graph and interpret an algebraic equation (Problem 20: graph y = 4x and describe the relationship) and find unit rates from graphs or point pairs (Problem 19: line through (0,0) and (2,10); Problems 34 and 17 require finding unit rates and writing equations from numeric data or tables). Several items require writing equations from proportional constants (Problem 18) and identifying constant of proportionality from a table (Problem 17).

Students compute and compare slopes for two functions given in different forms: Function A is presented algebraically (y = 6x - 2) and Function B is given as points (0,1), (1,5), (2,10); students find each slope and determine which has the greater rate of change. Students write functions from verbal situations (earnings per hour, bike rental with fixed fee) and interpret the slope and intercepts in context, connecting verbal and algebraic representations. Students compare starting values by comparing an equation (y = 5x + 3) to a separate graph and justify which function has the higher y-intercept. Multiple activities ask students to find slope from points, from an equation, from tables, and to explain increasing/decreasing behavior from graphs, so students practice comparing properties across algebraic, numeric, graphical, and verbal forms.

Students graph linear equations and find slopes (Activity 2: graph y = x - 3 and y = -2x + 1; Part B asks for slope from two points). Students solve systems where pairs of equations are given algebraically and also shown graphically (Activity 3: problems 1 and 2 supply graphs alongside y = x + 1, y = -x + 5, etc.). Students translate verbal situations into algebraic equations and solve them (Activity 4), connecting verbal and algebraic representations.

Students analyze scatterplots and describe relationships between two variables in Activity 3, including identifying positive/negative/no correlation and interpreting lines of best fit. The lesson includes a web link titled "Write an Equation for a Line of Best Fit" and Parent Plan notes that students will "use the equation of a linear model to solve problems... interpreting the slope and intercept." Students are also asked what it means when points are close to the line of best fit, which addresses interpreting linear trends graphically.

Students work with functions in multiple representations: they find an x-intercept from a table (problem 4), compute slope from point pairs (problem 5), graph an equation (problem 6), use the vertical line test to identify functions (problem 7), and translate verbal rules into algebraic functions (problem 8). Students also graph two linear equations (y = x + 2 and y = -x + 6) and identify their intersection (problem 32), which has them compare two lines graphically. These tasks show students practice reading, creating, and interpreting functions in tabular, graphical, verbal, and algebraic forms.