Students represent real-world quantities with positive and negative numbers in multiple activities: the scuba diver and digging/pile examples, the temperature scenario (representing −128.6 and +134.1 and answering what zero represents), the building floors activity (identifying −2 and −1 as garages and G as 0), and the finance problems (Marcus's debt as −$218.72 and other bank-balance tasks). Students plot and interpret positions on number lines and create a foldable that asks them to mark and compare opposites (additive inverses) such as +4 and −4. The Parent Plan skills list explicitly states the standard language about using positive and negative numbers to represent real-world contexts and explaining the meaning of 0, and students are asked to explain what zero means for atomic charge and financial examples.

Students are asked to represent and compare positive and negative quantities in multiple real-world contexts: temperature (47 and 14.5 below zero), elevation/depth (32.8 ft above water and 16 ft deep; balloon up 325 ft and submersible down 1,052 ft), and horizontal directions (drivers 58 and 73 miles east; hikers going opposite directions). The Parent Plan and Things to Know state that zero is neither positive nor negative and that zero is the starting/reference point for measuring directions; activities also require students to use number lines and absolute value as distance from zero. Students practice explaining comparisons in words (e.g., which temperature is colder, which number is greater because it is to the right on the number line) and are prompted to justify answers by referring to positions relative to zero.

Students plot and label points with both positive and negative coordinates across all four quadrants (Activities 1 and 2, Points in the Coordinate Plane). Students explore opposite numbers on number lines and show that opposites are equal distances from zero, then apply that idea to reflect points across the x- and y-axes (Activity 3 and Activity 5, Reflecting Points). Students use a city-map context (Deon's map with the bus station at the origin) and Explorer Eddie scenarios to place and move locations using positive and negative coordinates.

Students plot and identify points with both positive and negative coordinates on all four quadrants and use signs of ordered pairs to locate points (Activities and Parent Plan skills). Students use positive and negative coordinates in real-world map contexts: they plot bus stops, a town map, and a garden and compute horizontal/vertical distances in blocks or miles. Students reason about distances as absolute values (questions note that distances are always positive even when coordinates are negative) and practice moving specified numbers of units left/right and up/down from given points.

Students solve multiple real-world problems that require positive and negative numbers: they label 9 feet underground as -9 and explicitly answer what zero represents for the metal rod, compute temperature changes from negatives (e.g., -11 to -2, -10 to -3), and represent money before/after borrowing with negatives (Erica: 0, -2, 3). Students place positive and negative values on number lines, identify which plotted number is closest to zero, find distances between points located on opposite sides of a reference (Cassie 5 miles west and Tim 7 miles east), and are asked to classify numbers as integers or rational including negatives. The parent plan and skill lists explicitly state that students will use positive and negative numbers to describe opposite directions/values and explain the meaning of 0 in situations, reinforcing these student tasks.

Students plot and record ordered pairs on a coordinate grid labeled from -12 to 12 and call out x-y coordinates to indicate locations during gameplay. Students choose and write which quadrant to place their playing pieces and use the signs of x and y to determine and communicate quadrant locations. The parent plan and skills list explicitly state that students will "understand signs of numbers in ordered pairs as indicating locations in quadrants" and "find and position pairs of integers and other rational numbers on a coordinate plane."

Students are asked to place a set of numbers that includes negatives and zero in order (Basic Skills Review question: -4, |-9|, 2.5, 0, -1 1/2), so they practice comparing and ordering negative numbers and recognizing 0. The materials also include use of 0 in decimal and percentage problems (e.g., 0.09, 0.05, converting 0.83 to 83%), so students work with zero in numerical contexts. The presence of absolute value (|-9|) requires students to consider the magnitude of negative values.

In Basic Skills Review #7 students are asked to write "seventeen degrees below zero" and "eighty-two degrees above zero" in integer form, and the answer key shows (-)17° and (+82°). Several word problems require students to model increases and decreases with expressions (e.g., Jade spent $8 → n - 8; Benjamin earned $50 then spent some → 50 - n). Activity and evaluation tasks have students convert contextual quantities into signed integers or algebraic expressions and evaluate them for given values.

Students learn definitions of positive, negative, and zero and practice locating them on number lines (multiple number-line images and explanations). Students represent real-world situations with integers (bank balances, temperatures above/below zero, diving/submersible depths, digging below ground, game scores, money spent/owed) and write expressions that use positive and negative numbers (e.g., 0 - 5 - 4 = -9, -275 + 500 = 225). Students are asked to model situations starting at 0 (ground level) and to choose or write the correct integer expression for contextual scenarios (Antarctic temperature drop, builder digging, accounts and debts).

Students work with positive and negative numbers when the materials instruct them to "use the rules for adding positive and negative numbers" and when they change subtraction to adding the opposite (e.g., rewriting 6x - 3 + 2x + 7 as 6x + 2x + -3 + 7). Several practice problems and answer keys include negative values and subtracting negatives (for example, 14 - (-8) + 9p - 6p and exercises that simplify expressions with negative constants). The Basic Skills review also asks students to compare negative numbers and use absolute value (e.g., comparisons like (-)9 < (-)3 and |(-)8 1/2| = |8 1/2|).

The Parent Plan explicitly lists "Understand that positive and negative numbers are used together to describe quantities having opposite directions or values" and includes practice items on integer operations (e.g., problems like -14 - (-9), 20 - 36, 35 - (-7), and -27 + -8). Students are prompted to change subtraction to "add the opposite" and there are linked integer practice games (Orbit Integers, Fruit Splat, Town Creator) that have students add and subtract positive and negative integers. The Parent Plan also mentions showing distance between rational numbers as the absolute value of their difference, which connects to reasoning about positions on the number line.

Students are asked to work in Row 2, which focuses on "Like Terms and Positive and Negative Numbers," and several projects require use of integers in real contexts. The "Make a Quiz" task requires students to write five word problems using positive and negative integers (with a provided pelican above/below example) and to create an answer key. The "Design a Book Cover" and "Create a GoFish! Game" tasks explicitly require students to write and combine problems that add and subtract positive and negative numbers or use positive/negative coefficients, and the parent/teacher notes refer students to Lesson 4 for rules about adding and subtracting integers.

Students solve inequalities whose solution sets explicitly include negative numbers (for example, the explanation that −2 + 5 = 3 shows −2 is in the solution <et for n + 5 < 12). Students also substitute negative values when checking solutions and see that solution sets can extend to negative values (the lesson shows solution sets with ellipses that include negatives). Students write and solve inequalities in real-world contexts (shirts, pies, chickens, money, jelly beans) and determine which numeric solutions are reasonable for those contexts.

Students plot and read points on coordinate grids that include negative and positive axes (examples show axes labeled from -6 to 6 and points such as (-2,0), (-1,1)). Students compute table values that produce negative outputs (for example, 2x - 4 = y gives y = -4 when x = 0) and order negative numbers on the Basic Skills Review. Students apply negative values in real-world contexts such as freezer temperatures (y = x - 2) and see that hours worked and money earned are nonnegative in the Kevin earnings example.

Students solve and graph inequalities with negative solutions on number lines that run from -8 to 8 (for example, 4 < n + 6 leading to n > -2 with an open dot at -2, and n - 7 > -12 leading to n > -5). Students create and plot tables and graphs of linear equations that include negative coordinates (for example, x - y = 8 with a table entry y = -8 when x = 0, and rewriting y + 5 - 2x = 6 to y = 2x + 1 and graphing values including negative x and y). Students previously solve for variables where intermediate or final values are negative and place those values on number lines or coordinate grids throughout the activities and answer keys.

Students compute with negative numbers in a real-world temperature problem on the Basic Skills Review: they are given a starting temperature of -14° and asked to find the temperature after it drops another 7°, producing -21°. That item requires students to combine negatives (−14 − 7) and interpret the numerical result as a temperature. This is an explicit use of positive/negative values in a contextual setting.

Students plot and draw triangles using coordinates that include positive and negative values (e.g., (1,1), (-1,-1), (-3,-5), (4,2)). Activity 3 explicitly references the four quadrants and the origin (0,0), and students draw triangles in Quadrants I, II, III, and IV using negative and positive coordinates. Several student activity items require constructing and comparing figures given coordinates with negative signs.

Students work with negative numbers in the Basic Skills Review: they graph the inequality y < 2 on a number line (showing values less than 2) and plot points with negative coordinates such as (-3, 2) and (-5, -1) on a coordinate plane. Students also evaluate an expression involving a negative number (n^2 - (-5)) when n = 3, demonstrating arithmetic with negatives.

The Basic Skills Review includes Problem 4: 4 + 7 - 13 + 1 - (-2) =, and the answer key shows students change subtraction to adding the opposite, demonstrating computation with negative numbers. Problem 7 asks students to solve n + 3 < 5 and graph the solution on a number line ranging from -5 to 5, requiring students to work with negative values and locate them on a number line.

The lesson requires students to design statistical questions that use numerical responses and explicitly lists temperature as a suggested data attribute. Students are instructed to display numerical data on number lines (dot plots) and to create histograms and box plots, and to compute and report numerical summaries (mean, median, range, IQR, MAD). The project asks students to collect real-world numerical data, organize it, and interpret its shape and variability in context.

The Parent Plan lists "Understand ordering and absolute value of rational numbers," indicating students will work with negative and positive values. The Wrapping Up directs students to play a Balloon Pop game for ordering positive and negative numbers and to complete an online exercise practicing ordering and absolute value. Students therefore get explicit practice ordering positive and negative numbers and computing absolute values through the provided online activities.

Students are asked to plot and work with points that have negative and positive coordinates (for example, plot (-2, 4), (-3, -2), and start at (-3, 5)). Students reflect points over the y-axis and consider movement left/right and vertical/horizontal distances, and they play a graphing game that explicitly uses all four quadrants. The Parent Plan also states students will 'solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane.'

Students work with a number line that shows zero as the divider between positives and negatives and discuss contexts like temperature above/below zero, elevation, money owed/saved, and electric charge. In Activity 1 students write equations, draw number-line or symbol representations, circle zero pairs, and determine whether opposites combine to make zero (e.g., atom with equal protons/electrons resulting in total charge 0). Multiple activities require students to represent real-world situations (scuba diver, submarine, earning/spending money, temperature changes) using positive and negative numbers and to interpret the numerical result.

Students are asked to work with positive and negative numbers on the Review Quiz: they compute 5 + (-5), answer a context problem about a scuba diver descending 20 feet and ascending 15 feet, determine the temperature change for a drop of 2.5°F per hour for 6 hours, and compute products like (-3) × (-4). These items require students to perform operations with signed numbers and to reason about direction/change in simple contextual situations.

Students solve positive and negative number problems in the Review Quiz (e.g., 7 - (-3), -4 × 6, -24 ÷ 4) and complete word problems that use negatives in context (a money/debt question with an answer of -15 and a temperature drop problem with an answer of -12°F). The Real-World Problems activity includes a temperature scenario that frames a decrease as a negative change and the money problem frames borrowing as negative value. The Answer Key and activity pages show students computing negative results and working with negative exponents in calculator practice.

Students solve real-world problems that use positive and negative numbers: the scuba diver problem asks students to compute (-)30 + 18 to find a diver's position relative to the surface, and the temperature problem multiplies (-)2.5 by 6 to find a total negative change. Negative integers appear in number-sorting and classification activities (e.g., lists including -7, -11, -4) and students categorize numbers as integers, whole, rational, or irrational. Debt is represented with a negative value (Jamie borrows money and the answer is given as -$15).

Students work with temperatures given as -15°C and 5°C to find a temperature range and compute an average daily temperature. Students compute a submarine's final depth after diving from sea level to -450 meters and rising 175 meters, using sea level as the reference point. The wrap-up explicitly states that students practiced calculations with positive and negative numbers in real-world contexts.

Students solve explicit problems adding opposites (e.g., "What is the sum of -6 and 6?" and -12 + 12) and are asked to explain why opposites sum to zero. Students represent signed quantities in real-world contexts through submarine/diver depth problems (e.g., 300 feet below sea level ascending 125; 400 feet below ascending 150) and through a profit/loss net-profit problem. The Parent Plan repeatedly asks students to "describe situations where opposite amounts cancel each other out to make zero" and gives the hydrogen-atom example to connect positive/negative values and the meaning of 0.

Students work with real-world temperature values that include negatives (e.g., Average Temp: -10°C for Arctic Tundra and -4°C for Himalayan Mountains) and compute temperature differences to determine heater energy needs. The activity asks students to calculate energy based on those temperature differences (Task 1) and shows answer keys where negative outside temperatures are used to produce positive temperature differences and kWh calculations. The supplies and logistics tasks also require numeric calculations using these temperature-based results when comparing sites.

Students interpret the origin (0,0) in multiple contexts (e.g., 0 hours → $0 earned; 0 hours → 0 miles) and are asked to explain the meaning of the point (0,0). The materials include a negative-temperature example with points (−2, −4), (−1, −2), (0,0) and identify a negative unit rate (k = −2) to represent decreasing temperature. Several activities ask students to plot and interpret y-values from equations and tables, including contexts where y decreases as x increases.

Students are asked to explain what the point (0, 0) represents in a proportional relationship and what a point like (1, 4) represents in cost-vs-items contexts. The Parent Plan and review items explicitly instruct students to pay special attention to the points (0, 0) and (1, r) and include answer key language such as "The origin, meaning no items and no cost." Several problems require students to interpret graphs that pass through the origin and explain the meaning of those points.

Students create and interpret proportional graphs that begin at the origin and write equations in the form y = kx, with an answer key showing a line from (0,0) to (16,32). The lesson's skills list explicitly asks students to "explain what a point (0,0) … means in terms of the situation," and students label axes and plot cost and quantity starting from zero. Students set up tables and graphs (e.g., tablespoons vs. cups, number of lemons vs. total cost) that include the 0 point and reason about values relative to that origin.

Students work with subtraction and negative terms in multiple places (e.g., factoring examples like 15x - 10 and 9(z (-) 3)), and they solve equations that produce negative results (for example 6(x + 4) = 22 leads to 6x = -2 and x = -1/3). Activities ask students to manipulate expressions with minus signs and to distribute and combine like terms in forms such as ax + b = c and a(x + b) = c, giving repeated practice with negative coefficients and constants.

Students plot and analyze lines with negative values and negative slopes (e.g., Equation 3: y = -3x + 2 with negative y-values, and Activity 4 graph with point (6, -18) leading to k = -3). The Walk the Graph activity explicitly includes negative-slope equations (y = -2x, y = -3x, y = -4x) and directs students to move down for positive steps in x, so students practice representing negative y for positive x. The lesson also has explicit real-world context for zero (e.g., if you work 0 hours you earn $0) and emphasizes the origin (0,0) as meaning zero of both quantities in proportional relationships.

Students work with negative slopes explicitly: Activity 1 Part 3 provides a graphed line with points (0,0), (1,-3), (2,-6), (3,-9) and the equation y = -3x, and asks students to explain what the negative unit rate indicates about direction. The Calculating Slope notes and examples include y = -3x and state that the negative sign means the line slopes downward from left to right. Multiple activities ask students to plot and interpret the origin (0,0) in proportional contexts (e.g., "start with (0,0) since no weeks means no money saved" or "no time means no water collected").

Students identify and write intercepts as ordered pairs that include positive and negative coordinates (for example (3,0), (0,-3), (-4,0)). 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, 3x - 6y = 12). Students plot intercepts on coordinate grids and connect them to form lines, practicing placement of points in positive and negative regions of the plane.

The lesson tells students that "Moving up or to the right is positive" and "Moving down or to the left is negative," and instructs them to write a + or - when counting rise and run. Several practice problems include negative coordinates (for example, points like (-4, 1), (-5, -1), and (-3, -4)) and answer keys show negative rises/runs and negative slopes (e.g., (-)3/4, (-)1). The lesson also links slope to real-world situations such as ramps, hills, and an architecture ramp challenge, asking students to build and compare ramp steepness.

Students graph and compare positive, negative, and zero slopes (Day 3 Activity 3) and identify that m>0 rises, m<0 falls, and m=0 is flat. Students work with equations that have negative y-intercepts (e.g., y = 2x - 4, y = 2x - 8) and interpret those as lines starting below the x-axis (lemonade profit starts at -8). Students find cases with b = 0 (y = 2x) and are asked to recognize that this line passes through the origin and is a special proportional case.

Students plot and work with points that have negative coordinates (for example (3, -6) and (2, -8)) and compute slopes using differences that involve negative values (answer key shows slope = (-6 - 0)/(3 - 0) = -2). Students identify and work with negative slopes and equations that include negative coefficients (e.g., y = -4x + 2) and label lines as having positive or negative slope from graphs. Students also identify y-intercepts including cases where the y-intercept is 0 for proportional relationships (e.g., answers show y-intercept = 0 and equations like y = 10x).

Students identify and work with negative and positive intercepts on coordinate graphs (for example the example line with y-intercept (0, -3) and answer keys containing points like (0, -2)). Students find intercepts from tables and equations that include negative values and compute x- and y-intercepts algebraically (e.g., solving 2y+3x=4 and y=2x+3). Students interpret intercepts in real-world contexts and explain the meaning of 0 (for example, prepaid card starting amount (0,50) and x-intercept (5,0) meaning $0 left; water consumption with y-intercept (0,20) and x-intercept (10,0)).

Students identify and calculate positive and negative slopes (Activity 2) by determining whether a line goes up or down as x increases and by computing rise/run where rise or run may be negative. Students compute slopes from two points and tables that include negative coordinates and obtain negative slope values (examples show slopes like -2, -1, -3). Students classify horizontal lines as zero slope and vertical lines as undefined (Activity 3) and fill in rise/run/slope boxes for graphs with negative y-values and negative rises.

Students work with many examples that include positive and negative slopes and y-intercepts (e.g., equations and answer keys such as y = -3x + 5, y = 5x, y = -12x - 2). Students use real data for a subway context where they calculate slope from a table (miles per minute) and identify the y-intercept = 0 as the train starting at the station (origin). Instructions and worked examples explicitly have students find where x = 0 to determine b and interpret rise/run (e.g., "goes down 1 for every right 1") when the slope is negative.

Students identify and use negative rates of change in several contexts (answer key shows slopes of −3 for stickers, −2 for passengers, and −5 for battery percentage) and write linear functions that include negative coefficients (e.g., S = −3f + 30, P = −2s + 50, B = −5h + 100). Students interpret y-intercepts and explain the meaning of 0 in contexts (e.g., pages read at 0 hours = 0, bike rental cost at 0 hours = $5, Liam has $12 when c = 0). Students convert real-world situations into output = slope × input + starting value form and read slope sign as indicating increase or decrease.

Students calculate and interpret negative slopes in real contexts (e.g., Jordan's equation y = -3x + 100 and Taylor's table giving a slope of -3.5; Account A with slope -10) and are asked to compare rates of loss. Students identify the value of y when x = 0 as the starting point in multiple examples, using 0 to represent the initial amount (e.g., starting money, 0 minutes/0 gallons). These activities require students to compute and reason with negative numbers as rates (declines) and to determine initial values at x = 0.

Students identify and compute x- and y-intercepts and interpret starting values (e.g., tasks asking for the y-intercept of y=2x+3, y=4x−2, and determining which function has a greater starting value). Students work with graphs and coordinate points that include negative values and slopes (images and answer keys show points and intercepts like (1, -1), (0, -3), and negative slopes such as -2). Students analyze real-world numeric contexts such as water temperature experiments and earnings problems where they read and compare measured temperatures and write functions for total earnings.

Students are asked to analyze and create equations that include negative coefficients (example blue card: "What is the slope and y-intercept of y = -2x + 5?"). A yellow card example asks students to model a temperature that is "dropping 3 degrees each hour after sunset, starting at 70°F" and to write an equation and compute values, which requires using a negative rate of change. The unit repeatedly asks students to identify slope/rate of change and initial value from graphs, tables, equations, and real-world descriptions.

Students plot and translate points that have positive and negative coordinates (for example Q(4, -2), M(6, -2), exercises with X(0,0), Y(2,0), etc.) and label images like A'. They apply algebraic translation rules (x,y) → (x + a, y + b) where a and/or b are positive or negative (e.g., T_{-3,3}, T_{4,-1}, T_{-3,-3}), explicitly adding negative and positive values to coordinates. The checkers recording activity has students state start and end coordinates and describe moves as Ta,b, reinforcing use of signed numbers to indicate opposite directions on the plane.

Students plot and reflect points with both positive and negative coordinates repeatedly (e.g., A(4, -2), B(-3, 5), tasks involving points in all four quadrants). Students apply coordinate rules that change the sign of an x- or y-value for reflections across axes ((x,y) → (x,-y), (x,y) → (-x,y)) and use formulas for diagonal reflections that involve sign changes and swaps. Students use a digital tool to drag points across axes and observe how reflected coordinates change sign or overlap when a point lies on an axis.

Students plot and work with coordinates that include positive and negative values (e.g., points like H(-2,-1), A(1,1), and coordinates with 0). Students apply translation rules with positive and negative components (for example T_1,-1 and T_2,-1) and describe moves as "2 units right, 1 unit down," using signs to show direction. Students observe that reflections change the sign of a coordinate (reflection over the y-axis flips x from positive to negative) and use the origin (0,0) as a center for rotations.

Students work extensively with coordinates that include both positive and negative values (grids and problems range from -9 to 9 and many example/new coordinates include negative numbers). Students multiply x- and y-coordinates by scale factors and plot results from the origin O(0,0) (instructions tell students to draw lines from O and to compute new coordinates like (−1.5, −1.5) or (−8,4)). Several activities ask students to determine new positions and lengths after dilation using signed coordinates and to check x- and y-ratios to verify dilations.

Students plot and manipulate points with positive and negative coordinates on multiple coordinate grids (e.g., points like (−4, 1), (1, −3), etc.). Students perform translations by adding to x and y (e.g., T3,1 adds 3 to x and 1 to y), dilations by multiplying coordinates (e.g., scale factor 2 multiplies both coordinates), and reflections/rotations that explicitly change signs (discussion: "one coordinate changes sign" for reflections; rotation rule (x,y)→(y,−x) uses a negative). The activity pages and answer key use sign changes in coordinates as evidence of opposite directions and require students to reason from those signed values.

Students work with coordinate pairs that include positive and negative values (for example A(−1, 2) and B(2, −2)) and plot points on grids that span negative and positive x- and y-values. Activities ask students to form right triangles between points in different quadrants, count squares horizontally and vertically across the axes, and compute distances using those horizontal and vertical separations. Several student pages list point pairs with negative coordinates and provide space for students to calculate distances.

Students work extensively with coordinates that include positive and negative values (e.g., points like (-2,1), (2,1), and answers showing X' = (1,-2), translations T5,-3). Several problems require reflecting points across the x- or y-axis and rotating/ translating points using rules that change signs (rotation rules table: (−y,x), (y,−x), (−x,−y)). The practice and answer key show students computing and plotting images with negative coordinates after transformations.

Students solve many multi-step equations that include negative coefficients and negative constants (for example: "-7x + 4 + 6 = -4x + 6 + 9x", "-8 + 6k - 2k = -4 + 2k", and problems such as 8(x - 5) = -10). Activities require students to add, subtract, and move negative terms across both sides of equations and to isolate variables when negatives appear. Several solution steps explicitly show subtracting or adding negative values and dividing to find variables with negative results.

Students plot points and graph lines on coordinate grids that include both positive and negative coordinates (grids range from -16 to 16 and -7 to 7). Students write and manipulate equations that contain negative numbers and negative slopes (examples include y = -x + 6, y = -2x + 8, y = -2x + 6) and identify intersection points such as the origin (0,0) or other points with positive and negative values. Students solve systems algebraically (substitution and elimination) where they work with negative coefficients and constants to find (x,y) pairs.

Students perform algebraic steps that include negative numbers and negative coefficients (for example, multiplying an equation by -2 to get −6x−4y = −39.40 in the candy-shop elimination example and subtracting values such as −5 in the substitution steps). Students set up and solve real-world equations for motion, cost, and break-even analysis (distance = rate × time and total cost = rate × quantity + fixed amount) so they solve systems with addition, subtraction, multiplication, and division of signed numbers. Multiple worked examples show students manipulating signs while isolating variables and checking solutions.

Students identify and write linear equations that include negative coefficients (e.g., y = -4x + 20, y = -5x + 50, y = -8x + 40) and interpret negative slopes in real contexts (fuel consumption: slope = -6 means fuel decreases). Students explain the meaning of the y-intercept as the value when x = 0 in multiple contexts (e.g., at x = 0 the car has 180 liters, at x = 0 the ice cream shop sells 50 cones when temperature is 60°F). Several problem pages ask students to state independent/dependent variables, interpret slope sign, and solve using the equations in context.

Students identify zero pairs and write equations for hiking, banking, and submarine scenarios (e.g., 9 + (−9) = 0; −12 + 7 = −5) and state final positions or balances. Students compute with signed numbers in Sign Rules Control Room (multiplication and division problems) and explain why signs make sense. Students solve real-world signed-number problems such as a temperature dropping −2.5° per hour for 6 hours and interpret the total change, and they answer items about a gamer losing 45 points and a debt of −$72 split among friends. Students are asked to create their own real-world problems using positive and negative rational numbers and to explain their answers.

Students are asked to interpret the origin and intercepts in context (Activity 2 asks what (0,0) represents with the answer "zero hours earns zero dollars"), and Activity 3 asks students to identify y-intercepts and explain what the y-intercept represents as a starting value. Several problems and answer keys include negative numbers in equations or points (e.g., a line passing through (3, -4), equations with negative slopes or a y-intercept of −4), so students compute and work with negative values in graph/equation contexts.

Students compute with positive and negative integers (e.g., find the sum of -18 and 27; divide -48 ÷ 6; multiply (-6)(-4) and explain why the product is positive). Students apply integers in a real-world context when they solve the submarine problem (350 feet below sea level rising 125 feet to find its new position). Students work with negative values on a coordinate grid and identify positive and negative slopes and plot points with negative coordinates (e.g., graphing a line through (0,0) and (4,-8)).

Students work with coordinate graphs that include negative and positive axis values (axes marked from -5 to 5 and plotted points like (-1, -3)), and they find and interpret intercepts (e.g., graph y = 3x - 2 and identify the y-intercept of -2). Students calculate and interpret negative slope (e.g., y = -1.25x + 5 and explain it is decreasing) and use tables that contain negative entries (e.g., x = -5, y = -2). The activities ask students to identify x-intercepts as points where the output is zero and to explain what intercepts represent in context (e.g., which intercept represents when total cost is $0).

Students work extensively with coordinates that include positive and negative values (e.g., A(-3,4), P(1,-9), D(1,-2)) and are asked to write the coordinates of transformed points. Tasks ask students to reflect across axes, reflect across y = x, translate using rules (x,y) → (x+3,y-2), and apply rotation rules that change signs (e.g., 90° CCW changes (x,y) to (-y,x)). Several problems prompt students to "describe how the x and y values changed" or "explain the rule used," requiring attention to sign changes and opposite directions on the coordinate plane.

Students plot and work with points that have positive and negative coordinates (e.g., points like B(−3, 4), A(5, 10), the coordinate grid ranging from −6 to 6, and the line through (1, −5), (2, −3), (3, −1), (4, 1)). Students perform and interpret transformations that use negative values, such as translation rules T_{1,−3}, T_{−4,2}, and reflections that change signs (A(5,10) → A′(−5,10); B(−3,4) → B′(4,3)). Graphing, translations, and reflections require students to use positive and negative numbers together to represent opposite directions on the coordinate plane.