Students are given the definition of additive inverse and asked to show that a number and its opposite have a sum of 0 (Things to Know and Skills). In Activity 3 students use number lines and a foldable that asks them to mark opposites and complete the sentence "Opposites can be added to get a sum of __" (answer: 0). The atomic charge item and answer key ask students to interpret +1 and -1 combining to a charge of 0, and Everyday Integers (Paula earning +$8 then spending -$8) asks students to explain why the result is $0. The trivia scoring and Mark's predicted score problems require students to apply +10/−10 scoring to show how opposite quantities affect a total.

Students are asked directly about opposites (e.g., "What is the opposite of -12?", "What is the opposite of 18?", "What is the opposite of zero?") and complete practice items finding opposites. The lesson text defines the opposite as the additive inverse and explicitly states "the opposite of a number... is what we have to add to that number to get zero." Skills and activities connect positive/negative quantities to opposite directions or values (above/below ground, east/west, up/down).

Students plot pairs of opposite numbers on number lines (e.g., 6 and -6), draw segments from zero to each, and fold the paper at zero to see the points match up (Activity 3). Students are instructed that "opposite numbers are the same distance away from zero, just on opposite sides of the number line," and they practice finding opposites on both horizontal and vertical number lines. Later activities have students change the sign of coordinates (e.g., (-6, 4) to (6, 4)) and plot reflections across axes, reinforcing that paired opposite coordinates are mirror images across zero.

Students are asked to name the additive inverse of numbers (e.g., problems that ask for the additive inverse of 6, -3.5, and 0) and answer key entries show the additive inverse relationships. Student problems ask students to interpret zero in a real context (the metal rod question: nine feet underground as -9 and asking what zero represents). The Parent Plan and Skills lists explicitly include objectives such as "Show that a number and its opposite have a sum of 0 (are additive inverses)" and "Use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation."

The text explicitly states that combining a number with its additive inverse always results in zero and gives symbolic and number-line examples (e.g., 8 + (-8) = 0 and demonstration on a number line). Students practice finding opposites on number lines in multiple activities and compute sums that equal zero (for example, rewriting 9 - 9 as 9 + (-9) and evaluating to 0). The parent/skills sections and several activities list real-world contexts (temperature, elevation, credits/debits, electric charge, bank balances) where positive and negative quantities are used.

Students are directed to rewrite subtraction as "add the opposite" (e.g., rewrite 6x - 3 + 2x + 7 as 6x + 2x + -3 + 7) and then group and combine positive and negative terms. The lesson tells students to "use the rules for adding the positive and negative numbers" and includes practice problems that require combining positive and negative terms (for example problems like 14 - (-8) + 9p - 6p and guided steps changing subtraction to adding the opposite). Students also simplify expressions that involve opposite-signed constants (examples show combining -3 and +7).

The Parent Plan explicitly lists "Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q)" and "Understand that positive and negative numbers are used together to describe quantities having opposite directions or values." Student activities instruct: "Solve. Be sure to change subtraction to 'add the opposite.'" and include practice problems such as -14 - (-9), 20 - 36, and 16 - 35 + 2 (answer key shows 16 - 3 - 15 + 2 = 0). Several problems require students to convert subtraction to adding the additive inverse and compute sums of positive and negative numbers.

Students are instructed to "change subtract to add the opposite" in the Prove It! activities and the Parent Plan lists "Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q)". Students create word problems using positive and negative integers in the "Make a Quiz" task and solve real-world integer situations (e.g., the pelican example). Several activities (GoFish! game, Design a Book Cover, and rubric/skills lists) require students to combine positive and negative coefficients and to use positive and negative numbers together to represent opposite directions or values.

Students are explicitly taught about inverse operations (addition and subtraction) and shown that using the opposite operation can "cancel out" a number so that its numerical effect becomes zero (see Activity 3 discussion and the p + 3 = 8 example). The lesson states that "Additive inverses can be used to solve equations" and models cancelling a constant by performing the inverse on both sides (e.g., p + 3 - 3 = 8 - 3). Tape and hanger diagrams show removing equal amounts from both sides so that a constant is eliminated and the variable is isolated.

The Basic Skills Review includes integer arithmetic problems that require adding and subtracting negative numbers (for example, the problem 4 + 7 - 13 + 1 - (-2) and the provided solution that says to "change subtraction to add the opposite"). A number-line inequality problem (n + 3 < 5) asks students to solve and graph on a number line, giving opportunity to work with negative and positive values on a number line.

Students are prompted in Activity 1 to determine if opposites combine to make zero, write equations using one positive and one negative number (for example, 3 + (−3) = 0), and draw visual representations that circle zero pairs. The atom example (protons and electrons) is given explicitly to show equal opposite charges producing net zero, and the student activity asks students to identify zero pairs and decide what remains. Number line images and prompts ask students to illustrate movements to and from zero, reinforcing the idea of opposites canceling to zero.

The Review Quiz asks "What is the result of adding a positive number and its negative counterpart, such as 5 + (-5)?" with the answer key showing 0. The Review Quiz also presents a real-world context (a scuba diver descending 20 feet and then ascending 15 feet) that has students combine negative and positive quantities to find a net position. These items require students to compute and interpret sums of opposite-signed numbers.

Students work with positive and negative quantities in contextual problems such as the scuba diver who descends 30 ft and ascends 18 ft (calculating -30 + 18 = -12) and the Jamie debt problem (total -$15). The curriculum explicitly labels integers as positive and negative in the number-type diagrams and has students sort and classify negative numbers on activity pages. Number lines are used repeatedly for placing values (primarily square roots) which gives students practice with locating numbers relative to zero.

The Parent Plan explicitly states "Describe situations in which opposite quantities combine to make 0" and gives the hydrogen-atom example. The student instructions and review objectives repeat that students should "Describe situations where opposite amounts cancel each other out to make zero." Practice problems ask students to find and explain sums of opposites (e.g., Problem 1: sum of -6 and 6; Unit Test Problem 1: sum of -12 and 12) and to explain their reasoning. Additional contextual problems (profit/loss, depths/diver/submarine) require students to combine positive and negative quantities to find net results.

Students are instructed to label movement as positive when moving up or right and negative when moving down or left, and to write a '+' or '−' sign when they count squares for rise and run. Student activity pages and the answer key include examples with negative rises or runs (e.g., Rise: (−)3, Run: +4) and ask students to compute slopes using signed rise/run values. Activities require students to draw triangles on coordinate grids and compute signed changes in y and x between two points.

The lesson includes multiple problems that use the phrase "opposite of a number" and instruct students to multiply by -1 (e.g., Exercise 10: "The sum of six and the opposite of a number" with equation y = -x + 6). The answer key and worked examples compute outputs where an input and its opposite combine (for x = 6 the work shows -1 × 6 + 6 = 0). Several input/output tables and activities have students compute with negative numbers and apply the "opposite" operation to produce outputs.

Students practice adding positive and negative numbers when applying translation rules of the form Ta,b → (x + a, y + b). For example, the lesson works through M(6, -2) with T(-5, 3) showing 6 + (-5) = 1 and (-2) + 3 = 1, and the triangle example applies T(-4, 1) giving x + (-4) results. Multiple activities require computing new coordinates by adding negative and positive shifts (e.g., T_{-3,3}, T_{4,-1}, T_{-3,-3}).

The lesson includes explicit coordinate rules that change signs—e.g., (x,y) → (x,−y) for x-axis and (x,y) → (−x,y) for y-axis—and several practice problems where students reflect points and record coordinates (for example A(4,−2) → A′(4,2) and tasks asking whether A(3,2) and A′(3,−2) are reflections because only the y-value changes sign). Activity prompts and answer keys repeatedly point out that reflection changes the sign of one coordinate and show pairs of opposite numbers as image–preimage values. Students also use formulas to swap or negate coordinates for diagonal and axis reflections.

Students simplify equations by subtracting identical terms from both sides so the variable terms cancel (e.g., 3x+4 = 3x+7 → subtract 3x → 4 = 7). Students transform equations that have identical sides to show they are always true (e.g., 5y+2 = 5y+2 → subtract 5y → y = y or 2 = 2) and they complete activities that require filling blanks so both sides match (Creating Infinite Solutions). Activity 5 asks students to choose coefficients/constants so the variable cancels and leaves a false numeric statement, explicitly using cancellation of equal but opposite contributions.

Students are directed to add equations when "the variable terms are opposites" and the lesson explicitly notes y + (−y) = 0. In the elimination example (3x + y = 12 and 2x − y = 8) students add the equations and see the y-terms cancel to 0, then solve 5x = 20. The student activities require choosing "Add" or "Subtract" to eliminate a variable and to explain which action was used, providing practice that uses opposites combining to make zero.

Students perform elimination where they add or subtract equations to cancel a variable (for example: "y + x = 6" and "y - 2x = 0" are manipulated by subtracting one from the other to get "3x = 6"). The Activity 3 directions explicitly tell students to "combine equations to cancel out one variable" and provide worked examples that show y - y = 0 during elimination. Several practice problems require students to choose elimination and carry out the cancellation step in their solution work.

Students solve and analyze equations that produce identities or contradictions (for example, the problems 2x + 4 = 2x + 4 and 3x + 11 = 3x + 5 appear on the activity pages and answer keys). The parent plan explicitly requires students to transform equations until they reach forms x = a, a = a, or a = b, which requires subtracting like terms and seeing when identical expressions cancel. Several answer keys show the steps where matching terms are removed, resulting in statements like 4 = 4 (identity) or 1 = -5 (contradiction).

Students set pairs of equations equal and subtract one equation from another to eliminate a variable (Transportation Step 3–4 directs rewriting in standard form and subtracting (C−0.60x)−(C−1.25x)). In the housing and transportation tasks students solve 1200m = 1500 + 1050m and 225 + 0.60x = 1.25x by moving terms and cancelling like terms. In the phone-plans activity students simplify Plan B (10x+40)/2 to 5x+20, showing algebraic cancellation that produces identical expressions.

Students complete the 'Zero Pair Vault' activity where they identify zero pairs in real-world contexts (e.g., a hiker climbs 9 feet and then descends 9 feet; a deposit of $15 followed by a withdrawal of $15) and fill in equations and final positions. The student worksheet asks them to write equations for each scenario and interpret the final result, and the answer key explicitly shows 9 + (−9) = 0 and 15 + (−15) = 0. Students also solve related problems (e.g., submarine −12 + 7 = −5) and explain what the numeric result means in context.

Students practice elimination in Activity 3 where they add the equations 2x + y = 11 and 2x - y = 1 to get 4x = 12, which shows the y and -y terms combining to 0. The Answer Key explicitly records the step "Add equations: 4x = 12," demonstrating algebraic cancellation of opposite terms. The activities include multiple problems that require creating and using opposite coefficients so a variable cancels when equations are combined.