Students write and work with symbolic equalities such as a(b + c) = (a · b) + (a · c) and complete exercises that model the distributive property with both letters and numbers (e.g., 5(6 + 9) = 30 + 45). Students rewrite numeric multiplications as equivalent expressions (e.g., 46 × 27 as 46(20 + 7) and then as (46×20)+(46×7)) and verify the equal numeric results. Students are asked to represent '4 times n' in four equivalent symbolic forms (4×n, 4·n, 4(n), 4n) and to use commutative and associative property equalities written with letters (a, b, c).

In Day 2 the lesson explains that "equivalent means equal, and problems are equivalent if the same changes are applied to every part of the problem," and shows the numeric example 1.75 ÷ 0.25 transformed to 175 ÷ 25 by multiplying both dividend and divisor by 100. The lesson repeatedly instructs students to keep new division problems equivalent to the original when moving decimals and to multiply both dividend and divisor by the same power of ten. Students are asked to create and use foldables and practice problems that apply this procedure for making equivalent numeric division problems.

Students are given the distributive property in symbolic form a(b + c) = ab + ac and see numeric examples verifying equality (e.g., 5(6 + 9) = 30 + 45). They practice "un-distributing" numeric sums by factoring out common factors (e.g., 24 + 54 → 6(4 + 9); 28 + 36 → 4(7 + 9)) and complete activity problems asking them to rewrite expressions like 28 + 40, 63 + 54, and 48 + 12 in factored form. The activities ask students to check that the factored and expanded forms give the same numeric value, reinforcing the idea of equivalent expressions.

Students rewrite and factor numeric expressions using the distributive property (for example, converting 84 + 120 into 12(7 + 10) and writing undistributed forms such as 6(4 + 3) or 5(3 + 4)). Students use models and grids to show the distributive property and produce both distributed and undistributed forms (distributed: (5×3)+(5×4); undistributed: 5(3+4)). Students use prime factorization and GCF to express sums as factored products (for example, 560 + 480 = 80(7 + 6)).

Students are instructed to "show the total number of goody bag items using the distributive property" and are given the numeric example 12(5 + 3) = (12 × 5) + (12 × 3) = 60 + 36 = 96. The Parent Plan and Skills sections explicitly list "Use the distributive property to express a sum...as a multiple of a sum" as a skill students will use. The planning and spending activities require students to write equivalent numeric expressions when calculating totals and costs.

Students are shown that changing factor order does not change a product (e.g., the text asks "What if the problem were 2/3 × 5?" and cites the commutative property). Students compute and simplify fraction products (e.g., 2/3 × 1/2 = 2/6, which simplifies to 1/3) and complete cross-cancelling activities that ask them to rewrite fractions and multiply equivalent forms. The activities and answer keys include several explicit instances of recognizing equivalent numeric forms (e.g., 2/6 = 1/3, 10/3 = 3 1/3).

Students practice creating and recognizing equivalent numeric ratios in several activities. For example, Problem 6 asks students to write the ratio 12 red pens to 21 blue pens and find a smaller equivalent ratio (answer key: 4:7). Problem 5 has students scale a 2:5 ratio to 6:15 and explain how 2:5 and 6:15 are equivalent. Students also rewrite ratios in three forms (a:b, a to b, a/b), showing equivalence across representations.

Students practice recognizing alternative notations that name the same product (for example, the activity asks them to circle among 36 × n, 36 ⋅ n, and 36n and the text explains that (5)(2) means the same as 5 × 2). Students are given explicit descriptions of different ways to show multiplication (multiplication sign, dot, parentheses, and juxtaposition such as 3x or xy) and are asked to match algebraic expressions to verbal phrases. Students also work with the idea that a variable repeated in an expression has the same value each time (example: discussing whether p can be different values in 3p + 4p).

Students see visual examples that repeated addition of a variable is written with a coefficient (e.g., b b b b = 4b and x x x x x x x = 7x). The lesson explicitly states and shows that a lone variable equals one times the variable (1y = y) and that a variable printed alone has coefficient 1. Activities ask students to label coefficients and to write expressions from clues, reinforcing the connection between repeated items and coefficient notation.

Students practice evaluating expressions by substituting numerical values into variables (Activity 3, e.g., evaluating 5 + 4y when y = 6 and (n−3)^2 + 4 for several n values). Students translate word problems into algebraic expressions (Activity 2) and sometimes see alternate orders or notations given as acceptable answers (e.g., Braeden: n + 8 or 8 + n; Josie: 3n or 3 ⋅ n). Students complete tables that evaluate the same expression for multiple variable values, giving experience that different substitutions produce corresponding numerical results.

Students are given a clear definition that equivalent expressions have the same value even if they look different, with examples such as 4n = 2n + 2n and 3x + 6 = 3(x + 2). Multiple activities require students to rewrite and simplify expressions (combine like terms, use commutative/associative properties) and a matching activity asks students to pair expressions that are equivalent (e.g., n + 5n = 6n, x + x + 2 + 2 = 2x + 4). The lesson also includes practice pages and answer keys where students generate and recognize equivalent forms using visual models and algebraic manipulation.

Students are shown the definition that "Expressions are equivalent if a real number value substituted for the variable(s) produces the same answer" and are given algebraic examples such as 5(x + 3) + 2x - 12 simplifying to 7x + 3. Activity 3 gives explicit steps that tell students to simplify an expression, choose a real number, substitute that same number into both the original and the simplified expression, and compare results (example: evaluating both forms at x = 5 to get 38). Multiple student activity pages ask learners to evaluate pairs of expressions with specified substitutions and decide whether the expressions are equivalent, and area models/Distributive Property activities have students match visual models with two algebraic forms that represent the same area.

Students practice generating and recognizing equivalent expressions using properties of operations (e.g., Problem 14 asks students to use the distributive property to write 12(x + 6) as 12x + 72 and then evaluate both for x = 10). Students are asked to write equivalent expressions using the commutative property (Problem 13) and to decide whether pairs of expressions are equivalent after simplifying or evaluating (Problem 11 asks "Are these expressions equivalent?" after evaluation). The unit outcomes and activities explicitly list "Identify when two expressions are equivalent" and include matching/circle tasks (area model) and simplifying tasks that require students to produce or select equivalent forms.

Students simplify expressions using properties (distributive, commutative) and combine like terms in the "Prove It! Equivalencies" activity, with worked examples that transform expressions into a simplified form (e.g., 6(5 + n) − 24 − 2n → 6 + 4n). Students are then instructed to evaluate both the original and the simplified expression by substituting a numerical value for the variable to verify they produce the same numerical result. Multiple mini-projects (Make a Quiz, Design a Book Cover, Exponent Matching, Create a GoFish! game) require students to generate or match expressions that simplify or combine to equivalent forms. The parent plan and rubric explicitly list "Identify when two expressions are equivalent" and require correct labeling and use of the term "equivalent."

Students repeatedly substitute specific numbers for variables to see whether an equation becomes true (for example, evaluating 2x + 12 = 18 with x = 2 and x = 3, and evaluating 36 - 4n = 12 with n = 8 and n = 6). Activity pages ask students to choose which given value makes an equation true (multiple-choice guess-and-check substitution problems and practice pages). The lesson explicitly lists using substitution to determine whether a given number makes an equation true in the Parent Plan skills and models flipping sides of an equation using the symmetric property a = b ⇒ b = a.

Students represent expressions on both sides of an equation using tape diagrams and hanger diagrams and are taught that the two expressions on each side of the equal sign have the same value. Students are explicitly shown and told that n + 3 is the same as 3 + n (the commutative property) and are instructed to substitute found variable values back into expressions to check that both sides are equal. Students use diagrams to show that removing equal amounts from both sides preserves equality and practice rewriting simple addition expressions when solving equations.

Students substitute specific values into expressions to verify equality (for example, evaluating 2n = 8 with n = 4 and n/2 = 5 with n = 10). The Parent Plan explicitly lists the skill "Use substitution to determine whether a given number in a specified set makes an equation or inequality true." Tape and hanger diagrams represent expressions as groups (e.g., 2n shown as two equal n groups), which links expression structure to a numerical value for particular substitutions.

Students simplify expressions by combining like terms (e.g., 5 + 4n + 3 is rewritten as 4n + 8) and apply the distributive property to rewrite expressions (e.g., 3(n - 4) becomes 3n - 12). Students also substitute computed values into expressions to check equality (for example, substituting n = 3 into 2n + 5 = 11 to verify 11 = 11). The Parent Plan explicitly lists applying properties of operations to generate equivalent expressions as a targeted skill.

Students rewrite equations into equivalent algebraic forms (for example, rewriting x - y = 8 as x - 8 = y and y + 5 - 2x = 6 as y = 2x + 1). Students use the distributive property to expand expressions when solving (for example, 2(y + 1/2) → 2y + 1 and 3(y − 4) → 3y − 12) and solve resulting equations. Students draw tape and hanger diagrams for equations like n + 4 = 12 and 8 = 3 + n that show different expressions representing the same value for the variable.

Students write equations that set two algebraic expressions equal to represent angle relationships (for example, 5n = n + 92 for vertical angles and n + 2n = 90 for complementary angles). Students combine like terms and simplify expressions when solving (n + 2n is combined to 3n; 2n + 3n = 180 is used and solved). Students substitute a solved value back into both expressions to check that the two expressions give the same numeric measure (substituting n = 23 yields 115 for both 5n and n + 92).

The lesson has students use algebraic manipulation and substitution to rewrite and relate circle formulas (e.g., using inverse operations to derive C = πd and using d = 2r to show C = 2πr). Students are asked to replace C with 2πr in the area expression Area = (1/2) × C × r and simplify to A = πr^2, showing that two different-looking expressions represent the same quantity. The activities include numeric examples where students compute circumference using either the diameter formula or the 2πr formula for the same circle.

Students create and simplify equivalent ratios (for example, 8/2 simplifies to 4/1) and form equivalent fractions to convert to percentages. Students set up and use proportions with variables to solve for unknowns (for example, 1 inch/5 feet = 6 inches/n feet to find actual lengths). Students repeatedly use the language and procedures of creating equivalent ratios and equivalent fractions when computing scale factors and scaled measures.

Students set up and manipulate an algebraic equation when applying Euler's Formula: they write F + 5 (-) 8 = 2, rewrite it as F + (-)3 = 2, combine like terms, and add the inverse to both sides to solve for F. The lesson shows step-by-step algebraic work (changing subtraction to adding the opposite and isolating the variable) and has students perform those symbolic manipulations.

Students calculate surface area both by adding the areas of individual faces from nets and by using formulas (SA = 2LW + 2LH + 2WH and SA = 6s^2). The lesson explicitly states that the surface area is the same when computed with the faces of the net and when using the formula and shows a general derivation of the rectangular prism formula from pairs of congruent faces. Students also evaluate expressions at specific variable values in Basic Skills Review (for example, evaluating n^2 - (-5) when n = 3).

Students use and manipulate multiplication notation for variables (for example V = l (x) w (x) h and V = lwh) and rewrite repeated multiplication with exponents (for example V = 5 × 5 × 5 is written V = 5^3). Students verify that two computational methods produce the same numeric volume by counting fractional unit cubes and multiplying by the cube volume (e.g., 112 × 1/8 = 14) and then computing the volume with the formula using fractional values (e.g., 7/2 × 2 × 2 = 14). The lesson explicitly shows that s^3 equals s × s × s and converts mixed numbers to improper fractions before multiplication.

Students set up and solve equations with variables to find missing dimensions (e.g., 54 = L × 4 1/2 × 2, simplified to 54 = 9L and solved L = 6). Students rewrite and simplify algebraic expressions in practice problems (e.g., Basic Skills Review simplifies 4x + 3x - 9 + x + 16 to 8x + 7). Students manipulate formula expressions symbolically (e.g., SA = 6s^2, divide to get s^2 = 144, then identify s = 12).

Students practice using the distributive property to rewrite and factor numerical expressions (e.g., factoring 36 + 88 as 4(9 + 22) and rewriting 15 × 32 as 15(30 + 2) then expanding to 450 + 30 = 480). Activities also include problems that require prime factorization and manipulation of numerical expressions when finding GCFs and LCMs. Several items require rewriting expressions and applying arithmetic properties to show equivalent numeric forms.

Students simplify expressions using the distributive property and combining like terms (e.g., 3(n + 6) + 4n - 10 → 7n + 8 and x + 2x + 3 + 3x - 7 → 2x - 4). Students evaluate expressions at specified variable values (e.g., evaluate 32 - 3x when x = 6 and other evaluation problems). Students write expressions from word phrases and model situations with expressions (e.g., 4n + 5, 9n + 3).

Students write and compare numeric expressions that yield the same value (for example, 3 - 3 and 3 + (−3) are shown as the same), and they write equations for real-world scenarios using positive and negative numbers (e.g., 10 + (−8) = 2, (−)15 + 10). Students are asked to explain multiplication rules using the distributive property (the Challenge Problem asks them to show (−3)×(−4)=12 using the distributive property or a number line). Students also create rule lists and match examples of equivalent signed-number multiplication results (e.g., 2×(−3)=−6 and (−3)×2=−6).

Students record and use general exponent rules written with variables (e.g., a^m × a^n = a^{m+n}, (a·b)^m = a^m·b^m, x^0 = 1, x^{-n} = 1/x^n) on the "Properties of Exponents Notes" page. Multiple activities require students to rewrite expressions in exponent form and simplify them (Product of Powers, Quotient of Powers, Power of a Power, Power of a Product, and the Mixed Review worksheets). Examples show students transforming one form of an expression into an equivalent form (for instance (2·3)^3 → 2^3·3^3 and 2^2·2^3 → 2^5).

Students rewrite decimal measurements in scientific notation (Phase 1) and perform operations with numbers in scientific notation and decimal form (Parent Plan Skills). Students convert decimals to fractions (Phase 4: 0.375 = 3/8, 0.875 = 7/8), and classify numbers as rational or irrational (Phase 2), which requires recognizing equivalent numeric representations. The Parent Plan also states students will "generate equivalent numerical expressions" using properties of integer exponents.

Students are asked to simplify and rewrite exponential expressions such as 5^3 × 5^2, 4^5 ÷ 4^3, 3^4 × 3^2, and 4^{-2}, and to find reciprocals of expressions like 3^{-2}. The Parent Plan and review directions explicitly tell students to "use the rules for exponents (powers) to rewrite and simplify expressions," and several items require applying exponent rules to produce equivalent numerical expressions.

The Parent Plan Skills explicitly states students will "know and apply the properties of integer exponents to generate equivalent numerical expressions," which directs students to produce equivalent forms. In Task 3 students convert a scientific notation expression (7 × 10^2 kWh) to its decimal form (700 kWh), providing an explicit example of producing an equivalent numerical expression. The Parent Plan also references using properties of operations (including the distributive property) and working with scientific notation and square roots, indicating students perform transformations that preserve value.

Students rewrite complex fractional expressions as equivalent multiplication expressions using Keep-Change-Flip (for example, 3/4 ÷ 1/2 is rewritten as 3/4 × 2/1 and simplified to 3/2). Students set up and use equivalent fractions in proportion problems (for example, 2 cups/4 people = n cups/10 people) and manipulate algebraic equations with a variable (e.g., 2 × n = 150 × 5 → 2n = 750 → n = 375). Several answer keys show students transforming and simplifying numeric and fractional expressions to find equivalent values.

Students practice rewriting equations into the form y = kx and simplifying coefficients (for example, 4y = 8x → y = 2x and y = 6x/12 → y = (1/2)x). Students find the constant k by computing y/x in tables and graphs and classify equations as direct variation or not (sorting equations such as y = 17x, y = 3x^2, 8y = x, y = (1/4)x). Several activities require algebraic manipulation (dividing both sides, simplifying fractions) that show one algebraic form is equivalent to another.

Students write equations for proportional situations in the form y = kx and t = p × n (the lesson explicitly states y = kx is the same as t = pn). Students substitute numeric values into those equations and complete tables (e.g., multiply n by 12 to fill the movie ticket table, substitute t = 156 and c = 10 to solve 156 = 10m). Students test whether the equation matches actual measured or listed values (e.g., comparing expected cost from t = 15n to an actual discounted cost at n = 5).

Students set up and solve equations like 0.012 × V = $2,400 and 0.20 × I = $9,000, using a variable to represent an unknown and multiplying by a decimal percent. Examples show students writing total price as 1.08 × original price (and p × 1.06 = 374.40) to represent "original price + tax" in a single multiplicative expression. The lesson repeatedly uses formulas (Sales Tax = Price × Tax Rate, Gratuity = Tip Percentage × Original Bill, Commission = Sales Amount × Commission Rate) that require writing equivalent multiplicative expressions for percent situations.

Students set up and solve backward problems using a variable (e.g., "Let x be the original price: x × (1 - 0.40) = 18 → x × 0.60 = 18 → x = 30" and "x * 1.20 = 72 → x = 72 ÷ 1.20 = 60"). The activity pages and answer keys show students rewriting numeric expressions involving variables (for example replacing (1 - 0.40) with 0.60) and manipulating expressions to isolate x. Students also use the formulas Discount = Original × Discount Percentage and Final Price = Original Price − Discount, which are used numerically and with variables in worked solutions.

Students work with the algebraic formula I = Prt and substitute values to compute interest and balance (e.g., I = 600 × 0.04 × 5 = $120; Balance = 600 + 120 = $720). Students rearrange the formula to solve for unknowns (e.g., I = Prt → 250 = 2500 × r × 2 → r = 250/5000 = 0.05 and I = Prt → 108 = 600 × 0.06 × t → t = 108/36 = 3). Several practice problems require isolating a variable and performing equivalent arithmetic expressions to find rates or times.

Students repeatedly simplify and rewrite expressions using the Commutative, Associative, and Distributive Properties (e.g., examples and student pages that rearrange 3x + 5 + 2x to 5x + 9 and apply distribution in 2(n + 4) + 3n - 6 → 5n + 2). Several activities require students to expand, regroup, and combine like terms (Activity 1–5 and the Properties Quest) so that different-looking expressions are reduced to the same simplified form. The Properties Quest and answer keys map original expressions to a single simplified expression (and a letter), which requires students to recognize when an expression equals a particular simplified result.

Students repeatedly rewrite two-step computations (e.g., 40 + 0.05×40) as one-step products (40×(1+0.05) and 40×1.05) in the Sales Tax and Discount activities. The lesson states and uses the distributive property in reverse (A + AB = A(1 + B)) when converting Selling Price = Wholesale Price + (Wholesale Price × Markup Rate) to Selling Price = Wholesale Price × (1 + Markup Rate). Students write and manipulate expressions with variables (e.g., Total Cost = 25 + 3r and solving 25 + 3r = 58) and are asked to set up equivalent one-step and two-step expressions in practice problems.

Students practice rewriting expressions using the Distributive Property and its reverse in Activity 1 (e.g., 5(x+3) ↔ 5x+15 and factoring 6x+12 → 6(x+2)) and check correctness by distributing back. Activity 2 presents multiple algebraic forms of perimeter (7+7+5+5; 2×7+2×5; 2(7+5)) and students convert among them, showing the expressions give the same numerical perimeter. Activity 3 has students solve and compare pairs like 3x+2 and 3(x+2), demonstrating when structurally similar expressions are not equivalent and how parentheses affect equivalence. The Review Quiz and practice pages ask students to expand, factor, and simplify expressions, reinforcing rewriting expressions into equivalent forms.

Students are asked to rearrange equations into slope-intercept form (e.g., 3x + 2y = 8 → y = (-3/2)x + 4 and 5x - y = 5 → y = 5x - 5), performing operations like subtracting terms and dividing both sides. Activity 4 explicitly instructs students to isolate y on the left and shows step-by-step algebraic manipulation. Students then graph both forms and compare their results, reinforcing that the two forms represent the same linear relationship.

The Parent Plan explicitly tells students to "Rewrite and simplify expressions using the Commutative, Associative, and Distributive Properties" and gives the concrete example a + 0.05a = 1.05a, showing that two different expressions name the same quantity. Several review items require rewriting percent situations as multiplicative expressions (e.g., sale and tax problems) where students compute equivalent forms of price expressions. The unit repeatedly asks students to convert and simplify linear expressions and equations (for example, converting situations into y = mx + b), which involves generating equivalent algebraic expressions.

The Parent Plan explicitly states students should "understand that rewriting an expression in different forms" and gives the example a + 0.05a = 1.05a. Students write and use equations in the form y = mx (distance/speed) and y = mx + b (cost equations) on multiple activity pages. Students also set cost equations equal to find a break-even point (0.15x + 31.50 = 0.20x + 20.75) which requires recognizing when two expressions produce the same value for a given x.

Students rewrite equations into slope-intercept form (for example 4x + 2y = -8 is rearranged to y = -2x - 4) and perform distribution and simplification (for example y + 2 = -2(x - 3) is expanded to y = -2x + 4). Students use these algebraic manipulations to identify the slope and y-intercept from different equation forms (y = mx + b vs. standard form). The lesson includes examples showing that -2(x - 3) and -2x + 6 represent the same expression after distribution.

The lesson explicitly presents multiple algebraic forms of the same line (y = 2x + 3, y - 3 = 2x, 2x - y = -3) and asks students to rewrite equations into slope-intercept form, showing the same relationship in different expressions. Activity 3 has students isolate y (e.g., 2x + 3y = 6 → y = -2/3 x + 2), practicing algebraic manipulation that preserves equivalence. Several activities require substituting a known point and slope to solve for b (e.g., 4 = -1(2) + b), which has students perform algebraic steps that produce equivalent expressions/equations.

Students write function expressions in equivalent forms, for example converting P = 15h + 0 to P = 15h and noting the simplification. Students also rewrite the same function using different variable names (A = 6c + 12 versus y = 6x + 12), showing that the same mathematical rule can be written with different symbol choices. The lesson prompts students to simplify and substitute slope and intercept values into the output = slope × input + starting value structure.

Students derive the sphere formula by showing Volume(sphere) = (2/3) × Volume(cylinder) = (2/3) × πr² × 2r and then simplify that expression to 4/3 π r³, demonstrating an algebraic rewrite of one expression into an equivalent form. Students repeatedly write and use formulas (V = πr²h, V = 1/3 πr²h, V = 4/3 πr³) and perform algebraic steps like substituting values, multiplying out r² or r³, and dividing both sides to solve for a missing variable. Several activities require students to work backwards (e.g., given V and r find h or r), which has them rearrange and manipulate symbolic equations to isolate variables.

Students repeatedly rewrite and simplify expressions using the distributive property (for example 3(a+4) → 3a+12 and 4(y+2) → 4y+8) and combine like terms on multiple activity pages. The materials show students substituting solutions back into original equations to verify equality (for example plugging c = 6 into 12+15+3×6 = 45). Worksheets ask students to expand, simplify, and manipulate expressions as part of solving equations (many problems require distribution and combining like terms).

Students simplify both sides of equations until they are identical in multiple places (for example, the Student Activity Page that works through 5y + 2 = 5y + 2 and shows it simplifies to y = y with the note that "it doesn't matter what number we plug in for y"). In Activity 2 students fill missing numbers or coefficients so expressions on both sides match (e.g., 2(3x+4)+__ = 6x+8+10 where they find the missing 10), requiring distribution and combining like terms. Several practice items directly contrast equivalent expressions (for example 14y = y + 9y) and ask students to simplify and recognize that both forms name the same quantity.

Students simplify and combine like terms in problems such as 3w + 3w + w + w = 64 (leading to w = 8), showing they rewrite expressions into equivalent, simpler forms. The Parent Plan instructs transforming equations into simpler, equivalent forms (x = a, a = a, or a = b), and True/False items ask students to recognize when both sides simplify to the same expression (infinite solutions). Several examples ask students to substitute found values back into original expressions to check equality.

Students convert pairs of equations into the same form (for example, changing 4x − 2y = 8 and 2x − y = 4 into y = 2x − 4) and are asked to recognize that both equations simplify to the same equation. Several activities (How Many Solutions? Part 3, slope-intercept conversion tasks, and examples like 5x+10y=20 and x+2y=4) require students to rewrite equations and decide that the two forms represent the same line (infinite solutions). The lesson asks students to simplify equations and compare slopes and intercepts to determine whether two equations are identical or scalar multiples of one another.

The lesson repeatedly has students substitute one expression for a variable (e.g., y = x + 1 substituted into 2x + y = 7) and directs students to "swap in (2x+1) wherever y appears." Examples show students replacing y with an equal expression, solving for x, back-substituting, and checking the values in both equations. The substitution steps and practice problems require students to use an expression that is equal to a variable in place of that variable within equations.

Students plot and write equations from two points and determine when two lines overlap, intersect, or are parallel. Several problems present pairs of equations that are scalar multiples (for example, 3x - y = 6 and 6x - 2y = 12 or 2x - y = 6 and 4x - 2y = 12) and ask students to identify ‘infinite solutions,' indicating the two equations represent the same line. The substitution example explicitly sets two expressions equal (since both equal y) and solves for x, showing students equate expressions that represent the same value for a given x.

Students substitute one algebraic expression for another (e.g., S = J + 5 substituted into S + J = 65) and then combine like terms (J + 5 + J becomes 2J + 5). Students also set two expressions equal to each other to find break-even points (e.g., 30x = 20x + 50 and 5x = 2x + 15) and solve for the variable. Students check solutions by substituting numeric values back into the original expressions to verify equality (e.g., 35 = 30 + 5, 19.70 = 12 + 7.7).

Students practice simplifying expressions when they apply the distributive property (e.g., 4(2x - 3) = 20) and combine like terms (e.g., 5x - 2x + 6 = 12) while solving equations. Several problems ask students to identify special solution cases by recognizing identical expressions on both sides of an equation (e.g., 2x + 4 = 2x + 4 labeled as identity with infinitely many solutions). The parent plan explicitly describes transforming equations into equivalent forms such as a = a or a = b to show solution types.

Students write and simplify the phone-plan expressions (Plan A: y = 5x + 20 and Plan B: y = (10x + 40)/2) and are shown that Plan B simplifies to the same expression as Plan A. Students complete tables of values for both plans and plot both lines on the same graph, observing that the lines coincide and that the equations give the same outputs for every tested x. The Parent Plan and activity prompts explicitly ask students to compare equations, simplify expressions, and interpret when two equations graph to the same line (infinite solutions).

Students match tables, graphs, and equations such as y = 6x, y = 4x, and y = 10x to given value pairs and label relationships as proportional or not. Activities ask students to graph y = 6x, identify points like (1,6) and (0,0), and create their own proportional equation (for example y = 4x) and corresponding table and graph. Students also compare numeric ratios (e.g., 8:12 and 14:21) to determine if they are equivalent proportions.

Activity 1 explicitly directs students to "focus on recognizing when expressions are equivalent, even if they look different." The Student Activity Page for Activity 1 asks students to simplify expressions, use the distributive property, and factor expressions (problems 1–3), which are standard algebraic methods for producing and recognizing equivalent expressions. The Parent Plan and answer key reiterate examples of rewriting expressions (e.g., a + 0.05a = 1.05a and C = 9t + 6) and note that rewriting can make meaning clearer.

Students are asked to rewrite and simplify expressions (e.g., #26: 4x + 9 + 6x + 1 → 10x + 10) and to distribute and simplify (e.g., #27: 3(y + 5) − y → 5y + 15). Students also factor expressions (e.g., #28: 24m + 12 → 12(2m + 1)), which requires recognizing equivalent expanded and factored forms. The answer key shows these simplified and factored forms, indicating students produce alternative, equivalent expressions.

Students translate verbal descriptions into algebraic expressions (e.g., "The sum of a number and three times the number is 28" is written as x + 3x = 28 and then simplified to 4x = 28). In Section 6 students write the rule "Multiply by 2, then subtract 1" as y = 2x - 1 and complete a table of outputs for given inputs, showing they can evaluate different expressions at particular input values. Several tasks ask students to write equations from situations and to fill tables, which requires recognizing that different representations can produce the same numerical outputs for specific inputs.

Students practice expanding expressions with the distributive property and combining like terms in Activity 1 (e.g., 5(2x - 1) and 7x - 3x + 4), which requires rewriting expressions into equivalent forms. In Activity 2 Part A students determine when equations have infinitely many solutions and the answer key notes cases where both sides are identical, indicating recognition that two expressions can be the same for all variable values. The Parent Plan explicitly directs students to transform equations into equivalent forms such as x = a, a = a, or a = b, which involves creating and recognizing equivalent expressions during algebraic manipulation.

Students are asked to determine the number of solutions for equations such as 6x + 4 = 6x + 4 (problem 30) and 8x − 2 = 8x + 6 (problem 31). The answer key labels problem 30 as "Infinite solutions" and problem 31 as "No solution," which requires recognizing when the expressions on each side of an equation are identical for all x or never equal.