Unit 2: Solving One-Variable Equations
Students hone their skills of solving multi-step equations and inequalities, redefining their definition of "solution" to include cases such as infinite solutions, and interpreting solutions in context.
In Unit 2, eighth-grade students hone their skills of solving equations and inequalities. They encounter complex-looking, multi-step equations, and they discover that by using properties of operations and combining like terms, these equations boil down to simple one- and two-step equations. Students also discover that there are many different ways to approach solving a multi-step equation, and they spend time closely looking at their own work and the work of their peers. When solving an equation with variables on both sides of the equal sign, students are challenged with results such as 4=5, and they refine their definition of “solution” to take into account such examples. Throughout this unit, students use equations as models to capture real-world applications. They reason abstractly and quantitatively as they de-contextualize situations to represent them with symbols and then re-contextualize the numbers to make sense in context (MP.2).
In sixth grade, students developed the conceptual understanding of how the components of expressions and equations work. They learned how the distributive property can create equivalent forms of an expression and how combining like terms can turn an expression with three terms into an expression with one term. By the end of seventh grade, students fluently solved one- and two-step equations with rational numbers and used equations and inequalities to represent and solve word problems.
Students’ ability to manipulate and transform equations will be required again in Unit 5: Linear Relationships and Unit 6: Systems of Linear Equations. Furthermore, these skills will be needed throughout high school as students are introduced to new types of equations involving radicals, exponents, multiple variables, and more.
Pacing: 16 instructional days (12 lessons, 3 flex days, 1 assessment day)
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The following assessments accompany Unit 2.
Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.
Pre-Unit Student Self-Assessment
Have students complete the Mid-Unit Assessment after lesson 7.
Use the resources below to assess student mastery of the unit content and action plan for future units.
Post-Unit Assessment Answer Key
Post-Unit Student Self-Assessment
Use student data to drive your planning with an expanded suite of unit assessments to help gauge students’ facility with foundational skills and concepts, as well as their progress with unit content.
Suggestions for how to prepare to teach this unit
Prepare to teach this unit by immersing yourself in the standards, big ideas, and connections to prior and future content. Unit Launches include a series of short videos, targeted readings, and opportunities for action planning.
The central mathematical concepts that students will come to understand in this unit
Terms and notation that students learn or use in the unit
combine like terms
distribute a negative
properties of operations
To see all the vocabulary for Unit 2, view our 8th Grade Vocabulary Glossary.
The materials, representations, and tools teachers and students will need for this unit
To see all the materials needed for this course, view our 8th Grade Course Material Overview.
Topic A: Simplifying Expressions and Verifying Solutions
Write equivalent expressions using properties of operations and verify equivalence using substitution.
Define a solution to an equation. Solve and check solutions to 1 and 2 step equations.
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Topic B: Analyzing and Solving Equations in One Variable
Justify each step in solving a multi-step equation with variables on one side of the equation.
Write and solve multi-step equations to represent situations, with variables on one side of the equation.
Model with equations using a three-act task.
Solve equations with variables on both sides of the equal sign.
Write and solve multi-step equations to represent situations, including variables on both sides of the equation.
Understand that equations can have no solutions, infinite solutions, or a unique solution; classify equations by their solution.
Solve and reason with equations with three types of solutions.
Use equations to model a business plan and determine the break-even point.
Topic C: Analyzing and Solving Inequalities in One Variable
Solve and graph inequalities with variables on one side of the inequality.
Solve and graph inequalities with variables on both sides of the inequality.
The content standards covered in this unit
— Solve linear equations in one variable.
— Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
— Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
— Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Standards covered in previous units or grades that are important background for the current unit
— Write, read, and evaluate expressions in which letters stand for numbers.
— Apply the properties of operations to generate equivalent expressions.
For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
— Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
— Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
— Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
— Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
— Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.
For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
— Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.
For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
Standards in future grades or units that connect to the content in this unit
— Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
— Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
— Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
For example, rearrange Ohm's law V = IR to highlight resistance R.
— Analyze and solve pairs of simultaneous linear equations.
— Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
— Make sense of problems and persevere in solving them.
— Reason abstractly and quantitatively.
— Construct viable arguments and critique the reasoning of others.
— Model with mathematics.
— Use appropriate tools strategically.
— Attend to precision.
— Look for and make use of structure.
— Look for and express regularity in repeated reasoning.
Exponents and Scientific Notation
Transformations and Angle Relationships