Unit 4Equations and Inequalities

Lesson 1

Learning Focus

Solve multi-step linear equations using inverse operations.

Lesson Summary

In this lesson, we learned how to solve multi-step equations in the context of using operations to represent actions in a story. As we observed how to “un-do” the actions of the story, we developed a strategy for solving the equation by using inverse operations. The order in which inverse operations are applied when solving an equation matters, and we learned how to pay attention to the structure of the equation for clues to the order in which we should use inverse operations.

Lesson 2

Learning Focus

Use units to interpret and solve equations that contain primarily variables that represent quantities, such as a formula.

Lesson Summary

In this lesson, we learned how to solve literal equations for one of its variables using inverse operations with both variables and numbers. Literal equations are formulas for describing the relationships between multiple quantities. Interpreting the meaning of expressions involving quantities in terms of their units can be a tool for checking our algebraic work while solving equations.

Lesson 3

Learning Focus

Compare strategies for solving linear equations and literal equations.

Lesson Summary

In this lesson, we compared strategies for solving linear equations and literal equations and found that the processes for solving each are similar. We solve both types of equations by using inverse operations in the reverse order from the order used when evaluating the expression that involves the variable we are solving for. However, the answer to a linear equation is a number, while the answer to a literal equation is a variable or an expression. Sometimes we have to combine like terms, particularly if expressions containing the same variable occur on both sides of the equations. Properties of operations and properties of equality guide our thinking when solving equations and help us justify each step in our equation-solving process.

Lesson 4

Learning Focus

Justify properties of inequalities.

Use inequality notation.

Lesson Summary

In this lesson, we reasoned about inequalities to compare algebraic expressions. We found and justified the addition, subtraction, multiplication, and division properties of inequalities.

Lesson 5

Learning Focus

Write and solve inequalities to model real situations.

Write solutions to inequalities using set builder and interval notation.

Lesson Summary

In this lesson, we wrote inequalities to model contexts that had a range of solutions. We used the properties of inequalities to solve the inequalities and found that solving inequalities is very similar to solving equations, but we must be careful when multiplying or dividing by a negative number to reverse the inequality sign. We used interval and set notation to write solutions and learned that many of the solutions that we write are compound inequalities.

Lesson 6

Learning Focus

Understand similarities and differences in solving equations and inequalities.

Learn to avoid common errors and misunderstandings about inequalities.

Lesson Summary

In this lesson, we examined common mistakes and misconceptions about inequalities. We analyzed the similarities and differences in solving equations and inequalities, determining which properties applied to both and which properties were different for inequalities.

Lesson 7

Learning Focus

Organize data into matrices and use the structure to facilitate computation.

Lesson Summary

In the lesson, we examined reasons for organizing data into rectangular arrays or matrices. Each element in a matrix represents two characteristics or quantities, one by virtue of the row it is located in and one by virtue of the column it is in. Consequently, each element has units associated with it that describe both the characteristics of the row and column in which it is located. Paying attention to these units guides the ways we can combine matrices by addition, subtraction, and scalar multiplication.

Lesson 8

Learning Focus

Multiply matrices to model a context.

Lesson Summary

In this lesson, we learned how to perform the operation of matrix multiplication. The row-and-column structure of the two factor matrices facilitates this work since every element in the product matrix comes from adding together several partial products.

Lesson 9

Learning Focus

Model contexts with matrices.

Lesson Summary

In this lesson, we focused on writing and solving matrix equations to model different situations, including situations that involved matrix addition, matrix multiplication, the distributive property, and the scaling up of data. We found that the properties of operations can impact the way we write matrix equations, particularly when matrix multiplication is involved.