# Overview of linear equations and inequalities in two variables

Create and graph the solutions of linear equations and inequalities in two variables, and discuss their meaning in a real-world context (A-CED.A.2$^\star$, A-CED.A.3$^\star$, A-REI.B.3, A-REI.D.12).

This hook lesson provides an engaging context which allows students to exercise many of the skills they started to learn in middle school and will continue to apply in more sophisticated ways in this unit: setting linear equations in two variables to model a relationship between two quantities, solving for another variable, interpreting and graphing inequalities in two variables. It sets the stage for the units to come.

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#### Description

WHAT: In this introductory activity, students construct linear equations and inequalities to explore the mathematics of dating (A-CED.A.1$^\star$, A-CED.A.2$^\star$). Using the rule of thumb “half plus seven,” students determine the age of the youngest person a given person (“the dater”) can date, and use this to identify the range of permissible ages for “datees” that are not too young (A-CED.A.3$^\star$). They then invert the rule, create an equation to represent the age of the oldest person a dater can date and, as before, amend this to create an inequality, whose solutions, this time, whose solutions are the ages of all datees that are not too old. A datee must be neither too old or too young, so datee age must satisfy both inequalities. Graphically, this means that acceptable datee ages must lie in the half-plane determined by each inequality, namely the “Romance Cone” which is the intersection represents the total acceptable dating region, or the “Romance Cone” which is the intersection of the two half-planes (A-REI.D.12). Students use this to determine whether famous celebrity couples such as Tom Cruise and Katie Holmes were within the “RoCo” on their wedding date and, if not, use strategies to determine how long they would have to wait until they were (A-REI.B.3). This lesson is intended to be the hook of this unit; at this point, although students will solve the problems posed in the lesson, it is unlikely that they will all be view the process as “solving a system of linear inequalities.” Datelines can be revisited at the end of the unit when students have gained the full set of skills to solve systems of linear inequalities.

WHY: This lesson has several qualities that make it a good hook lesson:

1. The context is generally engaging.

2. The problem is easy to understand, but the answers are not immediately obvious, providing students an opportunity to reason abstractly and quantitatively (MP2); model with mathematics, and interpret their results in the context of the situation (MP4).

3. Students can begin work using tables to generate values for the “half plus seven rule” but will need to eventually graphs two linear inequalities on the same axes to represent and analyze the full situation. This is intended to motivate the work of the unit.

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