Unit 2 – Week 5 Reasoning with Linear Equations and Inequalities Lesson 3 Students describe the solution set of two equations or inequalities joined by either “and” or “or” and graph the solution set on a number line. Lesson 15 Story of Functions

Standards A.CED.1 – Create inequalities in one variable and use them to solve problems. (integer inputs only) A.CED.3 – Represent constraints by inequalities and interpret data points as possible or not possible solutions. A.REI.3 – Solve linear equations in one variable including equations with coefficients represented by letters.

Essential Questions What is a compound sentence? What is a declarative sentence? Does the word “and” mean the same thing in a compound mathematical sentence as it does in an English sentence? What is a compound math sentence?

Read, Write, Draw, Solve Determine whether each claim given below is true or false. Right now, I am in math class and English class. Right now, I am in math class or English class. 3+5=8 and 5<7-1. 10+2≠12 and 8-3>0. 3<5+4 or 6+4=9. 16-20>1 or 5.5+4.5=11 These are all examples of declarative compound sentences. When the two declarations in the sentences above were separated by “and,” what had to be true to make the statement true? When the two declarations in the sentences above were separated by “or,” what had to be true to make the statement true? Students will complete Exercise 1 from Lesson 15 from A Story of Functions

Discussion - Activator How does the word “and” mean the same thing in an English sentence and a math sentence? The word “and” means the same thing in a compound mathematical sentence as it does in an English sentence. In math if and separates two equations than the two equations must be true in order for the statement to be true. In English if two clauses are separated by “and,” both clauses must be true for the entire compound statement to be deemed true.

Discussion How does the word “or” mean a similar thing in a compound mathematical sentence as it does in an English sentence? The word “or” also means a similar thing in a compound mathematical sentence as it does in an English sentence. However, there is an important distinction: In English the word “or” is commonly interpreted as the exclusive or, one condition or the other is true, but not both. In mathematics, either or both could be true. If two clauses are separated by “or,” one or both of the clauses must be true for the entire compound statement to be deemed true.

Let’s look at some x + 8 = 3 or x – 6 = 2 4x – 9 = 0 or 3x + 5 = 2 X – 6 = 1 and x + 2 = 9 2w – 8 = 10 and w > 9

Exercise 2 Questions In order for the compound sentence x > -1 and x < 3 to be true, what has to be true about x? Where do the solutions lie on the graph? What are some solutions that are possible for this compound inequality? How many solutions are there to this compound inequality? x has to be both greater than -1 and less than 3. (Students might also verbalize that it must be between -1 and 3, not including the points -1 and 3.) Between -1 and 3, not including the points -1 and 3 Encourge students to look for solutions other than just whole numbers An infinite number of solutions that are between -1 and 3 (fractions/decimals/integers)

Ways to write you solution set X > -1 or x < 3 -1 < x < 3 Or displayed on the number line

Exercise Questions 3x – 4 < 17 or -2x + 8 ≤ - 2 It could either be less than -4, or it could be greater than 0, but x cannot equal -4 or 0 To the left of -4 and to the right of 0 No. Those symbols suggest that x must be greater than zero and less than -4 at the same time, but the solution is calling for x to be either less than -4 or greater than zero. Infinitely many except for numbers between -4 and 0 Encourge students to look for solutions other than just whole numbers

Exercise Question 1 – 4x ≤ 21 and 5x + 2 > 22

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Summarizer Consider each of the following compound sentence. x < 1 and x > -1 x < 1 or x > -1 Does changing the word from ‘and’ to ‘or’ change the solution set? Explain why. Create a number line graph for each compound sentence to support your reasoning. For the first sentence, both statements must be true, so x can only equal values that are both greater than -1 and less than 1. For the second sentence, only one statement must be true, so x must be greater than – or less than 1. This means x can equal any number on the number line.