I will let students work on this warmup individually and then share their work with a partner. Having students look for errors in another student's work requires more understanding than simply solving the problem themselves (MP3).
This set of questions also focuses on a common mistake made by algebra students which is forgetting to distribute a negative correctly when applying the Distributive Property. After students have had time to work through the error analysis with a partner, I have students do a non-verbal cue to assess which partnerships found errors in each solution. I will then project the work up on the board so that students can refer to specific lines when giving their method of solution.
Throughout this section of the lesson I will refer to slides from Solving Inequalities Add and Sub.
Slide 2: While this may seem slightly elementary, I always want to start off a mini-unit on inequalities by having students understand how to read the symbols. Students are often taught the inequality symbols in the lower grades in a very robotic way. What I mean by that, is when students are looking at this slide, ask them what the symbol in the top left means. Most of your students will say that it means "greater than." I want to push students to an understanding level so I ask the students if it could mean something else? I then have them do a turn-and-talk around this question.
I want my students to have more flexibility in understanding than that so I explain to them how to read the inequality in two ways. For example: x > 4 can be read as "x is greater than 4" or "4 is less than x". On slide #2, I do a few examples with my students of this variety so they get used to reading the inequality in two directions. Then, as I proceed through the next several lessons, I can pause occasionally to read an inequality in both directions.
Teacher's Note: I have also found it useful over the years to have students read the inequality starting with the variable each time. If you refer to the inequality above, "4 is less than x" does not tell you as much as "x is greater than 4".
Slide 3 provides each student an opportunity to write an inequality that is written in context. Students should work on these four examples with their partner and then share out as a class focusing particular attention on the statements which imply equivalence as well. (Example: Kayla wants at least 23 dollars means d>=23)
Slide 4 & 5
In Slide 4 and 5 we are working towards an understanding that the properties of algebraic equivalence (commutative, associative, and distributive properties) also apply to inequalities. The other point of emphasis is that inequalities have a range of values that make them true. By allowing students to substitute values into an inequality to show that it remains true will help them understand the concept in a concrete way.
In slide 5, students will be first making a conjecture about whether or not each statement should be true. This will require students to reason in an abstract way about each inequality (MP2). I have students do a think-pair-share around the collective list of four questions, making their own conjectures first and then checking their values and comparing their work with their partners. When summarizing these as a class, I am sure to discuss the properties for #1 and #2. In #3 and #4, I point out the value that is added to each side of the inequality.
Based on #3 and #4 from the slide before, I have students think about this question and then turn and talk about a way to justify their answer. Here answers will vary. As students are discussing with their partners I am listening for students who have an insightful way of explaining why this property works. For example, "Adding a value to each side would just make both sides bigger but they would still be the same distance apart on the number line."
Students will work with a partner for the independent practice section of solving_inequalities_add_practice. This practice set allows students to practice working with inequalities from three perspectives:
In order to help students work with precision I verbally remind students to take at least two values from their solution set and verify that they make the original inequality true. If the values do not make the inequality true, I encourage students to go back through their work step by step and try to identify the error.
Teacher's Note: I may go through question #3 as a class as students may not be familiar with the "not equal to" symbol. As with the other exercises, I will encourage students to plug the solutions back in to ensure the solution is correct. With this case they could also plug the symbol that x is not equal to back into show that it results in a statement that is false.
This closing activity not only allows students to apply an inequality in context but will also give me a pre-assessment of how students handle the concept of division with inequalities. Students will need to understand the problem by defining the variable (the number of rides) and write an inequality using the other given information (MP4).
When the inequality is solved the result is x<=8.57. I ensure that students understand that the typical rules of rounding do not apply here. If the answer is determined to be x<=9, I ask students why this could not be true? (When the value of 9 is substituted into the original inequality the statement would be false). Also, the answer must be rounded to a whole number because in the context of this question, decimals are not feasible (MP1).