Students will be working independently on an illustrative math problem that has them looking at an inequality to determine the solution. Students will use their knowledge of substitution to decide which groups of students can ride the ride without breaking the integrity of the inequality. This task will show students two inequalities, one written in words and one as a formula and they are given a set of possible solutions. My anticipation for this problem is that the students will substitute in the solution to determine if it can be a solution. I do not anticipate them trying to solve the inequality because we have not used this method before.
I really liked this task because students could get a good feel for the solutions to an inequality. They can see that there can be more than one solution.
As an extension to this task, students could determine the types of groups they could have to make the inequality true. For example, what combination of adults and children could be a solution.
The solution: Groups 1,2,4,5 can all safely ride the ride.
Be sure students are showing their substitution to prove their answer to be true.
This problem supports mathematical practices:
SMP1: Students will find an entry point to this problem
SMP2: Students will figure out what the numbers mean and how to substitute
SMP3: Students will prove their answer to be correct through the use of substitution.
Tools: Do now question
The students have seen this vocabulary page in a prior lesson (Writing Inequalities....The Solutions Are Endless). I want to revisit it since we will be using the graphing symbols today. I will go over the symbols, words and graphing symbols with the students. They will write this information in their notes. I like to tell the students that they can remember whether to use an open circle or a closed circle for graphing by thinking about the symbols they represent. I tell them that these symbols have a little something extra therefore they use the closed circle because it has a little something extra.
Tools: Vocabulary words
Today the students will be reviewing previously learned material to set them up for the learning today. I will use 2 examples that pertain to real life. Students should be able to answer and write the inequality. Once that has been established, tell them that they can use a number line to show their solutions. When I set up my number line, I tell the students they must have zero represented and they can use the positive and negative numbers as well. To set up the number line with all solutions becomes cumbersome and really not needed. To help students graph on the number line, I ask them, for example, “which numbers are less than 12?” Students should be able to say the numbers that go left on the number line. By making sense of the numbers, students will be using MP 2. Now we have to decide if we use an open or closed circle on 12? Is 12 included in our solution? The students should say no. Then what type of circle means not included? Students should say an open circle. Show students how the open circle is on 12 and the line goes to the left. Ask students how they could check to see if their graph is correct by using the number line? Students should be able to say that they can choose a point on the line and substitute it in for the variable to see if the inequality is true.
For students that struggle, have them write the inequality a < 12. Tell them that the inequality symbol points in the direction of the solution. Do not teach the students this trick. I only like to use it when students are struggling. Plus, students need to write a variable, symbol, and number to use this trick. Our goal is to get the students to understand the meaning of the written inequality to show it on the number line.
Finally, I’m going to have the students talk about the limitations of the solutions. For example, if children under 12 eat for free at what point does the solution become not reasonable? A sample response would be that there really is no such age as -6 years. So, the solution set could be represented as 0 < a < 12.
Tools: Inequality examples
Students will be practicing in their notes how to write and graph an inequality. They have seen these situations when writing the inequality. They will take it one step further to graph them. Students can use a Hands up, Stand up, Pair up to find a partner to share their solutions with. The inequalities may not be written the same way depending on where the variable is placed, but the graphs should all look the same.
Tools: Practice Questions.
Students will be working in pairs to complete an Around the Room. These are the same questions from writing inequalities, but the students will be graphing their solutions on a number line and then proving their solutions to be true. This will help them make sense of their solutions. (SMP 1 and 2). They will work independently, but share their responses with a partner to justify their solutions. (SMP 3)
As students are working, watch to see that they are checking their solutions.
The students will be writing a reflection about today’s learning. Their task is to write to an absent student about how to graph a solution on a number line. They are to use visuals and examples in their writing. If time permits, you could have students share their reflections with tablemates or out loud as a whole group
Tools: Closure Questions