How Long Will it Take? (Day 2 of 2)
Lesson 6 of 7
Objective: SWBAT solve quadratic equations of higher complexity and practice solving real world problems.
To begin the lesson I write the equation 4(x + 2) = 16 on the board and ask students to solve for x in their notebooks. I also ask that each step in the process be described. Students should have no problem with this. Most students will most definitely use distributive property to get 4x + 8 = 16 and go on from there.
I then proceed to ask students if there is another way of solving this equation. A student may quickly raise their hand, yet I allow some time for others to think before jumping into calling on the first to want to respond. Students will usually come up with dividing by 4 on either side first. If not, I lead them on saying, "Must we use distributive property first? Can we start by using the division property of equality first?"
Once chosen I ask the student to go up and show the work using the second route. I then write the equation 4(x + 2)2 = 16 and ask the students if our first method (distributive property first) works here and allow them to try it out and explain why it doesn't.
Finally I ask if our second method (dividing first) would work. I motivate the students to work it out at their desks before asking a volunteer to work it on the board.
After making sure students can solve the quadratic equation in the Launch, I inform the class that the "trick" is to think of (x + 2) as being one single number, say "n". In ther words the equation would be 4(n)2 = 16. Making this substitution, even mentally, makes solving the equation much like the way we solved the quadratics in Day 1 of this lesson. The importance of this type of reasoning is that if a property holds for all values of a variable, then it also holds for the expressions that have those values.
After this discussion, I ask the class to solve an equation like 3(2x – 11)2 = 75. I call on a volunteer to work on the board while his/her classmates work at their desks. I circle the room looking to see if students follow this new strategy and employ it in this case.
For the application section of this lesson I pair students up and hand each student in the class a a copy of the Application Worksheet Day 2. I ask students to make sure they show all work.
During this time I walk around assessing students working and answering questions. Students should be able to complete the first three questions without too much trouble.
Question 4 is a slight extension of the lesson because I ask students to graph the quadratic with their graphing calculator. Students should state that the y-intercept, in this case (0, 784) represents the beginning height at 0 seconds. The positive x-intercept (7, 0) represents the moment the stone hits the ground. We assume that by now students understand that the negative x-intercept represents an answer that can be discarded based on the context.
To finalize the activity I ask students to change pairs by finding the closest elbow partner to avoid wasting time. I ask the new pairs to compare their answers and discuss possible mistakes in their work.
Open the Desmos Calculator for all to see on the Smart Board and project the graph pertaining to Problem 2 of the Application section. In other words graph:
y = 4(x – 1)2 and y = 16.
I prepare this while students are working on their worksheet to save time. To end the lesson in a meaningful manner, I have a student volunteer go up to the board, and write the algebraic work done to solve this equation. I ask the student to discuss the process, including the key points of the graph in relation to the equation and its solutions. If the key points of the lesson are missed by the student, I pose questions about the process to help the student address these points.
Example questions that can be asked:
- What do the intersections of both graphs represent?
- Where in your work did you use reason by chunking (or substituting) as a means of solving?
- Are both equations meaningful here?