James Bialasik SWEET HOME SENIOR HIGH SCHOOL, AMHERST, NY
Algebra I : Unit #6 - Quadratic Functions : Lesson #21

Projectile Problems & Review

Objective: SWBAT use a quadratic function to model the motion of a falling object.
Standards: HSA-REI.B.4b HSF-IF.B.4 MP1 MP2 MP3 MP4
Subject(s): Math
60 minutes
1 Warm Up - 5 minutes

This brief Warm Up  gets students thinking about the previous day's lesson: how to analyze the graph of the function.  Students should work on questions #1-3 individually and then turn and talk with a partner about their results.

I encourage students to justify their reasoning when sharing their ideas with their partner (MP3). When students are sharing, I listen in to ensure that students are not interpreting the graph as the path that the ball travels.  If I hear this pop-up in an explanation, I ask the studnet to explain what the variables represent in this particular question.  This way, they discover for themselves that the graph is not representative of the vertical/horizontal path of the ball but rather the height versus time. 

2 Guided Practice - 10 minutes

The Warm Up leads students right into this first example (projectile_practice.doc).  They can now construct the graph based on determining the vertex of the parabola and then using their calculator to graph the remaining points. It is possible to have students calculate the points by hand, but in the interest of saving time, I will opt for having them use the calculator.  

Once students graph the parabola, they can answer the questions with their partner.  For Part (e), I will take the time to show students where the average rate of change is being calculated on the graph 

Next, I will have students begin question #2 on their own.  For this task, I encourage students to graph the function on their calculator to ensure that their answer for the turning point makes sense (MP2).  Finally, I let students work with their partners on Question #3.  Today, I post the correct answers so that students gain confidence that they are thinking about each question correctly.

I have included two video resources in this section.  Projectile Questions walks students through each of these problems.  The second demonstration shows how to use the graphing calculator to find the vertex of a parabola Use Calculator to Find Vertex).

3 Review for Assessment - 25 minutes

Due to the richness of the content, I plan to let students work through this review review at their own pace. After distributing Projectile_Practice_and_Review, I give students access to the Parabolas and Word Problems videos. These videos will help students check their understanding as they work.  

Below, I detail some of the key issues that I expect students to explore or need help with in selected review problems:

4a) Determine if student can apply the formula x=-b/2a.  The value of "b" was intentionally chosen here because students will often make an error in substituting in for "-b" in the formula.

4b) Ensure that students can determine f(3/2) algebraically.

4c) Students know that the roots can be found by setting the function equal to zero.  In this case, the greatest common factor of 2 should be factored out before factoring the resulting trinomial.  There should only be two roots.

4d) Students can find the y-intercept by letting x=0.

5) Two things to watch for here (both arithmetic issues).  (1) Ensure students are correctly substituting for "-b".  (2) when finding -4(a)(c) students get a positive result (because "c" is negative).

6) Students should realize that there could be up to three solutions due to the degree of the polynomial.  When they factor out the greatest common factor (x) that will give a third solution of x=0.

This is the section students typically struggle most with.  By this time, students are becoming fairly fluent with solving quadratic equations.  The issue here is reading slowly and for understanding (no skimming!)  Also, students should be defining their variables.

7) Students should draw a picture to help them define the length and width.  Ensure they are writing an equation for area rather than perimeter.

8) Students are defining both variables (as consecutive even integers) and writing an equation for product rather than sum.

9) Students should draw a picture to envision what is happening in this question.  When multiplying to find the area, students will need to multiply two binomials.

10) Ensure students have the correct equation to represent this word problem.

11) Students should define both variables (as consecutive integers) and then using these integers to substitute into the word problem.

Two more questions on quadratic functions:

12) Students should be able to use the vertex form of a quadratic function to write the vertex and the axis of symmetry.

13) Students should be able to use the vertex form of a quadratic function to determine how that function was shifted from the original parent function.  Students should also be able to determine the effect of the value of "a" on the shape of the graph (opening up/down, wider, narrower).

4 Closure - 2 minutes

Before the end of the period, I will post answers the following list on the board:

  1. Projectile Problems (#1-3)
  2. Quadratic Word Problems (#7-11)
  3. Solving Quadratic Equations(#5, 6) 
  4. Graphing Quadratic Functions (#4, 12, 13)

These are the broad categories that I would like students to use to organize their study plan for the assessment. Based on their work today, I encourage students to think about which topics they want to devote their time to studying.  This metacognitive portion of the lesson will help students focus their effort when reviewing for the assessment.