Applications of Quadratics Day 1

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SWBAT use a quadratic equation to model a mathematical situation.

Big Idea

Students use a graphic organizer to organize their thinking around solving quadratic word problems.


7 minutes

Slide 1

The logistics of this lesson are really interesting.  I want students to use white boards because I need them to be able to hold up their answers to slide 2 and 3 of the launch. However, before getting to those questions I ask my students to answer the question on slide #1 on their whiteboards.  Most students are able to "just think" of the number that will make the statement true (MP2).  However, some students will set up an equation and solve it.  I will have students from both camps share their answer and how they obtained it.  This simple categorization helps lead into today's lesson by giving students a glimpse of types of problems they will encounter in class today.

 Slide 2 & Slide 3

In my classes, it is often basic procedural mistakes that hold students back when using algebra to model a word problem (MP4).  Slides 2 and 3 of the launch will get students "warmed up" by asking them to write some simple algebraic expressions.  Again, students will use white boards and hold up their answer after each questions. I will use this "presentation" as an informal assessment of student readiness. My students will usually do okay on the first three questions.

On question Problem 4, some students will write "5-x" and others will write "x-5".  I will put both of these up on the board and have students turn and talk with their partner about which is correct.  Then, I will ask students to pick a specific number and imagine that number being decreased by 5.  I'll ask,"Which expression would give the correct answer?"  Usually after this intervention, Problem 7 will cause fewer issues.  However if quite a few students are still writing "7-x", I will do another mini-lesson to move things in the right direction.

Guided Practice

28 minutes

After the Launch, each student will be given a handout of the Guided_Notes.  As we work on the problems in the notes, I will ask students to continue to show their work on their whiteboards.  The reason I want both available is so that students can "mark up the text" of the word problem before trying to write their equation.  To do this, students can underline, write expressions, circle words, etc (MP1).  

For this part of class, I have students divide their mini-whiteboards into three sections with the middle section being the largest.  When reading a word problem, the first thing I want my students to determine is "what are our variables?" (aka, "what do we want to find?").  I ask my students to place this information in the left portion of the whiteboard.  The second thing I ask my students to do is to use the variables to write an equation or expression based on the problem description.  Finally, I encourage my students to make sure that answers make sense by solving and checking in the right section of the white board. After this introduction, we start working on the problems as a class.

I generally don't like to read problems to students.  So, I ask the entire class to read the first question to themselves either by looking at the projector or by reading the problem on their paper.  On the first read-through they should only be reading.  Then, I ask the class to read the problem a second time marking up the text as they read.  Now, on the third pass, I ask students to construct an equation.

Teaching Notes:

  1. In questions #1-4, "the setup" would mean that students are defining their variable as "let x = the positive number"  The work is shown in the middle section of the whiteboard.  
  2. One thing I stress is that students do their check using the original problem NOT the equation that they make.  Students should take their solution (in the case of #1 the solution is 2) and substitute that value wherever they see "a number" or "the number" in the original word problem.  I do this because if students construct the incorrect equation but then solve it correctly their answer still will not make the original problem true (MP6).

Questions #1-4 usually go fairly smooth.  We go through them 1-by-1 and students work on the whiteboards and hold up their responses once they are complete.  If some students are really excelling, I will have them work ahead using the guided notes.

In questions #5 and #6 the setup would include both expressions for both the length and the width of the rectangle (example: #5 would be "let x = width" and "let x+4=length").  For the check, their solutions should make sense for both dimensions (one is 4 more than the other) and should give the correct area.

Question #7 will require us to review how to set up variable representations of consecutive integers or consecutive even/odd integers before starting to write the equation to solve the problem.  


5 minutes

I designed this closing activity to help me to identify which students are grasping the idea of writing quadratic representations to solve word problems.  Before distributing it to the students I will remind the class that I still want to see all three parts in their solution (setup, work, and a check).