Standard Form

This is the form that every sees when they think Quadratic Functions. And it's this form that we get to apply the quadratic formula to!  That means we get to sing! I've been singing in math class for this topic for 25 years, so much fun!

Standard Form takes four class blocks! I will outline each lesson with excerpts from my lesson plans and the supporting ISN materials. Of course there are links at the very end for everything (word format, contact me if you want PDF)

lesson #1   Standard Form – calculating vertex and graphing using y-intercept & symmetry 

Objectives

Students will be able to

·         Calculate the vertex of a quadratic in standard from that cannot be factored

·         Use the y-intercept and the symmetry of a parabola to identify two other points and graph the parabola.

Learning activities

     What do we do when we have a quadratic that doesn’t factor? Discuss with first example, what can they find easily? Have students look on calculator to find vertex. Discuss “mini formula” used to find AOS and then vertex. Then use y-intercept and symmetry to find another point. Then do rough graph. Show how to check on calculator.

o   y = x² – 2x – 3                vertex (1, -4)  y-int (0, -3) other (2, -3)

o   y = -2x² + 12x – 7           vertex (3, 11)  y-int (0, -7) other (6, -7)

o   y = ½ x² + 8x – 9          vertex (-8, -41) y-int(0, -9)  other (-16, -9)

o   y = 6x² + 6x                  vertex (-.5, -1.5)  y-int (0, 0)  other (-1, 0)

o   y = 2x² + 7x + 21          vertex (-1.75, 14.875) y-int (0, 21) other (-3.5, 21)

o   y = 5x² + 10x – 3          vertex (-1, -8)  y-int (0, -3)  other (-2, -3)

o   y = -4x² + 4x + 8          vertex (0.5, 9)  y-int (0, 8)  other (1, 8)

o   y = - ½ x² + 6x + 3        vertex (6, 21)  y-int (0, 3) other (12, 3)

o   y = - ¼ x² – x + 1          vertex (-2, 2) y-int (0, 1) other (-4, 1)

·         ISN insert – Quadratics in Standard Form. 

If time - apply to "dropped objects" (I had to do this the next class block)

Lesson #2 Standard Form – when b = 0, finding intercepts using square roots.

Objectives

Students will be able to

·         Calculate the intercepts of a quadratic function when b = 0 using the square root method.

·         Identify both zeros – remembering both “positive” and “negative”.

·         Apply this method when solving dropped or falling object problems

Learning activities

·         Today we look at how to calculate those. We focus on standard form but when b = 0.

·         PPT lesson for initial discovery & examples.

·         ISN – insert for finding x-intercepts.

·         ISN – Dropped Objects

·         Exit slip – on PPT

ISN for this lesson:

More with Standard Form – finding the x-intercepts (quadratic formula)

Objectives

Students will be able to

• ·         Calculate the x-intercepts of a quadratic in standard form by using the quadratic formula. 
•         Focus this lesson is on rational solutions to get used to the formula

Learning activities

• ·         Warm up – movie stunt double problem (on PPT)
• ·         Incorporate ISN into lesson – start by gluing in title. Then show PPT
• ·         PPT: Solving any quadratic function in standard form. Especially necessary when you can’t factor and b ≠ 0. Remember ‘solving’ finds the intercept of the parabola. We use the quadratic formula! Very important formula used throughout higher mathematics. You must MEMORIZE the formula. But it’s easy to memorize with a catchy little song. See youtube videos on PPT. Then everyone SINGS, a few times and the next few classes too.  And next class we will start class with a quick memory quiz to be sure you have memorized it!
• ·         Then go back to ISN and fill in the 4 steps CAREFUL HOW YOU HOLD THE BOX WITH THE ARROWS (did it upside down in 2015)
• ·         Do example on bottom to show use.
• ·         Then add another page into ISN (first example from PPT) – call the page “Key Features” of a quadratic in standard form. Identify direction of opening, width of parabola, x-intercepts, y-intercept, AOS, vertex & draw a graph on little red graph insert to staple in.
• ·         HW worksheet – do #s 1-8 only and only finding x-intercepts.

  There are so many great quadratic formula song videos on youtube. I prefer the "pop goes the weasel" tune. And I'm kind of picky about the wording. I really think it should go "x equals opposite b..." NOT "x equals negative b...". Negative b can be confusing to students (what if b is already negative?) and isn't technically the correct vocabulary word to use. 

   Here is one of my favorites:

but I also show these if we have time....

and there is this funky one...

Here is my ISN page for the quadratic formula:

    We then get into analyzing all the key characteristics of a quadratic function when it is in standard form. We start with rational roots but then the next class get into irrational roots (simplest radical form!). And if time we explore (briefly this past school year) imaginary roots.

     Each ISN page is an example which we analyze completely and we staple a graph onto the page so it flips up revealing the characteristics and work used to find them.

     My ISN pages:

     example 1:

     example 2:

      
     example 3:

     example 4:

guess I didn't finish that page but it also would have a graph stapled on it with the irrational roots estimated as decimals for graph placement but shown in simplest radical form in the work. 

      Finally we summarize all of this with a neat little booklet foldable (which was a bit of a pain to create and assemble) with a "right hand reflect example" and then students did more practice for classwork and homework;

    

 right hand reflect example:

          And this folder has all my ISN templates for this unit (8 blog posts worth! I'll post this link on each blog). It's way easier to upload them all together.

Here are supporting materials I used for this topic.