Quadratic Data (Algebra One)

http://www.ats.ucla.edu/stat/mult_pkg/faq/general/y6.png

This was the last topic we had time for in our Algebra One quadratics unit. We did not manage to get to the discriminant anywhere, something I wanted to do but as we sketched out the end of the year I realized we just wouldn't have time.

We also didn't have time for a performance task, or a unit test. We just did quizzes and one big quest halfway through. And we had the final exam to prepare for.

Here is my lesson plan for introducing quadratic data:

Working with Data – Quadratic Functions

Objectives

Students will be able to

·         Apply quadratic regression analysis to data sets and analyze the data with the resulting equations.  They will do interpolation, extrapolation, find minimum or maximum values and intercepts – relating all to the problem situation. (same objectives over the entire lesson)

What will students do to learn this?

First Day:

·         Start class with quiz on vertex form

·         Activity  HIV data. Discuss – emphasis is on visualizing quadratic data and what it means in terms of a trend, creating a graph (both by hand & on calculator) and understanding the significance of the maximum value.

·         ISN insert – finding characteristics on the calculator

·         Considering a visual parabola:

 measuring a tunnel example (on ppt) – with provided scenario students will

a)    Input the data into their graphing calculator and draw a rough sketch of their graph (label axes)

b)    Write a quadratic function equation f(x) for the data (use STAT CALC #5).

c)    What is the R² value and what does this tell you about the fit of the function to the data?

d)    What is the vertex of this graph? What does it represent in this problem situation?

e)    Interpolation – Calculate f(4) and explain what your result means with respect to the problem situation.

f)     Extrapolation – Calculate f(10)  and explain what your result means with respect to the problem situation.

g)    What is the y-intercept of this graph and what does it represent?

h)    What is the positive x-intercept of this graph? What does it represent?

 Second day - work with data sets 

Here is the ISN insert

next year I want to do a "right hand reflect" example opposite this. 

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 supportingmaterials I used for this topic. 

Quadratics and Vertex Form (Algebra One)

The last form for us to go over is vertex form. It's probably one of  the easiest forms to work with as long as you are comfortable with the square root method of solving equations.

We started with an investigation worksheet - function equations shown in both standard & vertex form. They calculate the vertex from the standard form and then answer questions comparing the two forms.

We didn't get to converting standard form into vertex form using completing the square, but we did convert. We found the vertex using the old "-b/(2a)" formula and plugged that into the vertex form template and inserted the leading coefficient in for "a" in the equation. Really easy method, builds on something they already know. Oh well, sorry about skipping over completing the square (no time!).

Here are my ISN inserts

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. 

Projectile Motion (Algebra One)

Now we get to apply quadratics to projectile motion problems! Yay!

http://physics.tutorcircle.com/motion/projectile-motion.html

We do lots of examples, I have one of my old Algebra 2 PowerPoints that gives a decent introduction.

And then we do a lot in our ISN. We work with both ft/s and m/s. Here are my pages:

All the different types of questions that can be asked about a projectile motion situation can be confusing for students, so I made this ISN insert "Projectile Motion Deconstructed". It attempts to show all the different types of questions they might have to answer for any PM scenario.

The important thing is practice, practice, practice. My students work in their groups doing problems and then do more problems for homework. 

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. 

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. 

     

More with Quadratic Characteristics (Algebra One)

Before going into the next form I wanted to back track a bit and look at other characteristics of a quadratic function. My thought was to review domain & range and expose students to increasing & decreasing intervals, and end behavior. I didn't get to those last two but have the materials in my ISN for it. Maybe next year...

We started off by investigating width and direction of opening with this worksheet.

Then we summarized our findings with these ISN pages.

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.

Quadratics X-intercept Form (Algebra One)

After our initial investigations of Quadratic functions in general we start to look at the different forms of quadratic functions. We start with x-intercept form. This will pull in the factoring skill from the algebra skills mini-unit we just did.

Here is my lesson plan (which really ended up taking two class blocks)

Objectives

Students will be able to

• ·         Find the key features of a quadratic function when the function equation is in intercept form.
• ·        Convert standard form to intercept form and then proceed to determine key features.

Learning activities

·         Warm up – rocket problem. Students brainstorm together to answer questions about the rocket.

·         Let’s look more closely at graphs and key features of some quadratic function equations. WS “Explore Quadratic Functions” – students fill in table of values, graph & identify key features. They also factor and see if they notice anything about their factors and any of the characteristics. Should come to conclusion that the factors help us find the x-intercepts.

·         Let’s examine a few other function equations to see if we can come up with some rules to find key features using the equation, not the graph.

o   y = x² – 6x + 5  do together – what can we do to find key features? (factor – finds x-intercepts, discuss zero product property). How can we find the vertex? (give students some time to brainstorm without using graphing function of calculator - remember the property of symmetry. After a few minutes let them use graphing calculator graphs if necessary).

o   y = x² + 2x – 3

o   y = x² – 16

o   y = 2x² + 20x + 48

o   y = -x² – 4x + 32

o   y = x² + 6x + 9

o   y = -3x² + 12x + 63

·         Go back to original rocket problem. Relate the questions to the key features. Factor the original equation to see how those features can be determined algebraically.

·         Wrap up discussion of “intercept form” of quadratic functions and finding key features. Use ISN foldable.

·         Exit slip y = x² + 8x + 12 find x-intercepts, vertex, AOS, y-intercept and tell whether it opens up or down. Draw a rough sketch of the graph with parts labeled.

·         If time – do another application problem

o   The weekly profit function in dollars of a small business that produces fruit jams is

P(x) = 0.4x² + 40x + 360

where x is the number of jars of jam produced and sold.

a)      The small business “breaks even” when the profit is zero (cost & income are the same).  Determine at which point the business will break even (how many jars of jam produced & sold)

b)       Find the number of jars of jam that should be produced to maximize the weekly profit

c)       Find the maximum profit

·         Worksheet to start in class and finish for HW.

Here are my ISN pages

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 supportingmaterials I used for this topic. 

Quadratics Patterns and Differences (Algebra One)

Now this is something that I had not really done with my students before, looking at first and second differences of a sequence to determine whether it is quadratic or linear. Pretty cool and ties in with the sequence material from unit 1 and the linear functions in unit 4.

Here is my lesson plan

Objectives

Students will be able to

·         Identify if a pattern is truly quadratic by examining the second differences.

·         Create a quadratic pattern using second differences.

What will students do to learn this?

·

      Now we are going to look at some patterns – hand out patterns & differences WS. Ask students to do #1 – 3 (ignore blank spaces in the tables)

·         Find the differences in #s 4 & 5  discuss,

·         Extend pattern in #5, #2 use regression to find equations

·         ISN “patterns & functions” looking at differences.

·         Discuss “differences” between the f(x) terms (first, second etc).

·         Proving Quadratics Using Differences (WS – maybe skip #2 until next class)

·         HW  Patterns & Quadratics

and we went over into a second class to do the following:

·         Add “juggler” example to ISN

·         St. Louis Arch example

·         Other real life objects – banana & McDonald’s arches

·         Exit slip

·         HW – find an image on the internet that appears to be parabola shape. Print out and do the same as we did on banana & McDonald’s. (table of data, first & second differences, run quadratic regression with equation and R² value). this was cool but none of the images found were really quadratics.

Here are my ISN pages

left hand learning:

right hand reflect:

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. 

Introduction to Quadratics (Algebra One)

Our last unit of the year is Quadratic Functions. We've just finished our "mini unit" of algebra skills necessary for working with Quadratics. There are all sorts of concepts that the common core curriculum expects us to cover in Quadratics without being overly clear as to what goes in Algebra One and Algebra Two. I think the push is to cover it all in Algebra One now, but we ran out of time and didn't get to everything. And this was with an advanced level class that meets every day for 82 minutes in a block schedule for 180 school days. Phew!  But over time I think we will get to more of what we are supposed to as students come from classes in previous years that cover more of the common core, it's a slow evolution. So we do our best.

My approach to Quadratics is to treat them as just another type of function that we are trying to figure out, analyze and use when appropriate.

My lesson plan for this first day is as follows:

Objectives

Students will be able to

·         See that there are problem situations that are neither linear or exponential – in this case quadratic. They will initially see these in the context of area problems. They will examine two scenarios and analyze their results with the aid of a quadratic function equation.

What will students do to learn this?

Initiating activity

·         Go over HW sheet from after the test. Discuss how the equations and graphs are different from what we graphed so far this year (squared variable, U shape).

exploring parabolas

• ·         Explore – given fencing measuring 48 m, find all possible sized rectangular enclosures that can be formed (small group, blue graph paper – 2 each person). Create table of data IV = width and DV = area. Graph on one large first quadrant graph (blue graph paper) then do on calculator (scatter plot).  Is this graph continuous or discrete?
• ·         Discuss resulting shape (parabola) and brainstorm characteristics you see (U shape, opens downward, crosses x-axis (where),  has a highest point (where), symmetrical… discuss how this is the new function type we will be exploring.  It does have important characteristics which we will learn to identify and interpret in real world situations.
• ·         What really is a parabola? Paper folding (math=love) with patty pan paper. Full discussion. ISN page – what is a parabola. Define parts (vertex, axis of symmetry). Parabolas usually open up or down – but can also open sideways.
• ·         Go back to area problem - run quadratic regression on data. See equation and its perfect fit. y = -x² + 24x identify specific characteristics on this graph paper (vertex, x-intercepts, y-intercept, axis of symmetry)
• ·         Add to ISN – graph of data (on graph paper) with parts labeled. Characteristics of a Parabolas graphic organizer, Quadratic function frayer diagram, paper folded parabola with parts identified.
• ·         If time do “what is a parabola” foldable.
• ·         Do two class examples graphing with tables (one opening up or down, one opening right or left). Are they both functions?

·         HW – worksheet of graphing

Here are my ISN pages:

For the detailed instructions on how to do this paper folding parabola check out Sarah Hagan's blog math=love post Wax Paper Parabolas.

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.