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. 

Factoring Polynomials (Algebra One)

The last skill group in this pre-quadratics unit is factoring. 

The overall objective of this skill group is

"Given any polynomial, a student will be able to factor it completely". This includes factoring out any GCF, recognizing & utilizing special factoring patterns, and factoring trinomials.  We no longer cover factoring by grouping, primarily because of time constraints (and the Algebra 2 teachers tell us it is not necessary). Although, wait, we sneak in some factoring by grouping in my technique for factoring trinomials where leading coefficient is not one.

We use our interactive student notebooks (ISNs) to record our notes and to do examples. All documents are provided at the end of this post. I have them as word documents, if you prefer PDF contact me and I can send them to you in that format. 

GCF

When I taught 7th grade I used the "upside down double division" method of finding the GCF (also useful for LCM and simplifying fractions).  This is such a great method!  As a student I remember being totally confused by the method of writing the prime factorization of each number, etc, etc. This makes a lot more sense and is easy to remember.  (this goes on our left side learning page)

Introduction to Factoring

On the same day as we do GCF, we approach the idea of factoring. I focus on how it means to "undistribute". This foldable has them cut out and sort some polynomials - some are factored, some are not factored. First they have to make two piles (factored and not factored) and then glue them in the correct column with the corresponding "other" version of that polynomial across from it. Hard to explain, but check it out here:  

(I had students put this opposite the GCF, right side reflect page. But you could do just practice examples in calculating the GCF on the right page and put this on another two page spread. We were getting close to the end of our notebook and I was worried about running out of pages!). 

Factoring Trinomials when leading coefficient is 1 (after factoring out GCF)

To warm students up to the thinking they need for this topic I give them this puzzle sheet:

full sheet in WORD format at end of this blog post.

I start factoring by approaching the long way. I want my students to figure out their own shortcut. I use the box method to develop overall strategy (sort of backwards from box & multiplying with a little GCF thrown in). I show my students this video.   https://www.youtube.com/watch?v=X7Zedh6AQlI

He goes over three examples and then we do two more together:

• -x² + 5x + 6 (can’t have leading coeff of -1, must factor) = (-1(x – 6)(x + 1))
•  13x – 30 + x²  needs to be in standard form!   = (x + 15)(x – 2) 

I tell students that there is a shortcut that they should be thinking about trying to figure out when they do practice examples & their homework. 

We do this ISN foldable. I'm not sure how I feel about it. It shows how to factor this long way step by step with each step behind a door. Not sure if I'll use it next year. I was trying to be creative... (this goes on our left side learning page)

Right side reflect page:

actually some of those examples are ones we did together, so here is a better set of examples to use for "right side reflect":

• 3x² + 3x – 36         3(x + 4)(x – 3)
• -2x² -48 + 20x       -2(x – 6)(x – 4)
• -x² – 8x – 15          -(x + 3)(x + 5)
• 20x – 2x² – 48         -2(x – 6)(x – 4) 

Factoring Trinomials when leading coefficient is NOT 1 (after factoring out GCF)

I start the next class with a warm up of some factoring problems and while students do those I go around and check homework. I ask each student, did you notice a shortcut to factoring when you did your homework? Most students do and you can see it in their work. I ask them to think about how they might explain it. Then I have students go up to the board and work out the warm up problems using and explaining the short cut. It's pretty cool to see how they pretty much all figure it out. I never have to give them a set of rules to follow, they come up with their own way of thinking about it. 

Now we go into factoring trinomials where the leading coefficient is NOT 1, even after factoring out the GCF. The technique we use is the box method used above, It's a tad bit trickier because there is a leading coefficient. It's also called the British Method and you can find a PDF here that explains the method (or search on line for other examples).

For our ISN we do a long multi-step example (that I create with 8.5x14 paper) and then "right side reflect" they have 4 examples to work out.

Special Factoring Patterns

Now we revisit those factoring patterns that we saw earlier as special products. We start with a warm up that has them factoring the "long way" some of these.

o   x² + 4x + 4    (x + 2)²

o   x² – 25            (x – 5)(x + 5)

o   x² – 8x + 16    (x – 4)²

o   4x² – 9              (2x – 3)(2x + 3)

o   4x² + 4x + 1    (2x + 1)²

o   X² – 144         (x – 12)(x + 12)

o   4x² – 12 + 9   (2x – 3) ²

Students are good with difference of squares, I'm careful to use the correct vocabulary (conjugates). They have a harder time with perfect square trinomials and I'm okay with them factoring these the "old" way. But they must write their final factored result as a binomial squared. 

My ISN pages:
"left side learning" is another "sorting" insert where they must decide which each example qualifies as (plus finding some that don't fit either category and why they don't)

"right side reflect" is more explanation but with examples inside the booklet for them to work out.

Summarizing - all together now

Then we have a day to pull it all together. Sometimes this takes two days (I just give them the evens for HW on the first day, odds on the second day)

We start with a warm up 

o   12x² + 5x – 2       (3x + 2)(4x – 1)

o   -2x² – 6x + 56      -2(x – 4)(x + 7)

o   6x² – 11x + 4       (2x – 1)(3x – 5)

o   24x² + 34x + 12     2(3x + 2)(4x + 3)

o   36x² – 49        (6x – 7)(6x + 7)

o   25x²  – 30x + 9      (5x – 3)²

o   36a³b – 24ab + 60a²b         12a²b (3a – 2b + 5)

Which actually I think I'm going to use on the "right side reflect" next year and go right into doing the ISN insert which is a summary foldable booklet on all the different types of factoring:

This was actually a huge pain to make! But that was before I had gotten the instruction for making a booklet. I'll do that next year. But will include my old ISN document below. Each page on this is a separate page instead of folding three pieces of paper. (for a quick review of the cool booklet method see my blog post of July 9, 2015 FOLDABLE LOVE)

ISN Materials

Assorted worksheets and other documents

I use a lot of kuta materials too for HW and practice, both Algebra One and Algebra Two. And we don't be using our old falling apart textbooks this coming school year (there are not any real common core textbooks yet for high school) but I may scan pages from that book to use for HW and post them on my calendar (that's another blog post! coming soon in August 2015)

Polynomials and Polynomial Operations

More on establishing essential Algebra Skills for our Quadratics Unit.

A Frayer Diagram to define Polynomials:

And ISN insert to list vocabulary needed in talking about polynomials.

Hmmmm - I don't have an insert for adding & subtracting polynomials and think I should do one for next year. 

Multiplying Polynomials - I hate FOIL and don't mention it all all. I go over regular distributive property with a single multiplier and then show them what to do if you are multiplying two binomials (you've got two multipliers). There is also the cool "box" method that sets the stage for how I show factoring.

I expect my advanced students to recognize "special cases" (conjugate binomials and squaring binomials) and to be able to do them quickly, NOT the long way. (There are always a few students in an advanced class who are very hard workers but aren't especially intuitive so this drives them crazy but true advanced students should be able to do these quickly, In fact true advanced students kind of make up their own FOIL type method for multiplying other binomials, I just don't give them that word or show them that shortcut).

Here are my ISN originals.

Another Mistake that HORRIFIES Math Teachers

I tell my students that every math teacher dies a little inside when they see this mistake. That we want to run screaming from the room. That they won't get any kind of partial credit if they make this mistake, in fact I will question whether they should be in an advanced class. And I mention this over and over.  This meme may seem harsh but it shows how much this error horrifies me.

Simplifying Radicals

Before we officially begin our Quadratics Unit I do a little "mini-unit" where we build up our algebra skills. It was a bit alarming when we first worked with the common core and were developing our curriculum. No time was allotted to developing these algebra skills. It was assumed that students would seamlessly learn them while quadratics were being developed. Urp!  Much better to lay out these skills and then when we need them they are already in the student's toolbox of algebra for them to access (somewhat seamlessly). This mini-unit is divided into three sections.

1) Simplifying Radicals

2) Polynomials and Polynomial Operations (we skip division due to time constraints)

3) Factoring of Polynomials (we skip factoring by grouping)

So this blog post is on how we cover simplifying radicals. (topics 1 & 2 above were taught by my student teacher but I worked more closely with her on creating materials since I was finishing the unit with them).

Essentially we go over what it means for a radical to be in simplest form and then go over the process of simplifying, multiplying, dividing, rationalizing the denominator and adding/subtracting.

This could take a few blocks.

Then we summarize each process with this ISN page. This includes a cool foldable,

Introduction to Radicals (Algebra Foldable)

 that I had to purchase on teachers-pay-teachers from Lisa Davenport.

Sarah Hagan has some good simplifying radical ideas on her math=love page "Radical Radicals". I just wanted to have one summary booklet with all the techniques in one place, worth $3 I guess. 

Then the "right hand reflect" page has examples for students to complete.

Finally they have a packet of all kinds of problems to work on.