quadratic functions lesson 8 solving quadratic equations using...

Quadratic Functions Lesson 8Solving Quadratic Equations using the Quadratic Formula

Lesson Summary

Lesson Objectives

Common Core State Standards:

A.REI.4bx2 = ,

a ± bi a b

Time Allotment

Standards for Mathematical Practice:

MP.3

MP.4MP.6

Lesson Resources:

Lesson Breakdown: Day 1

Day 2

Day


Steps and Learning Activities Anticipated Student Responses and Teacher Support

Day 1 Quadratic Formula Opener

Students may need assistance setting up the quadratic equation. Remind them that they need to distribute the linear pairs to find the quadratic equation and set that equal to the new area.

Posing the Task

Remind students what happens when the denominator of a fraction is equal to zero.

Have the student’s research how the quadratic formula is derived or have the students derive it themselves. This concept will be addressed in depth during Algebra 2 or Integrated Mathematics III.

Remind the students that they must be careful when substituting the coefficients into the equation that they MUST be in the correct place or their calculations and solutions will be incorrect. Often times, if b is negative, the student may forget to make the value positive due to the formula.

Quadratic Functions Lesson 8

Solving Quadratic Equations using the Quadratic Formula


Day 2 Discovering the Discriminant Activity

“The definition is referring to a quadratic equation, not a function. The graphs represent the nature of a quadratic equation to connect that the real roots are shown by the function intercepting the x-axis. When the function does not intersect the x-axis, then the equation will have complex roots, or no real zeros.”

Posing the Task: Remind students to be careful when computing the discriminant that has negative values. This calculation may result in the wrong number of roots.

Day 3 Quadratic Formula Problem Set

Remind the students to rewrite the equations in standard form to determine the values of a, b and c. If not, this will cause a sign error and result in incorrect solutions.

Remind students that they can use the discriminant to check to see how many roots there will be for the quadratic equation. They can also substitute the value of x into the equation and solve to see if it is correct. Some students may need to write out the values for a, b and c to help them determine the discriminant.




Day 4 Performance Task

“Refer to the diagram that shows how far the pitchers box is away from the beginning of the platform, the hole, and where the platform ends.”

“What would it mean if the solution were negative?”

“Well let’s draw the parabola, what does it look like? What is its path?”

“So, is that the whole parabola?”

“Where would the rest of the parabola go?”

“Correct. So where is our negative solution?”

“Then, that means that the negative solution is extraneous in the context of this situation.”


Resources for English Learners

English Language Objective(s):

General Academic Terms (Tier 2) Content Specific Terms (Tier 3) EnglishLearners Unit Vocabulary:Every Day Terms/ Phrases (Tier 1)

EL Strategies:

Analysis of CCSS Implementation

conceptual understanding rigor

rigor application



Learning ProgressionsPrevious Knowledge 8.EE.2.

p

The solutions for quadratic equations using the quadratic formula will require the student to know how to represent and calculate expressions with the square root. Students will acquire this skill in an 8th grade math class. Current Knowledge A-REI.4b.

Students will solve quadratic equations by completing the square. Next Knowledge A.APR.2

Students will learn how to solve polynomial functions. Students will acquire this skill in an Algebra II or Integrated Math III course.

Materials & Attachments :

Quadratic Formula Opener:

Discovering the Discriminant

2( ) 5 6c x x x 2( ) 2 1v x x x 2( ) 2 5m x x x

two real roots

one real root

no real roots

Quadratic Formula Problem Set

2 0ax bx c

22 6 12 0x x

2 4 4 0x x

25 2 25 0x x

24 11 1 0x x

The Quadratic Formula Performance Task

d(these functions assume the toss is thrown at the center of the platform and is perpendicular to the foul line).

21( ) 0.25 7 3A d d d 2

1( ) 0.4 11 3.5K d d d2

2 ( ) 0.25 8 3A d d d 22 ( ) 0.45 12 3.5K d d d

23( ) 0.25 10 3A d d d 2

3( ) 0.4 13 3.5K d d d

1( )A d 1( )K d

2 0ax bx c

2 ( )A d 2 ( )K d

3( )A d 3( )K d

Teacher Notes – Quadratic Formula

Teacher Notes – Discriminant

Discriminant.

Discriminant

nonot

one

twotwo

Quadratic Formula Opener

2

2

(2 15)(2 8) 3604 46 120 3604 46 240 0

x xx xx x

246 46 4(4)( 240)2(4)

x

15.39,3.89x

Discovering the Discriminant

2( ) 5 6c x x x 2( ) 2 1v x x x 2( ) 2 5m x x x

25 5 4(1)(6)2(1)

x

3,2x

22 2 4(1)(1)2(1)

x

1x

22 2 4(1)(5)2(1)

x

2 4b ac

two real roots

one real root

no real roots

Quadratic Formula Problem Set

2 0ax bx c

22 6 12 0x x >6 F v=?L :x;6 F :v;:t;:Fsu;L sv r

2 4 4 0x x= L sÆ> L vÆ?L v

>6 F v=?L :v;6 F :v;:s;:v;L r

25 2 25 0x x

= L wÆ> L FtÆ?L tw>6 F v=?L :Ft;6 F :v;:w;:tw;

L Fv{ x

24 11 1 0x x

= L vÆ> L ssÆ?L s>6 F v=?L :ss;6 F :v;:v;:s;

L sr w

tT6 F ssT E sr L r

= L tÆ> L FssÆ?L sr

T LssG ¥:Fss;6 F :v;:t;:sr;

:t;:t;

T LssG vs

vT N vuwKNssw

T6 E svT E v{ L r

= L sÆ> L svÆ?L v{

T LFsvG ¥:sv;6 F :v;:s;:v{;

:t;:s;

T LFsvG r

tT L Fy

T6 E zT E s{ L r

= L sÆ> L zÆ?L s{

T LFzG ¥:z;6 F :v;:s;:s{;

:t;:s;

T LFzG Fst

t

T6 F xT E { L r

= L sÆ> L FxÆ?L {

T LxG ¥:Fx;6 F :v;:s;:{;

:t;:s;

T LFxG r

tT L Fu

tT6 F yT L sxtT6 F yT F sx L r= L tÆ> L FyÆ?L Fsx

T LyG ¥:Fy;6 F :v;:t;:Fsx;

:t;:t;

T L;G 5; ;

8

T N wrzKNF swz

uT6 L Fv:T E u;

uT6 E vT E st L r= L uÆ> L vÆ?L st

T LFvG ¥:v;6 F :v;:u;:st;

:t;:u;

T L?:G ?56 <

:

The Quadratic Formula Performance Task

d(these functions assume the toss is thrown at the center of the platform and is perpendicular to the foul line).

21( ) 0.25 7 3A d d d 2

1( ) 0.4 11 3.5K d d d2

2 ( ) 0.25 8 3A d d d 22 ( ) 0.45 12 3.5K d d d

23( ) 0.25 10 3A d d d 2

3( ) 0.4 13 3.5K d d d

1( )A d 1( )K d

2 0ax bx c

2 ( )A d 2 ( )K d

3( )A d 3( )K d

quadratic functions lesson 8 solving quadratic equations using...

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