March 30 April 3 - Great Hearts Northern Oaks, Serving ... · 3/8/2020 · Algebra 1 March 30 – April 3 6 Look at the following parabolas and follow the directions. The first one

Algebra 1

March 30 – April 3

Time Allotment: 40 minutes per day

Student Name: ________________________________

Teacher Name: ________________________________

Teacher emails: [email protected] and

[email protected]. Ms. Brintnall will be teaching Mrs. Chubb’s Algebra 1 class for

the remainder of the school. If you were in Mrs. Chubb’s class, you should email Ms. Brintnall for help if

needed!

mailto:[email protected]

mailto:[email protected]

Algebra 1


1

Packet Overview

Date Objective(s) Page Number

Monday, March 30 Students will be able to graph linear and quadratic

equations. (8-8)

3

Tuesday, March 31 Students will be able to graph quadratic equations

and calculate their axis of symmetry. (8-8)

8

Wednesday, April 1 Students will be able to describe transformations to

graphs of quadratic equations when a and c change.

11

Thursday, April 2 Review and quiz on graphing quadratic equations/

Review: Factoring perfect square trinomials

16

Friday, April 3 Students will be able to solve quadratic equations by

taking the square root of both sides. (12-1)

21

Dear Algebra 1 students,

Thank you for your hard work on our first remote learning packet last week. We enjoyed receiving questions

from many of you! We greatly miss seeing you each day in class and love hearing from you.

This week, you are about to reach the pinnacle of Algebra I: solving quadratic equations! This year, you already

mastered solving linear equations (for example, 3(𝑥 + 2) − 𝑥 = 2(𝑥 − 1), which is a first degree equation) and

have graphed solutions to linear equations in two variables (for example, 𝑦 = 7𝑥 − 4). There are some things

we know to be universally true about linear equations: they have one solution and they form a straight line

when graphed. Now, we get to apply our knowledge of linear equations to quadratic equations. Your knowledge

of factoring quadratic expressions will be helpful as well (i.e. factor 𝑥2 − 8𝑥 + 16). Using all of these skills,

you will discover things that are universally true about quadratic equations. While we wish we could explore

these new concepts with you in person, we are excited to journey with you as you explore quadratics in the

following packet.

As we begin studying quadratics, there will be some key vocabulary and notes that you will need to copy in

your looseleaf notes. These things will be highlighted throughout this week’s packet, so write down all

highlighted content. This week’s minor assessment (quiz) will be on Thursday, April 2. Keep in mind that,

unlike last week, you may NOT use your notes for this week’s quiz. However, writing down the key ideas as

you go will help you know what to review and make you well-prepared for this week’s quiz. Answers to all

activities (except the quiz) can be found on p.25 – 28 of the packet.

Know that we are thinking of you as you continue learning remotely! No question is too small, so please send

us an email if you are wondering about anything! We would love to hear how you are doing

With much love, and

Algebra 1


2

Just like last week, to stasrt, gather about pieces of loose-leaf and STAPLE them together (any kind of lined

paper or graph paper will do – you will only need about 10 sheets this week). Put your name on the very top of

EVERY PAGE (front and back) of these loose-leaf papers, just like you did last week.

This is the equivalent of your notebook during class, and we will refer to it throughout this packet as you

“loose-leaf packet.” We will ask you to write certain problems with particular titles, and all of this will be done

in that loose-leaf packet. At a later point, we will ask you to turn in that loose-leaf packet. Do not worry

right now about whether that will be online or in person, simply do the problems as we instruct with the

proper titles and labels.

I have gathered around 10 pieces of lined paper, put my name at the very top of every sheet on both the

front and the back, and stapled them. I am ready to go!

“Pure mathematics is, in its way, the poetry of logical ideas.” Albert Einstein

Academic Honesty

I certify that I completed this assignment

independently in accordance with the GHNO

Academy Honor Code. Right now in my

Algebra I class, this means that I will NOT

use a calculator except to check my answers

when I am finished with them.

Student signature:

___________________________

I certify that my student completed this

assignment independently in accordance with

the GHNO Academy Honor Code. Right now

in this Algebra I class, this means that the

student will NOT use a calculator except to

check answers when finished with them.

Parent signature:

___________________________

Algebra 1


3

Monday, March 30 Algebra 1 Unit: Quadratic Equations

Lesson 1: 8-8 Linear and Quadratic Functions (Graphing)

Objective: Graph linear and quadratic functions on a coordinate plane.

NOTES TITLE: (This will appear at the beginning of each lesson so you can title your notes for the day.)

Your Name

Lesson 1: Graphing Review

Bellwork: Review

Copy the list of names we have for the x-variable and y-variable into your notes. Spend 2-3 minutes

memorizing these terms. Close your eyes and say each pair of terms until you can list all of them.

x-variable y-variable

Domain

Independent variable

Input

Range

Dependent variable

Output

I have memorized the x-variable and y-variable terms above.

Complete the table of values for the equation 𝑦 =1

2𝑥 − 3 and graph on the coordinate plane.

Table of Values Graph

x y

0

0

2

-4

1

Algebra 1


4

Is this equation linear or quadratic? ____________________________

Is this equation first degree or second degree? ______________________

The equation you just graphed is LINEAR because it forms a LINE. It is a FIRST DEGREE equation.

What is the x-intercept of the equation 𝑦 =1

2𝑥 − 3? (____, ____)

*NOTE: Other names for the x-intercept are “root”, “zero”, and “solution”. So, if you are asked, “What is the

zero of this equation?” or “What is the root of this equation?”, you would give the same answer, (6, 0).

What is the y-intercept of the equation 𝑦 =1

2𝑥 − 3? (____, ____)

Now, read section 8-8 in your textbook:

NOTES:

𝑔(𝑥) = 2𝑥 − 3 is LINEAR

because it forms a straight LINE.

It is first degree because the

highest power on a variable is 1.

There is only ONE root/zero/x-

intercept.

ℎ(𝑥) = 𝑥2 − 2𝑥 − 2 is

QUADRATIC. When we plug in

values for x to get values for h(x),

it forms a shape called a

parabola. It is second degree

because the highest power on a

variable is 2. We see that there

are TWO roots/zeros/x-intercepts.

Algebra 1


5

Answer the following questions:

1) What is the name of the shape of the graph of a quadratic equation? ___________________________

2) What is a vertex? _______________________________________________________________________

Parabolas have an axis of symmetry that divides them into two symmetrical halves. This is a vertical line that

goes directly through the vertex.

Algebra 1


6

Look at the following parabolas and follow the directions. The first one is done for you.

1. Circle the roots.

2. Write down the coordinate for the vertex. Choose whether the vertex is a maximum or minimum.

3. Draw a dotted, vertical, line through the axis of symmetry.

4. Label the axis of symmetry with its equation.

5. Identify the y-intercept.

A)

Vertex: (3, -5) is a minimum point

y-intercept: (0, 4)

B)

Vertex: _____________ is a

__________________

y-intercept: ___________

C)

Vertex:

___________ is a

_______________

y-intercept:

_________

D)

Vertex:

___________ is a

_______________

y-intercept:

_________

x = 3

Algebra 1


7

If a parabola has a minimum point, we say that it is concave up (it looks like a smile ). If a parabola has a

maximum point, we say that it is concave down (it looks like a frown ). On p.6, A) and D) are concave up

and B) and C) are concave down.

Graphing Practice:

Make a table of values with at least 5 points for each equation. Graph the equation on the coordinate plane.

I have made a table of values for each graph and have completed these problems to the best of my

ability.

Algebra 1


8

Now, check your answers with the answer sheet at the end of the packet. If you got any wrong, try to find the

source of your error and correct it. This does not need to be done in a different color, unless that helps you.

At this point, check in with yourself – do you have any questions about this content or these problems right

now? If you do, write those questions here:

My questions at the end of Algebra 1 Lesson 1 (if any):

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

What action are you going to take to try to answer

these questions?

Ask my parent or family member.

Have my parent help me email Ms. Steger or

Ms. Brintnall.

I may have to hold on to this question for a later

time.

I have completed all parts of today’s lesson,

checked my answers, recorded my questions (if

any), and made a plan for answering my

questions if needed. I am finished with Lesson

1 of Algebra 1 for Monday, March 30th!

-------------------------------------------------------------------------------------------------------------------

Tuesday, March 31

Algebra 1 Unit: Quadratic Equations

Lesson 2: 8-8 Linear and Quadratic Functions

Objective: Graph quadratic equations and calculate their axis of symmetry.

NOTES TITLE:

Your Name

Lesson 2: Graphing Review/Calculating the Axis of Symmetry

Bellwork: Remember the standard form of a linear equation, 𝐴𝑥 + 𝐵𝑦 = 𝐶, and the slope-intercept form of a

line, 𝑦 = 𝑚𝑥 + 𝑏.

In the equation 2𝑥 − 7𝑦 = 4, A is ____, B is ____, and C is ____.

In the equation 𝑦 = −2

3𝑥 − 2.1, m is ____ and b is ____.

(Answers: For the first one, A is 2, B is -7, and C is 4. For the second one, m is −2

3 and b is -2.1)

Similar to our formulas for linear equations, we have a

standard form for a quadratic equation: 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄

Algebra 1


9

NOTE: a is the coefficient on the quadratic term (the one with the squared variable), b is the coefficient on the

linear term (the one with the variable to the first power), and c is the constant (the number not attached to a

variable). While these terms can be written in any order because of the commutative property of addition, we

typically write them in decreasing degree (the quadratic term first, then the linear term, then the constant).

In your loose-leaf notes packet, identify a, b, and c in the following equations. Then, rewrite the equation in

decreasing degree if it is not already written that way.

1. 𝑦 = 3𝑥2 − 5𝑥 − 2

3. 𝑦 = 8𝑥 + 2𝑥2 − 9

5. 𝑓(𝑥) = 𝑥 − 10𝑥2

2. 𝑦 = −1

2𝑥2 + 4𝑥

4. 𝑦 = 2𝑥 − 5𝑥2 6. ℎ(𝑥) = 6 −2

3𝑥2

I have written down my answers for a, b, and c in my notes for all equations above and am ready to

check my answers on the last page of the packet.

Review from yesterday: For the following equations:

1. Make a table of values to find at least 5 points for each graph.

2. Graph the equation.

3. Draw the axis of symmetry.

4. Label the axis of symmetry with its equation.

I have made a table of values for each and have completed these problems to the best of my ability.

Now, check your answers with the answer sheet on the last page of the packet. If you got any wrong, try to find

the source of your error and correct it. This does not need to be done in a different color, unless that helps you.

Algebra 1


10

Now, look at your graphs from p.7 and p.9. What do all the equations for the concave up graphs have in

common? What do all the equations for the concave down graphs have in common? HINT: Look at the a value!

When looking at a quadratic equation, how do you know whether it will be concave up or concave down? Write

your answer in a complete sentence. (Answer at the back of the packet.)

__________________________________________________________________________________________

__________________________________________________________________________________________

The axis of symmetry always passes through the _____________, which is also called the maximum/minimum.

There is an equation we can use to find the axis of symmetry without graphing first!

Axis of symmetry: 𝒙 = −𝒃

𝟐𝒂

Let’s take the equation from #5. How could we find the vertex without graphing?

Process Steps

1. Let’s look at the equation, 𝑦 = 𝑥2 − 8𝑥 + 13

2. a=1, b= -8, c=13

3. 𝑥 = −−8

2(1)

4. 𝑥 =8

2 , so 𝑥 = 4

5. 𝑦 = (4)2 − 8(4) + 13

𝑦 = 16 − 32 + 13

𝑦 = −3

6. The vertex is (4, -3)

1. Write down the equation.

2. Identify a, b, and c.

3. Plug in the values for b and a into the equation for

the axis of symmetry.

4. Simplify.

5. We know that the axis of symmetry always goes

through the vertex, so the x-coordinate of the vertex

matches the x-value in the axis of symmetry

equation. We can plug in the x-value into the original

value to get the y-coordinate of the vertex.

6. Now, we can write the vertex as an ordered pair!

Practice with Axis of Symmetry and Vertex: Copy the following equations in your loose-leaf notes. Find the

axis of symmetry axis of symmetry and vertex for each equation. Tell whether the vertex will be a maximum or

a minimum value. If you would like, you may try graphing them at https://www.desmos.com/calculator to see

what they look like!

1. 𝑦 = 2𝑥2 3. 𝑦 = −𝑥2 − 8𝑥 − 15 5. 𝑔(𝑥) = 6 + 6𝑥 −1

3𝑥2

2. 𝑦 = −𝑥2 + 2𝑥 4. 𝑓(𝑥) = 𝑥2 − 𝑥 − 6 6. 𝑦 = 9𝑥2 − 4

I have completed these problems to the best of my ability, using the example to help me.

https://www.desmos.com/calculator

Algebra 1


11






__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________


these questions?



Ms. Brintnall.


time.





2 of Algebra 1 for Tuesday, March 31st!

---------------------------------------------------------------------------------------------------------------------------------------

Wednesday, April 1

Algebra 1 Unit: Graphing Quadratics

Lesson 3: Quadratic Transformations

Objective: Describe transformations to graphs of quadratic equations when a and c values change.

NOTES TITLE:

Your Name

Lesson 3: Quadratic Transformations

Today, we will look at the parabolas of many different quadratic equations and see what causes them to have

different shapes. You will be filling out the following pages of this printed packet. However, you will need to

make tables of values and work on calculations. Use this notes page for your calculation work. It is important to

practice plugging in points by hand. Do not use a calculator to find values!

Algebra 1


12

1. Make a table of values and graph all of the following equations on the coordinate plane given. Write

the value of a in the box provided. If possible, use a different color for each.

𝑓(𝑥) = 𝑥2

-1

0

1

a = ____

𝑓(𝑥) = 2𝑥2

-1

0

1

a = ____

𝑓(𝑥) = −2𝑥2

-1

0

1

a = ____

𝑓(𝑥) = 3𝑥2

-1

0

1

a = ____

Graph these equations on the coordinate plane as well. Three inputs are given for you – choose two more

to complete each table with 5 points. If you’d like, check your graphs at https://www.desmos.com/calculator

𝑓(𝑥) =1

2𝑥2

-1

0

1

a = ____

𝑓(𝑥) = −1

2𝑥2

-1

0

1

a = ____

𝑓(𝑥) =1

3𝑥2

-1

0

1

a = ____

𝑓(𝑥) = −1

3𝑥2

-1

0

1

a = ____

CONCLUSION: What happens to the parabola as a changes?

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________


Algebra 1


13

2. Make a table of values and graph each of the following equations on the coordinate plane given. Write

the value for c in the box provided.

𝑓(𝑥) = 𝑥2 − 2𝑥 + 1

-2

-1

0

1

2

c = ____

𝑓(𝑥) = 𝑥2 − 2𝑥

-2

-1

0

1

2

c = ____

𝑓(𝑥) = 𝑥2 − 2𝑥 − 2

-2

-1

0

1

2

c = ____

CONCLUSION: What happens to the parabola as c changes?

__________________________________________________________________________________________

__________________________________________________________________________________________

OPTIONAL (If you’re running low on time, skip this for now and come back to it later.): Observe what

happens when b changes! Go to https://www.desmos.com/calculator and type in the standard form for a

quadratic equation:

1. Click on a, b, and c individually.


Algebra 1


14

2. Press “play” on b and see what happens! Can you describe what happens to the parabola when b

changes?

Function Review Practice: Finding x and y intercepts

Remember the other names for x-intercepts (we can also call these points zeros, roots, or solutions). In your

loose-leaf notes packet, complete #34 – 40 even. We’ve worked out #33 and 37 for you so you can see the

process clearly. Title your work for this section “Function Review Practice”.

Algebra 1


15

I have completed these problems to the best of my ability, using the examples to help me.






__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________


these questions?



Ms. Brintnall.


time.





3 of Algebra 1 for Wednesday, April 1!

Algebra 1


16

Thursday, April 2

Algebra 1 Unit: Graphing Quadratic Equations

Lesson 4: Review and Quiz

Your 40 minutes of Algebra 1 today will look roughly like this. These are estimates to help you manage your

time, not requirements.

A. 10 minutes of bellwork

B. 10 minutes of review

C. 10 minutes of quiz

D. 10 minutes of factoring practice

A. 10 minutes of bellwork

Graph 𝑓(𝑥) = 𝑥2 − 4𝑥 − 5.

• Write the roots as ordered pairs:

__________________________________

• Write the y-intercept as an ordered pair:

__________________________________

• What is the equation for the axis of

symmetry?

__________________________________

• Draw the axis of symmetry and label the

equation on the graph.

• Write the vertex as an ordered pair:

__________________________________

• Is this vertex a maximum or a minimum?

__________________________________

When finished, check your answers on the next

page.

Algebra 1


17

Graph 𝑓(𝑥) = 𝑥2 − 4𝑥 − 5.

• Write the roots as ordered pairs:

(-1, 0) and (5,0)

• Write the y-intercept as an ordered pair:

(0, -5)

• What is the equation for the axis of

symmetry?

𝑥 = 2

• Draw the axis of symmetry and label the

equation on the graph.

• Write the vertex as an ordered pair:

(2, -9)

• Is this vertex a maximum or a minimum?

Minimum

B. 10 minutes of review

The purpose of minor assessments (in our Algebra 1 classes we call these quizzes and sometimes Functions of

the Week), is to make sure you understood something. Take 10 minutes to review your loose-leaf notes and

this packet. Pay attention to highlighted notes. You may ONLY use your own mind on this minor assessment.

You may NOT get help from the internet, a calculator, this packet, your notes, or from another person on this

minor assessment. This is the time to ask your parent any questions that you might have BEFORE you

start the quiz!!

C. 10 minutes for the minor assessment.

Please read these boxes carefully before starting on the minor assessment.

I understand that I am NOT allowed to use this packet during my quiz.

I understand that I am NOT allowed to use my own loose-leaf packet during my quiz.

I understand that while Ms. Steger and Ms. Brintnall estimate that the quiz will take 10 minutes, it is okay

to spend the time I need.

I understand that I am NOT allowed to ask a parent, family member, or friend for help during my quiz.

I understand that I am NOT allowed to use the internet or any other resource to help with my quiz.

***Once you have read through the above statements and checked each box, you may turn the page to

begin your quiz. By signing the academic integrity statement on page 2 of this packet, you are saying that

you completed the quiz on your own and without use of your notes.***

𝑥 = 2

Algebra 1


18

Quadratics Quiz #1

1. Look at the graph to the right, pay close attention

to the scale, and answer the following questions:

a) This shape is called a

________________________.

b) How many roots does it have?

___________________

c) Write the root(s) as ordered pair(s):

_________________________________________

d) Write the y-intercept as an ordered pair:

____________

e) Circle the vertex. Write it as an ordered pair:

________

f) Is the vertex a maximum or a minimum?

___________

g) Write the equation for the axis of symmetry of

this graph on the line below. Draw it on the graph

and label it.

_____________________

2. Which of the following is NOT a synonym for the

others?

a) x-intercepts

b) y-intercepts

c) roots

d) solutions

3. For the equation 𝑦 = −3𝑥2 + 5𝑥 − 2, identify

the following:

𝑎 = ________ 𝑏 = _______ 𝑐 = ________

4. What value does y always equal at the roots?

___________

5. Write the formula for the axis of symmetry.

Algebra 1


19

5. Graph the equation 𝑦 = −𝑥2 − 2𝑥 + 3. Use this space to calculate your points and make your table of values.

x y

Graph here:

Algebra 1


20

D. 10 minutes of factoring practice

In your notes packet, factor #2-8 even. Remember, first, look for the common factor! Draw area models to

help you. If you do this in under 10 minutes, try #1 and 9 as well.

I have completed these problems to the best of my ability.

You may check your answers with those in the answer pages in the back of the packet.




__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________


these questions?



Ms. Brintnall.


time. (At this point in the week, it is probably

not a good idea to hold onto a question, unless

it is more of an extension question that you are

just curious about).





4 of Algebra 1 for Thursday, April 2nd!

Algebra 1


21

Friday, April 3 Algebra 1 Unit: Quadratic Equations

Lesson 5: 12-1 Quadratic Equations with Perfect Squares

Objective: Solve quadratic equations by taking the square root of both sides. (12-1)

NOTES TITLE:

Your Name

Lesson 5: 12-1 Quadratic Equations with Perfect Squares

Bellwork: List 5 different examples of a perfect square. Include options involving numbers, variables, and

quantities.

Examples: 64, 𝑥4, 4𝑎2, (𝑥 − 2)2

Recall: When we take the square root of a number, we write the principal square root as the answer (i.e. √25 =

5). Principal square roots are non-negative. However, when finding all solutions to an equation involving

variables, we must account for all possible solutions, for the mathematical definition of √𝑥2 is √𝑥2 = |𝑥|. We

would solve 𝑥2 = 25 like this:

1. 𝑥2 = 25

2. √𝑥2 = √25

3. |𝑥| = 5

4. 𝑥 = ±5

1. Original equation

2. Take the square root of both sides.

3. Note: √𝑥2 = |𝑥|, but √25 = 5

4. This means that |𝑥| = 5, so x could equal positive

OR negative 5.

Copy steps #1-4 from the left column above into your notes. Then, copy the examples below into your notes.

Algebra 1


22

Try one on your own! HINT: Factor the left side to begin:

Ex. 4 𝑦2 + 6𝑦 + 9 = 49

Solutions: {-10, 4}

Algebra 1


23

Copy Ex. 5 into your notes as well:

Assignment: Complete p.563 #5-30 mult. 5 on your notes page. Note: The directions ask you to write your

answers in simplest radical form. You do NOT need to estimate values like we did in the second part of Ex. 3.

I have completed these problems to the best of my ability, using the examples to help me.

Now, check your answers with the answer sheet on the very last page of this packet. If you got any wrong, try to

find the source of your error and correct it. This does not need to be done in a different color, unless that helps

you.

Algebra 1


24




__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________


these questions?



Ms. Brintnall.


time. (At this point in the week, it is probably

not a good idea to hold onto a question, unless

it is more of an extension question that you are

just curious about).





5 of Algebra 1 for Friday, April 3rd!

---------------------------------------------------------------------------------------------------------------------------------------

Have a wonderful weekend! 😊 😊 😊

~ Ms. Steger and Ms. Brintnall

Algebra 1


25

Answer Key for All Lessons

Lesson 1 –

Graphing

review

Lesson 2 –

Graphing and

finding the

axis of

symmetry

1. a = 3, b = -5, c = -2

3. a = 2, b = 8, c = -9

𝑦 = 2𝑥2 + 8𝑥 − 9

5. a = -10, b = 1, c = 0

𝑓(𝑥) = −10𝑥2 + 𝑥

2. a = −1

2, b = 4, c = 0

4. a = -5, b = 2, c = 0

𝑦 = −5𝑥2 + 2𝑥 6. a = −

2

3, b = 0, c = 6

ℎ(𝑥) = −2

3𝑥2 + 6

Axis of symmetry: 𝑥 = 4 Axis of symmetry: 𝑥 = −4

Algebra 1


26

When looking at a quadratic equation, how do you know whether it will be concave up or

concave down? Answer: If a is a positive value, the parabola will be concave up, whereas

if a is a negative value, the parabola will be concave down.

1. Axis of symmetry: x=0

Vertex: (0, 0)

Minimum

3. Axis of symmetry: x= -4

Vertex: (-4, 1)

Maximum

5. Axis of symmetry: x= 9

Vertex: (9, 33)

Maximum


Vertex: (1, 1)

Maximum


2

Vertex: (1

2, −6

1

4)

Minimum

6. Axis of symmetry: x= 0

Vertex: (0, -4)

Minimum

Lesson 3 –

Quadratic

Transformatio

ns

Algebra 1


27

Algebra 1


28

Lesson 4 –

Factoring

Review

Lesson 5 –

12-1 Quadratic

Equations with

Perfect

Squares

Documents

March 30 April 3 - Great Hearts Northern Oaks, Serving ... · 3/8/2020 · Algebra 1 March 30 – April 3 6 Look at the following parabolas and follow the directions. The first one