Complete Week 31
Package
Jeanette Stein
Table of Contents
Unit 12 Pacing Chart -------------------------------------------------------------------------------------------- 1
Day 151 Bellringer -------------------------------------------------------------------------------------------- 3
Day 151 Activity -------------------------------------------------------------------------------------------- 5
Day 151 Practice -------------------------------------------------------------------------------------------- 16
Day 151 Exit Slip -------------------------------------------------------------------------------------------- 29
Day 152 Bellringer -------------------------------------------------------------------------------------------- 31
Day 152 Activity -------------------------------------------------------------------------------------------- 33
Day 152 Practice -------------------------------------------------------------------------------------------- 36
Day 152 Exit Slip -------------------------------------------------------------------------------------------- 38
Day 153 Bellringer -------------------------------------------------------------------------------------------- 40
Day 153 Activity -------------------------------------------------------------------------------------------- 42
Day 153 Practice -------------------------------------------------------------------------------------------- 44
Day 153 Exit Slip -------------------------------------------------------------------------------------------- 49
Day 154 Bellringer -------------------------------------------------------------------------------------------- 51
Day 154 Activity -------------------------------------------------------------------------------------------- 53
Day 154 Practice -------------------------------------------------------------------------------------------- 56
Day 154 Exit Slip -------------------------------------------------------------------------------------------- 62
Weekly Assessment -------------------------------------------------------------------------------------------- 64
CCSS Algebra 1 Pacing Chart – Unit 12
Algebra1Teachers @ 2015 Page 1
Unit Week Day CCSS Standards Mathematical Practices Objective I Can Statements
12 – Solve Quadratic Functions
31 – Completing the Square
151
CCSS.MATH.CONTENT.HSA.SSE.B.3.B Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
The student will be able to complete the square in a quadratic expression and use it to find the maximum or minimum value of a function.
I can complete the square in a quadratic expression and use it to find the maximum or minimum value of a function.
12 – Solve Quadratic Functions
31 – Completing the Square
152
CCSS.MATH.CONTENT.HSF.IF.C.8.A Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
The student will be able to complete the square in a quadratic function to determine roots, extreme values, and symmetry.
I can complete the square in a quadratic function to determine roots, extreme values, and symmetry.
12 – Solve Quadratic Functions
31 – Completing the Square
153
CCSS.MATH.CONTENT.HSF.IF.C.8.A Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.
The student will be able to write a quadratic function in an equivalent appropriate form (i.e. standard form, vertex form, and intercept form) to highlight items of interest (zeros, extreme values, and symmetry).
I can write a quadratic function in an equivalent appropriate form (i.e. standard form, vertex form, and intercept form) to highlight items of interest (zeros, extreme values, and symmetry).
12 – Solve Quadratic Functions
31 – Completing the Square
154
CCSS.MATH.CONTENT.HSA.REI.B.4.A Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form.
CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.
The student will be able to derive the quadratic formula by completing the square from the standard form of a quadratic equation (ax2 + bx + c = 0).
I can derive the quadratic formula by completing the square from the standard form of a quadratic equation (ax2 + bx + c = 0).
CCSS Algebra 1 Pacing Chart – Unit 12
Algebra1Teachers @ 2015 Page 2
12 – Solve Quadratic Functions
31 – Completing the Square
155 Assessment Assessment Assessment Assessment
Day 151 Bellringer Name ____________________________________
Algebra1Teachers @ 2015 Page 3
Factor each of the following perfect square trinomials. In the last two problems, look for a greatest
common factor to remove first.
1. 𝑎2 + 4𝑎 + 4
2. 49 + 14𝑎 + 𝑎2
3. 𝑦2 – 8𝑦 + 16
4. 49𝑥2 + 28𝑥𝑦 + 4𝑦2
5. 𝑟2 + 24𝑟 + 144
Day 151 Bellringer Name ____________________________________
Algebra1Teachers @ 2015 Page 4
Answer Key.
Day 151
1. (𝑎 + 2)(𝑎 + 2)
2. (7 + 𝑎)(7 + 𝑎)
3. (𝑦 – 4)(𝑦 – 4)
4. (7𝑥 + 2𝑦)(7𝑥 + 2𝑦)
5. (𝑟 + 12)(𝑟 + 12)
Day 151 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 5
1. Use your calculator to graph the equation: 𝑦 = 𝑥2 + 6𝑥 + 9. Write the equation in vertex form.
2. Sketch algebra tiles to model the equation𝑦 = 𝑥2 + 6𝑥 + 9.
Day 151 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 6
Recall the values of each of the algebra tiles. The value of the tile is its area. We will only be working
with positive (red tiles) representations of algebraic expressions.
Our original equation was written in standard form,𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐. Since it is usually much easier to
graph a parabola if the equation is in vertex form,𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘, often we try to rewrite the
equation from quadratic form into vertex form. We will use algebra tiles to help us understand why this
procedure works.
When we are trying to write an equation in vertex form, we need to have a perfect square to make the
(𝑥 − ℎ)2 part of the equation. When the quadratic equation we are given is not a perfect square, we
arrange the parts to form a perfect square, adding what we need or keeping whatever extra pieces we
may get. This activity will help you discover how to start the process of forming the perfect square from
what you are given.
𝑥2𝑇𝑖𝑙𝑒
AREA = 𝑥 ∙ 𝑥 = 𝑥2𝑢𝑛𝑖𝑡𝑠
𝑥 𝑇𝑖𝑙𝑒
AREA = 1 ∙ 𝑥 = 𝑥 𝑢𝑛𝑖𝑡𝑠
𝑈𝑛𝑖𝑡 𝑇𝑖𝑙𝑒
AREA = 1 ∙ 1 = 1 𝑢𝑛𝑖𝑡𝑠
Day 151 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 7
3. Create a partial square with algebra tiles to represent𝑥2 + 2𝑥 + ____________ . a) How many unit tiles do you need to complete the square?
b) What are the dimensions of the completed square? L = W =
c) Replace c and ? with numbers to make the statement true;𝑥2 + 2𝑥 + 𝑐 = (𝑥+? )2.
4. Create a partial square with algebra tiles to represent 𝑥2 + 4𝑥 + ____________.
a) How many unit tiles do you need to complete the square?
b) What are the dimensions of the completed square? L = W=
c) Fill in the blanks with numbers to make the statement true: 𝑥2 − 4𝑥 + 𝑐 = (𝑥+? )2
5. Create a partial square with algebra tiles to represent 𝑥2 − 6𝑥 + ________________.
a) How many unit tiles do you need to complete the square?
b) What are the dimensions of the complete square? L= W=
c) Fill in the blanks with numbers to make the statement true: 𝑥2 − 6𝑥 + 𝑐 = (𝑥+? )2
Day 151 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 8
6. What is the relationship between the coefficient of x and the number of x’s you have down one side of your algebra tile diagram?
7. What is the relationship between the numbers of x’s down one side of the algebra tile diagram and
the question mark in your perfect square binomial? 8. What is the relationship between the coefficient of x and the question mark in your perfect square
binomial? 9. What is the relationship between the question mark of your perfect square binomial and the
number of blocks (or units) you had to add to make your diagram a perfect square? 10. In the expression = 𝑥2 + 𝑏𝑥 + 𝑐 , how do you use b to get the value of c to form a perfect
square? Use the examples above (3-5) to explain your answer. 11. Try these problems—Fill in the missing “c” and then rewrite the trinomial as a perfect square
binomial.
a) 𝑥2 − 10𝑥 + 𝑐 b) 𝑥2 − 4𝑥 + 𝑐
c) 𝑥2 + 12𝑥 + 𝑐 d) 𝑥2 − 12𝑥 + 𝑐
e) 𝑥2 + 7𝑥 + 𝑐 f) 𝑥2 + 𝑏𝑥 + 𝑐
Day 151 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 9
Represent each expression by sketching algebra tiles. Try to create a square of tiles. When doing so keep
the following rules in mind:
• You may only use one x 2 -tile in each square.
• You must use all the x 2 and x-tiles. Unit tiles are the only ones that can be leftover or borrowed.
• If you need more unit tiles to create a square you have to “borrow” them. The number you borrow will
be a negative quantity.
Standard form Number
of 𝑥2 Tiles
Number of x Tiles
Number of Unit
Tiles Sketch of the Square
Length of the Square
Area if the Square
(Length)2
Unit Tiles Left Over
(+) Borrowed
(-)
Expression Combining
Previous Two Columns
𝑥2 + 2𝑥 + 3 1 2 3
y 1
x
1
𝑥 + 1 (𝑥 + 1)2 2 (𝑥 + 1)2 + 2
𝑥2 + 4𝑥 + 1
𝑥2 + 6𝑥 + 10
Day 151 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 10
12. What is the name of the form for the combined expression in the last column?
Convert the following equations from standard form to vertex form by completing the square.
13. 𝑦 = 𝑥2 − 8𝑥 + 11 14. 𝑦 = 𝑥2 + 16𝑥 + 14
Steps: Example: 𝑥2 + 6𝑥 − 14 = 0 You try: 𝑥2 + 8𝑥 − 20 = 0
a) Move constant term of quadratic to the other side. Write the equation 𝑎𝑥2 + 𝑏𝑥 + ______ = −𝑐______ 𝑓𝑜𝑟𝑚
𝑥2 + 6𝑥 − 14 + 14 = 0 + 14 𝑥2 + 6𝑥 + _____ = 14 + _____
b) Complete the square by adding a constant to both sides.
𝑥2 + 6𝑥 + 9 = 141 + 9
c) Rewrite the left side of the equation as a binomial squared and simplify the right side.
(𝑥 + 3)2 = 25
d) Square root both sides. 𝑥 + 3 = ±5
e) Solve for X. 𝑥 + 3 = −5 𝑥 + 3 = 5 𝑥 + 3 − 3 = −5 − 3 𝑥 + 3 − 3 = 5 − 3 𝑥 = −8 𝑥 = 2
Solve the following equations by completing the square using the steps above.
15. 𝑥2 − 4𝑥 − 32 = 0 16. 𝑥2 + 8𝑥 + 7 = 0
a) a)
b) b)
c) c)
d) d)
e) e)
Day 151 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 11
If the leading coefficient≠ 1, we must first divide each term by “a” so that the coefficient of the 𝑥2term
is 1. Then complete the same steps above.
17. 2𝑥2 + 4𝑥 − 3 = 0 18. 3𝑥2 + 18𝑥 + 12 = 0
Day 151 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 12
Answer Key
1.
𝑦 = (𝑥 + 3)2 Vertex form
2.
3.
a) We need 1 unit tiles to complete the square. b) L=2 W=2 c) C=1 ?=1
𝑥2
x x X
X 1 1 1
X 1 1 1
X 1 1 1
𝑥2
x
X 1
Day 151 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 13
4.
a) We need 4 unit tiles to complete the square. b) L=3 W=3 c) C=4 ?=-2
5.
a) We need 9 unit tiles to complete the square. b) L=4 W=4 c) C=9 ?=-3
6. The number of x’s you have done one side is equal to half of the coefficient of x. 7. The numbers of x’s down one side of the algebra tile diagram is equal to the question mark in
your perfect square binomial. 8. The question mark in the perfect square binomial is equal to the half of the coefficient of x. 9. The square of the question mark of the perfect square binomial is equal to the number of units
you have to add to make your diagram. 10. To get the value of c, take half of b and square it. 11. a) 𝑥2 − 10𝑥 + 25 = (𝑥 − 5)2
b) 𝑥2 − 4𝑥 + 4 = (𝑥 − 2)2 c) 𝑥2 + 12𝑥 + 36 = (𝑥 + 6)2 d) 𝑥2 − 12𝑥 + 36 = (𝑥 − 6)2
e) 𝑥2 + 7𝑥 +49
4= (𝑥 +
7
2)
f) 𝑥2 + 𝑏𝑥 +𝑏2
4= (𝑥 +
𝑏
2)
2
𝑥2
x X
X 1 1
X 1 1
𝑥2
x x x
X 1 1 1
X 1 1 1
X 1 1 1
Day 151 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 14
Represent each expression by sketching algebra tiles.
Standard form Number
of 𝑥2 Tiles
Number of x Tiles
Number of Unit
Tiles Sketch of the Square
Length of the Square
Area if the Square
(Length)2
Unit Tiles Left Over
(+) Borrowed
(-)
Expression Combining
Previous Two Columns
𝑥2 + 2𝑥 + 3 1 2 3
y 1
x
1
𝑥 + 1 (𝑥 + 1)2 2 (𝑥 + 1)2 + 2
𝑥2 + 4𝑥 + 1 1 4 1
x 1 1
X
1
1
𝑥 + 2 (𝑥 + 2)2 −3 (𝑥 + 2)2 − 3
𝑥2 + 6𝑥 + 10 1 6 10
𝑥 + 3 (𝑥 + 3)2 1 (𝑥 + 3)2 + 1
12. the name of the form for the combined expression in the last column is a vertex form.
13. 𝑦 = (𝑥 − 4)2 − 5 14. 𝑦 = (𝑥 + 8)2 − 50
Day 151 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 15
Steps: Example: 𝑥2 + 6𝑥 − 14 = 0 You try: 𝑥2 + 8𝑥 − 20 = 0
a) Move constant term of quadratic to the other side. Write the equation 𝑎𝑥2 + 𝑏𝑥 + ______ = −𝑐______ 𝑓𝑜𝑟𝑚
𝑥2 + 6𝑥 − 14 + 14 = 0 + 14 𝑥2 + 6𝑥 + _____ = 14 + _____
𝑥2 + 8𝑥 − 20 + 20 = 0 + 20
b) Complete the square by adding a constant to both sides.
𝑥2 + 6𝑥 + 9 = 141 + 9 𝑥2 + 8𝑥 + 16 = 20 + 16
c) Rewrite the left side of the equation as a binomial squared and simplify the right side.
(𝑥 + 3)2 = 25 (𝑥 + 4)2 = 36
d) Square root both sides. 𝑥 + 3 = ±5 𝑥 + 4 = ±6
e) Solve for X. 𝑥 + 3 = −5 𝑥 + 3 = 5 𝑥 + 3 − 3 = −5 − 3 𝑥 + 3 − 3 = 5 − 3 𝑥 = −8 𝑥 = 2
𝑥 + 4 = −6 𝑥 + 4 = 6 𝑥 = −10 𝑥 = 2
15. 𝑥2 − 4𝑥 − 32 = 0
a) 𝑥2 − 4𝑥 + _____ = 32 + _____
b) 𝑥2 − 4𝑥 + 4 = 32 + 4
c) (𝑥 − 2)2 = 36
d) 𝑥 − 2 = ±6
e) 𝑥 − 2 = −6 𝑥 − 2 = 6
𝑥 = −4 𝑥 = 8
16. 𝑥2 + 8𝑥 + 7 = 0
a) 𝑥2 + 8𝑥 + _____ = −7 + ______
b) 𝑥2 + 8𝑥 + 16 = −7 + 16
c) (𝑥 + 4)2 = 9
d) 𝑥 + 4 = ±3
e) 𝑥 + 4 = −3 𝑥 + 4 = 3
𝑥 = −7 𝑥 = −1
17. 2𝑥2 + 4𝑥 − 3 = 0
𝑥2 + 2𝑥 −3
2= 0
a) 𝑥2 + 2𝑥 + _____ =3
2+ _____
b) 𝑥2 + 2𝑥 + 1 =3
2+ 1
c) (𝑥 + 1)2 =5
2
d) 𝑥 + 1 = ±√5
2
e) 𝑥 + 1 = √5
2 𝑥 + 1 = −√
5
2
𝑥 = √5
2− 1 𝑥 = −√
5
2− 1
18. 3𝑥2 + 18𝑥 + 12 = 0
𝑥2 + 6𝑥 + 4 = 0
a) 𝑥2 + 6𝑥 + _____ = −4 + _____
b) 𝑥2 + 6𝑥 + 9 = −4 + 9
c) (𝑥 + 3)2 = 5
d) 𝑥 + 3 = ±√5
e) 𝑥 + 3 = −√5 𝑥 + 3 = √5
𝑥 = −√5 − 3 𝑥 = √5 − 3
Day 151 Practice Name ___________________________________
Algebra1Teachers @ 2015 Page 16
Use the information below to answer questions 1 – 6. Using the relevant number of boxes, determine the constant term to complete the square.
1. 𝑥2 + 6𝑥
2.𝑥2 + 8𝑥
3. 𝑥2 − 2𝑥
Day 151 Practice Name ___________________________________
Algebra1Teachers @ 2015 Page 17
4. 𝑥2 − 10𝑥
5. 4𝑥2 + 4𝑥
6. 4𝑥2 − 4𝑥
Day 151 Practice Name ___________________________________
Algebra1Teachers @ 2015 Page 18
Use the information below to answer questions 7 – 13.
Complete the tiles to make a perfect square hence use it to write an expression representing
them if red tiles implies negative and green represents positive.
7.
8.
Day 151 Practice Name ___________________________________
Algebra1Teachers @ 2015 Page 19
9.
10.
11.
Day 151 Practice Name ___________________________________
Algebra1Teachers @ 2015 Page 20
12.
13.
Day 151 Practice Name ___________________________________
Algebra1Teachers @ 2015 Page 21
14.
Use the information below to answer questions 15 –17
Write the coefficient of 𝑥 of the quadratic equation if the tiles below are supposed to represent
a complete square.
15.
Day 151 Practice Name ___________________________________
Algebra1Teachers @ 2015 Page 22
16.
17.
Use the information below to answer questions 18 –20
Write the constant of the quadratic equation if the tiles below are supposed to represent a
complete square.
18.
Day 151 Practice Name ___________________________________
Algebra1Teachers @ 2015 Page 23
19.
20.
Day 151 Practice Name ___________________________________
Algebra1Teachers @ 2015 Page 24
Answer Keys Day 151:
1. 𝑥2 + 6𝑥 + 9
2. 𝑥2 + 8𝑥 + 16
3. 𝑥2 − 2𝑥 + 1
4. 𝑥2 − 10𝑥 + 25
5. 4𝑥2 + 4𝑥 + 1
6. 4𝑥2 − 4𝑥 + 1 7. 𝑥2 + 8𝑥 + 16
8. 𝑥2 − 12𝑥 + 36
Day 151 Practice Name ___________________________________
Algebra1Teachers @ 2015 Page 25
9. 9𝑥2 − 6𝑥 + 1
10. 3𝑥2 + 6𝑥 + 1
11. 𝑥2 + 4𝑥 + 4
Or 𝑥2 − 4𝑥 + 4
Day 151 Practice Name ___________________________________
Algebra1Teachers @ 2015 Page 26
12. 𝑥2 + 8𝑥 + 16
Or 𝑥2 − 8𝑥 + 16
13. 𝑥2 − 2𝑥 + 1
Or 𝑥2 + 2𝑥 + 1
Day 151 Practice Name ___________________________________
Algebra1Teachers @ 2015 Page 27
14. 16𝑥2 − 16𝑥 + 4
or 16𝑥2 + 16𝑥 + 4
Day 151 Practice Name ___________________________________
Algebra1Teachers @ 2015 Page 28
15. 4
16. 6
17. 6
18. 4
19. 9
20. 9
Day 151 Exit Slip Name ____________________________________
Algebra1Teachers @ 2015 Page 29
Draw color tiles representing the quadratic expression 𝑥2 − 4𝑥 hence complete it
Day 151 Exit Slip Name ____________________________________
Algebra1Teachers @ 2015 Page 30
Answer Keys Day 151:
𝑥2 − 4𝑥 + 4
Day 152 Bellringer Name ____________________________________
Algebra1Teachers @ 2015 Page 31
Express the following expressions in the form 𝑦 = 𝑎(𝑥 − ℎ) + 𝑘
1. 𝑦 = 𝑥2 − 4𝑥 + 1
2. 𝑦 = 5 − 16𝑥 − 2𝑥2
3. 𝑦 = 3𝑥2 − 9𝑥 + 7
Day 152 Bellringer Name ____________________________________
Algebra1Teachers @ 2015 Page 32
Answer Key
Day 152
1. 𝑦 = 𝑥2 − 4𝑥 + 1 = (𝑥 − 2)2 − 3 2. 𝑦 = 5 − 16𝑥 − 2𝑥2 = −2(𝑥 + 4)2 + 37
3. 𝑦 = 3𝑥2 − 9𝑥 + 7 = 3 (𝑥 −3
2 )
2+
1
4
Day 152 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 33
Consider the following expressions𝑓(𝑥) = 8𝑥 − 2𝑥2 − 3
1. Draw a table of value for 𝑥 and 𝑓(𝑥) for integer values in the range −3 ≤ 𝑥 ≤ 5.
𝑥
𝑓(𝑥)
2. Find the highest value of 𝑓(𝑥) from the above table.
3. Identify the value of 𝑥 from which the value in 2 occurs.
4. Express the expression above in the standard form of a quadratic function.
5. Enclose the first two terms in bracket and factorize appropriately so that the coefficient of 𝑥2
is 1.
6. Complete the square in the bracket
Day 152 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 34
7. Express the resultant in 6 above so that you have an expression of the form 𝑎(𝑥 − ℎ) + 𝑘
where 𝑎 can be positive or negative.
8. Compare the value of 𝑘 and ℎ in 2 and 3 respectively.
9. Make a conclusion based on the results from 8 above.
Day 152 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 35
Answer Keys Day 152:
1.
𝑥 -3 -2 -1 0 1 2 3 4 5
𝑓(𝑥) -45 -27 -13 -3 3 5 3 -3 -13
2. 5
3. 2
4. 𝑓(𝑥) = −2𝑥2 + 8𝑥 − 3 5. 𝑓(𝑥) = −2(𝑥2 − 4𝑥) − 3 6. 𝑓(𝑥) = −2(𝑥2 − 4𝑥 + 22) + 5 7. 𝑓(𝑥) = −2(𝑥 − 2)2 + 5 8. 𝑘 = 5, ℎ = 2
The values are the same 9. When a quadratic expression is expressed in the form 𝑎(𝑥 − ℎ) + 𝑘 where 𝑎 < 0, the value of
𝑘 is the maximum valueof 𝑓(𝑥) whilethe value of ℎ is the𝑥 −coordinate of the point where the maximum value occurs.
Day 152 Practice Name ____________________________________
Algebra1Teachers @ 2015 Page 36
Solve each equation by completing the square.
1. 𝑎2 + 2𝑎 − 3 = 0
2. 𝑎2 − 2𝑎 − 8 = 0
3. 𝑘2 + 8𝑘 + 12 = 0
4. 𝑎2 − 2𝑎 − 48 = 0
5. 𝑥2 + 12𝑥 + 20 = 0
6. 𝑘2 − 8𝑘 − 48 = 0
7. 𝑝2 + 2𝑝 − 63 = 0
8. 𝑝2 − 8𝑝 + 21 = 6
9. 𝑚2 + 10𝑚 + 14 = −7
10. 𝑣2 − 2𝑣 = 3
Day 152 Practice Name ____________________________________
Algebra1Teachers @ 2015 Page 37
Answer Key
1. {1, −3}
2. {4, −2}
3. {−2, −6}
4. {8, −6}
5. {−2, −10}
6. {12, −4}
7. {7, −9}
8. {5, 3}
9. {−3, −7}
10. {3, −1}
Day 152 Exit Slip Name ____________________________________
Algebra1Teachers @ 2015 Page 38
Determine the minimum or the maximum value of the following expression
𝑦 = 1 − 3𝑥2 + 18𝑥
Day 152 Exit Slip Name ____________________________________
Algebra1Teachers @ 2015 Page 39
Answer Keys Day 152:
−3(𝑥 − 3)2 + 28
The minimum value is 28
Day 153 Bellringer Name ____________________________________
Algebra1Teachers @ 2015 Page 40
1. Make 𝑥 the subject of the formula in the following
(i). 2𝑥 + 𝑡 = 𝑝𝑥
(ii). 𝑥
5+ 𝑠 =
4
2𝑥
(iii). 𝑡𝑥
𝑚+ 𝑟𝑥 = 1
2. Complete the square then factorize
(i). 4𝑥2 − 𝑥
(ii) 3𝑥2 + 2𝑥
(iii). 𝑝𝑥2 − 𝑟𝑥
Day 153 Bellringer Name ____________________________________
Algebra1Teachers @ 2015 Page 41
Answer Key
Day 153
1. (i). 𝑥 =𝑡
𝑝−2
(ii). 𝑥 =5
9𝑠
(iii).𝑥 =𝑚
𝑡+𝑚𝑟
2. (i). 4 (𝑥 −1
8)
2−
1
16
(ii).3 (𝑥 +1
3)
2−
1
3
(iii).𝑝 (𝑥 −𝑟
2𝑝)
2−
𝑟2
4𝑝
Day 153 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 42
Consider the following equation 4𝑥2 + 8𝑟𝑥 + 𝑘=0
1. Add −𝑘 on both sides of equal sign
2. Factorize 4 from the left hand side
3. Complete the square in brackets in 2 above
4. Simply the bracket so that you have an equation of the form 𝑎(𝑥2 + 𝑑𝑥 + 𝑒2) = 𝑓.
5. Factorize the square in the bracket using appropriate method
6. Make 𝑥 the subject of the formula.
Day 153 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 43
Answer Keys Day 153:
1. 4𝑥2 + 8𝑟𝑥 = −𝑘
2. 4(𝑥2 + 2𝑟𝑥) = −𝑘
3. 4(𝑥2 + 2𝑟𝑥 + 𝑟2 − 𝑟2) = −𝑘
4. 4(𝑥2 + 2𝑟𝑥 + 𝑟2) = 4𝑟2 − 𝑘
5. 4(𝑥 + 𝑟)2 = 4𝑟2 − 𝑘
6. 𝑥 =−2𝑟+√4𝑟2−𝑘
2
Day 153 Practice Name ____________________________________
Algebra1Teachers @ 2015 Page 44
Make 𝑥 the subject of the formula, simplify completely where possible
1. (𝑥 − 2𝑝)2 = 3
2.(𝑥 + 3𝑡)2 =3
5
3. (𝑥 − 4)2 = 5
4. (𝑥 + 3)2 = 1
5. (𝑥 −4
2𝑟)
2
= 𝑞
6. (𝑥 −4
25𝑝)
2
=1
9𝑡2
Day 153 Practice Name ____________________________________
Algebra1Teachers @ 2015 Page 45
7.(𝑥 − 𝑟)2 =3𝑟2
4
8. (𝑥 − 3𝑟)2 =9
25𝑡
9.(𝑥 + 4𝑤)2 = 4 +1
4𝑤
10.(𝑥 −3
4𝑡)
2
=𝑟
4
11. (𝑥 −3
4𝑠)
2
=𝑡
4+ 1
12.(𝑥 − 2𝑝)2 =1
2𝑡
Day 153 Practice Name ____________________________________
Algebra1Teachers @ 2015 Page 46
13. (𝑥 − 3𝑟)2 =1
4− 𝑡
14.(𝑥 − 3𝑑)2 =3
4+ 𝑟
15.(𝑥 −𝑟
5)
2
=𝑟
4
16. 𝑝 (𝑥 +4
9𝑞)
2
= 4𝑝 +𝑞
4
17.3 (𝑥 −5
9𝑡)
2
= 9
18. 2 (𝑥 −3
7𝑟)
2
= 4𝑟 + 3
Day 153 Practice Name ____________________________________
Algebra1Teachers @ 2015 Page 47
19. 4(𝑥 + 3𝑟)2 = 19
20. 𝑟2 (𝑥 −3
4𝑡)
2
= 𝑡2
Day 153 Practice Name ____________________________________
Algebra1Teachers @ 2015 Page 48
Answer Keys Day 153:
1. 𝑥 = 2𝑝 ± 1.732
2. 𝑥 = −3𝑡 ± 0.7746
3. 𝑥 = 6.236 𝑜𝑟 𝑥 = 1.764
4. 𝑥 = −4 𝑜𝑟 𝑥 = −2
5. 𝑥 = 2𝑟 ± √𝑞
6. 𝑥 =4
25𝑝 ±
𝑡
3
7. 𝑥 = 𝑟 (2±√3
2)
8. 𝑥 = 3𝑟 ±3
5√𝑡
9. 𝑥 = −4𝑤 ±√16+𝑤
2
10. 𝑥 =3
4𝑡 ±
√𝑟
2
11. 𝑥 =3
4𝑠 ±
√𝑡+4
2
12. 𝑥 = 2𝑝 ± √𝑡
2
13. 𝑥 = 3𝑟 ±√1−4𝑡
2
14. 𝑥 = 3𝑑 ±√3+4𝑟
2
15. 𝑥 =𝑟
5±
√𝑟
2
16. 𝑥 =4
9𝑞 ±
1
2√
16𝑝+𝑞
𝑝
17. 𝑥 =5
9𝑡 ± 1.732
18. 𝑥 =3
7𝑟 ± √
4𝑟+3
2
19. 𝑥 = −3𝑟 ± 2.179
20. 𝑥 = (3
4±
1
𝑟) 𝑡
Day 153 Exit Slip Name ____________________________________
Algebra1Teachers @ 2015 Page 49
Express 𝑥 in terms of 𝑙 and 𝑚 if (𝑥 − 𝑙)2 = 𝑚
Day 153 Exit Slip Name ____________________________________
Algebra1Teachers @ 2015 Page 50
Answer Keys Day 153:
𝑥 = 𝑙 ± √𝑚
Day 154 Bellringer Name ____________________________________
Algebra1Teachers @ 2015 Page 51
1. Use 𝑥 the subject of the formula in the following
(i). (𝑥 − 4)2 = 9𝑡
(ii). (2𝑥 + 𝑟)2 = 16
2. Find the coordinates of the vertex of the following 2𝑥2 − 24𝑥 + 9
Day 154 Bellringer Name ____________________________________
Algebra1Teachers @ 2015 Page 52
Answer Key
Day 154
1. (i) 𝑥 = 4 ± 3√𝑡
(ii). 𝑥 = −𝑟
2± 2
2. (−6, −63)
Day 154 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 53
Consider the following function 8𝑥2 + 4𝑥 − 1
1. Group the terms so that the first and the second term are in one group and the constant
term in its own group.
2. Complete the square if the expression in the first bracket
3. Factorize the expression in the first bracket (completed square)
4. Express the expression in 3 above in terms of 𝑟(𝑥 − ℎ) + 𝑘 where 𝑟 ≠ 0, ℎ and 𝑘 are
constants.
Day 154 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 54
5. Identify the relative maximum or the minimum value of the expression.
6. Identify the vertex of the function
7. Equate the function to zero then determine the value of 𝑥.
Day 154 Activity Name ____________________________________
Algebra1Teachers @ 2015 Page 55
Answer Keys Day 154:
1. (8𝑥2 + 4𝑥) − 1
2. 8 (𝑥2 +1
2𝑥 + (
1
4)
2
) −1
2− 1
3. 8 (𝑥 +1
2)
2
−1
2− 1
4. 8 (𝑥 +1
2)
2
−3
2
5. Relative minimum is −3
2
6. Vertex(−1
2, −
3
2 )
7. 8 (𝑥 +1
2)
2
−3
2= 0
𝑥 = −1
2±
√3
4
Day 154 Practice Name ____________________________________
Algebra1Teachers @ 2015 Page 56
Make 𝑥 the subject of the formula, simplify completely where possible from 1 - 5
1. (𝑥 − 2𝑑)2 = 9
2.(𝑥 − 6𝑝)2 = 25𝑝2
3. (2𝑥 + 4)2 =25
36
4. (2𝑥 + 3𝑠)2 =1
𝑛2
5. (𝑥 −𝑟
2)
2
=𝑞2
4
Day 154 Practice Name ____________________________________
Algebra1Teachers @ 2015 Page 57
Complete the following squares 6 – 13
6. 𝑥2 − 8𝑥
7. 𝑥2 − 22𝑥
8. 4𝑥2 − 40𝑥
9. 81𝑥2 − 36𝑡2𝑥
10. 4𝑥2 + 8𝑥
Day 154 Practice Name ____________________________________
Algebra1Teachers @ 2015 Page 58
11. 𝑝2𝑥2 + 2𝑝𝑥
12. 𝑎2𝑥2 − (4𝑝)2𝑥
13. 25𝑥2 +𝑎
2𝑠𝑥
14. 36𝑥2 − 4𝑟𝑥
Day 154 Practice Name ____________________________________
Algebra1Teachers @ 2015 Page 59
For questions 15 – 17, draw square tiles representing the square whose part is given below. Use
different colors for negative and positive coefficients of 𝑥.
15. 4𝑥2 − 4𝑥
16. 𝑥2 + 6𝑥
Day 154 Practice Name ____________________________________
Algebra1Teachers @ 2015 Page 60
Determine the minimum or maximum of the following expressions.
17. 𝑥2 + 4𝑥 + 2
18. 2𝑥2 − 9𝑥 + 3
19. −4𝑥2 + 8𝑥 + 3
20.−3𝑥2 + 7𝑥 + 8
Day 154 Practice Name ____________________________________
Algebra1Teachers @ 2015 Page 61
Answer Keys Day 154:
1. 𝑥 = 2𝑑 ± 3
2. 𝑥 = 6𝑝 ± 5𝑝
3. 𝑥 = −2 ±5
12
4. 𝑥 = −3
2𝑠 ±
1
2𝑛
5. 𝑥 =𝑟±𝑞
2
6. 𝑥2 − 8𝑥 + 16
7. 𝑥2 − 22𝑥 + 121
8. 4𝑥2 − 40𝑥 + 400
9. 81𝑥2 − 36𝑥 + 4
10. 4𝑥2 + 8𝑥 + 4
11. 𝑝2𝑥2 + 2𝑝𝑥 + 1
12. 𝑎2𝑥2 − 16𝑝2𝑥 +64𝑝4
𝑎2
13. 25𝑥2 +𝑎
2𝑠𝑥 +
𝑎2
400𝑠2
14. 36𝑥2 − 4𝑟𝑥 +𝑟2
9
15.
16.
17. Minimum value is -2
18. Minimum value −65
6
19. Maximum value 7
20. Maximum value 49
12
Day 154 Exit Slip Name ____________________________________
Algebra1Teachers @ 2015 Page 62
Express find the minimum or the maximum of 𝑦 = 2𝑥2 + 14𝑥 − 2
Day 154 Exit Slip Name ____________________________________
Algebra1Teachers @ 2015 Page 63
Answer Keys Day 154:
𝑦 = 2 (𝑥 +7
2)
2
−53
2
The function has a minimum at −53
2.
Algebra 1 Teachers
Weekly Assessment Package
Unit 12
Created by:
Jeanette Stein
& Phill Adams
©2015 Algebra 1 Teachers
65 Semester 2 Skills | Algebra1Teachers.com
Algebra 1 Common Core
Semester 2 Skills
Week Unit CCSS Skill
31 12 A.SSE.3b Complete the square
31 12 F.IF.8 Use completing the square to find maximum and minumum values
66 | Algebra1Teachers.com
Unit 12 - Solve Quadratic Functions
Weekly Assessments
67 Unit 12 - Solve Quadratic Functions | Algebra1Teachers.com
Week #31 - Completing the Square
1. Rewrite each quadratic in vertex format by completing the square
a. 𝑓(𝑥) = 𝑥2 + 16𝑥 + 71 b. 𝑓(𝑥) = 𝑥2 − 2𝑥 − 5
c. 𝑓(𝑥) = (𝑥 + 5)(𝑥 + 4) d. 𝑓(𝑥) = −9𝑥2 − 162𝑥 − 731
2. Complete the square to determine the minimum/maximum values for each quadratic.
a. 𝑓(𝑥) = 𝑥2 + 10𝑥 + 33 b. 𝑓(𝑥) = 𝑥2 − 6𝑥 + 5
c. 𝑓(𝑥) = 𝑥2 + 4𝑥 d. 𝑓(𝑥) = 6𝑥2 + 12𝑥 + 13
68 Unit 12 - KEYS | Algebra1Teachers.com
Unit 12 - KEYS
Weekly Assessments
69 Unit 12 - KEYS | Algebra1Teachers.com
Week #31 - Completing the Square Answer Key 1. Rewrite each quadratic in vertex format by completing the square
a. 𝑓(𝑥) = 𝑥2 + 16𝑥 + 71
𝒇(𝒙) = (𝒙 + 𝟖)𝟐 + 𝟕
b. 𝑓(𝑥) = 𝑥2 − 2𝑥 − 5
𝒇(𝒙) = (𝒙 − 𝟏)𝟐 − 𝟔
c. 𝑓(𝑥) = (𝑥 + 5)(𝑥 + 4)
𝒇(𝒙) = (𝒙 + 𝟒. 𝟓)𝟐 − 𝟎. 𝟐𝟓
d. 𝑓(𝑥) = −9𝑥2 − 162𝑥 − 731
𝒇(𝒙) = −𝟗(𝒙 + 𝟗)𝟐 − 𝟐
2. Complete the square to determine the minimum/maximum values for each quadratic.
a. 𝑓(𝑥) = 𝑥2 + 10𝑥 + 33
𝒇(𝒙) = (𝒙 + 𝟓)𝟐 + 𝟖 (−𝟓, 𝟖)
b. 𝑓(𝑥) = 𝑥2 − 6𝑥 + 5
𝒇(𝒙) = (𝒙 − 𝟑)𝟐 − 𝟒 (𝟑, −𝟒)
c. 𝑓(𝑥) = 𝑥2 + 4𝑥
𝒇(𝒙) = (𝒙 + 𝟐)𝟐 − 𝟒 (−𝟐, −𝟒)
d. 𝑓(𝑥) = 6𝑥2 + 12𝑥 + 13
𝒇(𝒙) = 𝟔(𝒙 + 𝟏)𝟐 + 𝟕 (−𝟏, 𝟕)