NUCC Secondary II Math 1

Unit 1.1

NUCC|Secondary II Math 1

Northern Utah Curriculum Consortium

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Weber School District

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Cache School District / Mike Hansen
Cache School District
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Cache School District / Sheena Knight
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NUCC| Secondary II Math 1

Table of Contents

5.1 Understanding Squares

Teacher Notes

Mathematics Content

Understanding SquaresA Develop Understanding Task 1

Ready, Set, Go!

Solutions:

5.2 Solve Using Square Roots

Teacher Notes

Mathematics Content

GravityA Solidify Understanding Task 2

Ready, Set, Go!

Solutions:

5.3 Solve by Factoring When a=1

Teacher Notes

Mathematics Content

Algebra TilesA Develop Understanding Task 3

Ready, Set, Go!

Solutions:

5.4 Quadratic Equations by factoring when a >1

Teacher Notes

Mathematics Content

Area of a PaintingA Practice Understanding Task 4

Ready, Set, Go!

Solutions:

5.5 Solve by completing the Square

Teacher Notes

Mathematics Content

Garden SpaceA Develop Understanding Task 5

Ready, Set, Go!

Solutions:

5.6 Solve with Complex Numbers

Teacher Notes

Mathematics Content

ReviewA Solidify Understanding Task 6

Ready, Set, Go!

Solutions:

5.7 solve Using the Quadratic Formula

Teacher Notes

Mathematics Content

Analyze Graphs of Quadratic FunctionsA Develop Understanding Task 7

Ready, Set, Go!

Solutions:

H5.8 Fundamental Theorem of Algebra

Teacher Notes

Mathematics Content

Investigating the DiscriminantA Develop Understanding Task 8

Ready, Set, Go!

Solutions:

5.9 Optional Catapult Lesson

Teacher Notes

Mathematics Content

Shape of Quadratic FunctionA Practice Understanding Task 9

Ready, Set, Go!

Solutions:

NUCC| Secondary II Math 1

Unit 5.1

5.1 Understanding Squares

Teacher Notes

Time Frame: 1 class period (90 minutes)

Materials Needed:Rulers

Purpose: Students will understand the concept of a square root. They will also learn to simplify radical expressions including rationalizing the denominator.

Core Standards Focus:A.REI.4 Solve quadratic equations by inspection, taking square roots,

completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a + bi for real numbers a and b.

Related Standards:Perimeter and area of a square.Solving equations.

Launch (Whole Class): Hand out the rulers and the task worksheet. Allow students 10 minutes to complete. Discuss their answers.

Draw a square on the board and write 64 on the inside of the square.If 64 is the area, what is the length of one side? How did you find the answer? Change the 64 to 100. What is the side length? Why didn’t you just divide 100 by 4?

Explore (Individual, small group or pairs):

Divide students into groups of 4. Have 1 student in the group be the ‘scribe’ and write down the group’s answers.

On the board change the square’s area to 18. Now what is the side length? How could you write this in radical notation? Draw a square with the area of 18 units. Draw in the units.

If time, change the area to 200 and have students discuss the side length and how to draw the units.

Discuss (Whole Class or Group):

Demonstrate simplifying a radical expression.

= = 3

= = 10

Now check to see if = 3

4.24 = 4.24

A number ‘r’ is a square root of a number ‘s’ if r2 = s.

The expression is called a radical. The symbolis a radical sign and the numbersbeneath the radical sign is the radicand of the expression.

Example 1:Simplify the expressions. (Use properties of square roots)

= = = 2

 = =  = 4

Remind students to refer to the root/square chart they created in the task for assistance.

Example 2:Rationalize denominators of fractions

= = =

b.Introduce ‘conjugate’. The conjugate of 5 + is 5 - . When multiplying conjugates together, the answer is a real, rational number.

(5 + )(5 - ) = 25 - 5 + 5 - = 25 – 2 = 23

=  = =

Assignment: Ready, Set, Go!

NUCC | Secondary II Math 1

Unit 5.1

Mathematics Content

Cluster Title: Solve equations and inequalities in one variable.
Standard A.REI.4: Solve quadratic equations in one variable.

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
Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a bi for real numbers a and b.

Concepts and Skills to Master

Complete the square.
Solve quadratic equations, including complex solutions, using completing the square, quadratic formula, factoring, and by taking the square root.
Derive the quadratic formula from completing the square.
Recognize when one method is more efficient than the other.
Interpret the discriminant.
Understand the zero product property and use it to solve a factorable quadratic equation.

Critical Background Knowledge

Factor
Simplify radicals
Understanding of complex numbers (Secondary II: N.CN.1)
Understand the real number and complex number systems (Secondary II: N.CN.1)

Academic Vocabulary
radicals, complex numbers solve, factor, discriminant
Suggested Instructional Strategies

Use of algebra tiles to demonstrate simple completing the square problems (see NCTM MTMS, March 2007, p. 403).

Skills Based Task: / Problem Task:
Solve the equation 6x2 –x –15 = 0 by factoring and by completing the square. Justify each method using mathematical properties. / Solve the quadratic equation 49x2 – 70x +37 = 0 using two methods. Describe the advantages of each method.
Some Useful Websites:
Illuminations: Proof Without Words: Completing the Square

NUCC | Secondary II Math 1

Unit 5.1

Understanding Squares

A Develop Understanding Task 1

Name______Hour______

Use a ruler and draw a square in the space below.

How did you draw the square?

Why is this shape a square?

What is the area of the square above?
Refer to the square that was drawn above and ‘show’ the area of the square.
Complete the following chart of roots, squares, and square roots for the numbers 1-20.

Root / Square / Square Root
1 / 1 / 1
2 / 4 / 2

NUCC | Secondary II Math 1

Unit 5.1

Ready, Set, Go!

Name______Hour______

Ready

Simplify the expression.

 4.

Set

Simplify the expression.

 6.  3

Go!

Simplify the expression.

6. 

What is the conjugate of 7 +? What is the answer when these conjugates are multiplied together?

Why is it necessary to multiply the denominator by its conjugate?

List all possible digits that occur in the units’ place of the square of a positive integer.

(Hint: Look at the root/square chart that was created earlier.)Use this list of digits to determine whether or not it is possible for each root to be an integer.

Guess / Calculator Answer / Is this a perfect square?

Is this a good rule for square roots? Explain.

Give an example of a 5-digit perfect square and its root.

Solutions:

NUCC | Secondary II Math 1

Unit 5.2

5.2Solve Using Square Roots

Teacher Notes

Time Frame: 1 class period (90 minutes)

Materials Needed: Pencils

Purpose: Students will learn how to solve quadratic equations using square roots while

using ± notation and to simplify the radicals when necessary.

Core Standards Focus:A.REI.4 Solve quadratic equations by inspection, taking square roots,

Related Standards: Simplifying radicals, solving linear equations

Launch (Whole Class): Hand out the Task worksheet. Allow students 10 minutes to complete.

Explore (Pair/Share):

Assign each student a partner to share their answers and to compare the way they solved question # 9. Have students show various ways that they solved the equation.

Also, explain that the equation of gravity is: h = - 16t2 + h0

What does h represent? (ending height, usually 0) t? (time, usually seconds) h0? (initial height)

Solve the equation when the initial height is 324.

Discuss (Whole Class or Group):

Ask: ‘How could you solve this equation x2 = 64?’

Ask: What numbers could you put in for x that would make that statement true?

Show - 8 would also work.

To solve quadratic equations, first isolate the x2 term. Then take the square root of both sides of the equation. Show ± notation for the positive and negative solutions.

Example 1:a. x2 = 100

x = ± 10

b. 2x2 – 3 = 87

2x2 = 90Add 3 to each side

x2= 45Divide both sides by 2

x = ± Take square roots of each side.

x = ±Product property

x = ± 3*Simplify.

c. Finding solutions of a quadratic equation

Find the solutions of ½ (w - 2)2 + 1 = 4

½ (w – 2)2 + 1 = 4

(w – 2)2 = 6Subtract 1 from both sides and multiply both sides by 2.

w – 2 = ±Take square roots of each side.

w = 2 ± Add 2 to each side

The solutions are 2 + and 2 -

*Remember to simplify the radicals and rationalize the denominator when necessary.

Assignment: Ready, Set, Go!

NUCC | Secondary II Math 1

Unit 5.2

Mathematics Content

Cluster Title: Solve equations and inequalities in one variable.
Standard A.REI.4: Solve quadratic equations in one variable.

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
Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a bi for real numbers a and b.

Concepts and Skills to Master

Complete the square.
Solve quadratic equations, including complex solutions, using completing the square, quadratic formula, factoring, and by taking the square root.
Derive the quadratic formula from completing the square.
Recognize when one method is more efficient than the other.
Interpret the discriminant.
Understand the zero product property and use it to solve a factorable quadratic equation.

Critical Background Knowledge

Factor
Simplify radicals
Understanding of complex numbers (Secondary II: N.CN.1)
Understand the real number and complex number systems (Secondary II: N.CN.1)

Academic Vocabulary
radicals, complex numbers solve, factor, discriminant
Suggested Instructional Strategies

Use of algebra tiles to demonstrate simple completing the square problems (see NCTM MTMS, March 2007, p. 403).

NUCC | Secondary II Math 1

Unit 5.2

Gravity

A Solidify Understanding Task 2

Name______Hour______

Simplify the expressions.

Give an example of a binomial and its conjugate.

What is the conjugate of 2 - ?

Simplify

Write down what you know about the Law of Gravity.

The height h (in feet) of a ball dropped from the top of a building can be modeled by h = - 16t2 + 256 where t is the time (in seconds). Solve the equation to find the time it takes for the ball to hit the ground, or when h = 0.

NUCC | Secondary II Math 1

Unit 5.2

Ready, Set, Go!

Name______Hour______

Ready

Solve the equation.

x2 = 16 2.x2 = 144

x2 = 1214.x2 – 4 = 0

x2 – 64 = 0

Set

x2 – 8 = 07.2x2 = 32

3x2 = 759.x2 = 12

x2 + 12 = 1311.x2 – 1 = 6

20 – x2 = - 29

Go!

2x2 - 1 = 714.3x2 – 9 = 0

½ x2 + 8 = 17

When an object is dropped, its height h can be determined after tseconds by using the falling object model h = -16 t2 + s where s is the initial height. Find the time it takes an object to hit the ground when it is dropped from a height ofsfeet.

s = 10017. s = 196

s = 48019.s = 600

s = 75021.s = 1200

From 1970 – 1990, the average cost of a new car C (in dollars) can beapproximated by the model C = 30.5t2 + 4192 wheretis the number of years since 1970. During which year was the average cost of a new car $7242?

Write a quadratic equation that has the given solutions.

± 24.± 3

-1 ±

Solutions:

NUCC | Secondary II Math 1

Unit 5.3

5.3 Solve by Factoring When a=1

Teacher Notes

Time Frame: 1 class period (90 minutes)

Materials Needed:Algebra Tiles – actual tiles, paper cut out tiles, or demonstrate with your computer using the site: any other site you may find.

Purpose: Solve quadratic equations by factoring where a = 1. Students have been introduced to factoring in Secondary Math 1, but they may need some review.

Core Standards Focus:A.REI.4 Solve quadratic equations by inspection, taking square roots,completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a + bi for real numbers a and b.

Related Standards: Factoring quadratics where a = 1

Launch (Whole Class): Distribute the Algebra Tiles Task worksheet to each student. Have them work on this for 10 minutes.

Remember factoring? What does it mean to factor a quadratic equation?

Explore (Individual, small group or pairs):

Have students share their task worksheet with another student. They can explain what their Algebra Tile diagram looks like. Have them draw one more equation on their paper and factor it.

x2 + 3x + 2

Discuss (Whole Class or Group):

Set each of the quadratic equations from the task worksheet equal to 0. (Have the equations in quadratic form first.)

How do you think we could solve for x? (Let them discover that factoring would be a good way.)

Demonstrate that when the equation = 0, then set each factor = 0 and solve each one separately. There may be 0, 1, or 2 solutions.

Have students factor the following and help them recognize the pattern with ‘c’ in each expression. (Remind students of the standard form y = ax2 +bx + c).

x2 + 7x + 12 = (x + 3)(x + 4) notice c > 0

x2 – 7x + 12 =(x – 3)(x – 4)

x2 + x – 12 =(x – 3)( x + 4) notice c < 0

x2 – x - 12 =(x + 3)( x – 4)

Example 1:a. Factor trinomials

w2 – 2w – 15

= (w + 3)(w – 5)

b. Special Patternsg2 – 20g + 100(Perfect square trinomial)

= (g – 10)(g – 10)

= (g – 10)2

z2 – 64(Difference of 2 squares)

= z2 - 82

= (z + 8)(z – 8)

After each equation is factored, then set each factored term equal to 0. Solve each factored term and those answers will be the solutions (also called zeros and roots).

x2 + 7x + 12 =(x + 3)(x + 4) = 0

so, x + 3 = 0 and x + 4 = 0

x = -3 and x = -4

Example 2:Find the roots of an equation

Find the roots of x2 – 13x + 42 = 0

x2 – 13x + 42 = 0

(x – 6)(x – 7) = 0Factor

x – 6 = 0 or x – 7 = 0Set each factor = 0

x = 6 or x = 7Solve for x

Example 3:Use a quadratic equation as a model.

A rectangular garden measures 10 feet by 15 feet. By adding x feet to the width and x feet to the length, the area is doubled. Find the new dimensions of the garden.

New area / = / New length /  / New width
2 (10)(15) / = / (10 + x) /  / (15 + x)
300 / = / 150 + 25x + x2 / Multiply using FOIL
0 / = / x2 + 25x – 150 / Write in standard form
0 / = / (x + 30)(x – 5) / Factor
x + 30 = 0 / or / x – 5 = 0 / Set each factor = 0
x = -30 / or / x = 5 / Solve for x

Reject the negative value. The garden’s width and length should each be increased by 5 feet. The new dimensions are 15 feet by 20 feet.

Example 4:Find the zeros (solutions) of quadratic functions

Find the zeros of y = x2 + 3x – 28
y = x2 + 3x – 28
= (x + 7)(x – 4) / Factor
x + 7 = 0 or x – 4 = 0 / Set each factor = 0
x = -7 or x = 4 / Solve for x

Assignment: Ready, Set, Go!

NUCC | Secondary II Math 1

Unit 5.3

Mathematics Content

Cluster Title: Solve equations and inequalities in one variable.
Standard A.REI.4: Solve quadratic equations in one variable.

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
Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a bi for real numbers a and b.

Concepts and Skills to Master

Complete the square.
Solve quadratic equations, including complex solutions, using completing the square, quadratic formula, factoring, and by taking the square root.
Derive the quadratic formula from completing the square.
Recognize when one method is more efficient than the other.
Interpret the discriminant.
Understand the zero product property and use it to solve a factorable quadratic equation.

Critical Background Knowledge

Factor
Simplify radicals
Understanding of complex numbers (Secondary II: N.CN.1)
Understand the real number and complex number systems (Secondary II: N.CN.1)

Academic Vocabulary
radicals, complex numbers solve, factor, discriminant
Suggested Instructional Strategies

Use of algebra tiles to demonstrate simple completing the square problems (see NCTM MTMS, March 2007, p. 403).

NUCC | Secondary II Math 1

Unit 5.3

Algebra Tiles

A Develop Understanding Task 3

Name______Hour______

Describe what it means to ‘factor a quadratic equation’.

Draw the algebra tiles: x, x2, and 1.

Draw what the quadratic equation x2 + 5x + 6 would look like using Algebra Tiles.

Write down the length and width of the above tile diagram. What does this represent?

Draw what the quadratic equation x2 + 8x + 16 would look like using Algebra Tiles.

Write down the length and width of the above tile diagram. What does this represent?

NUCC | Secondary II Math 1

Unit 5.3

Ready, Set, Go!

Name______Hour______

Ready

Factor the expression. If the expression cannot by factored, say so.

x2 – x – 22.x2 – 4x + 3

x2 + 8x + 154.x2 – 4

x2 + 2x + 16.x2 – 10x + 25

Set