Pythagorean/Intro Geometry Unit - Weeblyscottmatusicky.weebly.com/uploads/5/3/2/0/5320478/plane_geometry_… · Pythagorean/Intro Geometry Unit Day 4 Direct Lesson Topic: Solving

Pythagorean/Intro Geometry Unit

Day 2

Direct Lesson

Topic: The Area and Circumference of Circles.

Duration: One day.

Purpose:

The ideas of area and perimeter apply to circles as well. We need to know the anatomy of a circle in order to use the

formulas that get the area and circumference of circles.

Materials: Dry Erase Markers, ELMO, Circle notes, Worksheets.

Benchmarks/Standards:

G.GS.08.01 – Understand the definition of a circle; know and use the formulas for circumference and area of a circle

to solve problems.

Objectives:

SWBAT calculate the area of a circle.

SWBAT calculate the circumference of a circle.

SWBAT to manipulate the formulas to solve problems with circles.

SWBAT recognize the value and significance of PI.

Assessment(s) of Objectives: Problem-solving worksheet.

Anticipatory Set:

Tell a story about the history of the wagon train and how they would circle the wagons when in danger. Why did

they use a circle? Take two or three minutes to discuss the advantages of forming this shape.

Modeling/Input:

Every point on a circle is an equal distance from its center. The center of this circle is A. We call this circle A --

Referring to its center.

Diameter – The distance of the line across the circle through its center.

Radius – The distance from the center to any point on the circle.

***The radius is one half the diameter.

If you are given the diameter, your radius is the diameter cut in half.

PI is 3.14

Circumference: The perimeter of a circle.

C = 2 * PI * r (or) 2*3.14*r

Area: The amount of space inside the circle. If you know the radius of a circle, you can find the Area = PI * r^2 (r

squared)(to the second power).

Circumference has regular units. Area has square units.

Guided Practice: Hand out worksheets, assign which problems they are responsible for, and do one example of

each:

Examples: Find P and A of a circle with r = 15ft.

Find P and A of a circle with r = 3.7in.

Find P and A of a circle with d = 78m.

Find the radius of a circle with A = 100 sq ft.

Checking for Understanding: Each student must have the correct solution to the final example written down in their

notes in order to pass.

Closure: Remember that we are using 3.14 as a substitution for the PI symbol. If you know the radius you can find

the Area and the Circumference by using Pi r squared and 2 PI r.

Independent Practice: Skills Worksheet #13, #14.

Adaptations: Have the students either do the odds or the evens to cut down on workload.

Pythagorean/Intro Geometry Unit

Day 3

Direct Lesson

Topic: Reducing radicals.

Duration: One day.

Purpose:

More often than not, square roots won‟t turn out to be neat and tidy as perfect squares are. It is necessary to know

how to reduce a square root by the use of factoring the radicand.

Materials:

ELMO, Dry Erase Markers, “Square Rooty Hat” (A hat with squirrel ears sewn on)?

Benchmarks/Standards:

Standard (8.NS.2) – Use rational approximations of irrational numbers to compare the size of irrational numbers,

locate them on a number line diagram, and estimate the value of non-perfect squares.

Objectives:

SWBAT determine if a square root is reducible.

SWBAT follow the steps to reduce a square root to its lowest terms.

SWBAT recognize the perfect squares.

Assessment(s) of Objectives: Square Root/Reducing Worksheet.

Anticipatory Set:

http://www.ixl.com/math/grade-8/area-and-perimeter-word-problems (One example)

Do any of these numbers look familiar? What are their significances? 1, 4, 9, 16, 25, 36?

Input/ Modeling:

The not so perfect squares.

Step 1: Determine if the number under the radical is a perfect square. If so, you are done.

If not Step 2: Factor the number under the radical into two other factors.

Step 3: Determine if one of these factors is a perfect square.

Factor our its square root and keep the other number under the radical.

Example: Square root of 80.

80 is not a perfect square.

80 factors into 16 times 5. 16 is a perfect square so you can factor out a 4 and leave the 5 under the radical.

Try example on their own: Square root of 76.

Trick 1: If the number is even, it can always be divided by 2.

Trick 2: If you don‟t recognize the factor as a perfect square, keep factoring until you get a matching pair of

numbers. Consider these „married‟ and they can leave the house and become one. Those numbers left „unmarried‟

stay in the house (and work on their e-harmony profiles perhaps…)

Guided Practice:

Draw 8 cards for seat numbers to attempt their own reducing square root problem. Clarify any mistakes and use the

estimation on a number line using the perfect squares as an estimator.

Check For Understanding:

Students must have all 8 examples correctly written down to pass.

Closure:

Study your times tables if you have problems factoring easily. Remember that reducing square roots requires that

you know all of the perfect squares up to at least 225.

Independent Practice: Practice Worksheet 9.1C.

Adaptations:

The print on this worksheet is a little small. Print out a few that have larger writing for students with poor vision.

Pythagorean/Intro Geometry Unit

Day 4

Direct Lesson

Topic: Solving for X to the 2nd

power (Squared)

Duration: One day.

Purpose:

Often, we are presented with an equation that requires that we figure out what the unknown value of a variable. In

order to solve for X squared, we need to apply what we already know and add the finishing touch.

Materials:

ELMO, Dry Erase Markers, Worksheet.

Benchmarks/Standards:

N.FL.08.05 - Estimate and solve problems with square roots using calculators.

Objectives:

SWBAT use the inverses of addition, multiplication, and squaring to solve for unknowns.

Assessment(s) of Objectives:

Solving for X worksheet.

Anticipatory Set:

REVIEW PEMDAS BACKWARDS for solving for X. What mnemonic device do you have to remember this order

of operations?

Input/ Modeling:

The opposite of addition is subtraction.

The opposite of multiplication is dividing,.

The opposite of squaring is taking the square root. If you have an x squared equals a number, TAKE THE

SQUARE ROOT OF BOTH SIDES. We are solving for x, not x squared.

Examples: x^2 = 100

x^2 = 25

x^2 = 39

Guided Practice: Write example problems on the board and have students solve them for x in 30 seconds.

Checking for Understanding: Students will not be allowed to pass if they don‟t have the perfect squares written

down on their notes.

Closure: Remember that we are solving for x, not x squared. We have to take the square root of both sides in order

to finish the problem.

Independent Practice: Solve for x worksheet. 9.1 Practice B.

Adaptations: Make larger copies of the worksheet with bigger print and assign 10-12 of the worksheet problems.

Pythagorean Unit/Intro to Geometry

Day 5

Indirect Lesson

Topic: Right Triangles and the Pythagorean Theorem

Duration: 1-2 days.

Materials: Ruler (mm), Dry Erase markers.

Standards/Benchmarks:

G.GS.08.01 – Understand at least one proof of the Pythagorean Theorem; use the Pythagorean Theorem and its

converse to solve applied problems including perimeter and area problems.

Objectives:

SWBAT interpret the markings on a ruler.

SWBAT define the parts of a right triangle.

SWBAT manipulate the Pythagorean Theorem to solve for missing side lengths.

SWBAT interpret word problems into math problems.

Purpose: The Pythagorean Theorem is one of the easiest and most useful tools to use in math. This theorem can be

used to find the distance between two points when line of sight is broken.

Anticipatory Set:

Pythagorean Rap. http://www.youtube.com/watch?v=DRRVu-RHQWE

Explore concept/Find Patterns:

With the person next to you (in pairs or threes), as you listen to the rap, take note of any words that you recognize.

What do you think the Pythagorean Theorem states?

Input/Modeling:

Draw a right triangle. A triangle that has a right angle (90 degrees)

Label the legs and hypotenuse. The hypotenuse is always the LONGEST side ACROSS from the right angle.

The Pythagorean Theorem states that A^2 + B^2 = C^2

Where C = hypotenuse and A, B = legs. The legs are interchangeable. REMEMBER THAT C IS ALWAYS THE

LONGEST SIDE!!

Example: Is a triangle with side lengths of 30, 40, 50 a right triangle? Yes 2500=2500

What if the side lengths are 3, 5, 4? Yes 25=25. (C is always the longest side, not always the last one listed.)

Once you have your A, B, and C. You can use your solve for x squared skills to find the missing side.

Pass out Map worksheet. Have each student measure the distances using a ruler and have them write the answer in

millimeters.

Metacognition Opportunities:

How does your version of the Theorem differ from the Pythagorean Theorem. Write the answer down in your notes.

If you were correct, how might one prove that this theorem always works? We will discuss any questions you have

after bell work tomorrow.

Checking For Understanding:

Show me the answer to the Pythagorean Theorem problems 1-3 on the Worksheet.

Assessment(s) of Objectives:

Pythagorean Worksheet.

Closure:

The last step to solving the Pythagorean Theorem is taking the square root. If your answer is longer than C, then

you set up the problem incorrectly. C is the longest side.

Adaptations/differentiation: This lesson ideally should be spread out over two days so students have a chance to

work together to solve for missing sides.

Pythagorean/Intro Geometry Unit

Day 8

Direct Lesson

Topic: Finding the distance between two points. The Distance Formula.

Duration: One Day.

Purpose:

Every location can be represented by coordinates. Either latitude, longitude or by an X coordinate and a Y

coordinate. We know the Pythagorean Theorem finds the lengths of unknown sides of right triangles, but we can

use it to find the distances between two points.

Materials:

ELMO, Dry Erase Markers, Worksheet.

Benchmarks/Standards:

G.GS.08.02 – Find the distance between two points on the coordinate plane using the distance formula; recognize

that the distance formula is an application of the Pythagorean Theroem.

Objectives:

SWBAT calculate the error in measuring distances.

SWBAT use the distance formula to solve problems.

SWBAT connect the Pythagorean Theorem to the Distance Formula.

Assessment(s) of Objectives:

Map Worksheet, Pythagorean Theorem 2 Worksheet.

Anticipatory Set:

How far is it from Ypsi Middle School to Target 2 party store? How would you measure the distance when there is

a forest in the way?

Modeling/Input:

The Distance Formula is the same as the Pythagorean Theroem.

Ypsi Middle and Target 2 both have coordinates that are (X, Y)

Find A and B and then square root.

A is how far apart the X‟s are between coordinates.

B is how far apart the Y‟s are between coordinates.

Example 2 on notes sheet model for the students.

Checking for Understanding:

Draw cards and have each student volunteered give a step to solving the problem.

Guided Practice: Take out the map that you measured and calculate the exact distances using the Distance Formula.

Subtract the two distances and write how far off you were.

Closure: The distance between two points is just an application of the Pythagorean Theorem. First find A and B.

Independent Practice: Finish the back of the worksheet. Leave your answers as square roots or round them to the

nearest hundredth.

Adaptations: Depending on the location of the lesson (different school) I would have to choose two different

locations in the city.