unit lesson plan: measuring length and area: area of...

Unit Lesson Plan: Measuring Length and Area: Area of shapes Day 1: Area of Square, Rectangles, and Parallelograms Day 2: Area of Triangles Trapezoids, Rhombuses, and Kites

Day 3: Quiz over Area of those shapes. Day 4: Area and Perimeter of Irregular Shaped Figures Day 5: Area of Similar Figures Day 6: Quiz over Area of Similar Figures and Irregular Shaped Figures Day 7: Area of Circles and Sectors

Day 8: Area of Regular Polygons

Day 9: Geometric Probability

Day 10: Review for test Day 11: Test over Area of All Shapes and Geometric Probability

Area of a Square Lesson

Grade: 10th

/11th

Content: Mathematics- Geometry

Materials: pencil, paper, textbook, whiteboard, markers, Tiling floor worksheet

Standards:

Standard- Solve real-life and mathematical problems involving area.

Solve real-world and mathematical problems involving area of two- dimensional

quadrilaterals; square, rectangle, parallelogram and triangles.

Objectives:

1. TLW demonstrate their knowledge of area and perimeter.

2. TLW relate the area of parallelograms to rectangles and triangles to parallelograms

and derive a formula

3. TLW determine area of shapes put together. Triangles and parallelogram

Learning Activities:

1) Review the area of a square and a rectangle make sure everyone has a good understanding of

both of those. Give a examples.

a) Find the area of a square whose perimeter is 30.

i) Since p = 4s and s=7

. Then Area = s

2 = (7

) = 56

b) Find the side and perimeter of a square whose area is 20

i) Area = s2 = 20; Then s = . Then perimeter = 4s = 8

c) Find the area of a rectangle if the base has length 15 and perimeter 50.

i) P=50 b=15. Since P=2b+2h; 50=2(15)+2h so h=10

d) Find the area of a rectangle of the altitude has length 10 and the diagonal has length 26.

i) D=26 h=10. In , so 262=b

2 + 10

2 so b=24 Area = 240

2) Introduce the area of a parallelogram. Give them an example of how the area of a

parallelogram is similar to the area of a rectangle. A=h*b

3) Example problems Give them four points, tell them

to find the area and perimeter. A(-2,-1) B(-2,1) C(2,3) D(2,1)

4) Review properties of triangles

5) Demonstrate the area of a triangle is one half

of the area of a parallelogram.

6) Example problems: find area and perimeter height 8 base 21

Hypotenuse 15 leg 12; height 5 hyp 13

7) Draw figure 3 on the board and see if students can find the area

8) Distribute the Finding Area – Tiling the floor worksheet.

9) Give the students a quick exit slip 5-10 min before the end of class

To check for understanding 11

10

16

Assessment:

1. Students will be assessed informally thought the lecture.

2. Students will be assessed with the worksheet, exit slip, and with homework problems

Reflection:

I think this lesson might run a little short. Some students might already know some of this

content. I feel like it is a review.

Trapezoid Lesson

Grade: 9th

/10th


Materials: pencil, graph paper, straight edge, scissors tape,

Standards:

Standard- Solve real-life and mathematical problems involving area.

Solve real-world and mathematical problems involving area of two- dimensional

quadrilaterals; Trapezoids, Kites and Rhombuses.

Objectives:

1. TLW relate the area of trapezoids to parallelograms. Also the area of kites/rhombuses to

rectangles.

2. TLW derive a formula for trapezoids, kites, and rhombuses from those relationships

3. TLW determine area of trapezoids, kites, and rhombuses from those formulas


Direct instruction

1. Introduce the key terms of the section

Height of a trapezoid- the perpendicular distance between the basses

Diagonal- segment that connects two nonconsecutive vertices

Basses of a trapezoid- the parallel sides of the trapezoid

Guided instruction

Trapezoids

1. Fold the graph paper in half.

2. Then draw a trapezoid the graph paper.

3. Label the height with h and the bases with b1 and b2 within the trapezoid.

4. Now cut out the trapezoid. You should get two congruent trapezoids.

5. Label the second trapezoid with h height and bases b1 and b2.

6. Now have the students tape the trapezoids together to create a parallelogram.

7. Now ask them this question: how does the area of one trapezoid compare to the

area of the parallelogram formed from two trapezoids? Write expressions in terms

of b1, b2, and h for the base height and area of the parallelogram. Then write a

formula for the area of the trapezoid

Kites and Rhombuses

1) Next pull another out a piece of graph paper of a kite.

2) Then draw a kite and the perpendicular diagonals

3) Label the diagonal that is a line of symmetry d1. Then label the other diagonal d2.

4) Now cut the kite out. Then cut along d1 to form two congruent triangles.

5) Then cut one triangle along part of d2 to form two right triangles

6) Turn over the right triangles. Place each one with its hypotenuse along a side of

the larger triangle to form a rectangle. The tape the pieces together.

7) Now ask the students; how do the base and the height of the rectangle compare to

d1 and d2? Write an expression for the area of the rectangle in terms of d1 and d2.

Then use that expression to write a formula for the area of a kite.

8) If they haven’t figured out the formula yet give it to them.

a) Trapezoid

b) Kite/Rhombus

9) Then have them work on example problems. To turn in at the end of class.

a) Find the area of a trapezoid if the bases have length 7.3 and 2.7, and the altitude4 has

length 3.8

i) Here = 7.3; = 2.7; h = 3.8; then

= 19

b) Find the area of an isosceles trapezoid ABCD if the bases have length 22 and 10; the legs

have length 10.

i) Here = 22; = 10; AB = 10; In rectangle EBCF, EF = 10 and AE = .5(22-10) = 6

In , h2 = 10

2 – 6

2 = 64 so h = 8. Then

c) Find area of a rhombus if one diagonal has length 30 and side 17.

i) 172

=

2

+ 152 then

; d= 16; Area =

d) Find length of a diagonal of a rhombus if the other diagonal has length 8 abd the area

equals 52.

i) D = 8 A= 52;

and 52 =

d8 and d= 13

10) Assign homework.

Assessment:

Objectives

1. Through guided instruction cutting the shapes and seeing the relationships

2. Finding the formula from guided instruction

3. At the end of class having the problems due.

Reflection:

I really like this lesson because it isn’t just another boring lecture. It is a nice change up

to get the kids involved with hands on activities.

Quiz 11.1-11.2 Area of polygons Name______________________________

Graph the points and connect them to form a polygon. Then Find the area of the polygon.

1.) A(-3,-2) B(-3,3) C(4,2)

Find the area of the shaded polygons

2.) Trapezoid 3.) Parallelogram

13mm 3yds

24mm 15yds

8mm 12yds-

3yds 9yds

11mm

4.) Kite

5in 13in

5in 13in

5.) Triangle

8km 17km

12km

4

3

2

1

-4 -3 -2 -1 1 2 3 4

-1

-2

-3

-4

10mm

D1=8in

D2=18in

6.)

7.)

8.)

Answer Key

1.

= 17.5 units2

2.

3.

4.

5.

6.

7.

8.

Irregular Shape Lesson

Grade: 9th

/10th


Materials: pencil, graph paper, rubber bands, geoboards,

Standards:

Interpret representations of functions of two variables.

Generalize patterns using explicitly defined and recursively defined functions.

Make decisions about units and scales that are appropriate for problem situations

involving measurement.

Objectives:

4. TLW investigate several geoboard figures to determine the pattern between the number

of perimeter pins, the number of interior pins, and the resulting area.

5. TWL create an equation using symbolic algebra to represent the pattern.

6. TWL analyze precision, accuracy, and approximate error in measurement situations.

7. TLW determine area and perimeter of irregular shaped figures


Guided Exploration

1) Begin class talking about about how would you find the area of an irregular object like an

imprint of a dinosaur foot? Maybe even bring in a dinosaur imprint.

2) Check the students’ ideas. If nothing then guide them with prompting questions.

a) How could we get the measurements of the imprint?

b) What measurement should we use?

c) Could we start with the height and the width? Then what?

d) Could we find the area if we threw the imprint on a grid? Then what?

e) How would we find the area if it was on a graph?

3) Well let’s look at what happens when we graph polygons onto a square dot grid. The dots

are called lattice points.

a) Hand out Discovering Pick’s theorem

b) Hand out geo boards / geo paper.

c) For the last question on the hand out have the students use the dinosaur foot imprint

4) Tell the students that they have to approximate the area of the dinosaur foot?

5) Lead the students into the next topic of error due to rounding off.

Greatest Possible Error

1) Then talk about rounding error caused by rounding up and down

2) Introduce some vocabulary

a) Unit of measure is the increment to which something is measured.

b) Greatest possible error of a measurement is one half of the unit of measure.

3) Give the example that I am about 6ft tall. The greatest possible error is half of a ft or 6 in.

Therefore I can be .5 ft. from 6 ft. which means I can to be from 5.5 ft. tall to 6.4 ft. tall.

4) Explain to them that this is due to rounding off.

5) Give them the formula that shows the greatest possible error

a) b) Where m= the unit of measure

c) E= greatest possible error

6) Examples 7cm

a) Unit of measure is 1 cm

b) Greatest possible error is

1cm = .5 cm

c)

d) 7) Greatest possible error for area

a) For a rectangle h = 6 cm b = 8 cm

b) Greatest possible error is .5cm

c) Take the smallest lengths for the area

i) 5.5 * 7.5 = 41.25

d) Take the larger lengths

i) 6.5 * 8.5 = 55.25

e) Therefore the area of the rectangle is between 41.25 cm2 and 55.25 cm

2

8) Let them determine greatest possible error for the following problems

a) 14in

b) 6yds

c) 4.0cm

d) 16.4mi

e) 7.3mm

9) Give them an exit slip in last 5-10 min

a) Find the minimum and maximum areas of the trapezoid with height 5.8 m

base 3.6 m and 6m

Assessment:

Objectives

4. Informally through interaction with the geoboards

5. Students will be given Pick’s Theorem worksheet

6. At the end of class with the exit slip.

7. Questions during the lesson on their greatest possible error

Reflection:

Some reflection questions after teaching this lesson

Did you use the worksheet? If so, did it provide too much structure? Would you consider

using only the first-half of the worksheet in future?

Did students need more instruction on finding odd shaped areas? Would you include a

refresher of finding the area of triangles in future?

Did the students remain actively engaged with the mathematics? How did the use of

rubber bands or the computer affect classroom management?

Did it become necessary during the lesson to make adjustments to keep the students

moving toward the objective? If so, what could you introduce the activity differently to ensure

that all students understand the goal?

Quiz 11.4 - 11.5 Name___________________________

1) Find the area of the irregularly shaped object using Pick’s theorem.

Find the greatest possible error for the reported measurement. Then write and graph an inequality

that represents the range of values consistent with the measure

2) 25yds 3) 6.5mi 4)0.02km

In two similar triangles, find ratio of the lengths of

5. Corresponding medians if the ratio of the area is 9:49

6. Corresponding sides if the areas are 72 and 50

The areas of two similar triangles are in the ratio of 25:16

7. If the smaller length is 80. What is the length of the larger length?

8. If the perimeter of the larger is 125. What is the perimeter of the smaller?

The two polygons above are similar. Use these figures above to answer the questions.

9. The corresponding diagonals lengths are 4 cm and 5cm. If the area of the larger polygon

is 75cm2. What is the area of the smaller polygon?

10. The areas of the polygons are 80 and 5. If the smaller polygon has a length of 2. What is

the length of the corresponding larger polygon?

Answer Key

1)

2) Unit of Measure: 1 yd

a. Greatest Possible Error:

b. 25-.5=24.5

c. 25+.5=25.5

d.

e. 24 24.5 25 25.5 26

3) Unit of Measure: .1mi


b. 6.5 - .05 = 6.45

c. 6.5 + .05 = 6.55

d.

e. 6.4 6.45 6.5 6.55 6.6

4) Unit of Measure: .01 km


b. 0.02 – 0.005 = 0.015

c. 0.02 + 0.005 = 0.025

d.

e. 0.01 0.015 0.02 0.025 0.03

5) 3:7

6) 6:5

7) 100

8) 100

9)

(

)

A = 48

Area Ratio Lesson Plan

Grade: 9th

/10th


Materials: pencil, paper, textbook, whiteboard, markers, calculator

Standards:

o Understand measurable attributes of objects and the units, systems, and processes of

measurement



Objectives:

1. TLW discover the ratio of corresponding parts on similar figures is equal to ratio of

the area of those figures.

2. TLW use the ratios to find area of similar figures.


Direct Instruction:

1. Review that similar polygons

a. The ratio of their sides is equal to the ratio of their perimeter

i.

8

6

3 4

2. Then describe the ratio between the area

a.

b. Tell them that this is true for all polygons

3. Let’s Prove it

a. What is the ratio of the perimeter?

i. Sides of red = 10, 17, 21

ii. Sides of blue = 15, 25.5, 31.5

iii. 2:3

b. What is the ratio of the area?

i. height of red = 8 base = 21

ii. Heigth of blue = 12 base = 31.5

iii. 4:9

4. Therefore the ratio of area for triangles is also

5. Now let’s figure the area of similar polygons when we have the ratio of a

corresponding side

a. 5:2

b. 7:10

c. 1:

d. 9:x

e. a:5a

6. Let’s see if we can find the ratio of the corresponding parts when you have

the ratio of the area

a. 100:1

b. 400:81

c. 25:121

d. 9x2:16

e. 36:y2

7. Now let’s use the ratio to find unknown areas.

a. If the corresponding lengths are 2 and 6. And the area of the

smaller is 2. What is the area of the larger?

b. If the corresponding lengths are 15 and 20. And the area of the

larger is 240. What is the area of the smaller?

c. If the corresponding lengths are 7 and 9. And the area of the larger

is 210. What is the area of the smaller?

Assessment:

8. Objective 1 through informal class questions

9. Objective 2 through class problems and homework

Reflection:

I thought that the lesson is another basic lecture. The students do get to discover through

lecture, how ratio of area is related to ratio of corresponding parts. I am excited to see this leson

in action.

Sector and circle area lesson

Grade: 9th

/10th


Materials: pencil, paper, textbook, whiteboard, markers, compass, scissors, calculator

Standards:

Understand and use formulas for the area, surface area, and volume of geometric figures



Know the formulas for the area and circumference of a circle and use them to solve

problems Objectives:

3. TLW discover the area of circles and sectors

4. TLW analyze the ratio between area of a sector and area of the whole circle

5. TLW apply the formulas to find area and conversely to find the segments of the circle


Direct instruction

1. Review Circles properties.

a. Circumference of a circle is distance around the circle.

i. C = 2πr = πd

1. r= radius and d= diameter

b. Arc length is a portion of the circumference

i. Arc Length =

2. Then move into the Area of a Circle. Get the students to recall that area of a circle is

a. 3. Area of a Circle Activity

a. Use compass to make a large circle. Then cut out the circle

b. Fold the circular region in half. Fold in half a second time, then a third time and a

fourth time. Unfold the circle and cut along the folds to form 16 wedges

c. Arrange the wedges in a row, alternating the tips up and down to form a

parallelogram almost.

d. Write expressions in terms of r for the approximate height and base of the

parallelogram. Then write an expression for its area.

e. Explain how your expression can be used to justify the area of a circle

4. Example

a. Find the area and circumference of a circle

i. r = 5.2

1. Area = π(5.2)2 = 27.04 π

2. C = 10.4 π

ii. d = 12

1. Area = π

= 36 π

b. Find measure of radius and circumference

i. If Area = 144π

1. 144π = πr2

2. C =24π r = 12

c. Find area of given figure

i. Diameter = 7m

ii. Base = 5 m

7m

5m

iii.

iv.

5. Area of a sector

a. Sector of a circle is the region bounded by two radii of the circle and their

intercepted arc. Give drawing

b. The ratio of the area of a sector to the area of the whole circle is equal to the ratio

of the measurement of the intercepted arc is to 360

6. Examples of finding the area of sectors

a. Find the area of a 300 sector of a circle whose radius is 12.

i.

ii.

iii.

b. Find the area of the circle with a sector of 40 and area of 35m2

i.

ii.

iii.

iv.

= 315m

2

c. Find the measure of the central angle of a sector whose area is 6π if the area of the

circle is 9π

i.

ii.

iii.

Assessment:

10. Objective 1 students were assessed informally through the circle activity

11. Objective 2 during lecture students were introduced to the ratio.

12. Objective 3 Students were assessed during

Reflection:

Mostly lecture based but I threw in a discovery activity. I think that it is good to do some

hands on activity with math every once in a while to change it up. Also it is good for visual

learners to see how the area of a circle is derived from.

Regular Polygon Area

Grade: 9th

/10th


Materials: pencil, paper, textbook, whiteboard, markers, calculator

Standards:

Understand and use formulas for the area, surface area, and volume of geometric figures


involving measurement. Objectives:

6. TLW use the properties of regular polygons to find angle measures

7. TLW find the area of regular polygons by dividing it into congruent triangles

8. TLW analyze the three ways to find lengths on a regular n-gon


1. Review properties of a regular polygon of n sides

a. Each central angle c measures

b. Each interior angle i measures

c. Each exterior angle e measures

2. Introduce new terms for this section. Along with the definitions give the students

drawings of where these parts are on a regular polygon

a. Regular polygon is an equilateral and equiangular polygon

b. Center of a regular polygon is the common center of its inscribed and

circumscribed circles

c. Radius of a regular polygon is a segment joining its center to any vertex

d. Central angle of a regular polygon is an angle included between two radii drawn

to successive vertices

e. Apothem of a regular polygon is a segment from its center perpendicular to one of

its sides

3. Finding measures of angles an lines in a regular polygon. Draw a pentagon on the board

then ask students to find the following

a. Find the length of a side d of a regular pentagon if the perimeter p is 35

i. perimeter = number of sides x length of sides

ii. Pentagon has 5 sides with perimeter 35 then sides equal 7

b. Find measure of the center angle of a pentagon

i.

= 72

c. Find the measure of the angle between the apothem and the radius of a pentagon

i. Center angle of a pentagon =72 cut by an apothem is 36

d. Find the measure of the interior angle of a pentagon.

i.

4. Finding the area of a regular polygon

a. You can find the area of any regular polygon by dividing the polygon into

congruent triangles

b. A = area of one triangle number of triangles

A=(

)

i. Height of the triangle is a; the apothem of the regular polygon

ii. The base of the triangle is s; the side of the regular polygon

iii. Number of triangle is n; number of sides in the regular polygon

A =

iv. Commutative and associative properties of enequality

A=

v. There are n congruent sides of length s, so the perimeter P is

c. Area of a regular polygon is A=

or A=

i. Where P is the perimeter found by number of sides times legth of the

sides

5. Examples of area of a regular polygon

a. Find the area of a pentagon with sides 4.4 and apothem 3

i.

ii.

iii. 2

b. Find the area of a regular hexagon if the length of the apothem is

i.

1. a=

2. we can find the side of a regular polygon because the triangle made

with the apothem is a special right triangle 30-60-90

3. side equals 10

4. perimeter is 60

ii.

= 150 2

c. Find the area of an octagon with sides of length 10 and radius 13.

i. Step one find the perimeter 8(8)=64 B

ii. Find the apothem

1. BC= 13 EC=5

2. Using Pythagorean theorem BE=12

a. 52+x

2=13

2 x 13

b. x = 12 the apothem

iii.

iv.

16 A 5 E 5 C

d. Some time using trigonometry is the only way to find the area

i. Find the measure of a decagon with side length of 12

ii. Perimeter = 120

1. To find apothem AC

2. Start with the Center angle = 36 the apothem bisects it then that

angle is 16

3. Then use trigonometry to find AC

iii.

iv.

6. Those are the three main ways to find the sides of the triangle in a regular n-gon

a. Pythagorean Theorem

b. Special Right Triangles

c. Trigonometry

Assessment:

13. Assess students on participation in lecture

14. Assess students with quick summary at the end of the lesson to check understanding

15. Assess students on all objective on the homework

Reflection:

I think it was a good lecture. The key to a good lecturing is asking the students questions

during the lecture. This will keep them on their toes. The participation will let me assess their

understanding. And keep them focused.

CHAPTER 11 TEST Name_________________________________

Name the polygon. Then find the area and perimeter of the shaded polygons.

1.) 8m 2.) 7ft

12m 5m 5ft

5ft

25m 7ft

3.) 4.)

9in 18in

13 cm 7cm

9in 18in

8 cm

D1 = 10in

D2 = 26in 12cm

5.) Find the area of the shaded area to the nearest tenth.

6.) Find the shaded area of the sector AB in terms of π

7.) Find the area of the sector AB in terms of π

8.)Find the area of the circle in the terms of π. Find the radius to the nearest whole number.

10 yds

10yds

A

12 mm

C

B

A

8mi

C

B

70

A B

Area of sector= 49m2

Find the greatest possible error for the reported measurement. Then write and graph an inequality

that represents the range of values consistent with the measure

9.) 7.3mi 10.) 12yds 11.) 0.37km

12.) Find the area of the irregularly shaped object using Pick’s theorem.

Answer key

1.) .

(24+8)9=148.5 m

2

2.) 7(4.7) = 32.9 ft2

3.)

= 130 in

2

4.)

cm2

5.) = 21.4602

6.)

= 56 2

7.)

mm

2

8.)

2

9.) Unit of Measure: .1mi


b. 7.3 - .05 = 7.25

c. 7.3 + .05 = 7.35

d.

e. 7.2 7.25 7.3 7.35 7.4

10.) Unit of Measure: 1 yd


b. 12-.5=11.5

c. 12+.5=12.5

d.

e. 11 12.5 25 12.5 13

11.) Unit of Measure: .01 km


b. 0.37 – 0.005 = 0.365

c. 0.37 + 0.005 = 0.375

d.

e. 0.36 0.365 0.37 0.375 0.38

12.) A = 34/2 + 37 – 1 = 43

unit lesson plan: measuring length and area: area of...

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