Unit Lesson Plan: Measuring Length and Area: Area of shapes

Transcription

1 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

2 Area of a Square Lesson Grade: 10 th /11 th 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 = s 2 = 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 26 2 =b 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

3 Assessment: 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.

4 Trapezoid Lesson Grade: 9 th /10 th Content: Mathematics- Geometry 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 Learning Activities: 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 b 1 and b 2 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 b 1 and b 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 b 1, b 2, 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 d 1. Then label the other diagonal d 2. 4) Now cut the kite out. Then cut along d 1 to form two congruent triangles. 5) Then cut one triangle along part of d 2 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.

5 7) Now ask the students; how do the base and the height of the rectangle compare to d 1 and d 2? Write an expression for the area of the rectangle in terms of d 1 and d 2. 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, h 2 = = 64 so h = 8. Then c) Find area of a rhombus if one diagonal has length 30 and side 17. i) 17 2 = 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.

6 Quiz 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 ) Trapezoid 3.) Parallelogram 13mm 3yds mm 10mm 15yds 8mm 12yds yds 9yds -3 11mm -4 4.) Kite 5in 5in D 1 =8in D 2 =18in 13in 13in 5.) Triangle 6.) 8km 17km 7.) 12km 8.)

7 Answer Key 1. = 17.5 units

8 Irregular Shape Lesson Grade: 9 th /10 th Content: Mathematics- Geometry 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 Learning Activities: 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.

9 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 = d) Take the larger lengths i) 6.5 * 8.5 = e) Therefore the area of the rectangle is between cm 2 and 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?

10 Quiz 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

11 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 75cm 2. 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?

12 Answer Key 1) 2) Unit of Measure: 1 yd a. Greatest Possible Error: b =24.5 c =25.5 d. e ) Unit of Measure:.1mi a. Greatest Possible Error: b = 6.45 c = 6.55 d. e ) Unit of Measure:.01 km a. Greatest Possible Error: b = c = d. e ) 3:7 6) 6:5 7) 100 8) 100 9) ( ) A = 48

13 10)

14 Area Ratio Lesson Plan Grade: 9 th /10 th Content: Mathematics- Geometry Materials: pencil, paper, textbook, whiteboard, markers, calculator Standards: o Understand measurable attributes of objects and the units, systems, and processes of measurement Make decisions about units and scales that are appropriate for problem situations involving 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. Learning Activities: Direct Instruction: 1. Review that similar polygons a. The ratio of their sides is equal to the ratio of their perimeter i 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

15 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. 9x 2 :16 e. 36:y 2 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: Reflection: 8. Objective 1 through informal class questions 9. Objective 2 through class problems and homework

16 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.

17 Sector and circle area lesson Grade: 9 th /10 th Content: Mathematics- Geometry Materials: pencil, paper, textbook, whiteboard, markers, compass, scissors, calculator Standards: Understand and use formulas for the area, surface area, and volume of geometric figures Make decisions about units and scales that are appropriate for problem situations involving measurement. 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 Learning Activities: 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 = Area = π(5.2) 2 = π 2. C = 10.4 π ii. d = Area = π = 36 π b. Find measure of radius and circumference i. If Area = 144π π = πr 2

18 2. C =24π r = 12 c. Find area of given figure i. Diameter = 7m ii. Base = 5 m 7m iii. iv. 5m 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 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 35m 2 i. ii. iii. iv. = 315m 2

19 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.

20 Regular Polygon Area Grade: 9 th /10 th Content: Mathematics- Geometry Materials: pencil, paper, textbook, whiteboard, markers, calculator Standards: Understand and use formulas for the area, surface area, and volume of geometric figures Make decisions about units and scales that are appropriate for problem situations 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 Learning Activities: 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

21 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=( ) A = 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 iv. Commutative and associative properties of enequality 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. 2 iii. 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 side equals perimeter is 60 2 ii. = 150

22 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 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 = To find apothem AC 2. Start with the Center angle = 36 the apothem bisects it then that angle is 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.

23 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 D 1 = 10in D 2 = 26in 18in 13 cm 12cm 7cm 9in 18in 8 cm

24 5.) Find the area of the shaded area to the nearest tenth. 10 yds 10yds 6.) Find the shaded area of the sector AB in terms of π 8mi A C B 7.) Find the area of the sector AB in terms of π A C 12 mm B 8.)Find the area of the circle in the terms of π. Find the radius to the nearest whole number. 70 A B Area of sector= 49m 2

25 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.

26 Answer key 1.). (24+8)9=148.5 m 2 2.) 7(4.7) = 32.9 ft 2 3.) = 130 in 2 4.) cm2 5.) = ) = ) mm 2 8.) 9.) Unit of Measure:.1mi a. Greatest Possible Error: 2 b = 7.25 c = 7.35 d. e ) Unit of Measure: 1 yd a. Greatest Possible Error: b =11.5 c =12.5 d. e ) Unit of Measure:.01 km a. Greatest Possible Error: b = c = d. e ) A = 34/ = 43