AREA Judo Math Inc.

Transcription

1 AREA 2013 Judo Math Inc.

2 7 th grade Geometry Discipline: Blue Belt Training Order of Mastery: Area 1. Square units/area overview 2. Circle Vocab (7G4) 3. What is Pi? (7G4) 4. Circumference of a circle (7G4) 5. Area of a circle 6. Area of Polygons (7G6) 7. Area of irregular shapes (7G6) Welcome to the Blue Belt Area It s very important when we measure things to sure that we are using the right units. Ever heard your teacher say that before?! That s because units will be very different depending on whether you are measuring something that s a straight line (like the distance from here to the bathroom), the area of something that is flat (like this table top), OR the volume of something 3-dimensional (like this room). For each of these measurements, having an error in units will make your solution not make any sense! The way we can determine what type of units we need to use can be done by referring to the dimension that we are measuring. Check out the table below it will give you a quick overview of dimensions and the appropriate units for each. As you can see, the dimension directly relates to the exponent that is above the unit! So anything in 2 dimensions is in 2 and anything in 3 dimensions is in 3. What about the fourth dimension, you might ask? Well, the world we live in only has 3 dimensions, but sometimes people call time the 4 th dimension time and some people suspect that our brains can perceive the 4 th dimension even though we can t see it go ahead and google it, but watch out because it might start making your mind feel like its bending! In this discipline we will be working primarily with 2 dimensional space so nothing too crazy! Standards Included: Good Luck Grasshopper. 7.G.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. 7.G.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms Judo Math Inc.

3 1. Square units/area overview Area represents 2 dimensional space and is measured in units 2. That means that if you want to find the area of something, you are essentially seeing how many little squares will fit into it Squares like this one: To make sure we have the idea of 2 dimensional space, make a list in the box below of anything that you could find the area of (make sure to not include anything where you are measuring a straight line 1D or how much fits into anything 3D) Brainstorm of objects that would be measured in 2 dimensional space: Table top floor Practice finding area by counting the number of squares in each of the shapes below. You will have to do some estimating on some of these! Area of triangle: Area of rhombus Area of circle: Now find the area of this interesting shape by counting boxes 1

4 Square Units Investigation Jan has a board in the shape of a rectangle that is 60 inches long and 48 inches wide. 1. How long is the board measured in feet? How wide is it measured in feet? 2. Find the area of the board (in square feet). 3. Jan said, To convert the inches to feet, I ought to divide by 12. The board has an area of 48x60 in =2880 inches squared. If I divide the area by 12, I can find out the area in square feet. So the area of the board is = 240 feet squared What went wrong with her problem solving?! 2

5 Based on your investigation above, answer the following questions. Draw a quick picture to prove your answer to each: 1. How many square inches are in a square foot? 2. How many square feet are in a square yard? 3. How many square feet are in a square mile? 4. How many square cm are in a square meter? 3

6 2. Circle Vocab (7G4) Before we get into some complicated circle business, it is going to be very important that we have the circle vocabulary perfectly. Inspect this circle to review your circle vocabulary before completing the following exercises: 4

7 11. Now you draw a circle that has a circumference of 4.5cm below. If you have a compass, use that to draw it, if not, try to use other tools to make this drawing as ACCURATE as possible: 5

8 3. What is Pi? (7G4) Pi is a very special and irrational number. That means that it cannot be written as a fraction because the decimals go on FOREVER AND EVER AND EVER! Here s the first 400 digits of Pi: Ask your teacher if you can have a competition to see who can memorize the most! I bet you will win Even though Pi is irrational and cannot be written as a fraction, sometimes people approximate Pi to be 22 7 Turn 22/7 into a decimal using division. How close is it to what Pi really is? 6

9 One cool thing that you might not know about Pi : anyone can find it by simply dividing a circles' circumference by its diameter. Which circle, you may ask? ANY CIRCLE!!!! Measure the circles below to discover Pi for yourself! (you will have to ask your teacher for some string to measure the circumference!) Circumference (c)= Diameter (d) = c d = How close is this to Pi?! Circumference (c)= Diameter (d) = c d = How close is this to Pi?! The picture to the right is one final look at pi as you can see, the diameter fits around the outside of the circle 3 times with just a little bit left over ( to be exact) 7

10 Now try these: 1. The earth has a circumference of 24,901.5 miles and a radius of miles. How can you use these measurements to estimate pi? 2. If Jupiter has a radius of 44,423 miles, can you estimate it s circumference? What about a circle with radius n? Check out the cartoon to the left. What do you think it means? And one final parting Pi joke for you: Question: What do you get if you divide the circumference of a jack-o-lantern by its diameter? Answer: Pumpkin Pi! 8

11 5. Circumference of a circle (7G4) As we saw above the ratio of the circumference of a circle to the diameter of a circle is always This information can actually help us to understand how to find the circumference of ANY circle. = 3.14 Since C D = π we can multiply both sides by D to get the formula: C= π D And since the diameter is twice the radius, people also write the formula like this: C= π 2 R Example: 9

12 9. A circle has a diameter of 8 meters. What is the circumference of the circle? Use 3.14 for pi. 10. A circle has a radius of 7 feet. Find the circumference of the circle. Do not approximate pi (leave it your answer as pi) 11. James states The circumference of a circle is approximately 6 times the radius of the circle. Is James statement correct? Explain your reasoning. 10

13 5. Area of Circles (7G6) This initial activity may be done as a class with your teacher OR you can follow the steps below. Start by cutting out the wedges below then follow all of the instructions on the next page. You will be DISCOVERING one way to come up with the area of a circle! 11

14 (leave page blank for cutout on the previous page.) 12

15 Discovering the Area of the Circle! We are going to find the area of a circle by relating it to a rectangle (which, as you know, has an area of A=l*w). 1. Arrange your 16 cutout wedges just like the picture here: 2. When the circle is divided into wedges and arrange like this, does it look like another shape you know? 3. What do you think would happen if we cut the circle up into 32 wedges, or 55 wedges, or 1000 wedges?! Slice each wedge in half and then arrange in an alternating fashion again. What has changed? 4. What are the dimensions of the rectangle that is formed? 5. Since the top and bottom of the rectangle make up half of the circumference and the circumference (from the last section) is 2πr and half of that is. And the height is the radius of the circle 6. Therefore the area for this Rectangle aka a circle is: A = π r 2 13

16 With your new formula, practice finding the area of the following circles. Round your answer to the nearest tenth and use 3.14 for pi. 14

17 9. Find the area of a circle with a diameter of 10 meters. 10. What is the area of a circle that has a radius of 6cm? (use 3.14 for pi) 11. Sam drew a circle with a radius of 4 inches and Sally drew a circle with a radius of 8 inches. Sally said since the radius of my circle is twice the radius of your circle, the area of my circle is twice the area of your circle. Is Sally s statement correct. If not, explain what she could say isntead. 12. A circle inside of a square as shown in the picture here. The edges of the square each tough a point on the circle. If the square has an area of 16 square meters, what is the area of the circle? 15

18 13. A circular pizza can feed 4 people if it has an area of at least 200 square inches. A pizza from Joe s Pizza has a radius of 9 inches. Is it enough to feed a family of 4? 14. Juan wants to know the cross-sectional area of a circular pipe. He measures the diameter which he finds, to the nearest millimeter, to be 5 centimeters. a. To find the area of the circle, Juan uses the formula A=πr 2 where A is the area of the circle and r is its radius. He uses 3.14 for π. What value does Juan get for the area of the circle? b. Is there any possible error in Juan's measurement of the diameter of the circle? Explain how much it could be and how much this could affect your calculation of the area.. 16

19 15. 17

20 14. The figure to the right is composed of eight circles, seven small circles and one large circle containing them all. Neighboring circles only share one point, and two regions between the smaller circles have been shaded. Each small circle has a radius of 5 cm. a. Calculate the area of the large circle. b. Calculate the area of the shaded part and explain how you arrived at your solution. 18

21 15. Circumference/Area: If you are given the circumference of a circle, can you find the area? Below are a few circles for you to try this out with. Write your conclusion as a firm yes or no statement in a complete sentence. Circumference = 43.4 yd Circumference=17 yd 16. What is the area of a circle that has a circumference of 10π? 17. The circumference of a circle is 12.56ft was used to approximate pi, what is the area of the circle. 19

22 18. The circumference of the earth is approximately 25,000 miles. If you were to dig a hole to the center of the earth, how long would the tunnel be? 19. The redwood trees in California are said to have trunks so large that a far could drive through them. Estimate the circumference of the redwood tree. Use your estimate to find the area of the trunk if the tree was cut down. Explain all of your reasoning. 20. (final circle challenge!) What percent of the area of a circle is enclosed by an isosceles triangle one of whose sides is the diameter of the circle? Draw a picture to help show your solution. 20

23 5. Area of Polygons (7G6) In previous years, you have learned the basics of area, like the area of a triangle, rectangle, square, rhombus, parallelogram, and regular polygons. Fill in the blank boxes with the formula for each (if you need to look it up on a computer you can!) Shape Formula for area Triangle Rectangle & Square Rhombus & Parallelogram Trapezoid Regular Polygon 21

24 Find the area of the following shapes using the formulas from the previous page 22

25 4. If you forgot the formula for the area of a regular polygon, how could you figure out the area of the above shapes (think triangles!) Use one of the problems above to prove your solution. 23

26 5. 6. A triangle has an area of 6 square feet. The height is four feet. What is the length of the base? 6. Area of irregular shapes (7G6) Sometimes we will find ourselves needing to find the area of irregular shapes. What can we do in these situations? Well, break the shape up into pieces we can find the area of, of course! Check ou the track below. Find the area of it by breaking it up into shapes you know the area of. Area of track: m 2 24

27 Now find the area of these shapes by breaking them into familiar shapes. Make sure to always use the correct units: 25

28 Using any tools you need, draw a shape that has the following: 2 semicircles 1 rhombus 1 parallelogram 2 triangles 1 rectangle They should all be connected by at least on side. Label and find the area of your shape! Find the area given circumference: 26