5.1 and 5.2 By: John and Amelia

Pythagoras

POD The carpentry terms span, rise, and rafter length are illustrated in the diagram. A carpenter wants to make a roof that has a span of 20 ft and a rise of 10 ft. What should the rafter length be? c2 = a2 + b2 Use the Pythagorean Theorem. Replace a with 10 (half the span), and b with 10. c2 = 102 + 102 Square 10. c2 = 100 + 100 Add. c2 = 200 Find the positive square root. c = 200 Round to the nearest tenth. c 14.1 The rafter length should be about 14.1 ft.

What do we know? In section 2-3, we drew line segments with the lengths √2, √3, and the √4. We also learned the Pythagorean theorem, which occasionally gives these values for the hypotenuse of right triangles.

How can we use this? Theodorus’ Wheel Draw a right triangle with leg lengths of 1”. Draw another triangle, using the hypotenuse of the first triangle as the lower leg. - The upper leg should have a length of 1”. - Connect the legs to form the hypotenuse of the second triangle. Now, use blank paper, an index card, and a straight edge ruler to draw the two triangles Triangle #1 Triangle #2 1 Hypotenuse/Leg 1 1

To complete the wheel Repeat this process with a 3rd and 4th triangle. What value does the hypotenuse give each time? 1 √3 √2 1 1

Use the lab sheet 5-1 to complete the values

It should look like this √4 √5 √3 √6 √2 √7 √8 √9 It should look like this √10 √11 √12

Problem 5-1 Now that you have found all the values for the hypotenuses, you need to cut out the ruler from the lab sheet 5-1 and measure them. For each hypotenuse length that is not a whole number, give the two consecutive whole numbers between which the whole number is located (Ex. √2 lies between 1 and 2 on your ruler. For each hypotenuse length that is not a whole number, use your completed ruler to find a decimal number that is slightly less than the length and greater than the length.

Problem 5-1 Follow-Up Now use your calculator to find the length of the hypotenuses. Is this value close to your approximate values? Read and answer.

5.2 Representing Fractions as Decimals

Think About It Take the decimal approximation you found for √2 in Problem 5.1 Follow-Up and square it. What do you get? You either get a number that is slightly smaller or slightly larger than √2. Why? Do you think it is possible to find the exact value using a calculator? No! You will get similar results for √3 and √5. To better understand these decimals, let’s compare them to… When you found approximations, remember that their squares were always either smaller or larger than 2 Did anybody get anything different? Why is this?

Fractions! ¼ = 0.25 1/8 = 0.125 1/3 = 0.3333333… 1/11 = 0.090909090909… The first two fractions have decimal representations that end, or terminate. (Think about the ex-terminator who ends a bug’s life!) These decimals are called terminating decimals. The last two fractions’ decimal expansions do not terminate, but the decimals show a repeating pattern; they are called repeating decimals. After #1: Ask for any pattern they see in the decimal expansions.

Batting Averages

Your Turn!! Problem 5.2 Write each fraction as a decimal, and tell whether the decimal is terminating or repeating. If it is repeating, tell which part repeats. 2/5 = 3/8 = 3. 5/6 = 4. 35/10 = 5. 8/99 =

2/5 = 0.4; terminating 3/8 = 0.375; terminating Answers to Problem 5.2 2/5 = 0.4; terminating 3/8 = 0.375; terminating 3. 5/6 = 0.8333…; repeating; 3 4. 35/10 = 3.5; terminating 5. 8/99 = 0.08080808…; repeating; 08

Problem 5.2 Follow-Up #2 2. Find a fraction that is equivalent to the given terminating decimal. a. 0.35 b. 2.1456 c. 89.050 May not do this- play game!

Answers to Problem 5.2, #2 2a. 2b. or 2c. or

We Have, Who Has? Who has the first card?

ACE Problems page (60) Read “Think about this!  answer #’s 18-20 Your Turn Again! ACE Problems page (60) Read “Think about this!  answer #’s 18-20 Maybe do one per group or if time permits, let every group do each problem. Summarize afterwards

To find more info… http://mathforum.org/library/drmath/sets /select/dm_repeat_decimal.html http://www.mathematicalquilts.com/Quilt s_-_Page_2.html http://www.educationplace.biz/math/mat hsteps/7/a/7.rationals.ideas1.html http://hotmath.com/search/hotmath- search.jsp?term=repeat

How would you enhance this lesson if you taught it in your classroom? Tell Us What You Think How would you enhance this lesson if you taught it in your classroom? What would you repeat? What would you change?

I like a teacher who gives you something to take home to think about besides homework.  ~Lily Tomlin as "Edith Ann"