Parallelograms – Perimeter & Area Recall… What is the difference between one- and two-dimensions? What is a parallelogram?

Notes Parallelogram – a constructed from 2 pairs of parallel lines - Rectangles are a special type of parallelogram (non-parallel sides are perpendicular to one another) Parallelogram has 3 lengthwise measurements: sides (s), height (h), & base (b) Composite figures – a figure composed of different shapes

Area vs. Perimeter Perimeter – length around a figure - Expressed in units (such as ft, m, in, cm) - Perimeter of parallelogram expressed as, P = 2s + 2b. Area – number of square units in a figure Expressed in units-squared (such as ft2, m2, in2, cm2) Area of parallelogram expressed as, A = bh.

Try this… (see page 281) (1) On a coordinate plane, graph the following parallelogram given these coordinates: (0,0), (3,5), (7,0), and (10,5). (2) Measure the base, the height, and the sides of the parallelogram. (3) Cut out the parallelogram, then cut out the right triangle formed by the intersection of the base and the height. Translate the triangle so that the sides of the original parallelogram match up. What does this demonstrate about the areas of non-rectangular parallelograms with rectangles?

Example problems See the board & follow along (p. 283, ex Example problems See the board & follow along (p. 283, ex. 1-5, 8) Guided Practice (in pairs) On p. 283, ex. 9-13, 16 Put these in your notes, Names will be drawn randomly to show work on the board. Be prepared to explain your work/response! Homework – p. 283, ex. 17-29

Trapezoids & Triangles Trapezoids & triangles can be thought of as half sections of parallelograms.

Area & Perimeter of Triangles & Trapezoids (2) Perimeter computed similarly to parallelograms, summing each side of the triangle or trapezoid. (3) Area computed w/ the following formulae, Area (Triangle) = ½(base)(height) = ½(b)(h) Area (Trapezoid) = ½(base 1 + base 2)(height) = ½(b1 + b2)(h) Why is there the ½ factor in these formulae??? With a partner beside you, work ex. 2-10 (evens) on p. 287.

Example Problems & HW With a partner beside you, work ex. 2-10 (evens) on p. 287. You will be called on randomly to work these problems at the board! HW – p. 287, ex. 12-30

Right Triangles & Pythagorean Theorem

Right Triangles & Pythagorean Theorem On a sheet of paper, draw a line forming the hypotenuse of a triangle with edges of the paper serving as legs. Cut out the triangle along the line you drew. Record the length of the legs of the triangle. Square each leg length. [For example, if a leg of my triangle is 15cm, the square is (15cm)2 = 225cm2.] Add the two squared measurements together. Now, measure the hypotenuse of your triangle, and square this measurement. Compare the values you obtained from (5) & (6). What did you find?

Pythagoras lived in the 500s BC, and was one of the first Greek mathematical thinkers. He spent most of his life in the Greek colonies in Sicily and southern Italy. He had a group of followers who followed him around and taught other people what he had taught them. The Pythagoreans were known for their pure lives (they didn't eat beans, for example, because they thought beans were not pure enough). They wore their hair long, and wore only simple clothing, and went barefoot. Both men and women were Pythagoreans. Pythagoreans were interested in philosophy, but especially in music and mathematics, two ways of making order out of chaos. Music is noise that makes sense, and mathematics is rules for how the world works. Pythagoras himself is best known for proving that the Pythagorean Theorem was true. The Sumerians, two thousand years earlier, already knew that it was generally true, and they used it in their measurements, but Pythagoras is said to have proved that it would always be true. We don't really know whether it was Pythagoras that proved it, because there's no evidence for it until the time of Euclid, but that's the tradition. Some people think that the proof must have been written around the time of Euclid, instead.

Right Triangles & Pythagorean Theorem (1) Pythagorean theorem states, a2 + b2 = c2, where a & b are legs of a right triangle, & c is the hypotenuse. (2) There some special right triangles based on this: 3,4,5  32 + 42 = 52 5,8,12  52 + 82 = 122 8,15,17  82 + 152 = 172 7,24,25  72 + 242 = 252 These can be scaled up or down (e.g., 3,4,5  6,8,10) F.Y.I. – There are hundreds of proofs for this theorem, including one from former President James Garfield!

Practice (1) With a partner, work ex. 1-7 on p. 292. Be prepared to present at the board! (2) No homework!!! Have a fun, safe, restful winter break & be prepared to work hard when we get back! It’s been great getting to know y’all so far! (3) Check out the wiki for updates (write the website on the front cover of your agenda or somewhere you can easily find it).

Circles (1) Circumference – the distance around a circle (A) circle : circumference :: perimeter : polygon (B) Formula: C = 2r = d, where d = diameter, r = radius, C = circumference. (2) Area of a circle (A) Equal to  times the square of the radius A = r2. Let’s work ex. 1-6, p. 296 in our notes.

Practice & Homework With a partner, work ex. 7-16, p. 294 to your notes. You will be called randomly to the board to present. Homework Ex. 17-33, p. 296-297.