BIO 100

Lecture — Geometry By the end of this lesson, students will be able to: 1. Differentiate between, describe and name lines, rays and line segments. 2. Identify and describe acute, right, obtuse, straight and reflex angles. 3. Find parallel and perpendicular lines to a line when given the equation for that line. 4. Define reflection symmetry. 5. Identify lines of reflection symmetry on various shapes. 6. Predict the two-dimensional shape created when taking the cross section of a three-dimensional object. 7. Find the area and volume of various objects.

Lines, Rays and Segments In our everyday lives, if something is straight, we usually say that it’s in a line or is a line. However, in mathematics, a line has some specific requirements. Which of these is truly a line (mathematically)? Point to each “line” and ask students to raise hands or shout out when they think the one you are pointing at is a line. DON’T give them the answer yet.

Lines, Rays and Segments (Cont.) To understand lines, we should first understand what a point is. -A point represents a unique location in space, usually represented with coordinates, x and y. (We would also use a z coordinate if we were working in three dimensions.) We are familiar with the x and y coordinates, but the z one is new. This now brings us into the third dimension. Since we can now “move” in three different directions, we are in three dimensions (left/right, backwards/forwards, and up/down).

Lines, Rays and Segments (Cont.) Point (x, y, z) Can you find the 3 planes? Discuss how the point has the coordinates (x, y, z), which means an x-value of x, a y-value of y, and a z-value of z. Try substituting numbers in for x, y and z . . . like 3, 4, and 5 . . . to see if this helps students understand this better. Talk about what these values would mean. (We would move three steps in the x-direction, 4 steps in the y-direction, and 5 steps in the z-direction.) Point out that the y-axis is not vertical here—the z-axis is. Next, ask students what would happen if one of the number was negative—like instead of +3 for the x-value, where would -3 be? Then allow for some small group discussion. After a minute or so, ask a student to explain the direction one would move if the numbers were negative. If no students are able to, lead the discussion yourself. Finally, discuss the three planes—you can use a piece of paper to describe a plane, which has two dimensions. (A piece of paper does have three dimensions, but one of them is just really small.) Hold the paper up for each plane so students can see each one (the plane made by the x-axis and the y-axis, the plane made by the x-axis and the z-axis, and the plane made by the y-axis and the z-axis).

Lines, Rays and Segments (Cont.) To describe a line, mathematicians would say that a line is a series of points (infinite in both directions) that all lie on a single plane that make an infinitely small curve (meaning it’s straight). We name a line using any two points on the line. Name this line. A B C Give students a quick second to name the line on their own. You could name this line AB, AC, BC, BA, CB or CA, but you should put the line symbol above the two points chosen (you may need to write this on the board so students can see it). You can go back to SLIDE 6 now and discuss which one is the “line”.

Lines, Rays and Segments (Cont.) To describe a ray, mathematicians would say that a ray is also an infinite series of points (like a line), but it has one endpoint and is only infinite in one direction. We name a ray by starting with its endpoint, then using any other point on the line. Name this ray. A B C Give students a quick second to name the ray on their own. You could name this CB or CA, but you should put the ray symbol above the two points chosen (you may need to write this on the board so students can see it).

Lines, Rays and Segments (Cont.) To describe a line segment, mathematicians would say that a line segment is a finite series of points, with two endpoints. (Rays and lines both have infinite length, but a line segment does not.) We name a line segment with its two endpoints. Name this line segment. A B C Give students a quick second to name the line segment on their own. You could name this line segment AC or CA, but you should put the line segment symbol above the two points (you may need to write this on the board so students can see it).

Angles An angle can be described as the amount of turn or the number of degrees between two intersecting lines, rays or line segments. Since the human body has many moving parts, a working knowledge of angles is important.

Angles (Cont.) We’ll focus on six different types of angles.

Acute Angles Acute can mean two different things in medicine. If you have “acute pain”, that means that your pain is sharp. Or, if you have a condition that is not long-lasting, that can also be referred to as acute. With these definitions in mind, we can see how an acute angle has an appropriate name. You can talk about how an acute angle is sharp, like a knife—or short lived because it does not have very much turn. An acute angle is less than 90 degrees

Right Angles and Obtuse Angles Right angles are exactly 90 degrees. It does not always mean “turning right”. An obtuse angle is greater than 90 degrees, but less than 180 degrees. A right angle is exactly 90 degrees Point out the special way we denote a right angle. It is similar to a box, looking like it forms a small rectangle—which has all 90 degree angles. If you say the someone is obtuse, that would mean that they weren’t very sharp! An obtuse angle is in between 90 degrees and 180 degrees

Straight Angles and Reflex Angles Straight angles are exactly 180 degrees. Two lines, rays or line segments meet to make a straight line. A reflex angle is greater than 180 degrees. A straight angle is exactly 180 degrees A reflex angle goes beyond 180 degrees, a hyperextension

A Full Rotation When someone changes his/her mind, would you say that person did a 180 or a 360? Can you think of some parts of the body that can make these angles? A full rotation is 360 degrees If you do a “360”, you would be facing the same direction you were before. If you do a “180”, you are facing the opposite direction. Point out that the point where two bones meet is a joint—and these two bones form an angle. The hands and fingers can do most of the angles. Elbows (a hinge joint, which is a type of synovial joint) and knees (a compound or modified hinge, but also a synovial joint) should not be hyper-extended past 180 degrees, but the back can be (at little). The shoulder is a ball-and-socket joint (also a type of synovial joint), and some people may be able to do a full rotation with their arms.

Parallel and Perpendicular Lines Parallel and perpendicular lines are found on the same plane—like a coordinate plane. Parallel lines have the same slope, while perpendicular lines meet at a right angle. Point out the way we denote parallel lines (with arrows on the two parallel lines) and perpendicular lines (with the right angle symbol). Parallel lines Perpendicular lines

Parallel Lines Find and name the two parallel lines. Then, find the equation for each one (in slope-intercept form). Remind students that they can find the slope and the y-intercept in a variety of ways, and then they can put them into the y=mx+b format. Allow students to work at this for several minutes—hopefully, they will work with each other and fill in gaps on their own. Walk around and help students/groups individually. HAVE A FEW STUDENTS SHARE OUT THEIR METHODS AT THE DOCUMENT CAMERA.

Parallel Lines (Cont.) The equation for line AB is . . . y = 2x + 1 The equation for line CD is . . . y = 2x – 2 OR y = 2x + -2 What do you notice about these two equations? Give the equations for 5 other lines that are parallel to these lines. Graph a few. Other parallel lines include: y=2x y=2x+3 y=2x+5 y=2x-8 etc. As long as the slope is the same and the y-intercept is different, the line will be parallel.

Perpendicular Lines Find and name the line that is perpendicular to Line AB. Then, find the equation for that line (in slope-intercept form). Remind students that they can find the slope and the y-intercept in a variety of ways, and then they can put them into the y=mx+b format. Allow students to work at this for several minutes—hopefully, they will work with each other and fill in gaps on their own. Walk around and help students/groups individually. HAVE A FEW STUDENTS SHARE OUT THEIR METHODS AT THE DOCUMENT CAMERA.

Perpendicular Lines (Cont.) The equation for line AB is . . . y = 2x + 1 The equation for line EF is . . . y = (-1/2)x + 3 What do you notice about these two equations (specifically, the slope)? Give the equations for 5 other lines that are perpendicular to line AB. Graph a few. This is a great time to remind students that OPPOSITE is not the same as INVERSE. Opposite numbers sum to 0. Inverse numbers multiply to 1—some may say that inverse numbers are just flipped . . . and that’s OK too. (An inverse process is one that “undoes” the original process . . . put on your hat, take off your hat . . . multiply by 2, divide by 2.) Opposite numbers: 5 and -5 Inverse numbers: 5 and 1/5 The slope of perpendicular lines are opposite inverses. Other perpendicular lines include: y=(-1/2)x y=(-1/2)x+4 y=(-1/2)x+5 y=(-1/2)x-8 etc. As long as the slope is negative one half and the y-intercept is different, the line will be perpendicular to line AB. It may be a good idea to point out that another way to write (-1/2) is divide by negative 2. y=(-1/2)x is equivalent to y= (x/-2)

Symmetry Notice that this graph has a line of symmetry at x = 0. If you were to draw a line at x = 0 (really, there’s already a line there: the y-axis) it would create a mirror image on both sides of the line. Symmetry is an important concept in A&P. The graph of y = -x^2 would look like the graph of y = x^2, except it would open downwards. Where would the line of symmetry be? (Same: x=0)