Lesson Plan Ideas for Mathematics Classes
I. How to write the equation of a line:
A) Define slope
1) Do several examples. Math1 students need more examples. Math 3 and 5 need fewer examples.
2) Draw several lines on the board without a coordinate system. Ask the question “What’s my slope?” Ask “Which line has the greatest slope?”
3) Extend the concept in math 3 and 5 to examples where one of the points is variable and the slope is given. Have students solve. Draw lots of pictures. Make it real.
4) Give examples of lines that are perpendicular or parallel. Talk about a family of lines that share a common slope.
5) Talk about horizontal and vertical lines. Give lots of examples. Show students how they can recognize that a line is horizontal or vertical by quickly noting that the x coordinates or y-coordinates match.
6) Create a quiz that is open ended. You might have students write a paragraph about slope. Ask for at least 5 facts or details. Ask them to include several graphical examples.
B) Draw a line on an x-y plane. Label the y-intercept (0, b) and a general point (x, y). Calculate the slope of the line. Simplify the slope equation. Your final result will be the slope-intercept equation of the line: y = mx + b. Follow the derivation of the form with many specific examples. Again the higher level classes need fewer examples than the lower level classes.
C) Draw a line on an x-y plane. Label two points (x, y) and ( x sub 1 ,y sub1 ). Calculate the slope. Simplify. You now have the point slope equation of the line: y – y sub1 = m (x –xsub1). Again do many specific examples.
D) Given 2 points on a line create a flow chart that describes the steps necessary to write the equation of a line.
1) Do you know “b”? If so, calculate m and you are done.
2) Do the x coordinates match? If so the line is vertical. The equation is x = whatever the matching x-coordinate happens to be.
3) Do the y-coordinates match? If so the line is horizontal. The equation is y = whatever the matching y coordinates happens to be.
4) Finally, if none of the above applies, find m. Then write the equation of the line in point slope form and simplify. Alternatively find “b” using a specific point and slope and write answer in slope intercept or function form. Do lots of examples. Include several real world examples that would be of interest to the students. Depreciation after you have purchased a new car or piece of machinery; costs of renting a car where a fixed cost and a price per mile are given; salaries based upon commission would all be examples.
II. Extension of Ax+By = C into several other types of problems.
A) Students should be able to recognize the slope of a line from the equation. If the equation is in y = mx+b, the slope of the line is the coefficient of x. If the line is in format Ax+By =C the slope is –A/B. Derive this for the class. Then do lots of practice with the game “What’s my slope?”
B) If you have the problem of writing the equation of a line parallel to a given line and through a specific point there are many methods of solution. A method that students enjoy uses the fact that the answer line and the original line have exactly the same A and B. (Remember A and B determine the slope of the line, and the lines are parallel.) Replace the x and y in the original equation with the specific point that your answer line intersects. Find the new C. Write your answer using specific values for A, B, and C and the general x and y. Example: Find the equation of the line that passes through (3, 5) that is parallel to 6x + 7y = 12. To find C: 6(3) + 7(5) = C; 18 + 35 = 53 = C. The answer is 6x + 7y = 53.
C) If your answer line is perpendicular to the original line you can interchange A and B and switch one sign. Remind students that perpendicular lines have negative reciprocal slopes. Find the new C. Write your answer. For example: Find the equation of a line that is perpendicular to 4x +5y = 14 that passes through (6, 9). The new line will be of the form 5x-4y=C. Find C by 5(6) -4(9) = 30 – 36 = -6 = C. The answer is 5x – 4y = -6.
D) In Math 3 and 5 you can talk about distinguishing lines from equations of higher degree. Talk about depreciation. Is it usually linear? Talk about linear regression and other curve fitting techniques.
A) To describe domain one of my favorite examples is to use “My Closet”. Picture entering your closet. It is full of beautiful outfits. One red dress however is too small and can’t be used. Your domain of wearing apparel is everything in the closet but the red dress. Relate this to a rational function where the domain is the set of real numbers except whatever value makes the denominator zero.
Extend the closet example to a closet that is shared by a
husband and wife. Suppose that the clothes of the wife are on the right side of
the closet (call it the non-negative side) and the husband’s clothes are on the
left side (call it the negative side). If you have a radical function with an
even root your domain is the non-negative or wife’s side of the closet. Talk
about domain as the set of inputs that can safely be used in the function
Another example of domain is “The Birthday Party”. Your 5 year-old son has prepared a list of boys that he wants to take to the zoo to celebrate his birthday. Mom peruses the list; there is a problem with Johnny! Poor Johnny is not in Mom’s domain because he is
one holy terror Johnny is not an acceptable input for this particular function (ha ha). Have fun!
B) Describe domain using the example of a child beginning his or her life in his domicile or domain. As the child matures he or she travels out on the range.
C) When I talk about function notation I always emphasize
that f(x) notation is really
f ( ). You are in need of an input to put in the open parentheses. The x is simply a place holder for whatever your input happens to be. This seems to help students deal with problems of the f(x+h) category. You can extend this to evaluating polynomials. If you are to evaluate 3x+5 rewrite the problem as 3( ) +5. You can then replace the spot in the open parentheses with a specific value of x. This also helps students when their inputs are negative.