1) Introducing a new function? Start with a picture or video! Ask ... Where's the math? Students generate many, varied thoughts and questions about the math in the picture. A few of my favorites include the blob jump (quadratics), ferris wheel (trig functions), tire tracks on road (square root functions), bouncing ball (exponential functions), and Grand Canyon Skywalk cantilevered walkway (rational functions). Record students' questions and use them to get into the math of the day. Other sources of videos include the 3act resources!
2) Midunit, use a vocabulary activity to get students thinking. Give each student in class a word or a phrase; ask them to mingle to form sentences that represent mathematical fact. Here are some phrases to use (I'm sure there are more  this is just a quick brainstormed list!). Notice, the list has descriptive phrases  they can be applied to a wide variety of functions. Write them on half sheets of paper to distribute. Students who don't receive one of these phrases writes in the needed words that finish the sentences. Give students a graph or don't  let them generate the graph that would illustrate their sentence. Use often during the year to see proficiency increase with the phrases. Use their sentences to segue into the lesson.
3) Take the numbers out of the word problems that you plan to use. Replace them with a variable or even a box, circle, etc. Ask students to show how to solve the problem without the numbers! This emphasizes process over solution.
... Original Problem: The radius of each wheel of a car is 15 inches. If the wheels are turning at the rate of 3 revolutions per second, how fast is the car moving? Express your answer in inches and in miles per hour.
... Updated without numbers: The radius of each wheel of a car is x inches. If the wheels are turning at the rate of y revolutions per second, how fast is the car moving? Express your answer in inches and in miles per hour.
... Students would explain ...
2 * pi * x
(y) (2 * pi * x) = inches per second
(60) (y) (2 * pi * x) = inches per minute
(60) (60) (y) (2 * pi * x) = inches per hour
[(60) (60) (y) (2 * pi * x)]/ [(5280*12)] = miles per hour
4) Post a random number generator (or use dice) to create unique problems for practicing graphing and transformations! Give students a KEY ... see below ...
Stretch or Compress
1: V Stretch by factor of 2
2: V Stretch by factor of 3
3: V Compress by factor 1/2
4: Reflect over xaxis
5: H Stretch by factor of 1/2
6: H Compress by factor of 2

Horizontal Translation
1: Horizontal left 1 unit
2: Horizontal right 1 unit
3: Horizontal left 3 units
4: Horizontal right 3 units
5: Horizontal left 5 units
6: Horizontal right 5 units

Vertical Translation
1: Vertical up 1 unit
2: Vertical down 1 unit
3: Vertical up 2 units
4: Vertical down 2 units
5: Vertical up 3 units
6: Vertical down 3 units

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