In the last few years I worked with Pre-AP students. We were encouraged to emphasize critical thinking, developing students' understanding and use of abstract notation ... in preparing students for AP Calculus.
Here are a few questions that we used in various units:
Create equations for parabolas that fit these criteria:
a. Vertex in the first quadrant, concave down, and 2 roots with an axis of symmetry at x = 2.
b. Vertex in the second quadrant, concave up, and no roots.
c. Vertex on the y-axis, 2 roots, and concave up.
Explain how you determined each of your equations.
Find the value of b that will make the function pictured have only 1 zero at -6.
Find the value of k that makes the linear expression, x - 2, a factor of the cubic expression pictured. Explain your thinking.
We used multiple representations (word descriptions, tables, equations, and graphs) in comparison activities. I believe this one is from a textbook we used:
Consider the function, f(x) = log base b of x where b and x are real numbers. Can the domain include 0? Explain you thinking.
There are more questions/activities.
Parent functions are a huge topic in Algebra 2. In this post I describe some ways we try to encourage critical thinking as well as just memorizing the functions.
I describe some other problems I considered rich tasks in this blog post. I share a problem for several units in Algebra 2.
In addition, I used critical thinking structures in class. Here are a few examples:
All of the above questions/activities are noted in my blog somewhere :)
I do have a few additional activities for sale on TpT that address critical thinking in a few units of instruction.
How do you get students to analyze, compare, contrast, and problem solve?