As always, the exit ticket is important to help all students focus on some of the big ideas of the lesson.
To answer the first question, it is great for students to articulate the fact that roots help us solve equations involving exponents when the base is the unknown. In a later unit, students will solve exponential equations using logarithms when the exponent is unknown, so it is important for them to articulate this idea now.
The second question is a great opportunity for students who have not used equation or who have not used both sets of equations to look at a different method. Both equations would enable a student to find the multiplier. An interesting higher level discussion is to talk about the pros and cons of each method. The first method is faster in that there is only one equation, but it doesn’t give any information about the starting value. The second method takes a little longer because the equations need to be manipulated, but then it is easy to use the original equations to find the starting value.
Hopefully, students are able to say that the function is decreasing if the outputs are getting smaller as the x-values get bigger, but they may need some coaching about how to actually write this sentence. This is a good time to give the sentence frame, “A function is ______________ if as ___________, _____________.” Though it seems very simple, it can help them organize their thoughts articulately.