Instead, I've been taking more time for almost everything.

I've learned that the extra time is often better spent on the front side of a new concept, rather than after the fact like "Oh no, they didn't understand that so let's just repeat it for a couple more days".

I try to start with what they know, then gently stair-step them into the new thing.

Case and point, our recent venture into solving exponential equations (without logs) like these:

On the first day, I equipped students with white boards and said we were going to work on some puzzles. (Don't you just love the word "puzzle"?). Try to write an expression that equals 8, using a base of 2 with an exponent.

Step 1: We did a whole bunch of those, starting with positive integer exponents and working up to the use of negative integers and even a few fraction exponents. Students needed some reminders here and there, but we kept practicing until they were answering them all with accuracy. Then . . .

Step 2: We continued with more, but now students needed to choose what base number to use. There were some great moments here to capitalize on, like when students tried to use a base of 9 to get 243 but realized it wouldn't work and had to use a 3 instead. Or, when two students each used a different base and both answers were correct. I made a huge deal about these when they happened. Next . . .

Step 3: I gave students TWO goal numbers. They must write an expression for each, using the same base number for both.

And now, without realizing it, they've really learned how to do the hardest part of solving the equations for tomorrow's lesson.

Step 4: (Same as step 3, really) To finish the class period, I grabbed a copy of the practice set I wanted them to do the next day and pulled sets of goal numbers directly off of it.

On day 2, students easily transitioned into solving these.

On day 3, extra practice with Add Em Up.

BONUS! From here I introduced logarithms, and found that students were completely prepared to answer the question "What exponent goes with this base number to make this value?" The extra day spent on the front side of solving exponential equations ended up having a double payoff.

Smart! Working on the front side is a great thing to remember.

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