The standard curriculum has students analyse arithmetic and geometric sequences. It usually has students establish recursive formulas and explicit formulas for each of these sequences.

Now that that remote teaching actually offers a lot more thinking and puzzling time, perhaps introduce some non-traditional powerful thinking first: some experiences on BELIEVING IN PATTERNS and NOT BELIEVING IN PATTERNS and what you can do in each case. 

It could give powerful context to the traditional expected work and actually help make it all “click.”

STEP 1: WHAT TO DO IF YOU BELIEVE IN PATTERNS: Finding Formulas for Sequences

Have students watch the video here and try as many of the practice problems as they feel like exploring in the text that follows it. Zero pressure.

STEP 2: BUT CAN WE TRUST PATTERNS?

Discuss the DOTS ON A CIRCLE puzzle at the very end of the text in the STEP 1 link. One just cannot trust patterns.

STEP 3: WHAT TO DO IF YOU DON’T BELIEVE IN PATTERNS.

Have students watch the video, and read the text perhaps too, found here.

They can also play with their Personal Polynomial at the site here (and watch the videos there). So much to discuss!

Okay … now time for …

STEP 4: DO THE CURRICULUM 

Arithmetic sequences are now straightforward in light of the work we did in Step 1.

Geometric sequences are now curious as they fail the methods of step 1 and they seem to be a different type of formula from step 3. They could now be seen as objects worthy of study: they have a simple structure but defy the all the work of steps 1, 2, and 3.

Comment: If students are curious about the other material they see connected to the highlighted lessons here (which are about identifying quadratic curves) by all means let them explore and discuss!  

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