*This week should have been my first official – third unofficial – week back. Instead, I’m starting this school year as I ended the last – walking the picket line. I haven’t been up to blogging since this started. Below is a draft from June. I never got around to finishing it. The ending has a “pack up your personal belongings” feel. I left it as-is; seems fitting that this post should come up short… I mean, 10% of my pay – and my colleague’s – was being deducted at the time. *

Recently, I invited myself to a colleague’s Math 8 class to try out Always, Sometimes, Never. In this formative assessment lesson – originally by Swan & Ridgway, I think – students classify statements as always, sometimes, or never true and explain their reasoning.

Because it’s June, we created a set of statements that spanned topics students encountered throughout the course. Mostly, this involved rephrasing questions from a textbook, *Math Makes Sense 8*,* *as well as from Marian Small’s *More Good Questions*, as statements. That, and stealing from Fawn Nguyen.

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To introduce this activity, I displayed the following statement: **When you add three consecutive numbers, your answer is a multiple of three.**

Pairs of students began crunching numbers. “It works!”

“You’ve shown me it’s true for a few values. Is there a counterexample? What about *negative* numbers? Does it *always* work? How do you know? Convince me.”

Some students noticed that their calculators kept spitting out the middle number, e.g, (17 + 18 + 19)/3 = 18. This observation lead to a proof: take one away from the largest number, which is one more than the middle number, and give it to the smallest number, which is one less than the middle number; each number is now the same as the middle number; there are three of them. For example, 17 + 18 + 19 = (17 + 1) + 18 + (19 – 1) = 18 + 18 + 18 = 3(18).

I avoided explaining my proof: *x* + (*x* + 1) + (*x* + 2) = 3*x* + 3 = 3(*x* + 1). This may have been a missed opportunity to connect the two methods, but I didn’t want to send the message that my algebraic reasoning trumped their approach. “Convince me,” I said. And they did.

To encourage students to consider different types of examples, I displayed a ‘sometimes’ statement: **When you divide a whole number by a fraction, the quotient is greater than the whole number.** Students were quick to pick up on proper vs. improper fractions.

Next, students were given eight mathematical statements. We discussed some of the statements as a whole-class. Some highlights:

**A whole number has an odd number of factors. It is a perfect square.**

I called on a student who categorized the statement as always true because “not all of the factors are doubled.” We challenged *doubled* before she landed on square roots being their own factor pair. For example, 1 & 36, 2 & 18, 3 & 12, 4 & 9 are each counted as factors of 36, but 6 in 6 × 6 is counted only once.

**The price of an item is decreased by 25%. After a couple of weeks, it is increased by 25%. The final price is the same as the original price.**

Like the three consecutive numbers statement above, students began playing with numbers – an original price of $100 being the most popular choice. I anticipated this as well as the conceptual explanation that followed: “The percent of the increase is the same, but it’s of a smaller amount.” I love having students futz around with numbers; it’s so much more empowering than having them “complete the table.”

The number 25 was chosen carefully in hopes that some students might think fractions: (1 + 1/4)(1 − 1/4) = (5/4)(3/4) = 15/16. None did. There’s a connection to algebra here, too: (1 − *x*)(1 + *x*) = 1 − *x*². Again, I didn’t bring these up. Same reason as above.

**One side of a right triangle is 5 cm and another side is 12 cm. The third side is 13 cm.**

All but one pair of students classified this as always true. That somewhat surprised us. More surprising was how this one pair of students came to realize

*BTW, the blog-less Tracy Zager has a crowd-sourced a set of elementary Always, Sometimes, Never statements.*

**Update:** I stand corrected.