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Thursday, June 24, 2010

Absolute Value Both Rigorous and in Context

I know I said I was done for the year. SORRY. I am literally sitting around school twiddling my thumbs today. I am ripping this idea off from Dan, but trying to extend it to be appropriate for Algebra 2. Absolute value is one of the first lessons of the year, and in the past my students neither understand it conceptually nor remember an algorithm for solving equations and inequalities with anything like reliability. This feels more like an Algebra 1 lesson to me, but I think it will be necessary.

This is my version... peanut M&Ms were the cheapest/most voluminous things I could find. There are about 230 in a large bag, by the way. Yesterday I polled 50 faculty and staff. In the fall I am going to have to get my butt into overdrive within a day or two to collect at least as many data from students.

I have yet to nail down the details, but the flow will go something like this:

Put up a picture like this.

Ask how far away the houses are from school. Get a few volunteers to describe the mental procedure they used to determine distance from school. Point out that everyone naturally used a difference and absolute value to express distance. And that further, if we can represent distance as absolute value with an equation, we will be able to use it to ask and answer more interesting and difficult problems than our intuition can handle alone. Graph by hand y = |x| by making a table of values. Note the characteristic V shape.

Questions to Answer
Bust out laptops and distribute excel file. As per Dan's original plan, kids will have some choices about what questions to explore and time to flail.

- Who won?
- Rank everybody.
- Top 10 Guessers.
- Any ties?
- Worst guesser?
- Which grade guessed best?
- Which job guessed best?
- Calculate percent error.
(Maybe some/all kids can present aspects of the results on posters we can display?)

Once that's all squared away, I want everyone to explore:
- On average, how good were the guesses?
- Create the scatterplot that displays the characteristic V shape.
- What is the equation of the connected graph of that plot and what do the variables represent?

(This popping up on my screen should not have been, but was, the best part of my day yesterday:)

Follow-on problems once equation is achieved. Solutions using both the graph and the equation.

- What guess corresponds to the average distance from the correct guess?
- What did the worst guesser guess? The best?
- In what range did the better-than-average guessers guess?
- In what range did the worse-than-average guessers guess?

New problems and generalization:
Write an equation/inequality that models the scenario. Make sure to define your variables.
- Today’s temperature is 10 degrees off from the usual temperature.
- Today’s temperature will be within 10 degrees of the usual temperature.
- Today’s temperature will be more than 10 degrees off from the usual temperature.
- If the usual temperature is 68, find values for the three forecasts above using algebra. Show all work at every step.
- Graph the scenario. Indicate the three different forecasts on the graph.

- Write a general expression for the distance between a changing value and a known value. Define your variables.

- Put this equation into words: |x – 10| = 3
- Solve it, showing all work at every step.
- Write down/discuss a procedure for solving any absolute value equation.

- Put this inequality into words: |x – 10| < 3
- Solve it, showing all work at every step.
- Put this inequality into words: |x – 10| > 3
- Solve it, showing all work at every step.
- Write down/discuss a procedure for solving any absolute value inequality.

Feel free to poke holes in this or let me know how you would implement it differently. Also I need to get them solving and graphing more complicated equations and inequalities like say 10 = 2 |3x - 4| + 7, so I'd love to hear if you see any natural ways to make that happen. I haven't been able to think of any yet.