Math Don't Break

Integer operations are always an interesting endeavor with 7th grade students because they come pre-loaded with so many rules.  So. Many. Rules.

We've been talking about making our own rules, so we have this sequence of products and I ask students to discuss what patterns they notice.

-3 (3) = -9

-3 (2) =  -6

-3 (1) = -3

-3 (0) =  0

-3 (-1) = ??

Stuff we noticed:

"It starts with a -3 every time."

"It goes down by 1."

"It changes by 3."

I zero in to the apparent contradiction in going down by 1 and changing by 3 so we can clean up the language a bit.  This starts an nice little exchange about whether or not going from -9 to -6 is an increase or decrease.  We conclude it's actually an increase.  I have to remember to take my time here because this isn't an insignificant point:  Kids seem to think in absolute value.  

So what comes next? 

I wrote down everything I heard.  

"3".  "-3".  "4".  "-4".  

"Wow!"  I say.  "We've got a great argument about to happen.  This is awesome!  So many different opinions.  So which is it?"

Some minds change when groups start to discuss.  The students who thought 4 or -4 were thinking of sums and not products.  That leaves 3 or -3.  

"Ok, so which is it?"

If I had a dollar for every time a student said "A negative times a negative is a positive" followed by "because my teacher told me", I'd have all the dollars.  

But then Isaac offers a reason worth looking at. 

"I think it's -3, because positive 3 times positive 1 is positive 3, so negative 3 times negative 1 is negative 3."

So I write the following on the board:

(pos) (pos) = pos

(neg) (neg) = neg

We talk about this pattern Isaac. has noticed.  "Does this work for you all?"

Jordan speaks up, "I don't think so.  It has to be positive three so that it doesn't break the pattern."

"Which pattern is that?"

"The pattern goes from -9 to -6 to -3 to 0.  It's increasing by 3 each time so the next answer has to be 3."

"Why would that be so?" I ask. 

Then Vanessa chimes in.

 "Because math don't break."  

More Fraction Multiplication

My 7th grade students are in the middle of exploring fractions.  They are currently researching the question:

What are fractions and how can I add, subtract, multiply and divide them?

I'm not sure how much instruction I want to give here. Is this intuitive? What can I do to make this more usable?

 Applet with more instruction is here.

Update
John Golden:

@dcox21 @danielpearcy worth letting students adjust estimate afterfirst fraction is shown?
— John Golden (@mathhombre) November 8, 2012

Good idea.  Updated version.

Ticket Prices

The other night, Dawson and I were doing some math together and we ran across a problem that asked to interpret a scatterplot for the average movie ticket prices for the past 10 years. We poked around online until we found this image:

http://www.the-numbers.com/market/

We used the information to come up with an average price and dropped it into Excel and came up with a best fit line and all. Personally, I find Excel clunky, but since we could use it to quickly calculate average price, we went with it. But it just so happens that I'm doing linear relationships with my 7th graders, so I get a two-fer with this one.

Step 1: Calculate average ticket prices and created a scatter plot.

This went pretty quickly, but one group mistook the raw number for the average ticket price. For example: from 2003 to 2004 the revenue went down, but so did the number of tickets sold. The raw numbers led this group to believe that the prices went down. After a quick conversation, they realized that if the number of tickets sold decreases as well, the price can actually increase.

Step 2: Predict the price of a ticket in 2020 and justify answer.

Most groups used the trend of the graph to predict, but one group actually calculated the average rate of change from year to year and came up with $.24/year. They used this to extrapolate a price in 2020.

Step 3: Decide what type of curve best fits the data.

Two camps on this one: Those who thought it was linear and those who thought it would be like "the graph we got when we did compound interest." Nice work kids.

Note: We will explore that exponential thingy but we ran out of time today.

Step 4: What line would best fit the data?

They were creating a group graph so they took a meter stick and just plopped it down where they thought the line should go.

Step 5: Estimate the equation of that line.

We played with this applet yesterday, so students had a pretty good idea how to use the graph to predict what the equation should look like. I was pleasantly surprised at the fact that all groups decided the rate of change was somewhere between $.22 and $.25 and all said the initial condition was $4.34.

Step 6: Let GeoGebra work her magic.


Add the ordered pairs and use the "Best Fit Line" tool to calculate the regression.

Step 7: How good was our guess?

Proper-tays

I don't usually enjoy teaching properties because they seem so math-y.  I like asking my kids to justify what they do, but for many, the properties just seem to be vocabulary that is forced upon them.  Necessary evil, I guess?  They are great for doing mental math tricks and kids use them without thinking of them, so I suppose there is no harm in giving a name to the stuff they already do. 

Raise your hand if your kids mix up associative and commutative properties? 

No more.

The Process

1.  Give example of property with respect to addition.

ex: Associative: (2 + 3) + 4 = 2 + (3 + 4)

2.  Ask students to write another example of their own. 

3.  Ask for a rule using a, b and c.

4.  Ask students to write down the key characteristics of the associative property in Tweet form. (very few words)

Now here's the kicker:

5.  Can you guess what the property for multiplication is going to look like?

This worked great for the associative, commutative and identity properties.  A great discussion on the inverse property ensued and I ended up telling them that we want a multiplication problem that equals 1. 

Done.


None of these properties are worth anything if we don't apply them. Next step is getting them to put words to all that stuff they "just do in my head."


6.

Let's synthesize this a bit more. 

7. Now write a similar problem using the multiplicative properties. 

I told the class to keep an eye out for times when we will use the inverse and identity properties--which will happen daily once we start solving equations.