# Halfway

### Algebra 1 - Last Week

We are about to begin working on the symmetry of parabolas and finding the vertex if we know the x-intercepts. In order to find the axis of symmetry, students must find the x coordinate that is halfway between two other coordinates.

To prepare for this, I start class with the Number Talk "What number is halfway between 44 and -7?" My students were split between 18.5 (the correct answer) and 25.5 (the distance from -7 to 18.5, and from 18.5 to 44).

After several students share their strategies, most are convinced that 18.5 is the answer. But two students, who are among my strongest Number Talks participants, remain unconvinced. One of them, MC, speaks up: "Mr. Haines, I know my answer is wrong but I still can't see why."

### Pre-Algebra - Two Weeks Ago

We are checking homework and students are debating the problem below.

To prompt discussion, I ask one table "How tall was Ilana at age 3?"

"30 inches"

"And how tall was she at age 5?"

There is a pause...

I am discussing rounding with Joe Schwartz. To me, this seems like such an intuitive idea. Is 81,866 closer to 81,000 or 82,000?

Joe, who actually teaches this stuff, is kindly informing me that rounding can be far from intuitive. I ask him what the hardest part is - knowing the correct place value? Finding the upper and lower bounds of 82,000 and 81,000?

Joe says "Finding the middle."

For his students, the hardest part of rounding is figuring out what number is halfway between 81,000 and 82,000.

Telanna (whom everyone needs to follow @TAnnalet) posts a picture from her class

### Bedtime - Last Night

J picked his number book for bedtime, so I decided to see what his thoughts were on finding the middle.

### Me, At Home, Tonight

You know when you buy a new car and suddenly you see silver Camrys everywhere? I am having the exact same experience with the skill of finding the midpoint between two numbers.

I want to be clear: I'm not talking about finding half of a number. I'm talking about finding the midpoint between two numbers, neither of which is 0. And as it turns out, this skill appears everywhere, at all grade levels.

In stats, you might need to find the median of a data set with an even amount of data points. In algebra or geometry, you might need to find the midpoint of a line segment. In 4th grade, you might need to model a problem on a number line that doesn't begin at 0. What should be in the middle of your line?

I love this type of question because there are two distinct strategies that you can use to find it, and both strategies can be useful. If I am finding the number halfway between 234 and 452, I can add the two numbers to get 686 and divide by two to find the mean, which is 343.

But if I want to know the midpoint of 68 and 88, I don't add them and divide. Instead, I find their difference (20) and add half of it to 68 to get 78. Two totally different strategies, each equally valid and useful depending on the problem.

The other reason I'm intrigued by this problem is, I don't know when and where it is taught explicitly. To me, it feels like one of those skills that we assume kids will develop over time, but we never spend a day talking explicitly about these strategies.

I wonder what other skills and concepts in math class are like that...