This was an Algebra II class and the second half of the period was going to consist of an assessment on parent functions, transformations, and domain and range. I walked in during the first half and the teacher was reviewing a few homework problems. The problems had been done up on the board by students.

3. Use multiple representations of mathematical entities.

Leinwand encourages frequent use of the number line. The students had graphed and our teacher was meticulous about the accuracy of their graphing. She touched on scale, the need to put labels when a point was graphed so that we could understand where that point was in relation to the other parts of the graph.

Leinwand encourages frequent use for students to draw or show and then describe what is drawn or shown. As she went through each problem she dialogued with the student about what the graph looked like, why a point was here, why it curved like that. Again, the students had drawn the graphs.

Leinwand encourages teachers to use pictorial representations to help students visualize the mathematics they are learning. Our teacher used her arm to represent this:

Of course she had the kids stand up and do it themselves. She didn't stop there. She offered two other way for the kids to quickly remember the shape of this graph, one of them being, "Think of an eyebrow." I bet the kids remembered the shape of this graph on the assessment!

4. Create language-rich classroom routines.

Leinwand emphasizes that students and teacher explanations should make frequent and precise use of mathematics terms, vocabulary, and notation. As our teacher reviewed the domain and range of the parent functions she reiterated that

*the kids chose*what notation to use. She made sure to dialogue with the students about the names of the notation (set-builder notation versus interval notation). "What does a bracket represent versus a parentheses? What is another method we could use to represent this range?"
5. Take every available opportunity to support the
development of number sense.

This was SO COOL! Leinwand calls for frequent discussion and modeling about how to use number sense to "outsmart" the problem. A student had graphed the function:

This is simply a linear graph shifted square root of 2 down on the y-axis. Square root of 2 is approximately 1.4 (which our teacher allowed the student to use as the estimation, another small shift Leinwand encourages). The student had used a table to create values for her graph:

This created a bit of a messy graph, though not inaccurate. Our teacher emphasized that students could choose ANY values they want for x. Our student chose those five values for x. Our teacher then asked the class what would happen if we chose these two values:

Which of course leads us to this graph:

Math problem, consider yourself outsmarted.

She used several of the other small shifts throughout the time I was in the room. And this was only homework review/assessment prep! This was teaching at its finest and I am so proud of our teacher for taking these small shifts and intentionally putting them into her daily practice. I know the kids are benefitting because of it.