Number Transformers: Understanding Functions

Overview: We sought to work with students to generate the concept of each input to a function having its own unique output. Prior to our lesson, students had been examining various patterns as they change over time, making connections between iterations. Through this habit of thinking, they were primed and ready to notice commonalities between different inputs and their respective outputs for a certain "number transformer," the function.

On the day of the lesson, students were given an input and its respective output and made conjectures for the many possible functions that could fit for the input and output. We then gave students other inputs to try and then revealed the output for the number transformer, guiding them to figuring out that each function has a unique set of inputs and outputs.

An equity goal that we had was to elevate students' voices by allowing them to share their ideas to the class on the white board. Further, they were able to take ownership of their ideas by putting their names next to them and having the class reference them by the name of the student that shared them. In this lesson, the class was able to build on each other's ideas in order to co-create knowledge that they will use for the rest of the functions unit.

The Lesson

Content Goal: Students will understand how to create their own expression for a set of given inputs and outputs. Then, students will be able to restate another classmate’s thinking to demonstrate understanding and provide evidence of the function.

Structure: Our lesson began with a Number Challenge as Launch to grab students' attention; we told them that there are an infinite number of functions to connect the first input to its output and challenged them to think of as many as possible. The class then explored other inputs and tested if they could use the same function to get the series of inputs and outputs. Then, we moved into an Explore phase to allow students to explore a table with two inputs and respective outputs given, allowing them to find a common function and test out what would happen with different inputs. Finally, we brought students' ideas together in the Discuss phase by having them share their ideas of the function that they chose and getting a chance to build on others' thinking.

Student Impact

Lesson Debrief

Thoughts on the Lesson

Julie: We are excited that we were able to complete the lesson. We were able to see the full picture of the lesson study process. Now that we were able to complete the process we have a solid foundation on what we could do for next trimester. Understanding the logistics of lesson study will allow us to focus on building the equity theme.

Morgan: Being able to experience the full cycle of lesson study allowed us to build a foundation to move forward. We heard a lot of student voices - all three of Julie’s focal students shared! I hope that the lesson communicated to the students that it is ok to come up with different strategies.

Freddy: We were able to see that students were engaged and that Julie has built a positive classroom culture. Every student that I spoke to had something to say about mathematics.

Missed Opportunities

Questioning the why on ideas that students share

Pushing students to justify their thinking

Helping students not to assume that one way is correct

Leaving sufficient time for a rich discussion

Next Steps

1. Keep up the positive classroom culture

2. Practice making cold calling into warm calling

3. Continue with students airing their ideas or a classmates idea

4. Make final connections with how each piece is connected, reflect on entire lesson

5. Add more layers onto the discussion conversations

6. Give students more time to talk intentionally in the classroom