##
* *Reflection: Checks for Understanding
Transformation of Functions Day 1 - Section 4: Closure

I really liked how this lesson allowed the students to create their own meaning around the structure of a function and the resulting transformation. Students learned this content very deeply because they were able to use the calculator to experiment, make predictions and make generalizations. All of these different facets of the lesson led to a much deeper level of engagement and deeper understanding.

On the exit ticket I noticed that most students were able to match up the graphs with the equations. During the investigation, many students commented on the fact that the horizontal shifts seemed to be "backwards" from what they would expect (subtracting a constant moved the graph to the right and vice-versa). Some of these same students made an error on the ticket out by using the same logic for a vertical shift. This is something that should be addressed in a future lesson through further investigation of transformations.

*Closure Reflection*

*Checks for Understanding: Closure Reflection*

# Transformation of Functions Day 1

Lesson 14 of 18

## Objective: SWBAT determine a connection between the structure of a functions equation and the appearance of the graph of the function.

## Big Idea: Students will use structure and patterns to determine the "rules" for translating functions.

*40 minutes*

#### Opening

*3 min*

This opening allows students to connect the topic of transformation of functions to a concept that they have already learned in previous grades. Once you project transformations_open1, have students do a Quick Write regarding what happened to move from the first image to the second image. Tell them to be as specific as possible when describing what happened. Once students have a minute to jot down their thoughts have them turn-and-talk to describe the transformation and give a rationale for their description (MP3).

At this point I have a couple of students share out how they knew that the triangle was shifted 7 units to the right. Most students will say that they tracked one of the points (either a, b, or c) to find out how much it moved. Really focus on this idea because when the students investigate the movement of parabolas they will also be focusing on one point, the vertex of the parabola.

*expand content*

#### Launch

*7 min*

In the transformations_launch presentation, students will first see a slide which structures the process of describing linear transformations of functions. In the subsequent slides, students are given an original parabola (in blue) and asked to describe the transformation to the new parabola in red.

I have students do a Think-Pair-Share as they look at the first transformation. As students share with their partners, I listen for those who are discussing how far the vertex moved. This tells me that they are using the important idea of following one point on the graph

**Note: **Unless students have studied parabolas they may not use the word vertex. Many students say "bottom point". You can introduce the term during the share out so that students can use it in their explanations during the investigation

We go through the second transformation a little more quickly as students will now be catching on to the idea. Point out to students that the parabola is not getting more narrow (many students will think this). Show them that if you shifted the red parabola back down it would match up with the blue parabola.

*expand content*

#### Investigation

*25 min*

The transformations_investigation walks students through transformations of a parabola in all four directions on the coordinate plane. Let students work through the investigation with a partner using a calculator to graph each function.

**Note: **I had students use a TI-84 graphing calculator. Each student graphed y=x^2 in the y1 line and I had them bold this function. This can be done by keying over to the far left of the equation and then hitting enter one time. Here is a link to perform this function if you are not sure how.

Ensure that students are describing both the direction and number of units for each transformation. Students should notice relatively quickly that the constant that is being added either inside or outside of the parenthesis will be the amount of movement. The next challenge is to determine a pattern for making the graph move up, down, left and right. Students will need to look at the structure of each equation to make these generalizations (MP7).

*expand content*

#### Closure

*5 min*

For the closing activity students should write down all five functions on a half-sheet. Once they write down all five functions, put up the next slide and have the students match each function with the appropriate graph. After they match the graphs I also have the students write what the transformation was for each function (example: the graph shifted 8 units to the left). I want to continue to have them describe the transformation to deepen the connection between the structure of the equation and the resulting transformation.

#### Resources

*expand content*

Hi Haley! Great to hear these lessons are having an impact in your classroom. Answers to both of your questions:

1) The graphics are made in Geogebra. Awesome program that I believe has a free version (that's all I use). For some graphics in other lessons I also use Desmos online calculator. For student use on investigations, Desmos is second to none in my opinion. The ability to easily set up sliders for various parameters of the function really makes the investigation dynamic!

2) I differentiate for the most part through scaffolding. I take a lot of time before the lesson determining what misconceptions students may have and determine what scaffolds students may need to help them resolve those misconceptions. The scaffolds serve to build students up to the content. That said, another way to differentiate is to choose tasks that have a high "ceiling". This would mean that the task offers students the opportunity to extend their thinking and make new and interesting observations. In other words, the task is more "open" rather than "closed". Students do more than just find an answer.

Let me give you an example. In this lesson, the content is scaffolded for all students because I knew most of my students would not bring much prior knowledge on transformations to the task. However, more sophisticated students will be able to make more cohesive observations about how changing various parameters affect each function. If you look at a lesson like "The Tower Task," I knew that some students would be able to really take the pattern and run with it while others will have trouble getting started. The scaffolds in that lesson help students to list out the first few patterns in an organized way. I would also say that the "Tower Task" has a relatively high ceiling and so students can really make some interesting observations about the pattern which will eventually lead them to both explicit and recursive formulas.

In the end, differentiation is all about access. The key to differentiation is to allow each of your students to access the content at their level in a way that will make sense.

I hope this helps!!

| 2 years ago | Reply

Also, what do you use to create your graphics? For example, the parabolas in the two different colors and line types.

| 2 years ago | Reply

I love this lesson and pretty much every lesson I have seen of yours. One question I have is how do you differentiate in your lessons?

| 2 years ago | Reply*expand comments*

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- LESSON 1: PRE-ALGEBRA: Evaluating Expressions
- LESSON 2: Defining Functions Recursively
- LESSON 3: Tower Task: Exploring Explicit Formulas
- LESSON 4: Function Notation
- LESSON 5: Understanding Domain and Range
- LESSON 6: Multiple Representation of Functions
- LESSON 7: Piecewise and Step Functions
- LESSON 8: Mirror Task: Understanding Equivalent Functions
- LESSON 9: Modeling with Functions
- LESSON 10: Functions Practice and Assessment
- LESSON 11: Introduction to Piecewise Functions: Dance-a-Thon Question
- LESSON 12: More with Piecewise Functions
- LESSON 13: Evaluating Functions Day 2
- LESSON 14: Transformation of Functions Day 1
- LESSON 15: Transformation of Functions Day 2
- LESSON 16: Transformations "How To" Guide
- LESSON 17: Functions Review Assignment
- LESSON 18: Functions Unit Assessment