Transformations of Graphs

Description of the Lesson

Overview

Students will use technology to discover how changes to a function affect its graph.

ISTE Standard

1c, 9-12 Grade Level Expectation:  Students use technology to seek feedback that informs and improves their practice and to demonstrate their learning in a variety of ways.

NE Standard

MA12.2.1.a(AT) Analyze and graph non-linear functions (e.g., quadratic, trigonometric, square root, logarithmic, rational, higher-order polynomials, exponential, absolute value, piecewise, and sinusoidal).

MA12.2.2.d(AT) Create new functions out of existing functions using addition, subtraction, multiplication, division, translation, dilation, and composition.

Rubric Used for Assessment

Teacher Instructions:  Rubric for grading graphs on quiz and test:

2 - The student response:

*Shows a mathematical understanding of the problem and offers a correct solution

1 - The student response:

*offers a correct solution with no supporting evidence or explanation

*offers a partially correct answer to the problem

*may contain flaws indicating an incomplete understanding of the task or concept

may demonstrate a poor understanding of relevant mathematical procedure or concepts

0 - The student response:

*gives an incorrect response with no work shown

*offers no mathematical understanding of the problem

Example Student Artifact(s)

NA

Lesson Design Reflection

Step by Step Procedures:

HOOK/ATTENTION GETTER:  Sample ACT question (Link)

Teacher Instructions:  Project ACT question on board or share in google docs and give students a couple of minutes to work.

Student Instructions:  Answer the question. Work alone for a minute or so, then compare your answer with one or two other people.  Justify your answer. We will come back to this question later and go through the answer.

INDEPENDENT AND GUIDED REVIEW:  Students complete Parent Function worksheet. (Link)

Teacher Instructions:  Hand out worksheet. This should be review from Algebra.  Give students several minutes to complete this. If they don’t remember some of them, remind them that they can make an x|y chart and plot some points.  After students have compared answers with a partner come back together as a group and summarize.

Student Instructions:  Complete the worksheet by yourself, without technology.  Once complete, find a partner to compare and justify answers.  When you are finished we will come back together as a group to summarize.

GUIDED PRACTICE:  Students complete transformation worksheet using Desmos and/or GeoGebra.  (Link)

Teacher Instructions:  We are on a 90-minute block schedule.   I split this into a two-day lesson - vertical/horizontal shifts and vertical/horizontal shrinks and stretches the first day (the first four pages); reflections and compositions the second day (the last two pages).  If you are on a traditional schedule you may need three or four days to complete the lesson or cut down the number of transformations per page.

For the most part, the students are working independently, however I did encourage them to ask a partner if they weren’t sure how to summarize what they discovered.  When students finished the first two pages, we revisited the ACT question. They should be able to see by now that the correct choice is “b” as the parent function should be moved right 2 and up 3.

I did need to give them a little more guidance on the vertical and horizontal stretches and shrinks.  Most of the students just wrote “it gets narrower” or “it gets wider”. I nudged them along by asking them to put the numerical constant into the rule to be a little more specific, for example “the y-value is being multiplied by 2.”

Students had very little trouble using GeoGebra and Desmos.  I had to show a couple of students where to find the cube root function, but for the most part they needed very little guidance with the technology.

Student Instructions:  Complete all but the last page of the transformation worksheet using Desmos and/or GeoGebra.  After we discuss the “rules” of transformations as a group when everyone is finished, I will have you go back and complete the last page, without technology.  Compare those graphs with a partner and/or use Desmos or GeoGebra to check.

INDEPENDENT PRACTICE:   I assigned several problems from the textbook.  Graphing to be done by hand on graph paper using the transformation “rules” they discovered from the guided practice.

Teacher Instructions:  Assign the most relevant problems from your textbook.  The first day I assigned only problems over what we covered that day (vertical/horizontal shifts and vertical/horizontal stretches and shrinks).  The second day I assigned problems over what we covered that day (reflections and compositions).

Student Instructions:  Follow the directions in the book.  Graph the functions on graph paper using transformation “rules” - no technology.

Assessment:  1) I had the students hand in their homework problems from the textbook.  I made corrections on their papers, but I didn’t put a grade in the grade book for this as I usually treat homework as practice.  When I handed it back the next day I asked if they had any questions and showed a couple on the board. If they needed more independent help they were to come to me either during our Access and/or study hall time.

2)  A day or two after the assignment I gave a quiz over the most recent 2-3 lessons. - no technology

*the transformation question is #7 on the 1.4, 1.5, 1.6 QUIZ in the scanned exemplary work examples

3)  Students were assessed again at the end of the chapter on the chapter test. - no technology

*the test question is #9 on the last page of the scanned exemplary work examples

Personal Reflection

 I don’t allow technology for the homework, quiz, or test as the purpose of the lesson is to use the technology to speed up the graphing process and to compare the parent function with the transformed function to develop the “rule”.   Once students recall the parent functions and know the transformation rules, graphing by hand should be almost as quick as using technology to graph the function. Also, technology won’t help on a question in the form of the Lesson Hook similar to what would be on the ACT or on my test question #9.