Piecewise Functions for Trig/Math Analysis

We are about to start functions and transformations in trig/math analysis, so I made some piecewise functions in Desmos to use for the intro.

The above function is named f(x).  Below is what I entered to graph the function.  

Then we can view transformations of f by typing the transformation on an input line to the left of the graph.

You can show or hide each graph by clicking on the colored circle to the left of the equation.  Below is 2f(x).

And f(2x).

Here are the links to the Desmos graphs.  Graphs 3 and 4 include questions for students to answer.   Hope some of you can use them!

Piecewise 1

Piecewise 2

Piecewise 3

Piecewise 4 

All of these functions were taken from Calculus AB free response questions, hence the link back to the website.  I also made a table of values to use for by-hand practice.

Watch this video by Meg Craig for a fantastic demo of how to use the table above.  Scroll down to the embedded video to view.

SBAC Test Resources 2015

Updated links.  Also available on the mvla website under the common core tab, as well as on our common core resources page from 2013-2014.

Smarter Balanced Question Types

Smarter Balanced Practice and Training Tests

Smarter Balanced Practice Test Scoring Guides

SBAC Grade 11 Released Items

SBAC Grade 8 Released Items

Clovis Unified School District SBAC Page.  Organized by claim for easy navigation.

SBAC Calculator

Square Root and Cube Root Graphing: Alg2 Section 6-5

Yep, it's that time again.  We are back to graphing, and we are back to transformations.  The book that we use for Algebra II starts in Chapter 2 with transformations of the absolute value function. Chapter 4 includes quadratics, Chapter 5 is cubics, and now in chapter 6 it is square root and cube root functions.  

Since I won't be able to help out in all of the classes this time around, I am hoping this blogpost can be helpful in getting ready for the lesson (which I taught in Jennifer C's class).  

I began the lesson by looking back at the Desmos pre-made sheet for parabolas in vertex form for a reminder of how a, h, and k transform a graph.

 I decided to take a different approach for the intro to 6-5.  Instead of starting with functions and table of values, I started with a transformation given in words.

I showed the Geogebra applet below.  Directions are to move points A, B, and C so they are first ON the square root graph, then shift each point up from the parent graph by two.  Students could pick which of the points from the parent graph to look at, as long as A, B, and C are two units up from a point on the parent graph of y=sqrt(x).

 Step 1:

Step 2:

Next I typed the y=a*sqrt(x-h)+k into the input bar, and we talked through the transformations that had taken place.  I changed a, h, and k as we decided on values.  For this example a=1, h=0 and k=2.

If you have entered the correct function into the input bar (hit enter when finished), the transformed function will turn green if correct.

Students practiced a few problems with both horizontal and vertical translations, and then problem 5 asked students to reflect y=sqrt(x) about the x-axis.  About 90% of students were able to move points A, B, and C to the correct location, and about 50% correctly guessed that the correct transformed function was y=-sqrt(x).  This was with NO direct instruction for this part of the lesson.  I was hoping for this moment, as one of my goals for this approach was for students to think about how we were transforming a set of points, and specifically to think about what happens to a set of points before we think about the transformed function.  I did call the class back together to finish up problem 5.  I also walked them through problem 14 so that they would get a chance to see a cube root transformation modeled by teacher.

You can access the problem set here, as well as from the email attachment.  There is space to record transformed points and the transformed function.  Answers can be checked using the applet, which is set up to indicate a correct answer as long as your transformed function went through points A, B, and C.  Since students were checking their own answers, I got to walk around and asked questions to check for understanding.  No doubt it is too many problems, and would have been much better on a block day, but it still was a great experience for students.  Overall, I like this approach for introduction to graphing a new type of function in Algebra II, and I can see adapting this lesson for the other function types (with tweeks of course!).

One last note is that it is possible for students to get the wrong "answer" using this applet.  The floating points A, B, and C begin at (1,4), (4,4) and (7,4), so if a student enters the function f(x)=4 the applet will indicate that they are correct.  This has happened a few times since I started using applets like this one, and it's made for great conversation and reinforces the need to check the reasonableness of an answer.  

I Spy II: Class Activity

This post gives a few samples from the follow up activity for the I Spy introduction to congruent triangles activity.  The directions that will be given to students are:

I Spy

Directions: Each diagram has a pair of triangles.  One triangle has been either translated, reflected, or rotated to create the second triangle.  As you view each diagram, see how many of the following objects you can find.

-Angle Bisector

-Midpoint

-Isosceles Triangle

-Parallel Lines (or parallel segments)

-Right Angles

Team Directions:  Each team member should use a different color marker to record responses to the problems.  Switch roles for each problem.

There are 6 problems to total to give the groups.  The first two are easier than those shown below, with only one conclusion (I spy item) to make from the diagram.  The photos below are provided as a preview for this activity.  The full problem set with directions is available here.

I Spy: Introduction to Congruent Triangles

Below is an introduction to congruent triangles that makes use of student prior knowledge of transformations.  It also encourages students to deduce information from a diagram and to justify their reasoning. 

Show students the following diagram, and go through the steps below.  Some questions/prompts are shown in blue.

1.  What type of transformation is shown in this diagram?

2.  Reveal the measures of ABC.  Ask students, “Who thinks they know another measure in the diagram?  Be ready to explain your answer.  Show me ONE other measure that you know, and hold up your whiteboard when you are ready.”  Teacher can show answers from students that used notation correctly.  Reveal the measures of AFE.

3. Show students the statement below.

Ask them how we might fill in the blank, and why.

4.  Move point C to the horizontal line (you can do this in the Geogebra applet).  

Leaving the measurements showing, ask students to tell you what they “spy”.  They can write on their whiteboards, “I spy __________________.”  Answers must be defended.  Make sure to have the list of possible answers written on the whiteboard.  

The choices are angle bisector, midpoint, isosceles triangle, parallel lines, and right angles.

3.  Move point B to F. Move point A down the vertical line so your diagram looks like the one below.

Leaving the measurements showing, ask student to tell you what they “spy”.  They can write on their whiteboards, “I spy __________________.”  Answers must be defended.

I SPY CHOICES:  angle bisector, midpoint, isosceles triangle, parallel lines, and right angles.

4.  Turn the measurements off.  Move point A back up to the horizontal line, and make sure point B is on point F.  Ask student to tell you what they “spy”.  They can write on their whiteboards, “I spy __________________.”  Answers must be defended.

I SPY CHOICES:  angle bisector, midpoint, isosceles triangle, parallel lines, and right angles.

Extension if there is time:  Since we haven’t seen parallel lines yet in this dynamic diagram, ask students how we can modify the diagram to form parallel lines (and explain why).

There is a second page in the Geogebra Book that has a triangle rotated around a vertex by 180 degrees to form the second triangle.  

Part II will have the I Spy Class Activity if you have time for it.

Symmetry Artist

I was observing a teacher today, and one of the lesson resources was Symmetry Artist from Mathisfun.com.  I was so excited by this resource that I could hardly focus on the lesson.  If you've never seen this resource before, it looks like the screenshot below:

 My five year old created the drawing above.  When she was finished I told her to pick a different number of "petals", and she chose 9 (shown below).

What an amazing resource!

Pierce, Rod. "Maths is Fun - Privacy Statement" Math Is Fun. Ed. Rod Pierce. 27 Feb 2012. 20 Oct 2014 <http://www.mathsisfun.com/Privacy.htm>

9-5 Review Activity

LINK TO GEOGEBRATUBE BOOK WITH ALL 5 REVIEW PROBLEMS

Below is the Geogebra applet for problem 1 showing three line segments.  One of the segments is independent and the position of the other two depends on the first.  Which line segment is independent from the others?  You can drag line segments or points in the applet below.

For this diagram the blue segment maps to the orange segment (reflection about the y-axis), and the orange segment maps to the purple segment (translation right 3 and up 2).  We can also think about which points are independent/dependent, and about the mapping between the points.  Point A maps to point F, which is then mapped to point D.  The type of transformation that maps the points is the same as the transformation that maps the segments (reflection then translation).

The objectives for student use of this lesson are to identify the types of transformations for each problem (2 per problem), write the coordinate rule for each transformation, find the image of a point given the pre-image, or find the pre-image of a point given the image.  Since there are two tranformations per problem, vocabulary might get in the way of understanding.  The organizer below can facilitate conversations about the objectives, and can help students organize their thinking.  I would model the thinking and fill out this entire organizer with students before they get started with practice on their own.

Completed Organizer for problem 1:

This activity is still in the draft stage, but my plan is to have students use the exact same organizer for each problem.  I suspect that rotations might be hard for students to see, so I might model problem 2 as well so we can do the first problem with rotations together.  

There is no answer key yet, but the transformations for each problem are:

1.  Reflect about y-axis, then translate 3 units right and 2 units up. Begin with segment AB.

2.  Translate two units left and 5 units down, then rotate 90 degrees counterclockwise about origin.  Begin with segment CF.

3. Rotate 180 degrees counterclockwise about the origin, then translate 7 units left and 3 units up. Begin with segment BD.

4.  Translate 6 units left and 10 units down, then reflect about the x-axis.  Begin with segment AF.

5.  (This is supposed to be problem 6, looks like I missed an upload).  Dilate by a factor of 3 with center at origin, then translate 8 units left.  Begin with segment BF.

LINK TO GEOGEBRATUBE BOOK WITH ALL 5 REVIEW PROBLEMS