Transformations 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

Translations with Geogebra

Unit planning for transformations is almost complete.  We have a set of notes/practice worksheets from the county office of ed (really great!), an FAL (Transforming 2d figures), and a common test.  The next step is thinking about how to use Geogebra to help with demonstrating the concepts and with practice.

There are a few different ways you can use Geogebra to look at translations.  The first way is to use a vector to translate a point or a figure.  The diagram below has a point and a polygon, so my next step will be to add a translation vector anywhere on the screen.  It doesn't matter where you put the vector.  You can find the vector tool under the dropdown menu for lines.

 For this example I'll use a vector to translate my point/figure 3 units right and 2 units down.  You can see the vector below, at the origin.

It might make more sense for students to see the vector starting at the point that will be translated, but then you need to construct a new vector each time if you want to stay consistent.  If I put the vector away from my diagram, I can use it each time I want to translate an object, and explain to students that the translate tool works by selecting the object first, then the vector that describes its translation (see pic below, located in a dropdown menu).

The diagram below shows my point and my polygon after translation.  Notice how the program automatically names the points in the image using the prime notation.

If you want students to focus on using the coordinate rule (x,y) -> (x+3,y-2), then you can do the transformations without using a vector by making use of the coordinates of your pre-image point(s). Before you give this a try, it is best to open a new window and add point A.  The way Geogebra names the coordinates of point A is (x(A),y(A)).  To translate point A 3 units right and 2 units down, create a point with coordinates (x(A)+3,y(A)-2).  Create this point by typing into the input bar at the bottom of the screen.  Better yet, give this new point the name A'.

Select the "move" tool, then drag point A around and watch how point A' moves.  Another helpful strategy is to turn the trace feature on for both points.  Do this by right clicking on the point and selecting "trace on".

Make sure the "move" tool is selected, then drag point A and watch as point A' follows along and traces out a figure that is congruent but translated 3 units right and 2 units down.  To clear the traces from the screen, type ZoomIn[1] into the input bar at the bottom of the screen.  (This command zooms in your screen, making it 1 times as large as it was before.  Clever trick to get the traces off the screen).

Geometers Sketchpad has lots of presentations on transformations that can be viewed on the Dynamic Number Project website.  My understanding is that students can access these presentations with the Sketchpad Viewer on an iPad.  I haven't thought too much yet about how I can use these in my class.  We don't have Sketchpad, and we also don't have ipads for each student.  For now I am trying some of their ideas on Geogebra, though I have to admit the experience isn't as smooth.  One strategy that Sketchpad uses is to attach a point to the perimeter of an object.  Geogebra has a similar tool, which is an option in the dropdown menu under polygons.  Make sure you have the object and a point created first.  Then when you select the "attach/detach point" tool select the point first, then the object to attach it too.  I found out the hard way to select the interior of the object.  By selecting the perimeter of a polygon, the point was confined to the line segment that created that side only.  

Once I had point A attached to the polygon below, I dragged point A to the edge and around the perimeter.  This created a congruent the congruent shape in orange traced out by point A'.

More to come on transformations with Geogebra as I prepare for a Thursday meeting.

Links you may find useful:

Geogebra Tools from Making Math Visual blog.
Geogebra Wiki.  Includes tutorials on how to get started with Geogebra.
Some Things I Wish I Knew When I Started Using Geogebra.
More practice with Transformations

#mtbos challenge week 6: SBAC Performance Tasks

Sherrie's list of links for this blogpost is amazing, and I can only imagine the lists we are going to end up with when all of the #mtboschallenge participants share over the course of the next week.

I don't feel like I have tons of new info to contribute here, so I'm just going to add one website that has come in handy during the past year.

Clovis Unified School District SBAC-Math Page

I live in California, and we are one of the 22 states that will be taking the Smarter Balanced Assessment Consortium (SBAC) test for the first time in the Spring of 2015.  This will be the first standardized test for California that is completely online, and it is an adaptive test.  This means that the level of difficulty is not predetermined, but rather adjusted as you get answers correct or incorrect during the testing session.  Part II of the SBAC test will be a performance task, which will take at least one hour to complete and counts for a big chunk of the test score.

Smarter Balanced has released plenty of sample test items, but I love Clovis Unified School District's website because it allows you to search for SBAC released test item by claim.  As far as I can tell, claim 4 (modeling and data analysis) refers to performance tasks. There are 3 performance tasks released for grade 8, and 4 performance tasks released for grade 11.  You can see a few more performance tasks through the Smarter Balanced website by viewing the sample items, the practice test, or the training test.  I don't usually go to the Smarter Balanced  website because it takes longer to navigate to the problems that I want to see, and they are not provided in pdf version.  However, if you want to see how students will interact with these problems then the Smarter Balanced website is the place to go.

Click below to take a look at a couple of these performance tasks:
High School Grade 11
Middle School Grade 8

#mtboschallenge week 5: Trashketball and Ricochet Robots

I was so busy this week I thought for sure that I would have to skip the weekly blogging, but here I find myself with a bit of free time and a fun prompt to write about (math games and activities).

While cleaning out my inbox today, I ran across an old email from a colleague sending me to The Exponential Curve blog for rules on how to play the review game trashketball.  I was pretty excited about this find, especially since I also have been following The Exponential Curve blog by Dan Wekselgreene.

Many of the teachers in my math department have used a modified version of Dan's trashketball to help review for tests.  One big difference for our game is that we print the problems and cut them apart, and give one problem at a time to each group.  Our procedure is below.

1. Hand out problem 1 to each group.

2.  All students show work on paper, and come to consensus on answer

3.  Students take turns bringing me their work, along with the piece of paper that the problem is on (quarter sheet, so not too wasteful).  If the answer is correct and work is shown, I will let them keep the quarter sheet to use later when we take shots into the recycle bin.  If the answer is incorrect I keep the quarter sheet.  In either case, they pick up problem 2 and bring back to the group.  I can also ask for an explanation before they move on, but I'll admit this can cause problems if other groups are waiting for answers to be checked.  I keep the quarter sheets organized in a flipbook envelope.

4.  When all groups are finished with the problem set (8 or so problems), we crumple the quarter sheets into balls and take shots into a recycle bin to determine the winning group.  

A couple of notes:
- I agree with Dan that you can only do the same review game so many times.  I'd stick to 2 or 3 times for the year.
-If groups finish early there is a review assignment to work on.
-After reading Dan's trashketball post (with a 2014 lens), I would love to implement his version with one student per group up at the board at the same time showing the work.  Students will have had time to work on the problem with the group, and then the chosen group member can bring up the work with them and rewrite on the board.  They take the shots and keep score at the end of each round.  

Have you checked out Nick Romero's math basketball review game?  Mini-whiteboards AND basketball!  Sounds like a lot of fun.  
I've also purchased lots of games over the years to use in my math classes.  My two most recent purchases have also turned out to be my favorites.  They are Set and Ricochet Robots.

#mtboschallenge week 4: 3-2-1 Summary

My week was hectic, so a 3-2-1 Summary will be the more enjoyable writing prompt for me this week.

3 things I got in the mail this week

-Powerful Problem Solving by Max Ray

-Cooperative Learning and Geometry; High School Activities by Becky Bride.  I’ve been hearing about this book for awhile (usually called Kagan Geometry) and so far it has lived up to the hype.  There are great problem sets along with participation structures you can use for each problem set so that there are no "hogs" or "logs" during groupwork activities.

-Mentoring Matters by Laura Lipton, Bruce Wellman and Carrlette Humbard.

Now I just need to find time to read these books.  Making Thinking Visible is also on its way.

2 things that happened this week

-I taught two demo lessons.  The first was a Calculus lesson on limits.  We started this lesson by exploring gravity on different planets, and ended by discussing how to use limits to find the velocity of a falling object on a given planet.   The short video below shows the effect of gravity when you throw a ball on different planets.  This Geogebra presentation is by Michael Richardson (mrrichardson username on geogebratube.org), and was originally designed for an Algebra I class.  I liked it so much I've kept it in mind when planning for other classes. .



The second demo lesson was an Algebra II lesson on transformations of absolute value graphs.  This is year 2 for me as an instructional support teacher, and it is encouraging to have quite a few invites to do demo lessons so early in the year.  Great to be back in the classroom with kids too.

-Final inspection from the city for the two bathrooms that have been remodeled in my house.  When my husband and I bought our 50 year old house two years ago, we knew that there was extensive water and termite damage due to leaks, so we started saving and planning right away.  The first day of summer our front bathroom was demolished, and nearly three months later we are finally finished!  What a job!!!

1 thing I am looking forward to

-Trying out some of Nat Bantings Whiteboard activities.  Hoping to try some of the quadratic functions whiteboard questions in Algebra II this week.

Don't forget to add your Sunday Summary to the linky at @druinok's blog, Teaching Statistics.

Dinner Table Math

My friend recently shared with me his method for multiplying two numbers.  He talked me through the example below, which is for 321 x 123.  We had a good time figuring out why this method works, and I think others might enjoy this too.

It might not make as much sense if you don't experience it, but I'll list my friend's process below.

1.  First he drew the three sets of lines with negative slope.  You can see that there are three lines for the hundreds place, two lines above for the tens place, and one line on top for the ones place.

2.  Next he drew three sets of lines with positive slope.  The first line started at the bottom left and intersected all lines with negative slope.  This line was for the hundreds place in the number 123. Then in the middle there are two lines with positive slope, representing the tens place.  Finally there are 3 lines with positive slope representing the ones place.  The order/orientation of the lines matters.

3.  Last, he started counting intersections, and wrote the number of intersections at the bottom.  You can see that some sets of intersections are in the same column as other sets of intersections.  Count these sets together.

4.  If there are more than 10 intersections per set, you carry the tens digit to the column on the left, and add to the existing number.  You can see in the picture that there are 14 intersections for the middle column, so he changed this number to 4 and made the number to the left (8) increase by one (to 9).  So 321 x 123 = 39483.

I tried one more problem for fun.  You can scroll down to see the diagram and solution for 384*231.

After I figured out what was going on with this method, I decided to start from the right.

Method 1: with carrying                                                          Method 2: without carrying

ones: 4                                                                                        4  ones

tens: 0 and carry the 2                                                           + 200 (for twenty tens)

hundreds: 7 and carry the 3                                                   +3500 (for 35 hundreds)       

thousands: 8 and carry the 2                                                +25000(for 25 thousands)

ten thousands: 8                                                                  +60000(for 6 ten thousands)

Answer: 88704