Google Docs in Math: Error Analysis

Looking for ways to incorporate more Claim 3 problems into your curriculum?  Error analysis problems provide students with an opportunity to communicate their reasoning and focus on the Math Practice Standards "attend to precision" and "construct viable arguments and critique the reasoning of others."

A screenshot of my first digital error analysis assignment is below.  I made this on a Google Doc by adding pictures of the problems worked out in my own handwriting.  You can make a copy at this link.

There are many ways to get images into a Google Doc.  One way is to open Google Drive on your phone, navigate to the file that you want you image to be in, and then add an image by pressing the plus sign and taking your photo.  From your Google Doc you will locate and insert the image, and crop as needed.

I used Google Classroom to distribute this assignment to students.  We talked through the first problem together.  Then color coded each error according to its type.  I provided the following key at the bottom of the assignment.

This entire process took about 10 minutes.  With just a few minutes to spare before the end of class, students got started on the second problem and submitted their assignment.  Below are some of the results.

You'll notice that we didn't always agree on the type of error represented in the problem.  This was our first try at this type of assignment, and over time we'll continue to discuss the types of errors that we are making and hopefully have more agreement.  The amount of writing and discussion that took place in just 10 minutes was impressive, and I can see the results of this type of assignment improving over time.

Please let Shelley know if you are interested in similar lesson for your own class.

Geogebra/Desmos Functions Resource Page

I came across this resource page that I made for a presentation back in January.  I thought it was going to continue to get filled out, but I had to move on to other projects.  I wanted to share this in case it inspires any summer learning, and I hope the resources can help give an idea of how dynamic graphing software can enhance the student learning experience for functions.  A lot of these links are to pre-made Desmos or Geogebra pages by other authors, and some I created on my own.  I linked to blogposts for a few of the activities as well to give a better idea of how the activity was used in class.



You can also open this document in Google Drive and make your own copy at this link.  I've spent a lot of time over the past couple of years thinking about how to use Geogebra (and now Desmos) to improve student understanding of functions.  Most of the textbooks that I've used in Algebra and Algebra II in our district don't go very far beyond "graph this function" for practice problems in these sections.  Geogebra and Desmos offer us a chance to change the task and the types of questions that we ask when teaching functions.  

For the project above I included links to sliders, flashcards, and transformation activities when I could find/create them.  Students can play with sliders when first being introduced to a new function type to help them attend to the structure of the function with its parameters, and they can build on the background knowledge from previous function types.  Flashcards can be used to review vocabulary, and to check for understanding of the values of the parameters.  More on Geogebra Flashcards here.  Transformation activities or "other" activities mostly consist of Geogebra pages where students will find the equation for a given function.  

Though I haven't used all of these pages, the ones that I have used in Algebra II and Pre-Calc facilitated richer conversations, and overall better understanding of functions and transformations in my own class.

Statistics Modules from Engage NY

Engage New York is a Common Core Curriculum available online for free, and it includes a Statistics Module for Algebra, Algebra 2, and Precalculus.  You can preview the table of contents for each module below, and download for easy use.  Consider using a lesson or two in your statistics unit if you haven't already made plans to teach from another resource.

Need a recommendation?  Want a demo lesson?  Or would you prefer a hard copy of the curriculum? Please let Shelley know how she can help.

NOTE: This blogpost was crashing some browsers. I am going to repost with screenshots of table of contents and links to download modules from the Engage NY page.  Sorry about that!

SBAC Practice Activities: The Basement

This activity is modified from SBAC released performance task called "Home Office", and includes portions of parts A, B, and C.  The flooring options were also changed to allow students to complete the task in a 50 minute class period.

As with other practice performance tasks, students will show work on a Google Doc.  The top of the page includes links to the various pieces of information and tools that the students will need to solve the problem.

In terms of showing work, it has been helpful to remind students to think about each problem as being worth 3 points (1 point for showing work, 1 point for explaining thinking, 1 point for answer with units).  We don't know ahead of time what rubric will be used, so giving students explicit directions on what to show on their Google Doc is helpful.

Once students navigate to the "Introduction and Specifications for Work" page, they will see the diagram below.  The first task is to determine how to split the basement into a home office and workshop.  The budget for this part of the project is $30,000 and the contractor charges $50 per square foot to finish the home office.  The workshop will be unfinished. 

 The first task asks students to finish the maximum area, and to draw a vertical line to separate the home office from the workshop.  The interactive diagram (image below) allows students to use the segment tool to add in the vertical line.

You can also show students how to use the text tool to add labels to the diagram.  When introducing the activity it will be helpful to let students know that the black points are 2 units apart.  

Students struggling to find the various dimensions of the workshop can turn on the grid (see image below), which is set up to scale by 2 units as well.

Students should add a screenshot of their image into the appropriate place on the Google worksheet.  

The second task asks students to pick a flooring for the home office from the table below.  The budget is $1500 for flooring, which is in addition to the $30,000 for the contractor.  Task 2 gives students an opportunity to justify their reasoning.  There are many flooring options to pick from that come in under budget.  Students receive full credit for choosing an option and providing an explanation for how they know that they are staying within the budget.

Task 3 asks students to find the dimensions of a bathroom that takes up 10% of the home office and is 3 feet longer than it is wide.  They are asked to do this by writing and solving a polynomial equation.  I included a link to Desmos so students can use it to solve the polynomial equation (quadratic), which is not factorable.  If this activity is given during a 50 minute class, students will most likely not get to task 3, or it can be completed by our early finishers that need the extra challenge.

The rubric and correct answers for this task are here.  The Google Doc for this assignment is here.

SBAC Claim 4: The Performance Tasks and Classroom Activities

Claim 4 is Modeling and Data Analysis:

This is the part of the test where students will complete a performance task. Students will complete an in-class activity prior to the performance task.  Smarter Balanced has already provided the classroom activities that will be used this year.  You can view the list of activities and the schools assigned to each activity.  Mountain View will be using Rooftruss, and Los Altos will be using Zipline.  More information will be available from administration as we approach the testing window.

 The chart below shows the various content standards that may appear on the performance tasks. We also have many sample performance tasks to look at. Visit this page from Clovis Unified School District to view sample math performance tasks for grade 11.  

SBAC Practice Activities: Regression and Desmos

Links:
Assignment (Google Doc)
SBAC High School Calculator
Desmos

This task is for students in Algebra II or Trig/Math Analysis.  The problems on the Google doc are modified from the textbook for Trig/Math Analysis.  The mathematical goal for this lesson is to find the equation of a linear regression for a set of data, and use the equation to make predictions.  The technology goal was to find needed information and solve problems by navigating between various windows.  Below is one of the problem sets we asked students to work on.

We had students open a separate link to access data so that they could easily copy and paste the data into Desmos in order to find the regression equation.  Below is a snapshot of the data stored in a Google Sheet.

After you copy the data from a spreadsheet, you can go to the first input line in a Desmos sheet and paste the data.  It's as simple as control-C and control-V copy and pasting.  In line 2 you type y1~ax1+b.  Desmos automatically makes the subscripts for you.  Note that we use "~" instead of "=" when finding a regression equation.  See the regression parameters on line 2.

You might be asking why we used Desmos for regression when SBAC has its own regression calculator.  The answer is that Desmos makes regresssion sooooo easy, and it also automatically scales your graph to fit the data (see below).  We didn't want students to be focusing on too many new technology skills for this lesson, so we stuck to Desmos for the regression piece.

Once students find the regression equation, they need to use the equation to make predictions/answer the questions on the worksheet.  For this part we encouraged students to have two browser windows open side by side.  They can also use paper to do some of the work, though final answers will be scored from the Google Doc. Below is a screenshot of what students see when they are working with the SBAC calculator to answer questions 9 and 10.

One of our findings from this activity was that students were not clear on how to show their math work on the computer, or they weren't sure how much work to show.  We have started coaching students on the expectations by adopting a 3-point rubric for each problem.  Students can earn credit on each problem for the following items:

-1 point earned for showing work

-1 point earned for explaining thinking (when the problem specifies)

-1 point for final answer with units.

Below is a sample response for problem 9 that earns full credit.  

SBAC Practice Activities: Patterns and Formulas

This activity will provide students with a chance to review recursive and explicit formulas for arithmetic and geometric sequences.  Students will build formulas to model both patterns and real world situations.  They will check their formulas by using Desmos, and answer questions relating to each situation.  The Google Doc is available at this link.

Problem 1 is below, and asks students to use a recursive formula to identify a pattern from this page.

All patterns are from Fawn Nguyen's Visual Patterns website.  A few of the patterns that students will be looking at are below.  You can see that pattern 7 fits our recursive description for problem 1.

Once students have identified the correct pattern, we send them over to Desmos to add their data to a table.  When we are talking about patterns in math, we describe them as starting with stage 1, and then progressing to stages 2, 3, etc.  In the table below, we are saying that stage 1 has 5 blocks, stage 2 has 7 blocks, and so on.  You can see the each stage has two more blocks than the previous stage.

Students love using Desmos to see their pattern data plotted in real time.  Once students have the points plotted, we discuss how a recursive formula is limited in its usefulness because it doesn't have predictive power.  For example, I can't use a recursive formula to predict how many blocks will be in stage 100.  To do this we need an explicit formula.  Students can input an explicit formula into line 2, and immediately see whether or not they have the correct formula based on whether the line goes through the data points.

After practicing a few more problems with patterns, we move on to real world scenarios.  The scenario below was adapted from a worksheet found on betterlessons.com by Colleen Werner.

As with other SBAC practice activities, teachers coach students on how to answer the questions in order to receive full credit. We never know what the rubric will look like on standardized tests, so we've created three areas of focus:

-1 point for showing work
-1 point for explaining your thinking
-1 point for giving the answer with units

Part III of the lesson is a mini-performance task, modified from the SBAC Practice Test performance task. This is great if you have an extended period of time for the lesson (90 minute block).  Otherwise we found parts I and II to be great practice activities to help students get ready for SBAC.