Engage NY Algebra: A Graphing Story Interactive Tool for Review

For Module 1 Lesson 5.  The tool below is embedded, so you can interact directly from this page.  Full size version available here.



We are going to have our first test on lessons 1-5, so this is a good time to start reviewing and continue to make connections between concepts.  The introductory example of lesson 5 is below:

This context isn't as easy to visualize as an elevation-vs-time graph.  I'm hoping the activity described in this post (below) using the applet above will help students generate their own graphs for example 1.  I also want to continue to incorporate strategies to support English Language Learners since the Engage NY curriculum is heavy on the reading.  For this lesson I want to use Example 1 to incorporate a structured opportunity to write.  When students share their writing with a partner we are also incorporating a structured opportunity for all students to speak.  Below is a list of directions for students, and some description for how I see this lesson playing out in my class.

1.  Check the box for graphing story A. Then move slider t from 0 to 50. Using what you know about graphing stories, write this story on your paper. What could the units represent for the x and y-axis?  (Note: We'll use this Google Doc for our writing)

Students will see the graph below

and they will write in the yellow highlighted space provided on the Google Doc

2.  Turn off Story A and reset the time. Now check the box for graphing story B. Move the slider from 0 to 50, and write the story on your paper (or on Google Doc).

Students will see the graph below and write the story on their paper.

3.  Reset the time and go through the process again with both stories checked and the labels revealed. Write a new story on your paper using the new information. 

Students will see the graph below.

Students will have time to reinterpret their story within a new context, and then share their interpretation with a partner.  

4.  How does this story compare with your original stories?

The last part of the writing activity will be to help students reflect on the difference between their assumptions of what the information represented and the true story.  

A powerful consequence of using an interactive visual is that you can continue to use the tool to review old information and connect this understanding as you learn new information.  When reviewing for the test I can see pulling up this applet and asking a different set of questions.  (When is Earl moving away from his door?  What does it mean when Earl's graph has a positive slope?  What are the coordinates of the point of intersection when Maya and Earl first meet? etc.)

Note: This lesson is part of one of the summer work projects at my school, and includes making a few interactive lessons for our Algebra class.  This is a new skill for myself, and I'd appreciate feedback if you have experience or thoughts in the area of interactive lessons.  My district is piloting the Engage NY curriculum for the first time next year.  We are also in the Google Domain and will have access to Google Classroom for the first time, in a one-to-one environment.  We'll primarily use Google Docs and other Google Apps for Education as we learn how to meaningfully incorporate technology into our math classes. 

Useful Links

Engage NY Algebra Module 1 Lesson 5

Engage NY Algebra Lesson 3 Interactives

Engage NY Algebra Notes on Scaffolding Instruction for English Language Learners

Engage NY Algebra Math Curriculum

Engage NY Algebra: Module 1 Lessson 3 interactives

Below is a modified version of Engage NY Algebra Module 1 Lesson 3 (modifications in blue, see teacher version here).  I am adding links to Desmos or Geogebra in order to make the lesson more interactive.  I am still learning the level of technology that would make these lessons better than paper and pencil, so feel free to chime in on the comments.   As you read through the lesson it is helpful to know that Engage NY treats their lessons like a choiceboard rather than a ready to go lesson.  The intent is not that every problem is finished by the end of the day, but rather that teachers choose the problem set strategically to best teach the content to students.   For this lesson my best guess is that we could go through example 1 and the exploratory challenge during class, and then the homework can come from the problem set.  



For example 1 students might use the embedded Geogebra link to add points of interest to a graph and use the pen tool to connect the points of interest.  I'm not convinced this is better than paper and pencil for this particular problem, but at the same time I might use this example as an opportunity to teach students these two Geogebra skills that we will use throughout our course.  If we want to compare sketches I can teach students to take a screen shot and embed on a given slide in this Google Presentation.  This will be day 3 of students looking at elevation versus time graphs, so introducing additional tech skills might be appropriate with students feeling comfortable with content.  The screenshot below shows the pen tool, as well as the input area where students would add points.

Given the time it takes to teach students to use new technology, I am more likely to spend class time on the exploratory challenge problems (1-3).  For these problems I want students to do all of the graphing on Desmos.  My question is whether I am changing the task too much by moving to Desmos.  In the paper and pencil version of problem 1, students watch this video (I couldn't load on Chrome, but Safari worked), scale the y-axis, and then plot by hand.  They will notice that at 0 seconds, there are 2 bacteria, at one second there are 4 bacteria, at two seconds there are 4 bacteria (etc.).  I want students to go to this prescaled Desmos graph and plot the points in pre-loaded table of values on a pre-scaled graph.

For problem 1 I am more concerned with students' ability to interpret the situation and extend the pattern.  If this is my goal, then asking students to choose the y-axis scaling and draw their own table of values seems like a lot of extra steps.  They can generate the graph below in about 30 seconds if starting with the blank graph above.  

Problem 2 is an extension of problem 1, with students noting that 20 minutes of real time are covered by 1 second in the video (this info is not included in the student version of text, so you have to refer to the teacher version for how to lead the lesson).  They could go back to their Desmos graph and show understanding of this fact by changing the times in the x-column of the table of values.  Then we can walk through how to change the scale of the x-axis so the data points are visible.  Adding additional points necessitates a rescaling of the y-axis if you want to see all of the points.  We can rescale by clicking on the wrench icon towards the top right and changing the minimum and maximum values for each axis.  

Using Desmos to explore the scale of a graph is helpful because the graph updates its window real time, so you can immediately see the effect that an x-y min/max change has on the graph.  Students may lose out on a bit of the reasoning for how to label the tickmarks, but in the end I believe the more valuable skill is determining the domain and range values and interpreting them in the context of your problem.  

What do you think of these digital additions to the lesson?  Are they too much for students?  Does this change the task in a way that takes away from the mathematical goal? (Students choose and interpret the scale on a graph to appropriately represent an exponential function. Students plot points representing the number of bacteria over time, given that bacteria grow by a constant factor over evenly spaced time intervals.)

Download or make a copy of the Google Doc here.  Original source is here.  Engage NY is a great resource even if you aren't using it as your primary curriculum, and you can modify this document to use in your class.

This lesson is part of one of the summer work projects at my school, and includes making a few interactive lessons for our Algebra class.  We are piloting the Engage NY curriculum for the first time next year.  We are also in the Google Domain and will have access to Google Classroom for the first time, in a one-to-one environment.  We'll primarily use Google Docs and other Google Apps for Education as we learn how to meaningfully incorporate technology into our math classes. 

Engage NY: Notes on Scaffolding Instruction for English Language Learners

Link to document

Sample lessons (exemplars) found in this document.  
Proof of Laws of Exponents is an 8th grade lesson, page 45 (module 3 lesson 6)
The Power of Exponential Growth is an Algebra I lesson page 63 (module 3 lesson 5)

Strategies:

Teach Academic Vocabulary
-use engaging informational texts
-choose a small subset of vocab for in-depth study
-practice by writing, speaking, and listening
-student friendly definitions
-teacher should introduce by pronouncing clearly and students can respond chorally
-note when the word has another meaning in a different context, or is a homophone (some and sum)
-identify part of speech (a perfect square, to square a number, a square tile)
-meaningful way for students to record the word, such as a graphic organizer or glossary
-Multiple opportunities to review the words, such as flashcards or foldables
-Structured opportunities to use the words across the 4 modalities (reading, writing, speaking and listening)
-word banks or word walls organized by concept.  Teacher refers to wall either before or during use of the words.

Strategies with Details:

Integrate Oral and Written Language Instruction into Content Area Teaching
-strategically use instructional tools to anchor the instruction (videos, visuals, graphic organizers)
-explicitly teach the concept specific vocab (see above) during instruction
-daily opportunities for students to talk about content in pairs and small groups
-provide writing opportunities to extend learning and understanding
-use structured approaches to Socratic discussion

Concrete and Visual Models includes manipulatives, illustrations or other (think Desmos/Geogebra) ways for students to have a more hands-on experience of the content.

Graphic Organizers and Foldables
-graphic organizers include concept maps
-can be provided partially completed, or students can construct from scratch
-these support ELLs because they help display complex text succinctly and graphically

Multimedia can enhance instructions, especially if subtitles are available in students' first language.

Structured Opportunities to Speak With a Partner or Small Group
-pair with more proficient English speakers
-skills to use include elaborate and justify, support ideas with examples, paraphrase, and synthesize conversation points.
-sentence frames

Provide Regular, Structured Opportunities to Write
-Writing assignments should be anchored in content
-Can include language based support such as graphic organizers and sentence frames and starters

Build Background knowledge
-will be helpful before starting a performance task (like SBAC in class activity before Math performance task). Make sure all students are on the same page with respect to the situation before starting the task.
-Do no rely on the affirmative to gauge student understanding of context

Clarify Content Delivered in a Second Language
-reword text by using present tense, shorter sentences, graphics, color, etc.

Teacher Modeling and Explanation
-model and explain the thought process that will be included in the lesson activity.  Clearly explain each task and model an expected student response.

Cueing
-Include a clear focus at the beginning of each lesson by stating standards, objectives, and agenda for the day in student friendly language

Capitalize on Student's Home Language Skills and Knowledge
-glossaries and side-by-side texts that include home language translations (Note: Google Docs has an easy translate feature that created a new copy of the doc in the chosen language.  Can easy put these side by side in separate windows)
-pair ELL student with bilingual partner so discussions can occur in home language and in English

From the Math 8 Exponents lesson
Note:  This lesson is about proving the exponent laws.  Some of the highlights include
-Explicitly introduce new vocabulary.  Give students a chance to review relevant vocab from previous lessons.  Can use a foldable organizer.
-Keep important information statements visible throughout the lesson for reference.  From past experience, this is an extremely valuable strategy and helps student retrieval.  Think anchor charts.  
-The HW for the night before can include a reading to help students prepare for the lesson and a Socrative discussion.  They will take write down notes, quotes, questions, and a summary.  Can write these in the margins of the text.
-Give students a partially completed statements and reasons table for first proof
-Keep the initial example done with concrete numbers visible so students can relate the symbolic argument to the the work just completed (I didn't actually see an example in there done with concrete numbers.  Might be in the original lesson)

-From pages 58-59

The Power of Exponential Growth is an Algebra I lesson, page 63 (module 3 lesson 5)
-Cueing: Introduce objectives, student outcomes, and key vocabulary for the lesson.  Write these on the board so they can be referenced later.  During the initial conversation, give students time to make meaning and have partner talk.
-Use a matchbook style foldable to record vocabulary information
-rewrite task prompt for introduction example so it is present tense, shorter sentences, and the first few rows of the table of values pre-filled with question marks for the entries that students should fill out on their own.

After example 1, AIR suggest the following structured opportunity to speak with a partner or small group

Example 2
-use multimedia to enhance comprehension (video How Paper Folding Can Get You To the Moon.  Watch it, very interesting)
- Reworking the wording of the problem (without sacrificing rigor or content) is essential.
-Provide a blank table of values for the exit ticket problem (exponential growth problem)

Homework
-AIR gives us the HW assignment for this lesson rewritten with the same rigor but scaffolded language load. See page 80.  Embed relevant images and have students create tables of values to help them solve problems.  For original HW, see page 6.

AIR suggests teachers work in groups to go over the lessons and consider how to apply the strategies to any given lesson within the Engage NY curriculum.  My district is piloting the Engage NY curriculum for Algebra 1.