## Desmos: Gr 6 Inequalities

I recently co-taught Andrew Stadel’s 6th grade Desmos Activity Builder on Iequalities with Monica, Joe and Allison. This post is a summary of what happened in the different classes.

The AB was facilitated using the Desmos Classroom Conversation Toolset.  In general, I either activate Teacher Pacing or Pause Class immediately after creating the class code.  If the first slide has key information that I don’t want students to interact with just yet, I’ll use Pause Activity otherwise I’ll use Teacher Pacing.  For this activity, I used Teacher Pacing.

Classroom Conversation Toolset

## Day 1

Slide 1:  Students read the directions.  I then gave them a 10 second countdown before activating Pause Class. I warned them ahead of time to avoid over dramatic reactions.  Using Graph Overlay, I revealed the collective responses. My last step prior to the class discussion was to turn off  the Show Original feature.  By doing this, all the points on the number line correctly represent the given instructions.

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What happens when the overlay looks like this…

Me:  There’s something on the number line that makes me go hmm.. I’d like you to talk with your elbow partner and find the part of the Overlay that concerns me.

I circulated, listened and mentally selected a few students to call on when I called back the class. Once the students revealed my concern that a student graphed the 8, I say …

Me: I don’t want names because it doesn’t matter who graphed this point.  I would like the person who did slide the black dot to the 8 to regraph the point.  I’ll unpause the activity and give the class 5 sections to make changes.

Once the pause button is turned off, many students took this opportunity to change their answer including the student who made the error. I counted down from 5 and then pressed Pause Class. Now that the number line represents correct solutions, the class discussion begins.

Most students mention

• There are a lot of dots
• I see all of our answers
• A lot of kids picked number ___
• No one picked ____

Me:  Why are there so many dots (solutions) on the number line? Talk to your partner.

Students: (summary)

• We were told to pick a number greater than 8
• There are a lot of numbers greater than 8
• There are a lot of students in the room picking numbers

My Pitfall:  At first, I didn’t incorporate the vocabulary word, solution, enough before moving onto slide 2.  This over site became evident when students were working on slide 10.  I adjusted my plan for future classes.

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Slide 2: Using Teacher Pace, I advanced students to slide 2.  To foster mathematical discussion, I apply the following format.

• I gave students 20 seconds to read through the slide.
• Students are instructed to discuss the slide prompt with either and elbow or face partner.
• I like to press, Pause Class, at this point.  If I don’t, some students jump right into typing their response thereby ignoring their partner and skipping the discussion.  To develop mathematical understanding, students need daily opportunities to interact with math both orally and auditory.
• The co-teacher and I circulated, listened to conversations, asked questions etc…
• The Pause Class feature was turned off and students were given time to write and submit their response.
• We circulated and read responses.  I also scanned the teacher dashboard for responses.
• I like to circulate and read over students’ shoulders to provide editing support. Some students need help transferring their mathematical reasoning to written form. I’ll often ask a student who’s stuck to explain their thought.  If it makes sense, I’ll ask the student to type what they shared with me.  If their thought is jumbled, I’ll ask clarifying questions to flesh out a clear response.
• A few students are selected to start and add to the discussion.

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Slide 3:  I used the Think-Pair-Share format again.

• Silently read for 20 seconds.  Everyone is silent, even the teachers.  Some students are easily distracted, so I don’t talk or answer questions.
• Pick a piece of information to share and share it with your partner
• We circulated, listened and selected students to start the conversation and others to keep it going.  I’ll often prep students ahead of time.  I’ll ask them if they’re willing to share with the class what they told me.  If they are nervous, I suggest they practice by repeating their comment to their partner.

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Slides 4 – 6:  These 3 slides are a repeat of slides 1 – 3 using the less than and less than or equal to symbols.

Slide 7:  I used the Think-Pair-Write-Share  format on this slide.  Writing about math isn’t an easy task which is why I asked students to process with a peer prior to typing.

## Day 2

Prior engagements conflicted with the co-teaching of Day 2, therefore I was only able to run Day 2 with Allison’s students.  We started off the day with some A/B partner work.

I wrote an equality sign on the board and asked the A’s to say it to their B partner.  If I hear any discrepancy (half the students saying greater than and half saying less than), we practice one more time as a whole group before moving on.  This process is repeated for all 4 inequality symbols.

To practice vocabulary, I ask students to show me (using their fingers) a solution to x < 5. I scan and jot down what I see.  I’ll ask why I didn’t see the solution of 5 before moving on. This process is repeated for the next inequality:  x is greater than or equal to 1.

Slides 8 & 9:  In my 20+ years of teaching, I was my first time teaching 6th graders about inequalities. I quickly realized that a clear distinction between the purpose of Day 1 and Day 2 needed to be made. During Day 1, students graphed a single point to collectively show a range of responses.  For Day 2, they were learning how to graph ALL possible solutions.

When first responding to slide 2, almost all students moved the circle to a number greater than 10 – like they did the day before.  I took the time to discuss the concept of a boundary number, ask for solutions and mark a few with red arrows.

Next, we discussed which inequality symbol to select.  The distinction came full circle when students mentioned that all 4 suggested solutions were found in the green shaded region.

Slides 10 & 12:  By incorporating the word, solution, from the beginning, students zipped through this slide.

Slides 11, 13  & 14:  For these slides, students practiced graphing 3 more inequality statements.  I picked 1 student per group to be the Table Leader/Captain.

Me:  Groups, each group member is to graph the given inequality.  Captains, when your group agrees on a graph, please raise your hand.

The teacher and I circulated the room and supported collaborative groups.  Here are some statements we said:

•  (To a Captain)  I see that your hand is raised, but not everyone is finished. Please check in with your group.
• I see both red and green shading.  That’s interesting.
• I see 3 different graphs.  Will you all compare your graphs and discuss.
• I still see 2 different graphs at your table.  Please discuss.

When a group is ready to be checked, we start questioning.  I tend to direct my questions to a specific student within the group.   This is to avoid multiple people talking over each other as well as allowing for quieter students to have a voice. Some questions are:

• Explain why the 10 is shaded in?
• Is 10.1 a solution? Why or why not?

Slide 15: I love this slide.  Teachers, please spend a chunk of time on this slide exploring, uncovering and clarifying misconceptions.

Here’s how it flowed for me…

• Me: Repeat after me:  5
• Students: 5
• Me:  Is less than or equal to
• Students:  is less than or equal to
• Me: x
• Students: x
• Me:  Now show me, using your fingers, a solution to the inequality 5 is less than or equal to x

I was planning on jotting down some of their solutions.  When I looked around the room I saw students displaying a variety of answers ranging from 1 to 10. Uh Oh!  This was critical moment.  I could either acknowledge only the correct solutions or I could set the stage for conversation by acknowledging both correct and incorrect.  I chose the latter.

The minute I jotted down 3 and then 3.5, I could sense students questioning my move. YES! I could hear students mumbling comments and see others wildly raising their hands. After soliciting a few more solutions, I stood back from the board and said:

Me:  If someone disagrees with a given solution, please state which one you’re questioning and explain why.

Student 1:  I disagree with 3 because 5 is not less than or equal to 3.

I’d then draw a line through the 3.  This process repeated until all solutions were deemed acceptable by the students.

The Activity Builder is displayed onto a whiteboard, which allows to me to highlighted the correct solutions.  See the recreated picture below.  Once I mark up the number line, I ask students to determine which inequality would shade the section covering the correct solutions.

Even with the above visual displaying, I saw the following:

Typically there was a mix of right and wrong answer at each group, therefore I used my go to statements:

Me:  I see different answers.  Please discuss.  I’ll be back to check in.

At this point, the students do all the work. Group members help each other out.  They explain & discuss. Upon my return, I check for understanding.  After I’ve checked in with many groups, we discuss whole class.

The confusion surrounding this problem was eventually clarified.  The best part:  The students clarified the confusion on their own.  When I stepped back and listened, I heard (paraphrasing):

• You have to shade the bigger numbers because x is bigger than 5.
• 5 is less than or equal to x is the same as x is greater than or equal to 5
• If 5 is smaller than x, then x is bigger than 5

The Exit Ticket:

I walked up to each group and said:

• Me:  5 is less than or equal to x is the same as:

then I pointed to them, and they responded with:

• Group: x is greater than or equal to 5.

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## Closing Thoughts

Students enjoyed the flow of choral response, picking a solution, silent reading, A/B partner share, writing a response, reading the other responses, discussing the difference between less than and less than or equal to, circling the boundary number and determining which side to shade.

I spent 2 days working through Andrew’s Inequality Activity Builder and we reached slide 15.  Inequalities are abstract for 6th graders.  Students needed a lot of processing time. Time to articulate the new vocabulary and notation.  Time to discuss the meaning of an inequality and its solution.  Time to graph inequalities.  And finally time to clarify misconceptions surrounding  x < 4 and 4 > x.