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.

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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.

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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…

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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.

Me:  Talk with your face partner about what you see.

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.capture

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.

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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. Capture.PNG

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.

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

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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?
  • What made the group decide to shade to the right?
  • 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.

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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.

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

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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.

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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.

 

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Teacher Moves – Soft Skills

 A long term substitute, who’s currently in a credentialing program, recently made this observation.
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“In my program, I’m shown all these activities and my professors assume students want to participate.  They (professors) don’t tell us what to do when students don’t want to engage.”
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One of the more challenging responsibilities a teacher has is creating a classroom culture where students want to participate and feel safe participating.  To attain this climate, teachers can commit to teaching their students soft skills. Some soft skills
are (but not limited to) screen-shot-2017-01-13-at-8-36-40-pmhelping students:

  • Communicate effectively
  • Manage time
  • Make decisions
  • Self-motivate
  • Work as a team member/group dynamics
  • Problem solve
Teaching soft skills is a daily commitment.  Some teacher moves required are overt and others subtle.  Consistency is key.
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Attending to Soft Skills

I was assisting an 8th grade teacher implement cooperative groups.  For cooperative groups to be successful, commitment to soft skills is necessary. The posed question was:   What can we do to help kids get along?
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Background: I had spent 2 days with this teacher. Day 1 focused on introducing the concept of cooperative groups.  Day 2 started off with a team building activity, followed by a content activity.  On the second day, we were both busy circulating the room working with groups.  Although we touched base regularly, we were too focused on students to observe the other in action.  In response to the question, What can we do to help kids get along?, I compiled a list of concrete examples from the second day.  All names have been changed.
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Angela’s group:  They were the first group to finish listing their 5 commonalities.  I mentioned this and applauded their cooperative behavior.  Discussed how productive they were when they were cooperating instead of antagonizing each other. They took pride in their accomplishment.  Soft skill:  Working together as a group
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Bill (and Carlos’ group).  The girl who sits across from him always answers in a condescending tone.  Bill mentioned this when I told him to ask his group for help instead of asking me.  At that point, I spoke with the group about Bill’s (and as it turns out Carlos’) hesitation with asking for help.  The girl apologized on her own.  I thanked her for apologizing.  I hope she has more patience with her group in the future.  Soft skills: Working together as a group & communication skills
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Diana kept asking me for help.  I sat with her group and discussed how they have to work together.  I said, “Diana has a great question” I looked at Diana and said,”Will you repeat it to the group?” Diana repeated the question.  I then stepped in and said to the team, “I’d like you to help your group member. I’ll be back to check in.”  When I checked in, Diana was progressing.  I thanked the entire group.  Soft skills:  Working together as a group & communication skills
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Everett/Frankie/Gabe:  I tactfully called Everett out on his behavior. He made the decision to participate appropriately.  I praised him and his group every time I saw them be productive. Soft skills:  Working together as a group & making decisions

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Henry/Isaac/Jared:  Isaac can be fragile at times.  If he gets irritated, then he wants to be left alone. Henry wants to ask if he’s ok, which irritates Isaac more.  Henry doesn’t back off. Isaac gets angrier.  We talked about giving Isaac space and what that means.  We discussed that it would be best not to talk to Isaac until he feels better, therefore he should work with Jared for now.  I thanked Henry for being flexible and working with Jared.   Soft skills:  Working together as a group & communication skills

Keith:  Sat with Keith’s group multiple times.  Always highlighted positive behavior and addressed distracting behavior. If we are consistently highlighting positive behavior and calmly addressing negative behavior, I believe Keith will turn his behavior around.  Soft skills:  Making decisions & working together as a group

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Laura:  I noticed that a group member graphed the line for her, so I asked Laura to describe how it was graphed. She couldn’t. I gave the group instructions to help her and told them I’d be returning to assess their progress.  I had to return three times because the group was off task.  Laura was able to explain how the line was graphed the third time I returned. I thanked the group for completing their assignment.  Soft skills:  Working together as a group & making decisions.

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These examples describe covert ways of teaching soft skills (in my opinion).  Although I was direct when speaking to students, only the targeted students received the message. Each message was personalized to address the presented concern.
Facilitating academic skills and nurturing soft skills simultaneously is not easy at first. One must consciously work to make the practice a habit.  Like most practices, some days are easier than others.
When tired or irritable, I’ve learned to identify and acknowledge positive student actions. My go to sentence starter is, “Thank you for …”
  • Thank you for getting out your materials
  • Thank you for helping your group member
  • Thank you for being patient
  • Thank you for being productive
  • Thank you for being flexible
  • Thank you for sharing your thinking.
When it comes to giving thanks, NO action is too small. This gesture turns my mood around, works wonders with students and reminds me that attention to soft skills maintains the cohesive nature of my classroom.
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A No Fail New Year’s Resolution

My To Do list often looks like this:

  • Laundry
  • Clean closets
  • Finish garden project
  • Fix clogged sink
  • Breathe

At the end of the day, if I didn’t tend to the laundry, closets, garden or sink, I can still cross off, breathe and feel good about it. For 2017, I wanted a to make a no fail resolution. One that is guaranteed to be successful – Like adding, breathe, to a To Do list.

After some thought, I decided my  2017 New Year’s Resolution is to make mistakes. The more I let this decision sink in, the more I likto-doed it.  Here’s my updated To Do list:

  • Laundry
  • Clean closets
  • Finish garden project
  • Fix clogged sink
  • Breathe
  • Make a mistake

At the end of each day, I can cross off 2 items. I’ll be twice as productive!

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On a serious note:  

I’m not afraid of mistakes.  Mistakes have aided in my personal and professional growth and I’m grateful for them. For years, I’ve been activity promoting Carol Dweck’s & Jo Boaler’s work with growth mindset.  First at home with my children, then in the classroom and recently as a middle level math TOA/coach.

Although I joke about being twice as productive, there’s some truth to the statement. If I acknowledge the learning connected to the mistake, then each error in judgement, each incorrect move, each trial was not in vain.  They’re experiences to be valued for they’ll help me make better decisions in the future.

As 2016 draws to a close and the new year begins, I wish everyone a happy 2017 filled with joy, love and mistakes!

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Desmos: Percent Challenges

Inspiration

These days, I’ve been a bit obsessed with the double number line.  I love using it to teach proportional reasoning – especially percents.  Why? A double number line…

  • Allows for flexible thinking.
  • Expands on a single idea to present a bigger picture.
  • Compares percentages
  • Builds number sense

Ex.  75% of $36 is $27.

screen-shot-2016-12-03-at-6-17-27-pm

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Also,

  • A proportion is a section of a double number line.
  • The double number line shows students where the percent proportion originates..
  • The picture on the right shows 4 different proportions that could answer the question, What is 5% of $36?
  • All 4 proportions were pulled from the double number line.

 

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The Desmos Activity Builder

Given my current focus on percents, I wanted to create percent challenges where students

  • Answer more than one question within the given situation.
  • Compare the relationships between percents.
  • Visualize the problem as a whole before working on the separate parts.
  • Can use a variety of strategies to solve the problem (including a double number line if desired)

This Desmos Activity Builder holds my first set of challenges.  Percent – Bar Modeling

The Activity Builder is broken into 3 parts.

  • Part 1:  Learning to use the Percent Bars
  • Part 2:  The Challenges
  • Part 3:  Card Sort

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Part 1:  The Percent Bars

The purpose, of the percent bars, is for students to visually model a given situation. To accommodate varied approaches, the bars stretch to a desired length as well as move about freely. Students acclimate to the percent bars during the first 3 slides.

Slide 1:  First, I ask students to address the answer.  Most students create the this:

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I then ask them to change the look of the arrangement.  The criteria:  The bars must still represent the same percentages, but they’re connected horizontally instead of stacked.

Once the criteria has been set, I start circulating the room.  The most common arrangement I see is pictured below.  This misconception is the reason I spend a chunk of time teaching students how to use the percent bars.  When I come across a student with this arrangement, I’ll start asking questionsCapture.PNG

  1. What is the percentage of the blue bar?        Student:  50%
  2. Will you drag the blue band under the red?  Student:  Ok

As the student moves the blue bar, I’ll ask more questions.
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  • What do you notice?
  • What is the percentage of this blue bar?
  • How can you fix it?
  • What percentage should the green bar be?
  • Is it 40% in your picture?  Prove it.

 

Here are a few successful combinations

In a whole class format, I’ll highlight 1 or 2 solutions and ask students to prove the percentages for each bar.

Ex:    The red bar starts at 0% and ends at 10%.  The green bar starts at the 10% and stops at the 50%.  50% – 10% =40%.  The blue bar goes from 50% to 100%.  100% = 5-% = 50%

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Slide 2:  I asked students to stretch the percent bars to match the given criteria BUT their arrangement had to be different than their shoulder partner’s arrangement.

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Slide 3:  Students assign their own percent values to the bars.

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Part 2:  The Challenges

Slide 4:  We start Challenge 1 together. It’s the only challenge worked on as a whole group. Before students are let loose to work at their own pace or with a partner, I’ll point out 4 features.

Feature 1:  Setting up Visual Model  

I learned from experience that many students skim the percentages and start randomly placing marbles.  To avoid this behavior and to promote strategic thinking, I required students to first read the directions and set up the percent bars accordingly. To build capacity for flexible thinking, I’ll highlight a few different arrangements.

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Feature 2:  Turning on the marbles 

Now, that the students have created a visual model of capturethe percentages,  I’ll show them how to turn the marbles on and  off.

 

 

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Feature 3:  Moving Marbles

If you click on the center of the black point (marble), you can drag it into position.

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Feature 4:  Checking Your Solution

I’ll enter in incorrect values for the red, blue and green marble and explain that if the green check doesn’t pop up, then they must rethink their solution.  At this point, students are free to work on their own or with a partner.  I do not go over the correct solution to Challenge 1 with the class.  I’ve noticed many students choose to start on their own, but start collaborating with their group members quickly.

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Example of correct solution:

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Part 3:  Card Sort

The final slide is a card sort.  There should be 3 stacks of 5 cards.

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Teacher Moves while Students are Working

Teacher Move 1: Ask questions

Challenge 1 sample questions:

  • Does 50% mean a quarter or a half?
  • What is the relationship between the red and green percentages?
  • If the red and green marbles both represent 25%, should they have to have the same number of marbles?

Challenge 2 sample questions:

  • What do you know so far?
  • How do the green and red percentages compare?
  • If you had 2 red marbles, how many green would you have?
  • Can you have an odd number of green marbles?
  • How many blue bars(10%) fit inside the red bar(30%)?
  • If you had 1 blue marble, how many red would you have?
  • I noticed you changed your thinking.  What inspired your change?

Challenge 3  sample questions:

  • Tell me what you know?
  • Which bar represents Anne?
  • Would you drag Anne’s money value into her bar?
  • What do you notice?
  • How did you know that?
  • What percentage amount is 1/4?
  • Who has more money?  Paul or Anne?

Challenge 4 sample questions:

  • How do Lilly’s and Chloe’s percentages compare?
  • Who has twice as much money as Chloe?
  • Tell me something about Lilly and Andrew?
  • Did you set up all the percentages?

 

Teacher Move 2: Facilitate Collaboration

  • Sara, Jeremy has the same question you just had.  Will share with him what you learned?
  • (after a few minutes)  Jeremy, what did Sara share with you?
  • Miguel and Aaron, I see you two are stuck on the same challenge.  Will you share with each other what you’ve done so far?
  • Group,  Crystal has a question.  Crystal, will you ask the group your question?  I’ll be back to hear about your discussion.
  • (If 2 students are working together) Alyssa, Pablo just made a good observation.  Will you repeat what he said?

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Desmos: Ordering Fractions on a Number Line

A couple of months ago, my team and I ran a Desmos workshop for elementary teachers.   Two activities were highlighted.  The first was an individually paced challenge AB (activity builder).  The second activity shared can be run as a whole class discussion.  Both activities were tested in a 6th grade math intervention class.

  1. Self paced challenge:  What Fraction Am I?
  2. Whole Class Activity: Ordering Fractions on a Number Line.

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Lesson Guide for Ordering Fractions on a Number Line

Slide 1:  This is a partner activity. One person uses the fraction model on slide 2 to compare fractions.  The other student estimates the placement of the given fractions on slide 3.  The students switch roles for each challenge.

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Slide 2:  Comparing Fractions.  Click and drag on the blue point in the lower left hand side to move the fraction.

Screen Shot 2016-10-29 at 2.22.12 PM.png                       Screen Shot 2016-10-29 at 2.22.02 PM.png

 

Slide 3:  Ordering Fractions

Screen Shot 2016-10-29 at 2.32.55 PM.pngOnce you give students time to work together and decide on where to place their fractions,

  • Pause the activity
    • You don’t want students to change their answer once they see the class overlay
  • Click on the Overlay
    • The Overlay reveals the collective answers of the class, therefore highlighting misconceptions and areas of need.
    • Discuss the results as needed.

Screen Shot 2016-10-29 at 2.40.50 PM.png                      Screen Shot 2016-10-29 at 2.51.10 PM.png

  • Promote self reflection by un-pausing the activity and giving students a brief period of time to adjust their answers.  Then pause, show the overlay again and ask if anyone would like to share if they moved a fraction and why.

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Revealing the Answer

Slides 3 – 10:  There are 7 Number Lines altogether.  Each number line has a code to reveal the correct placement of the given fractions.  For Number Line 1, the code is n = 1.  Instruct students to enter n = 1 in row 1 on the left hand side.  The correct placement of the fractions shows up under the number line in red.

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You could also instruct student to enter the code while the overlay is displayed.

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The Reveal Codes

The 7 reveal codes are listed below.  They are located within the teacher tips for each slide.

Screen Shot 2016-10-29 at 3.30.12 PM.png

  • Slide 1:  n = 1
  • Slide 2:  n = 2
  • Slide 3:  n = 3
  • Slide 4:  n = 4
  • Slide 5:  m = 1
  • Slide 6:  m = 2
  • Slide 7:  m = 3

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Checking Student Progress Using The Teacher Dashboard

The Teacher Dashboard holds a lot of valuable information beyond the class overlay.  In the picture below, the students are listed in alphabetical order.  You can click on any of these thumbnails to see a student’s individual work.

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A couple of class overlays:

Slide 5:

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Slide 6:

Screen Shot 2016-10-29 at 3.29.06 PM.png

 

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Sneak Peak into What Fraction Am I?

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*************************************************************************Other fraction related posts:

 

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3-Act : Dried Mango

Percent of change falls within the 7th grade standard domain of Ratios  & Proportional Reasoning. Knowing that students often struggle with this topic, I wanted to create a 3-act to aid in their understanding. I initially wanted to show the percent of decrease when making beef jerky. Due to aesthetics and cost, I dropped the beef jerky  idea and moved forward using the more cost effective mango.

Act 1Dried Mango – Act 1

Act 2:

Before

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Length of mango piecescreen-shot-2016-09-18-at-9-20-19-am

 

 

 

 

 

 

Width of mango piece

 

 

 

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Height of mango piece                            

 

After

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Length of mango piece

 

 

 

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Width of mango piece

 

 

 

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Height of mango piece

 

 

 

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Act 3: The Reveal

Once finished with the first 2 acts, I sat looking at my computer wondering about how to organize The Reveal.  Desmos to the rescue.  I created a Desmos double number line to track the percent of decrease.

0-0       12-5y

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Multiple Representations

screen-shot-2016-09-18-at-12-11-13-pm

In anticipation of

  • student responses
  • exposing students to various strategies
  • and preparing to lead a whole class discussion by sequencing and  connecting differing strategies

my colleague, Mari, created the following summary.

 

Click here to see my other 3-act lessons.

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#ObserveMe: A Professional Development Opportunity

Screen Shot 2016-09-04 at 12.12.56 PM

With 2 weeks of wonderful experiences to share, I sat down at my computer ready to blog.  However, my mind kept jumping from story to story. I was unable to pick one experience to highlight.  I decided, instead, to discuss the reason why I have so many experiences to share.  The inspiration for this post stems from my journey as a TOA/Coach and Robert Kaplinsky’s Call to Action, #ObserveMe.

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My Journey (Abridged)

2 years ago, I stepped out the classroom to be a middle level math TOA/coach.   The 3 other middle level math coaches and I were sent to a variety of  trainings focusing on 3-Acts (Dan Meyer), Number Talks (Sherry Parrish & Ruth Parker), CMC-South and Mindsets (Jo Boaler).  That first year, our teachers were pulled out of classes for 5 district PD sessions.  The 5th to be lead by the TOA team.

My Dilemma:  As a classroom teacher, I’d already been incorporating Carol Dweck & Jo Boaler’s work on mindsets but I’d never run a 3-Act or a Number Talk.  How can I provide a training session on a strategy I’ve never implemented with students?

Solution:  Get practicing!  First, I practiced with my coaching team.  Second, I asked teacher’s if I could hone my skills with their students. I practiced Number Talks, 3-acts and Desmos based lessons.  Each experience broaden my understanding of teaching.

  • I learned the nuances of Number Talks, 3-Acts and Desmos
  • I learned which questions elicited student conversation and which ones didn’t.
  • I learned how to better facilitate whole group and small group conversations.
  • I learned how to react to a student question or strategy that I didn’t anticipate.
  • I learned how to navigate the unknown with more confidence.

Since I’m a TOA/Coach, I didn’t have the luxury to practice in private. Every misstep was witnessed by a classroom teacher.  Sometimes, I walked away frustrated and embarrassed for the lesson didn’t go as planned.  Other times, the lesson flowed better than expected. In both cases. I gained insight into student learning.

A Year 1 Experience  (We have a rotating schedule)

  • Period 2:  With Mrs. M:  The lesson didn’t meet the discussed expectations. I felt horribly.
  • In between classes:  Explained what happened to Mrs D. Asked if I could borrow her period 4 exploratory students to refine the lesson.  She agreed.
  • Period 3:  Analyzed lesson.  Made changes.
  • Period 4:   Taught the revamped lesson to Mrs. D’s exploratory class.
  • Period 1: Back with Mrs. M. Shared changes. Taught lesson – again.  Everything went as originally intended.

Summer of 2015

  • Attended TMC15.
  • Was introduced to the Desmos Activity Builder feature and the Vertical Classroom Model. 2 more areas that I needed practice implementing.

Year 2

  • I asked teachers if I could test out my Desmos Activity Builders tasks with their students.
  • Teachers understood that it was a learning process for me.  Designing an activity was easy, but I needed student interaction in order to understand it’s effectiveness.
  • While students were working, the teacher and I would reflect on student learning. The Desmos ABs were refined.
  • I valued this collaborative process. It helped me grow as an educator.

Towards the end of the second year

  • I was continually testing out student collaboration and Desmos ideas.
  • Therefore I was constantly being observed.
  • I began analyzing and refining activities with teachers during the lesson.
  • I reached a point where making mistakes in front of teachers wasn’t a concern anymore. The insight gained surpassed any embarrassing moments.

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Year 3 – Transparency

Year 3 kicked off a month ago. Due to 2 years of conversations, demo and co-teaching lessons, planning together and trainings, I believe my teachers understand my teaching philosophy. They’ve watched me…

  • Interact with students.
  • Test out new strategies.
  • Practice strategies
  • Flop
  • Reflect
  • Rebound
  • Succeed
  • Grow

My teaching skills are constantly on display.  There’s no place to hide and it’s liberating. The transparency provides a level of freedom I haven’t felt before as an educator. My wish is for other teachers to experience the same freedom – which brings me back to the Twitter hashtag, #ObserveMe.

Robert Kaplinsky’s  #ObserveMe campaign encourages teachers to embrace collaboration and actively request constructive feedback.  My favorite part – Teacher requests are clear and specific.  As pictured, signs outline the skills each teacher is focusing on.

Screen Shot 2016-09-04 at 12.14.09 PM

 

Screen Shot 2016-09-04 at 12.12.36 PM

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Closing Thoughts

The past 2 weeks have been incredible. So wonderful, I couldn’t decide which experience to share first. I attribute this positive dilemma to being observed for 2 straight years – whether I wanted to be or not.

As a result, every teacher knows how I work in a classroom.  They know my strengths and weaknesses. They see my passion – My need to push boundaries and try new strategies. Teachers see my excitement when they discuss wanting to branch out and they know I’ll support their efforts. Mutual trust has been established.  I believe I earned their trust because I allowed myself to be observed regularly.  I’m an open book.

As the school year progresses, find colleagues that will support your efforts.  Ask them to work in your classroom during their prep period.  Ask them to provide feedback on a specific strategy. Reciprocate the favor.  Be transparent in your teaching practices and collaborate.

  • Explore together
  • Practice together
  • Observe together
  • Reflect together
  • Question together
  • Learn together
  • Improve together
  • Grow together
  • Succeed together
  • Celebrate together

 

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Comparing Fractions w/ Desmos

For the past few days, I’ve been hanging out with Mrs. Becker’s (Peytra) 6th grade exploratory class.  We both like experimenting with new strategies and often use her exploratory kiddos as guinea pigs.  Our recent experiment – Fractions.

Last year I started creating various fraction activity builders. These activity builders are slightly different than your average task for they don’t walk a student through a series of guided questions. I designed the slides so teachers and students have the flexibility to create their own problems.  It’s more of a general or open tool to be use as desired.

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That said, I’d like to highlight 2 moments from today…

Part 1 – Comparing Fractions

Directions:

  1. Go to slide 2.
  2. Shade 1/2 of the red square and 1/4 of the blue
  3. Insert the appropriate comparison symbol
  4. Capture    Capture1
  5.                              Original Slide                                         After Directions

 

Eric’s Ah-ha Moment

Even though the blue 1/4 clearly looks smaller than the red 1/2, a handful of students used the less than sign.

  • Me:  I noticed you used the less than sign, therefore saying 1/2 is smaller than 1/4. What’s your reasoning?
  • Eric:  The denominator is bigger.  4 is bigger than 2.
  • Isaac:  (Isaac joined the conversation) Look at the colors.
  • Me:  What do the colors tell you?  Eric still wanted to say that 1/4 was bigger than 1/2.
  • Isaac:  Which color is shaded more?
  • Me:  (to Eric who is still hanging onto his belief that 1/4 is bigger than 1/2).  Do you see the blue dot?  Click on it and drag it over to the red 1/2.

Eric grabbed the move-able dot and dragged the blue 1/4 over to the red 1/2.  When Eric lined up the pictures, his face looked puzzled – he began doubting his first answer.  Time for a story…Capture

  • Me:  Let’s say I baked your favorite type of cake.  Which would you prefer:  The red 1/2 piece or the blue 1/4 piece?
  • Eric and Isaac:  (eyes wide picturing yummy cake) The 1/2!
  • Me:  Why? (to Eric)
  • Eric:  I’d get more.
  • Me: OK.  Let’s go back to the denominator.  When the denominator gets bigger, what happens to the piece of cake?
  • Eric:  The piece gets smaller.
  • Me:  Would you repeat that statement?
  • Eric:  When the denominator gets bigger, the piece of cake gets smaller.

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Part 2: Equivalent Fractions

Directions:

  1. We now know that 1/2 is greater than 1/4.
  2. Change 1/4 to represent a shaded region that is equal to 1/2.  You are not allowed to use 1/2.

Aiden’s Ah-ha Moment

After class, Peytra shared that one table had a great ah-ha moment.  I pressed for details!

Peytra:  Aiden had no idea how to start. He stared at the screen for a bit.  I watched him drag the blue and place it on the red.  He then began making adjustments (to the fraction) until the blue and red matched. He figured out how to find the equivalent fraction on his own!!!

Capture

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Desmos & the Whole Class Discussion

The Teacher Dashboard can be used to facilitate whole class discussions.  Pictured below is the dashboard.  The blue column on the right houses all the names.  The Anonymous Icon (the person) switches student names to famous mathematicians. When the person is hanging out in the white circle, the Anonymous Icon is in play.

Screen Shot 2016-08-24 at 8.18.16 PM.png

Screen Shot 2016-08-24 at 8.19.50 PM

To view student progress, click on the desired slide. When the word Responses, is in blue text and underlined then you’re on the student response page. On this page, teachers can view all student responses and decide which ones to highlight. Simply click on the student’s individual response to enlarge.

Screen Shot 2016-08-24 at 8.24.34 PM

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Closing Thoughts

Peytra and I are well aware that comparing and finding equivalent fractions are not 6th grade standards.  We also know that there are many students who don’t understand fractions conceptually. While some students immediately created an equivalent fraction for 1/2, others struggled. The conversations were insightful and supported our desire to integrate more opportunities for fraction exploration and sense making.

The Activity Builder:  Exploring, Comparing & Finding Equivalent Fractions

If you use this activity builder, I’d love to hear about your experience!

Fraction Related Posts

 

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Rethinking Classroom Design: Collaborative Stations, VNPS & Desmos

Twitter Math Camp ’15 sent me down the vertical classroom path (Click here to read post). Throughout the 2015-2016 school year, I promoted vertical non-permanent surfaces (VNPS) whenever possible.  Lucky for me, Joe, a 7th/8th grade teacher, was interested and additional white (shower) boards were installed in his room.  As I observed his students using VNPS, my support of the vertical classroom grew. By years end, I was convinced that implementing collaborative stations would naturally shift instruction to an inquiry and problem based classroom.

By May, 6th grade teacher, Kim, jumped on board. We planned out her redesign and over the summer white (shower) boards, cut to size, were installed. School’s not in session yet, but here are some pics. (In my original design, the white boards were higher.  They can be raised if needed)

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Collaborative Stations in College

Part of my summer was spent attending college tours for my son.  The tour guide at UC Riverside in Riverside, California walked us through their library. In addition to the typical library contents, I noticed collaboration stations!  Stations consisted of a flat surface, monitor to project a computer screen and a white board. This multi-medium approach to learning and collaborating is becoming the norm.

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Gearing up for 2016 -2017

Implementing Collaborative Stations

As mentioned earlier, I believe collaborative stations naturally cause a shift in instruction. It’s hard to compete with direct teaching when students have access to computers and a group white board.  Once students experience working in this structure, they’ll ask for more lessons using this format.

Therefore, my coaching team and I began this school year by modeling the collaborative approach. We ran a 7th and 8th grade professional development workshop which focused on solving linear equations using collaborative groups.

The Task:  Discuss

  1. the process of solving,  -8 + 2(x + 5) = -3(x – 1) + 4 

  2. the meaning of the equation’s solution.

 

Part 1:  Establish ResponsibilitiesCapture

Arrange students in groups of 4. Give each student a number from 1 – 4. Each number represents the individual’s responsibility within the group:

  • 1’s: Desmos component
  • 2’s: Vertical white board: Equation
  • 3’s: Vertical while board: Properties
  • 4’s: Desmos component

Part 2: Establishing Curiosity

 Display the equation and ask

  • 1’s to enter the equation into the Desmos graphing calculator
  • 2’s to write the equation on the board
  • 3’s to grab the property descriptions
  • 4’s to enter the left side & right side expressions into Desmos (on different rows)

 

              View for the 1’s                                                           View for the 4’sCapture1

Capture

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Capture2

At this point, ask the Desmos students to show the group their screens and describe what they see. Someone is bound to click on the intersection point, toggling the coordinate (1, 4).Students are naturally inquisitive. They’ll ask questions and search for meaning. A stage for discovery has been set.

Part 3: Working Through the ProcessCapture5.JPG

Students at the white board are responsible for recording an informal 2-column proof. The teachers to the right, are working on a different problem as described above, but you can see the development of their 2 column-proof.  Left side lists the computation.  Right side lists the respective justification.

The steps written on the board are also entered into Desmos. Since the students already know the answer to the equation, their focus is directed on the process. Desmos provides constant feedback.  As each version of the equation is entered, students immediately know if their current step was correct.  If the value of x doesn’t show the number 1 (for this case), or if a 3rd line pops up then a mistake was made and error analysis kicks in.

The computer views:

Capture    Capture1

Part 4:  The Teacher as a Facilitator

During the task, the teacher circulates the room listening to and facilitating conversations.  Transitioning from the Sage on the Stage to a Teacher Facilitator can be tricky at first.  It takes practice. Here are a few practices to keep in mind:

 1:  Transitioning  To avoid groups separating into 2 subgroups (computer and white board), teachers first need to monitor the transition. Scan the room and observe each group.  First, confirm that all members are executing their individual responsibility and then assure group members are talking to each other.  To refocus students or aid in collaboration, I ask questions.

  • Who’s in charge of writing the justification? Who had the responsibility to enter the equation into Desmos?
    • With these questions, groups begin to self-monitor
  • (To a student who looks confused) Which job were you assigned?  (To the group):  Can one of you help out … 
    • These questions are to place the responsibility back onto the group.
  • Have you coordinated with your group members working on Desmos? On the white board?
    • I ask these, when I notice a lack of communication between group members

2:  Formative Assessment – Ask Direct Questions  As groups progress through their task, visit each group, and ask questions to assure collaboration and to check for understanding. Examples of questions:

  • To the Desmos person:
    • In which row was the distributive property used?
    • Explain what happened from row 2 to row 3.
  • To the white board person
    • What does it mean when the value of x changes in the computer? Is it good or bad? Why?
    • Explain where you used the addition property of equality?

If a student is unable to respond, I’ll prompt the group to step in.

  • Teacher:  Could I have everyone’s attention?  The question I’d like everyone to answer is: Explain all the places where the addition property of equality was used? Please make sure everyone in the group can answer this question.  I’ll be back in a minute.

When I return, I typically call on the original student but not always. If he/she responded appropriately, then I may ask a second member to restate their response.  If the original student could not answer correctly, then I’d say…

  • It seems you need 30 more seconds.  Work together.  I’ll be back.

Usually by the second time, the original student is able to answer the question.

3:  Highlight Different Methods. During the training, the two groups below approached their given problem differently.  The group on the left began by multiplying both sides of the equation by 2, whereas the group on the right combined like terms first.

The same experience is destined to happen with students. When it does, take time to acknowledge and embrace alternate paths to a solution. VNPS allow students to review and critique each other’s work from a distance – providing both planned and unplanned opportunities to absorb multiple strategies.

4:  The Question Bomb:  A question bomb occurs when a teacher asks a question and walks away leaving the group members to rely on each other for the answer.

  • Teacher:  I can tell that you (the group) clearly understand the relationship between the equation’s solution of 1 and the intersection point.  What’s the significance of the 4?  I’ll be back to check in with you.  (Teacher smiles and walks away giving students space to discuss)
  • Once a question is dropped, the teacher circulates the room before heading back to check in. It’s alright if the group doesn’t have a complete answer as long as they provide evidence of a discussion. Based on the evidence, the teacher chooses the next guiding question.  Without teacher follow through, the initial question bomb will not push student thinking. Teachers: Keep moving, ask questions and connect with students.

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Getting Back to the Training

The coaching team reworked the content to create more inquiry within the lessonCapture2

Equation 1:  We introduced the collaborative processed described above.

Equation 2:  We switched roles and practiced again.  Those working with the computer now wrote on the white board and vice versa.

Equation 3:  Teachers entered the equation and noticed the twist.  For students, the missing x value and parallel lines create curiosity.  Why are the lines now parallel? Why is the x value not displayed? What about the equation causes parallel lines? Some students may even search the earlier equation for answers.

Capture Capture1

Equation 4:  The last equation involves the final twist – A missing x value matched with a single line. Another mystery. A missing x value but only one line.  Why??

Capture Capture1

Students can organize their information in the template.  To create mathematical intrigue, the first row intentionally left blank. As students record information and make connections, the three categories of solutions will become apparent..

Capture.JPG

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Closing Thoughts

 A summary of main ideas.

  1. Collaborative stations involve desk space, technology (Desmos), manipulatives (etc…) and a VNPS
  2. Present students with a mathematical situation to explore – incorporate mystery 
    1. In this case, students were to discover the 3 types of linear solutions.
  3. Incorporate explorations that require students to use multi-medium approach such as white boards, technology and/or manipulatives simultaneously. 
    1. Students collaborated on creating a 2-column proof on the white board.
    2. They checked their process on Desmos.
    3. They also explored the meaning of the graphical representations on Desmos.
  4. Let students discover connections.  The discovery process creates a more meaningful experience.
  5. Practice facilitation techniques. Work the room
    1. Transitioning
    2. Formative Assessment – Ask Direct Questions
    3. Highlight Different Methods
    4. Drop a Question Bomb
    5. Keep moving, ask questions and connect with students
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Desmos, Progressions & Supplementary Angles

Summer!  The time to catch up on sleep, projects around the house and blog posts.  This post is the 3rd in a 3 part series I started back in February. I spent two days working with Candace and her 7th grade students on complementary and supplementary angles.  I wrote posts on the complementary angle activity builder (day 1) and the opening activity on day 2.  Finally, here’s the post on the Desmos Supplementary Angles Activity Builder.

In this post:

  1. Lesson guide for the  Supplementary Angles Activity Builder
  2. Extension question
  3. Connection to grade 8 standards/highlighting common core progressions.
  4. Closing thoughts & ticket out the door

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Part 1:  Lesson Guide

Desmos Supplementary Angles Activity Builder

Slide 1 : Since it was the second day moving angles with sliders, students jumped right in and got to work.

Supp S1b.png

 

Supp 2eSlide 2:  As students are writing, the overlay allows you to read their responses in real time. Take this opportunity to address SMP #3 & 6 when possible.  Ask students to

  • Refine their explanation by using academic language (precision of language)
  • Fix grammatical mistakes.
  • Elaborate on a thought. (mathematical precision)
  • Connect their thinking to other aspects of mathematics.

Student Responses:

Supp 2c

Supp S2a

Supp 2d

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Slide 3 & 4:  These 2 slides were designed to connect geometry with linear equations. This process was also done the day before with complementary angles, which is why one student stated the equation, x + y = 180, in her response above.

By definition, supplementary angles occur when the sum of 2 angles equals 180˚ Therefore when a student creates a 0˚ angle and a 180˚ angle, the supplementary angle sign doesn’t pop up.   This situation should be addressed during the whole class discussion on slide 5.

Supp NOT

Since students ran through a similar question series with complementary angles, many just skip to slide 4, fill in the chart and input the equation.  A few students are compelled to enter in ALL the possible points, as shown below. 🙂

Supp 6A

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Slide 5:  This planned stop prepares students for the upcoming discussion. This is where I’ll write the shared supplementary angle pairs on the board for all students to see.

Supp StopScreen Shot 2016-07-01 at 6.01.18 PM

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Slides 6 & 7: For the learners who require more scaffolding, The slide 5 class discussion combined with slides 6 & 7 will walk them through the rest of the activity.  For students who have already finished this section, you may wish to present them with the extension question given in part 2 of this post.

Supp S7

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Slide 9 & 10:  When searching Teacher.Desmos.com/browse, I came across Kate Nowak’s activity builder, Measuring Circles.  It included the slide to the right.  I loved it!!

  1. Proportional reasoning is a main topic in 7th grade.  Any opportunity to spiral back to it is welcomed.
  2. I love the overlay aspect of displaying the class consensus.
  3. In my adaption, I used rectangles as opposed to circle.
  4. Now this activity integrates geometry, linear equations and proportional reasoning.
    Screen Shot 2016-06-30 at 8.37.40 PM

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Supp w: contraintsPart 2

Extension Question

When I c0-taught this lesson, we discussed that the example, 0 + 180 = 180, does not represent supplementary angles but didn’t connect this fact to the graph of x + y = 180. I wish I had. So…

Extension Question:  The line representing supplementary angles, x + y = 180, goes on forever in 2 directions.  As discussed earlier, not all points that add up to 180˚ describe supplementary angles.

  • What part of the graph represents supplementary angle pairs?
  • What part of the graph does not represent supplementary angle pairs?

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8th Grade Connection

Common Core Standards have been developed in a specific sequence or Progression. Proportional reasoning in 6th grade leads to discussing the constant of proportionality in proportional data in 7th, and then expands the conversation to include understanding the slope of both proportional and non-proportional relationships in 8th.

Here’s a list of areas to discuss in an 8th grade classroom.

  • Function or non-function
    • Function:  There’s one output for each input.  If 30˚ is the input, the only output could be 150˚
  • Proportional or non-proportional
    • Not proportional:  As one angle increases the other angle decreases
  • Linear or non-linear
    • Linear:  The graph is a decreasing line
  • Discrete or Continous
    • Continuous:  The angles 20.08˚ and  159.92 are a possibility.
  • Identify the x and y intercepts.
    • Y-intercept:  (0, 180) When the first angle is 0˚, the second angle is 180˚
    • X-intercept (180, 0)  When the first angle is 180˚, the second angle is 0˚
    • These 2 situations do not represent pairs of supplementary angles.
  • Determine the slope and explain what it means
    • Slope is -1.  As the first angle increases by 1 degree, the second angle decreases by 1 degree.
  • Identify the independent and dependent variables
    • X:  angle 1, independent variable
    • Y:  angle 2, dependent variable
  • Equation forms
    • Standard:  x + y = 180
    • Slope-intercept:  y = -x + 180
    • Point-slope:  (y-150) = -(x-30)
    • Test your equations on Desmos
    • Screen Shot 2016-07-02 at 8.55.05 AM
  • Write a compound inequality to display the supplementary pairs
    • See below

Supp close up Screen Shot 2016-07-02 at 8.38.47 AM

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Closing Thoughts

Shedding light on the overlapping parts of math can build and strengthen students’ conceptual understanding of math.  By the end of this lesson, the 7th graders clearly understood the equation, x + y = 180 and it’s connection to supplementary angles.  Their ticket out the door was to describe the meaning of the x, y & 180.

I went into designing the supplementary and complementary activity builders for 7th grade.  The goal was to discuss the 2 relationships while connecting them to linear equations.  The more I mulled over the connection, the more I thought about 8th grade. With a little tweaking, the general idea behind this activity can be applied it to the 8th grade function and equation standards.

 

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