Communicating Mathematically

3-Act : Dried Mango

Posted on September 18, 2016 by jgvadnais

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 jumped from beef jerky to dried pineapple and finally decided on dried mango.

Act 1: Dried Mango – Act 1

Act 2:

Before

Length of mango piece

Width of mango piece

Height of mango piece

After

Length of mango piece

Width of mango piece

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.

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Multiple Representations

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.

Posted in 3-Act, Uncategorized | Tagged 3-Act, math education, Percent of decrease, percents, proportional Reasoning, SMP#4 | Leave a comment

#ObserveMe: A Professional Development Opportunity

Posted on September 4, 2016 by jgvadnais

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.

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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

Posted in Uncategorized | Tagged #ObserveMe, Carol Dweck, desmos, math education, professional development, Robert Kaplinsky | Leave a comment

Comparing Fractions w/ Desmos

Posted on August 24, 2016 by jgvadnais

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:

Go to slide 2.
Shade 1/2 of the red square and 1/4 of the blue
Insert the appropriate comparison symbol
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…

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:

We now know that 1/2 is greater than 1/4.
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!!!

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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.

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.

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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

Posted in desmos, fractions, Uncategorized | Tagged Comparing Fractions, desmos, elementary, equivalent fractions, fractions, grade 4, grade 5, Grade 6, math education, SMP #2 | 2 Comments

Rethinking Classroom Design: Collaborative Stations, VNPS & Desmos

Posted on August 6, 2016 by jgvadnais

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

the process of solving, -8 + 2(x + 5) = -3(x – 1) + 4
the meaning of the equation’s solution.

Part 1: Establish Responsibilities

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’s

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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 Process

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:

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 lesson

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.

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??

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..

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Closing Thoughts

A summary of main ideas.

Collaborative stations involve desk space, technology (Desmos), manipulatives (etc…) and a VNPS
Present students with a mathematical situation to explore – incorporate mystery
1. In this case, students were to discover the 3 types of linear solutions.
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.
Let students discover connections. The discovery process creates a more meaningful experience.
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

Posted in desmos, Math Education, TMC15, Twitter Math Camp | Tagged classroom design, collaboration, collaboration stations, desmos, equations, math, math education, Vertical classroom, VNPS | 1 Comment

Desmos, Progressions & Supplementary Angles

Posted on July 2, 2016 by jgvadnais

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.

Day 1: Desmos, Complementary Angles & SMP#3
Day 2:
- Opener: Using Assessing & Advancing Questions
- Supplementary Angles

In this post:

Lesson guide for the Supplementary Angles Activity Builder
Extension question
Connection to grade 8 standards/highlighting common core progressions.
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.

Slide 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:

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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.

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.🙂

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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.

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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.

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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!!

Proportional reasoning is a main topic in 7th grade. Any opportunity to spiral back to it is welcomed.
I love the overlay aspect of displaying the class consensus.
In my adaption, I used rectangles as opposed to circle.
Now this activity integrates geometry, linear equations and proportional reasoning.

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Part 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
Write a compound inequality to display the supplementary pairs
- See below

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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.

Posted in desmos, Uncategorized | Tagged common core, desmos, education, math, progressions, SMP, Standards of Mathematical Practices, Supplementary Angles, technology | 1 Comment

Building Academic Language w/ a Desmos Box & Whisker Polygraph

Posted on June 14, 2016 by jgvadnais

Before school ended, I had the pleasure of running a Box & Whisker Polygraph created by 2 colleagues, Tim Rochester and J.J. Martinez in Carmen’s 6th grade classes. She had not yet introduced Box & Whisker Plots. Afterwards, I brought the activity to Peytra’s 6th grade classes. The following is a summary of what happened in both classes.

Round 1:

Word Bank: Box, Whisker

To introduce the activity (& topic), I drew a box & whisker plot on the board and labeled the box and whiskers. Carmen and I then required students to use the vocabulary of box & whisker within their questions.

We circulated the room, monitoring students, reading questions and pointing out opportunities into integrate vocabulary. When students were comfortable using the given word bank, we introduced more vocabulary. First, by adding labels to the box and whisker diagram that was on the board. Second, by requiring them to use the new words within their questions.

Below is a color coded box and whisker plot diagram. Each color represents another vocabulary round.

Round 2:

Work Bank: Median, Box, Whisker

Round 3:

Word Bank: Median, Maximum, Minimum, Box, Whisker

Round 4:

Word Bank: Lower & Upper Quartile, Median, Maximum, Minimum, Box, Whisker

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Closing Thoughts

We did have a student who only spoke Spanish. Many of the students in this class are bilingual, therefore Leo was able to fully participate.

Day 1: Box and whisker plot vocabulary was discussed.

Day 2: Carmen expanded the discussion to include finding the value of each vocabulary word.

Carmen and Peytra’s students were fully engaged in playing Polygraph.

Time well spent!!

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Click here for other Polygraphs
Click here for Desmos’ Ploygraph site

Posted in desmos, Uncategorized | Tagged desmos, math, math education, ploygraph, technology, vocabulary | 5 Comments

Nets, Surface Area & Desmos

Posted on May 21, 2016 by jgvadnais

Before explaining the lesson, I’d like to thank Monica, Allison and Peytra for allowing me to field test this Desmos Activity Builder with their students. If it wasn’t for their flexibility, I wouldn’t have the opportunity to refine the activity. I value their innovative spirit and feedback.

The initial purpose in creating the interactive rectangular prism net on Desmos was for students to investigate nets & surface area by first creating a net on the computer and then recreate it as a 3-d model. Embedded in the process was a discussion of surface area.

My initial plan was successful. Monica’s students were able to recreate their prism and turn it into a 3-d model. Although, we agreed the surface area discussion required more attention. I went back to work and built in a few challenges.

The challenges uncovered a new deficit – Spacial reasoning.

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Day 1: Exploring Nets

For the day 1 challenges, I suggest the student to computer ratio be 2:1

Slide 1: This slide was challenging for some of the 6th grade students. The reason? Students counted the small boxes instead of looking at the number lines. Many students stated the length and width of the red rectangle as 8 by 4 units. This lead to a whole class discussion on reading the axes/number lines.

I had tried multiple ways to format the grid.

Since the zoom buttons cause the grid squares to change size, I turned off the Desmos grid and used lists to establish a set grid. Students could determine dimensions easily but the list generated grid lines slowed the movements of the prism. Students and teachers were frustrated. My gratitude goes out to Allison who could see the merit of the activity through the computer glitches.
For the next day, I used the number line. No delays. Yay! Students ignored them. Boo!

Slide 2: Interacting with the Net.

Figure 1 Figure 2

Figure 1: You can’t force the conversation. For the earlier classes, my plan was to emphasize how the sides connected to each other. My words fell on deaf ears. Conversations within groups were non-existent without multiple prompts. Clearly students weren’t ready for this discussion.

Figure 2: By the time I ran the lesson in Peytra’s class, I simply asked students to describe how the dots moved. This request worked.

Slide 3: Stop Sign. This is when all the discussions occurred.

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Slides 4 – 8: Challenges 1 – 5

Students progressed through 5 challenges. When they created the desired net, a little congratulations guy pops up.

Student reflection: At first, I wasn’t sure what to do. But once the little guy popped up, I studied the picture to figure out what I did right.

In the works: My colleague, J.J., threw out the idea of programming the net to fold into its respective 3-d prism. Since those programming commands currently escape me, I programmed the folded cube to pop up once the challenge was met.

At the moment, only 1 challenge has the 3-d image. The others are in the works.

My aha moment: As the students moved through the challenges, discussions increased. Peytra and I noticed 2, 3 and occasionally 4 students standing around a computer discussing their strategies. The vibe in the room was upbeat and academic.

The challenges uncovered and addressed a deficit in student learning. Their need to explore spatial relationships of 2-d nets was apparent . I turned to Peytra and said, “This is what day 1 needs to focus on – Exploring interactive nets and nothing else!” Finally, the progression of the concepts, to best serve students, became evident.

Day 1: Investigating 2-d nets (Desmos)
Day 2: Discussing Surface Area (Desmos)
Day 3: 2-d nets to 3-d models

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Wrapping up day 1:

Oh, they went there….

While computers were shutting down, students were given 30 seconds to compose a response to the following prompt: What did you learn today? Discussion starters spoke first. The others shared going in a clockwise direction. Then, the sharing turned whole group. The results are pictured below.

First I heard:

I learned about nets.
I learned how to use the number line to find the length size
I learned how to move the dots.
I learned about dimensions.

Then I heard:

It was difficult.
I solved challenge #2 – Mind Blown!!
It was hard to match up the sides.
The challenges made me think. There was more to it than just moving the dots up and down
I had to create a plan.

My intent from the beginning was to discuss how one side of a prism effects another. Before exploring the challenges, students weren’t ready for this conversation. I tried forcing it without luck. I dropped the conversation and moved on.

After navigating through the challenges, interacting with nets and observing the patterns, students not only were ready but they started the conversation. Students were able to explain which sides effected another and how.🙂

This experience inspired me to create slide 9:

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Days 2 & 3- Discussing Area & Creating 3-d Models

Slide 10: Starting the discussion of area

Slides 11 & 12: Expanding the area discussion to include surface area

Slides 13 & 14: Challenge 8 & 9 reverses the area conversation.

Slide 15: Time to transition from computer work to partner/group/whole class discussion. This is the teacher’s opportunity to discuss the intricacies of surface area and clarify misconceptions.

Slide 16 & 17:

Students create their own rectangular prism on the last slide and then recreate the net on the design worksheet. Students may use the same or different color combination
Students complete the math calculations
Once the surface area has been calculated, students are to recreate their net one
last time on graph paper.
This last net is then cut out, folded and taped together.
Students, happily, take home the 3-d model.

Posted in Uncategorized | Tagged desmos, nets, rectangular prism, surface area | 12 Comments

Progressions: Fraction Fact Families

Posted on May 7, 2016 by jgvadnais

This, my district’s second year implementing Common Core strategies, has been the year of PROGRESSIONS. As the months flew by, the use of the word progressions increased. Being a math coach, I’ve spent time studying and discussing the progression of standards across the middle level grades with my colleagues.

The 6th grade problem to the right kicked off a rather meaty conversation. The more we questioned, discussed, tested, reflected and questioned some more, connections between elementary and middle level division structures became apparent. This post reflects the thinking resulting from our discussion. You can see some of our notes below.

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In this post:

Fraction fact family investigation.
1. Integrating elementary division structures with middle level concepts (6th grade standard of dividing fractions)
2. Using visuals to model the multiplying and dividing fractions.
Connecting fraction fact families with the initial problem 2/3 ÷ x = 8/9

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Part 1: Fraction Fact Family Investigation

Elementary: Fact family triangles, number bonds & the traditional division bar are typically found in elementary math classes.

3 • 4 = 12 12 ÷ 3 = 4

4 • 3= 12 12 ÷ 4 = 3

Middle Level: What if we adapted these structures for fractions??!!!?!?

2 • 1/4 = 1/2 1/4 • 2 = 1/4

1/2 ÷ 2 = 1/4 1/2 ÷ 1/4 = 2

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Let’s Investigate

Fact Family Discussion 1

1/2 ÷ 1/4 = 2

If I can write 12 ÷ 3 using a division bar, then I can do the same with 1/2 ÷ 1/4. But why use this format?

I rewrote the problem using the division bar due to the language that it’s associated with. Student’s naturally say, “How many times does 3 go into 12?” When working with fractions, we want our students to react similarly and ask, “How many 1/4ths fit into 1/2?”

Division Bar Visual Model (Created on Desmos Activity Builder)

The Desmos created visual model is interactive. The green dot allows students to drag the divisor (1/4) and place it on top of the dividend (1/2) to determine the number of 1/4th groups that fit inside 1/2. In this case, 2 groups of 1/4 fit inside 1/2.

The visual model helps students predict or estimate the answer. One can see that the red 1/4 can easily fit inside the purple 1/2, therefore our answer must be greater than 1. When we physically move the divisor, the answer will become evident. 2 groups of 1/4 fit inside 1/2.

1/2 ÷ 1/4 = 2

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Fact Family Discussion 2

1/2 ÷ 2= 1/4

Division Bar Visual Model (Created using Desmos Activity Builder)

To reiterate:

By rewriting the horizontally written problem using the division bar, students are more apt to ask the question, “How many 2’s go into 1/2?”. Using the division bar strengthens the conceptual understanding of dividing fractions.
After awhile, we hope our students can also apply the, “How many”, question when the division problem is written in the horizontal form – Therefore increasing the flexibility of division structures.
Interchanging forms brings awareness of the overarching structure of division (SMP 7).
The purple dot allows the divisor (2) to move and cover the dividend (1/2).
Only 1/4 of the 2 fits inside the dividend. Or Only 1/4 of the 2 covers the 1/2

1/2 ÷ 2 = 1/4

Another perspective: 1/2 divided into 2 groups means that each group will represent a size of 1/4. This strategy sets up a great segue into Fact Family Discussion 3.

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Fact Family Discussion 3

2 • 1/4 = 1/2

2 groups of 1/4 equals 1/2 or 1/4 of 2 groups equals 1/2

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Summarizing the Investigation

Top row: Finding the quotient through division
Middle row: Finding divisor through division
Bottom row: Finding the dividend by multiplying

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Part 2: Connecting Fraction Fact Families

& the Inspiration Problem

As the summary acknowledges, the inspiration problem can be written with the division bar. The new arrangement offers a different perspective and possibly more clarity for the student. The division bar format highlights the missing value as the divisor, therefore to find the missing value students can divide 2/3 by 8/9.

Figure 1 Figure 2 Figure 3

Figure 1: How many groups of 8/9 fit inside 2/3? Since there’s more blue than red, we know that the blue (8/9) will not completely fit inside the red (2/3). It’s difficult to visually see how many 8/9 exactly fit inside 2/3 but we can estimate that the answer will be less than one.

Figure 2: A common denominator is applied. The 2 fractions are split into equal sized sections. There are 24/27 shaded blue and 18/27 shaded red. To answer the question, we need to determine how many of the 24 blue sections fit into the 18 red sections.

Figure 3: Two blue sections are moved to cover the remaining red region. Only, 18 of the 24 blue 27ths fit inside/cover the red region. 18/24 reduces to 3/4. 2/3 ÷ 8/9 = 3/4

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Fact Family Algorithm

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Fact Family – Visual

2/3 ÷ 3/4 = 8/9

Figure 1 Figure 2 Figure 3

Figure 1: 2/3 ÷ 3/4 = x Again we can see more blue than red, therefore the answer will be less than 1.

Figure 2: A common denominator is applied. 9/12 are shaded blue. 8/12 are shaded red. To find the answer we have to fit 9 blue sections into 8 red sections. Since it’s impossible to fit all the blue sections into the red, the answer will be less than 1.

Figure 3: Blue sections are moved to cover all the red sections showing that only 8 out of the 9 blue sections cover the red region. Therefore the answer is 8/9. 2/3 ÷ 3/4 is 8/9.

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Fact Family – Visual

8/9 • 3/4 = 2/3

Figure 1 Figure 2

Figure 1: 8/9 • 3/4

Figure 2: 8/9 is overlaid on to 3/4. The purple section represents the answer. Out of the 36 sections, 24 are purple. The answer is 24/36 which is reduced to 2/3. 8/9 • 3/4 = 2/3

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Closing Thoughts

The conversation that inspired this post was a catalyst for further exploration of fractions. Exploration that’s aiding my understanding of Common Core Progressions and proving the importance of multiple representations.

Throughout this post, I weaved in multiple representations. For the variable, I used a box, an x and sometimes I left the space blank. I wrote multiplication and division problems horizontally as well as in a fact family triangle, number bonds and with a division bar. Problems were solved using the algorithm and a visual model.

When students interpret mathematics through multiple representations, they’re displaying a deeper understanding of concepts. I believe this is an underlying theme of Common Core. Awareness of the Progressions highlights the teacher’s decision to purposely build on concepts throughout the grades by connecting previous strategies and content with current material.

For 6th graders, fractional fact families, writing a division of fraction problem with a division bar and solving them with a visual model are examples of incorporating previous strategies to understand current material.

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Why Do We Invert & Multiply? An Explanation

Posted on April 9, 2016 by jgvadnais

Ours is not to reason why, just invert and multiply. WHY!

Keep Change Flip WHY!

My district’s focus, this year, has been conceptual understanding. Therefore my coaching colleagues and I have been searching for a clear explanation to why we invert and multiply. In the process, beside our numerous in depth geeky conversations, we’ve become proficient at solving division of fraction problems multiple ways – Drawing pictures being my favorite method. I can describe which parts of the picture represents the division process. But if someone were to ask me, “Why do we have to flip the second fraction?”, I’d still stumble over the explanation.

My colleagues: Tim, Mari and Karon

Recently, a series of experiences connected with earlier conversations have me believing I’m getting close to a clear explanation. Here’s my attempt to shed light on the situation …

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Groups vs Group Size

Synapses started firing when using Nat Banning‘s website, FractionTalks.com. While leading a FractionTalks activity, the language of groups and group size started swirling around in my head. My “aha” moment: Groups and group size are reciprocal concepts.

Groups: 2 Groups: 3 Groups: 4

Group Size: 1/2 Group size: 1/3 Group size: 1/4

The given FractionTalks.com prompt: How many ways can you divide into … lead into other questions.

How many ways can you section the rectangle into halves? More than 1
How many groups did you end up with? (2)
How many squares make up each group? (6)
How many squares are found in 1/2 the rectangle? (6)
How many squares are found in the group size of 1/2? (6)

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Classroom Example

How many ways can you divide the rectangle into thirds? (Many)
How many groups did you end up with? (3)
How many squares make up each group? (4)
How many squares are found in 1/3 of the rectangle?(4)
How many squares are found in the group size of 1/3? (4)

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Equal Sharing

At a recent Anni Stipek training, we were presented with the following problem:

Patrick had 1/4 of his birthday cake left. If he shared it equally among himself and 2 friends, how much of the birthday cake will each one get?

Step 1: I drew my picture and wrote the expression, 1/4 ÷ 3. When a fraction is divided by a whole number, the concept of equal sharing is applied. The shaded part represents 1/4 of the cake.

Step 2: The remaining 1/4 of the cake is split into 3 sections b/c we have 3 people. Each section represents a group size of 1/3. I get to eat 1/3 of the remaining 1/4 of cake.

I split the 1/4 into 3 groups which means my share represents the group size of 1/3. This acknowledgement addresses why we write the reciprocal of the second number.

1/4 ÷ 3 (3 groups) now becomes

1/4 • 1/3 (group size of 1/3 of the 1/4 remaining cake)

Step 3: The portion of the original cake that I get to eat is 1/12

1/4 • 1/3 = 1/12 of the original cake

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Understanding what’s truly being asked.
I’d like to go back to the original problem and refine the question:

Original Problem: Patrick had 1/4 of his birthday cake left. If he shared it equally among himself and 2 friends, how much of the birthday cake will each one get?

Rewritten: Patrick had 1/4 of his birthday cake left. If he shared it equally among himself and 2 friends, What portion of the remaining cake will my piece represent when compared to the whole cake?

Understanding the question: When I rewrote my question, I’m acknowledging that I need to find how my share compares to the whole cake. I have to compare the same ideas, which in this case is group size of my share to the size of the whole cake.

Step 1: 1/4 ÷ 3. Divided the remaining cake into 3 groups.
Step 2: Step 2 inverts the language. Groups and group size are reciprocal concepts. To answer our question, our answer needs to be in the context of size. Therefore we have to invert the concept of the groups to use the language of group size.
Step 3: 1/4 of the remaining cake • my share (1/3) = the part of the whole cake that my piece represents. (1/12) or 1/4 • 1/3 = 1/12

This process emphasizes the importance of relating the answer back to the whole.

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Repeated Subtraction

Let’s continue with the cake theme to discuss the repeated subtraction approach.

Denise brought her remaining 3/4 cake, to work, to share with friends.

Given this scenario, let’s tackle the problem: 3/4 ÷ 1/8

If we use the equal sharing method, this problem says Denise is going to split the remaining 3/4 of cake among an 1/8 friend. Huh? That statement doesn’t make any sense. There has to be at least 1 whole person in which to share the cake. Since we don’t have at least 1 whole person, the equal sharing method isn’t appropriate in this scenario.

The equal sharing approach is used when dividing by a whole number. 1/8 is not a whole number. It’s a fraction, therefore we have to apply the repeated subtraction process.

In this problem, the 1/8 represents the size of the piece in comparison to the whole cake. Therefore, we need to determine how many groups of size 1/8 fit into 3/4 of the cake. The process of determining the number of groups makes this is a repeated subtraction problem.

Let’s start over using the repeated subtraction approach. Denise brought to work her remaining 3/4 cake to share with friends. If each person gets a piece representing 1/8 of the whole cake, how many people can get a piece?

3/4 ÷ 1/8

Step 1: I drew a rectangle and shaded in 3 out of the 4 sections

Step 2: Now to find sections of 1/8 …

Question: But how do I determine how much is 1/8?
Answer:
- We need 8 groups to clearly see a group size of 1/8. Here’s the reciprocal aspect coming into play. By inverting the size of 1/8 to 8/1, we are now talking about groups
- Once the number of groups is acknowledged (8), we have a clear visual of the size, 1/8

Step 3: 3/4 of 8 groups is 6 groups

3/4 ÷ 1/8 means, How many groups of size 1/8 are found within 3/4?
3/4 • 8/1 By establishing 8 groups, we can determine how many of those 8 groups is represented by 3/4. Or … What is 3/4 of 8 groups?

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Visually – Drawing a Picture

The explanation starts off the same …

3/4 ÷ 1/8

Step 1: Draw a rectangle and shade in 3 out of the 4 sections

Step 2: Now to find sections of 1/8

Question: But how do I determine how much is 1/8?
Answer: Since we are looking for groups of size 1/8, common denominators are needed. Therefore 3/4 ÷ 1/8 becomes 6/8 ÷ 1/8

Step 3: Now it’s clear how to determine the size of 1/8. When the number of groups needed is applied, the process of finding groups of size 1/8 is easier to see.

Back to our question: How many 1/8’s fit into 6/8?
Answer: 6 groups of 1/8 fit inside 6/8

Analysis:

A: The work shown in A doesn’t show a reciprocal Here’s my thinking… By converting to common denominators, the group amount issue is addressed.

B: The work in B shows the reciprocal action of switching from group size (1/8) to number of groups (8).

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Summary

Dividing by a whole number 2/3 ÷ 4

Equal sharing
Division means separation into equal groups
Answer is a size compared to the whole

My answer of 2/3 ÷ 4 represents the portion of the whole cake that I receive. 4 people are eating the remaining 2/3 of the cake.

Each person received 1/6 of the cake.

1/6 • 4 = 4/6 = 2/3

(Size of a piece) • (Number of people) = Amount of cake shared

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Dividing by a fraction 2/3 ÷ 1/4

Repeated subtraction
Division means finding the number of groups of the given size
Answer represents the number of groups found.

Mari brought in 2/3 of her remaining birthday cake. If the 2/3 is split into a portion size of 1/4? Is it possible for Karon, Tim, JJ & Jenn to all get cake?

Method 1

2/3 ÷ 1/4: How many groups of 1/4 fit inside 2/3? Can Karon, Tim, JJ and Jenn all receive cake?

Since we are dividing by a fraction (group size 1/4) we need to determine the group number we are working within. Converting to common denominators will do that.

We now have 8/12 ÷ 3/12, which means: How many groups of 3/12 fit inside 8/12?

The context of 12 has been established, I can look for groups of 3/12ths. I circled 2 full groups of 3/12. There were 2 sections of the remaining 3 needed, resulting in an answer of 2 2/3 groups

Answer: 2 2/3 people (groups) can get a piece of cake.

For this problem, Karon and Tim will each receive cake. J.J. will receive a partial portion. No cake for Jenn.

Karon and Tim’s pieces represent 1/4 of the cake. J.J piece represents 1/6 of the cake.

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Method 2

Because the above method, used common denominators to acknowledge the group context. I did the problem again to clearly show the reciprocal action of changing the 1/4 size to 4 groups.

This format clearly shows the division of cake among Karon, Tim, JJ and Jenn. Karon and Tim get full pieces. JJ gets a partial portion. Again, no cake for Jenn.

Conclusion

The conversations with my colleagues, the trainings I’ve attended, the search to understand keep, change, flip conceptually and the writing of this post have all helped me unravel some of the fascinatingly complex and intriguing aspects of fractions. I can confidently say, my understanding of fractions has grown but I still have a lot to learn.

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Fraction Related Posts

Posted in Uncategorized | Tagged division, fractions, invert, math education, Reciprocal | 2 Comments

The Zeroth Power: Desmos & VNPS

Posted on April 5, 2016 by jgvadnais

Inspiration

Negative exponents can be a confusing topic for many students. Due to time constraints, I’ve focused more on the rules in lieu of introducing negative exponents conceptually. As a result, students become confused on how the negative exponent effects the base. I wanted to change this outcome.

One night, while my family and I were watching, Creed, I began tinkering with Desmos. (Side note: Sylvester Stallone should have won the Oscar for Best Supporting Actor). I created a variety of slides. The next day, when I sat down to organize my work, the glaring need to incorporate a zeroth power activity was apparent.

The following activity was therefore created: The Zeroth Power

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Part 1 – Desmos

Slide 1

Slide 1 is packed with information and needs to be broken down. Students will need to understand

The meaning of the slider
The meaning of the x/y chart
How the slider effects the graph
How the slider effects the x/y chart
How the x/y chart and the graph are connected.

Slides 2 & 3

Slide 2 is designed to kick off the conversation surrounding slide 1

Slides 4 & 5

Now that students are aware of the connections in slide 1, it’s time to set up the discussion about the power of zero.
Before moving onto slide 6, call on students to share out their coordinates. Record coordinates on the board
Reinforce the meaning of the x and y columns. The x-column represents the exponent. The y-column represents the value of the base raised to the given exponent.

Slide 6 & 7

Slide 6 requires students to describe, in their own words, the meaning of each coordinate.

Slides 8 & 9

Give the students 10 seconds to choose their base value and move their slider. Then go to the overlay. Ask students to discuss their observations, with a partner. Then submit their answer in slide 9

At this point, students should’ve arrived at the conclusion that the graph for each base has one point in common, (0, 1). With that conclusion, the discussion medium changes from Desmos to a vertical non permanent surface (aka over-sized white board)

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Part 2 – Vertical Non Permanent Surfaces (#VNPS)

I love VNPS and the concept of a vertical classroom. In my ideal school, all classrooms would have white boards as walls. As a coach, I feel it’s my duty to expose teachers to the magical powers of a vertical classroom. How? I bought three 4ft x 8ft shower boards at Lowe’s. The cost – roughly $13 per board.

Lowe’s will cut the boards for free!! All 3 boards were cut into 2ft x 2ft squares. Since I work at 2 schools, I’ve stored half at each school.

Back to the lesson

At this point, the purpose of the activity is to understand why the exponential graph of each base crosses at the (o,1). Why is the value of any base raised to the power of zero is equal to 1.

Ask each group to pick a base value
Using their base value, the group’s are to generate an x/y chart similar to the one in slide 1. All work displayed on their white board (#VNPS)
As a group, they are to find a pattern that explains why (their base)^0 =1
Group members Prep their Rep, by requiring each member practice explaining the pattern.
The teacher circulates through the room, listening to group discussions and asking questions.
The teacher picks a few groups to share their work and pattern. Due to the size of the white board, groups can easily share their work with the class.

Conclusion

The zeroth power is the gateway concept to understanding negative exponents. It’s necessary for students to visually see that any base raised to the zeroth power equals 1, then discover the pattern and explain their reasoning.

The 2 mediums, Desmos & VNPS, support a student’s development of the concept. Desmos allows students to interact and explore this concept which nicely segues into a conversation about negative exponents and eventually asymptotes. The VNPS provides a big space to process and clarify thinking.

*My original goal was to create working with negative exponents. Here it is…

Negative Exponent Exploration

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My Desmos Page: Desmos Activities

Posted in desmos, Uncategorized | Tagged base, desmos, exponent, math, math education, negative, Vertical classroom, VNPS, zeroth power | 1 Comment

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Before

After

Multiple Representations

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My Journey (Abridged)

Year 3 – Transparency

Closing Thoughts

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Part 1 – Comparing Fractions

Eric’s Ah-ha Moment

Part 2: Equivalent Fractions

Aiden’s Ah-ha Moment

Desmos & the Whole Class Discussion

Closing Thoughts

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Collaborative Stations in College

Gearing up for 2016 -2017

Implementing Collaborative Stations

The Task: Discuss

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

the meaning of the equation’s solution.

Part 1: Establish Responsibilities

Part 2: Establishing Curiosity

Part 3: Working Through the Process

Part 4: The Teacher as a Facilitator

Getting Back to the Training

Closing Thoughts

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Part 1: Lesson Guide

Part 2

Extension Question

8th Grade Connection

Closing Thoughts

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Round 1:

Round 2:

Round 3:

Round 4:

Closing Thoughts

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Day 1: Exploring Nets

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Wrapping up day 1:

Oh, they went there….

Days 2 & 3- Discussing Area & Creating 3-d Models

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Part 1: Fraction Fact Family Investigation

3 • 4 = 12 12 ÷ 3 = 4

4 • 3= 12 12 ÷ 4 = 3

2 • 1/4 = 1/2 1/4 • 2 = 1/4

1/2 ÷ 2 = 1/4 1/2 ÷ 1/4 = 2

Let’s Investigate

Fact Family Discussion 1

1/2 ÷ 1/4 = 2

1/2 ÷ 1/4 = 2

Fact Family Discussion 2

1/2 ÷ 2= 1/4

To reiterate:

1/2 ÷ 2 = 1/4

Fact Family Discussion 3

2 • 1/4 = 1/2

Part 2: Connecting Fraction Fact Families

& the Inspiration Problem

Fact Family Algorithm

Fact Family – Visual

2/3 ÷ 3/4 = 8/9

Figure 1 Figure 2 Figure 3

Fact Family – Visual

8/9 • 3/4 = 2/3

Closing Thoughts

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Understanding what’s truly being asked.
I’d like to go back to the original problem and refine the question: