Spiralling MDM 4U - Term 1 review

It's now been a full term of spiralling through MDM 4U and I thought this would be a great time to reflect on what has gone well, what were some struggles, and where I'm heading with the course. 

HIGHLIGHTS:

The students' involvement in classroom discussions and willingness to try even if they are unsure if they are correct have been highlights of the course thus far. The students have been engaged in the activities we have done in class and were able to clearly show their knowledge of games of chance in their "Unfair Game" project. 

It has also been interesting to easily connect terms and concepts that would be separate in the textbook in different sections. In particular, it was great to see the connection between permutations and combinations and how their formulas are so interconnected.

We ended the term with a review sessions where students were asked to work in pairs and write down all the terms and concepts (with formulas) that we had covered since the beginning of the term. Here is an example of one list:

After the list was made, students were asked to self-assess their confidence with each topic. They were asked to put a check mark beside concepts they knew, a - beside concepts that they needed a bit more clarification with, and a 'x' beside concepts that they were still struggling with. It was interesting that most of the concepts that had 'x' beside them were concepts that we had just recently introduced in class. 

STRUGGLES: 

One of the most common comments from students is that they are struggling to keep track of what we are learning in each class. In previous mathematics courses, students would have been given worksheets to fill out and that dictated what they were learning and what they needed to know for assessments. One of my dilemmas with spiralling has been how to keep track of our learning without filling in a worksheet. It's not that I am completely opposed to worksheets (I use them in my other courses) but I feel they defeat the purpose of discussion based learning. With a worksheet, students only want to know what to fill in on the worksheet and not explore the concepts being taught. With my approach to MDM 4U, I want the learning to be authentic and the students to be in the learning driver's seat. 

I do write key terms on the board and we often work through examples as a class but I believe the concern is around what students should write in their notes and quite simply how to make a proper note in math that is not a worksheet. I believe I may have a solution to this dilemma that I will be implementing in the new year upon our return from the holidays. My plan is to use a sheet like this:

Students would have access to these sheets at the beginning of class and would then fill it in as they need as the concepts arise. A summary would then be emphasized at the end of the lesson or the beginning of the next lesson so that it is clear what concepts are being covered. 

WHAT'S NEXT:

The remainder of the year will continue to be activity-based and discussion based learning in this course. We are moving towards more statistics based concepts and not as much of a focus on probability. My goal is to create at least one 3-Act math task to use in a lesson to either introduce a new concept or reinforce a concept.  

Students have also selected their topics for their year-end project. It will be exciting to see them collect their data (either primary or secondary data) and apply all their skills from the entire course to analyze and answer their key questions. 

Creating Art in Math

We decided to let our students demonstrate some of their creative abilities for our inverse assessment in MCR 3U. The key skills we covered were
- what is an inverse?
- relationship between domain and range of a function and its inverse
- relationship between any point on a function and the corresponding point on the inverse
- relationship between the transformations applied to a function and the corresponding transformation on the inverse

Instead of assessing these skills in a typical tests, the following assignment was used.

Though the instructions are quite straight-forward, it does take some thinking and understanding of functions and transformations. Having just completing an entire unit on transformations, this assignment was a chance for students to apply these skills to a unique situation.

Here is my attempt at an exemplar:

The biggest challenge was making sure that the functions they created were close enough together to create a closed figure as their final product. Most students got around this by creating more of each type of function to create the image they envisioned. I could not complain as a teacher as students were further practicing their skills and providing even more evidence of their understanding of functions and transformations by creating several equations of each type.

The art segment of this project was marked using a rubric.




Following this, students came to class and were asked to determine the equation of the inverse of one of their functions (quadratic, radical or rational). This created a unique assessment for each student as they had each created different functions in their art project. 

Are you game?

Spiralling in MDM 4U continues and we've reached the end of the second spiral. With a great focus on probability and counting techniques, students have had the opportunity to experience and demonstrate in these two spirals knowledge of permutations, combinations, factorials, binomial and geometric distributions, probability and odds.

As a summative assessment, students were asked to create an unfair game. They could use any materials available in the classroom or provide supplies from home as needed. In addition to creating the game, students were also asked to analyze the game - determine the probability of winning, show that their game was unfair, along with a few other questions.

Most students chose to revolve their games around dice and cards. These were manipulatives that we had used repeatedly in class and they were comfortable determining probability with. 

Some examples included:

Game #1: 
A player pays 3 tickets for the dealer to roll a 20-sided die on a flat table.

- If the die lands on an even number (red) excluding 20, the player loses the 3 tickets they bet with
- If the die lands on an odd number (black) excluding 1, the player loses 1 ticket they bet with and 2 tickets are given back to them
- If the die lands on either a 1 or 20 (green), the player wins 10 tickets, but paid 3 to play so overall gains 7 tickets

Game #2:

The name of the game is Cross the River, this is because the pathway takes the students across a beautiful river. It costs the student 2 tickets each time they play this game. It guarantees that the student wins every time! He or she will start on the start sign and will start off by rolling the two dice once. If he/she rolls an odd sum then he/she will move up two spaces, but if he/she rolls an even sum then he/she will move up one space. They will get four rolls in total. This game guarantees that they win because if they roll four even sums, which is the minimum, they will at least get to the fourth space, and this will give them one ticket back. Technically they are winning, but in reality they are losing one ticket in the sense that they pay 2 tickets to play, but only get 1 ticket back.

Game #3: 

The name of my game is “The Role Of Fortune”. I choose this name because I was hoped that the familiarity of the name, derived from “The Wheel Of Fortune” would grab the students attention and draw them towards my game. My game is a very simple game which will captivate and interest younger students because of its ease to play and enjoyment from playing. My game costs 3 tickets and involves rolling 2 die. The rules are simple, your objective is to roll a sum of 2, 3, 4, 7, 10, 11, or 12. Any other role will result in the loss of your 3 tickets. A sum of 2 or 12 results in the grand prize of 3 tickets. A roll of 3 or 11 results in a prize of 2 tickets and finally a roll of 4 or 10 results in the prize of 1 ticket.

We then had the opportunity to host the grade 8 math class for a Games Day. Students used the opportunity to collect experimental data for their games. Following the Games Day, students were asked to analyze their experimental data and compare it with their theoretical calculations.

When students began planning their game, they focused primarily on making it unfair - as stated in the expectations of the project. After the Games Day, students were then contemplating how to make their games more "interesting" and more fair. As part of their final reflection about the project, students commented on improving their game by making it straight forward (not too many steps) but also fun and not too obvious.

Finally, students were asked to create a sign for their game so that players knew the rules of the game, the cost of the game, and the probability of winning the game. They struggled with how to state the probability of winning their game without giving away that it was unfair. It was another example of a reassuring theme in the course that we must be analytical with data to ensure we have the whole picture and do not simply believe all that we read. 

A sweet 3-Act Math Task

The other day we had the opportunity to host our Grade 8 students in the Senior School to showcase what Grade 9 math is all about. To give them a taste of linear systems, John Doma (@Domanator19) and I created a sweet 3-Act Math Task.

Act 1:
Act 2:
Act 3:

Though I have used several 3-Act Math tasks in my MPM1D class, this seems to have been one that created the most opportunities for different solutions. Though unintentional, the data that we provided in Act 2 allowed for multiple solutions. This lead to the majority of students being engaged in the activity and willing to contribute their ideas (maybe it was also because we gave them a small box of Smarties to use in their problem solving!).

Here are some of the ideas that students brought forward:

a) 2 cm = 215 Smarties. Therefore 1 cm = 107.5 and so 15 cm = 15(107.5) = 1612.5 Smarties
b) 5 cm = 485 Smarties. Therefore 15 cm = 3(485) = 1455 Smarties
c) 11 cm = 1147 Smarties. Therefore 15 cm = 11 cm + 2 cm + 2 cm = 1577 Smarties

My take away from this activity was making sure to truly think about what data to provided. Going forward, I will definitely be reconsidering the data I do provide in Act 2 taking into consideration multiple pathways to creating an answer.

Developing a classroom of inquiry

My MDM 4U course has been an adventure this year. My goal, as mentioned in previous posts, has been to spiral the course or at the very least deliver the content through activities. It's now been two months and I thought I would share some highlights thus far.

1. Students are more engaged and willing to participate.

• They absolutely love playing games - even if its as simple as "pick a number from 1 - 6 and once your number is rolled, sit down". They are starting to ask questions that help lead the lesson as opposed to being told what they need to know.

2. Students regularly contribute and offer answers, even if they are unsure if they are correct.

• Most activities require students to contribute their findings or results of an activity. I found that by having each begin the course by contributing in this way, I have created a space where students are more willing to contribute.
• Its okay to say "I don't know". It's even okay to offer an answer and when asked why you chose that number to say "I don't know" but I thought...

3. Students offer solutions without definitely knowing the correct formula.

• I have students who have prior knowledge and are stuck on having a formula. It's been great getting them away from the need for a formula and focusing more on explaining why something is happening. Students in my class are getting better at communicating their understanding.
• Students are making connections between activities and again are not so focused on just memorizing formulas.

So until the next post, here is to creating more activities to discover statistics.

Favourable outcomes

We have spent a lot of time rolling dice in my MDM 4U classes so far this year. One day we played 3 different short games all involving being in pairs and rolling a pair of dice.

Here are the details of the three games.

Game #1:Roll 2 dice and consider the sum of the dice.
Player A wins 1 point if the sum is EVEN. Player B wins 1 point if the sum is ODD.

Game #2: Roll 2 dice and consider the product of the dice.
Player A wins 1 point if the product is EVEN. Player B wins 1 point if the product is ODD.

Game #3: Roll 2 dice and consider the sum of the dice.
Player A wins 2 point if the sum is EVEN. Player B wins 1 point if the sum is greater than or equal to 7.

This lead to a discussion of which player would you choose to be in the game and why.

For Game #1, I had a justification for 3 different outcomes:
1. Choose Player B because the mode sum is 7 so there would be more odd outcomes.
2. Choose either one because Even and Odd are equally likely (but not sure why they are equally likely)
3. Choose Player A because Even outcome more frequent
Even # + Even # = Even #
Even # + Odd # = Odd #
Odd # + Odd # = Even #

It was great that students were willing to share their thought process and we had supporters and opponents of each theory. We then developed a chart to show all possible outcomes of the sums and saw that both Even and Odd outcomes are equally likely. Probability was introduced.

In Game #2, based on their results of playing the game, each student would choose to be player A. Similar to game #1, by examining a chart of the possible products, it was clear that Even outcomes are more likely.

We started to discuss the idea of fair games and mutually exclusive events.

Homework for the class was to determine how to chance Game #2 so that it was equally likely for Player A or Player B to win the game. Students came up with some alternate version of one of the following:

- Player A gets 1 point for an EVEN product and Player B gets 3 points for an ODD product (changing the point value of each player)
- Player A gets 1 point for a product of 1 and Player B gets 1 point for a product of 36. All other products result in no points for either player. (only selecting some of the products)

Further discussions in upcoming classes will focus on fair games.

Spiralling MDM 4U - Day 1

I've made it through my first day adventuring down the path of pseudo-spiraling with MDM 4U. And it was actually quite successful and fun! My first disclaimer is that what I am doing may not be true spiraling but I can definitely say that it is activity-based learning. The students have been warned that each class may look disorganized but there is a master plan! My biggest leap of faith was leading a lesson without having a nice handout for students to fill in as we went through the activity.

Today's lesson began with each student being given a dice and asked to roll it 30 times. They were asked to keep a tally of each value rolled in a chart. Students were then asked to chat about their outcomes where they discussed why a few of one number and more of another arose and the discussion led to each value should be represented equally.

We then tallied everyone's responses in one large table and discussed these findings. There was some excitement when the first two columns (# of 1s and # of 2s) were the same and then further intrigue when there were almost double the number of 3s. Again, the discussion led to why each value was not represented equally.

A comment was made about the size of our sample (287 total rolls) and if this was large enough. This is where we distinguished between a population and a sample and started to lead to the idea that the closer the sample represents the population, the better representation it is of the population.

The data was then graphed in a histogram (first individually by each student and then together with the large group). One question that arose was "why can't we draw a scatterplot?" I was really excited that this came from the group and I wasn't telling them right from the start that a histogram was the way to go because of ...

Again, we discussed what should the histogram look like in an ideal situation and why use a histogram over other graphs. The terms discrete random variable and continuous random variable were introduced. Outcomes and events were discussed.

Students were then given a second die that was different than 6-sided and asked how their results would differ if the same activity were repeated with the new die. The idea that the larger the number of sides on the die, the less uniform the histogram would be for 30 rolls. Furthermore, the larger the number of sides in the die, the more rolls would be required to see a uniform distribution.

We conlcuded the lesson by reading a current article and discussing the need to critically assess the numbers we read. In the future, I will give the article for homework before discussing it to ensure all students have enough time to fully read the article.

Key Terms Introduced in the Lesson:
Population; Sample; Discrete Random Variable; Continuous Random Variable; Event; Outcome

Plinko: A summer project

This upcoming year, I will be teaching MDM 4U for the first time in quite a while. My ultimate goal is to have students explore most of the concepts through activities before practicing key skills.

After perusing the textbook and doing a quick Internet search, I came up with the idea of building my own Plinko Board to help with some of the probability concepts. This became one of my summer projects.

What you will need (see below for a photo of the final product):

·            2’ by 4’ sheet of peg board

·            2’ by 4’ sheet of plywood

·            5/16^th dowels

o   We purchased 3 feet long dowels and cut them into 3’’ pieces. I did see you could have bought craft dowels that were precut and about the same size.

o   My board has a total of 300 dowels (216 (18 x 12) white dowels and 84 (7 x 12) black dowels). 

·            Wood glue

·            1/2’’ plywood cut 3.5’’ wide for the sides and bottom edges

·            2 hinges to secure the stand

·            stand (2x3’s)

·            hooks to secure the stand when in use

·            ribbon (to create the zig zag walls)

o   I sized out how much ribbon I needed and then sewed loops at each end. I then slipped one loop at one end over a peg in the first row and then looped it down along the sides and slipped the other loop on the last peg in the last row. This was done on both sides.

·            paint

Every summer, I spend a few weeks up in Northern Ontario visiting my parents and I am very fortunate to have a retired father with a creative mind and a workshop that he lets me invade over the summer for a project or two. One of the first days that I was home, I was watching “The Price is Right” with my dad after lunch and I said “Dad, I am going to build a Plinko Board this summer. You can do it using peg board and some dowels.” This now became a joint project – me for education purposes and my dad for interest sake.

Disclaimer: my dad did the building of the board and I did most of the aesthetics of the board. My dad is great at being given a description and he magically creates the idea with the materials he has lying around his workshop. So I know what materials were used but have little details to provide on how all the materials came together (especially with the stand – it seem to magically appear over night and I have no idea how it was created).

Some things I did note during the building process was that it is a great idea to paint the board and the pegs before it is all put together.  Also, you will need to create a barrier along the sides so that the chip doesn’t get stuck in the sides. My first thought was to use elastics but could not find elastics that were wide enough. My final decision was to use ribbon. See the list of materials for more details on how I put that together.

I decided to leave the bottom values blank so that I could adjust them as I saw fit. My idea is to begin the highest score in the middle of the board and working outwards. I would create small paper slips that I would place at the bottom and could replace if need be.

It was now play time and after some trial and error it was discovered a single poker chip was a bit too light and sometimes stopped along the drop. The weight of 2 poker chips seemed like a better tool to drop. I also found foam golf balls and they seem to work the best – they did not get stuck along the way. My current thought is to have a few different items to use to drop in my class.

My thought at the moment is to have the board in my class and use a few minutes each class for the first few weeks to collect data. I would have the students play and record their result on a common chart. They would keep track of what location they dropped from and where it landed. We would then use this data to drive the discussion in a future lesson.

Stay tuned for how this all unfolds in my class in the upcoming school year.

My year in review

As we approach the end of another school year, I take this opportunity to reflect on my teaching over the past 10 months. Though this was my 12th year of teaching, I realized that I had many firsts in the 2015-16 school year.

Primarily, this year I
1. Tweeted for the first time (I'm sure this isn't the correct terminology - I'm still learning this Twitter world)
2. Created my first blog post.
3. Attended my first EdCamp in Windsor (I found out about this on Twitter)
4. Used 2 Truths and 1 Lie thinking activity (again, a Twitter find)
5. Used VNPS
6. Participated in my first TwitterChat

Yes, I guess you could summarize this by saying I finally discovered Twitter this year (Though I created my account in the last school year, I became a more active user of Twitter this year - not just reading items but also posting and using what I read). I'm still amazed at Twitter's power to universally connect teachers to collaborate and easily share ideas.

Though I may not always walk away from browsing Twitter (yup, pretty sure all the Twitter pros are laughing at my incorrect terminology) with an activity that I can use in my classroom, I readily find statements or articles that make me think - my bookmarks have grown greatly this year. This thinking often leads to a discussion within my department. I know I have greatly appreciated these department discussions - they can sometimes be lengthy but passing ideas back and forth and having colleagues that challenge my ideas has been extremely valuable.

I'm not exactly sure what my specific teaching goals will be for next year but I know that Twitter will continue to be part of my daily routine! Looking forward to more learning and more discussions next year.

Eggstreme bungee

We summarize our linear relations unit with an "eggstremely" fun activity - one of my favorite days of the year!

Based on the "Barbie Bungee" activity, our goal is to have an egg have the best bungee jump experience possible (i.e. get as close to the ground as possible without hitting the ground). An elastic band is taped to the egg and a bungee cord is created using elastic bands. We use three different sizes of elastic bands so that each pair of students had to do their own calculations and couldn't just rely on the first group's answer. By the time we do this activity, students are familiar with finding the slope of a line using two points, graphing lines, and determining the equation of a line given two points.

At the beginning of the activity, students (working in pairs) are given an egg and 10 elastic bands and told the following as the main idea:
 
Using their 10 elastic bands, they create a table of values with number of rubber bands and length of drop. Because the drop happens so quickly, we have them film each drop with a camera that has a slow motion option so that they can get more accurate drop lengths.

Students then determine an equation to represent the drop length based on the number of rubber bands. Because this activity was done over two days, this was what students were expected to complete by the end of day 1. We ensured each student left that class with their data collected and their homework was to graph their data and determine the equation, if not done in class.

Class #2 began with partners comparing equations. We then headed outside to the drop zone.

We measured the drop zone and then students were to use their equation to determine how many elastics they needed for their egg to have a safe jump. They then tested their predictions. To speed things up, we had created chains of 20 elastics for each type of elastic. Once a group has tested their prediction, they would then pass along the chain of 20 elastics. Each egg was then dropped and some were successes, other were not. Here is a video of one of the very successful drops.

We do have prizes (fruit snacks) for the winning team so there is an extra incentive on getting the drop closest to the ground. In order to help them determine how low their egg goes, we have them film their drop and having the background board with the different colours, allows them to distinguish how close their got to the ground.

Finally, after the activity is complete, we have the student complete a reflection piece. We use this reflection piece as part of their summative evaluation of these skills (finding the equation of line, interpreting the meaning of slope and y-intercept, extrapolating using an equation). 

Why I like this activity:
Most of the math that students experience has one right answer and usually has "nice" numbers to deal with. Because the weight of eggs varies slightly and the amount of stretch in elastics also varies, there is no definite correct answer for this activity. Because the height of the drop also varies from year to year, an answer can't be passed down from year to year. Students must work out their own prediction using their data.

It was interesting to see students question their answers when its based on a real world example. Students will trust their mathematical solution (they know the steps to find an equation and solve for a variable) but will doubt that the number they found is correct. They then watch their peers egg drop and then start to question their solutions even more. Some groups often take one elastic off just to be sure their egg won't hit the ground.

2 Truths and 1 Lie - Encore

I had the chance to re-try the activity "2 Truths and 1 Lie" in my Grade 12 Calculus (MCV4U) course today and had very positive results. When I did this activity before, I did not get the depth of answers I was hoping for (you can see my previous post on my first attempt and what the activity is about) and so I modified the delivery of the activity slightly. I don't think I could have envisioned the amazing outcome that I got today.

Just to give context of what my students know, we are just finishing up the calculus section of the course and will be starting Vectors next week. The final Calculus test is at the end of this week and covers the derivative of all functions (polynomials, trigonometric functions, exponential functions, logarithmic functions), and problem solving.


I started the activity by displaying this image to the class and told them 2 of these were true and 1 was a lie.

I gave them 1 minute to individually determine which of the statements was the lie (they didn't have to write anything down, just think and come up with an answer they could justify). After the one minute, they worked in pairs with the person sitting beside them and came up with a group decision on which one was the lie. I then asked them to justify why the other two options were true.

As they did this, I walked around the room and listened to some of their thinking. I checked in with each group and asked some sort of follow question. We then came together as a big group and I asked, by a show of hands, who thought each was the lie. At this point, I would ask certain students why they thought it was a lie (it was interesting that some chose the left-most option as the lie as it was the one they least understood and thought it must just be wrong). It was then revealed that the middle option was the lie and that the other two were truths (we also spent a few minutes explaining what the first option meant).

Now the students got to be creative. Similar to my first attempt, I gave each student three pieces of paper and asked them to create 2 truths and 1 lie based on some calculus concept we covered in the course. I also asked that the statements could not be direct facts, they each had to require some thinking to decide if it was a truth or a lie (It could not be "the derivative of y=sinx is y'=cosx but could be finding the derivative of a y=sin(ln(x-3))). They were allowed to look through their notes for inspiration. All their contributions were then posted on the front board (they were asked to put their initials on the back so I could see what they came up with).

I now asked each student to go up to the board and select one of the notes (as long as it wasn't their own) and return to their desk. We then went around the room and each student read their note and stated whether it was a truth or a lie and how they figured it out. The truths were collected in one part of the room and the lies were in a separate location.

I ended the activity here. We had many more contributions to still sort through and my plan is to continue sorting through them as a class before the final exam in June. I'm not sure if I will have each of them pick another note and justify if its a truth or a lie or if I will collate the remaining options and have the class sort through them electronically (not sure how I would do this yet).

Why I am thrilled with the outcome:
- their answers all required some thinking to figure out
- each student willingly participated
- because they had access to their notes, each student had an entry point to the activity
- I was able to further see their understanding when they chose someone else's note and had to determine if it was a truth or a lie.

My next steps:
As we start to think about end of year exams and making productive use of review time, my current plan is to try this again with my class by assigning each student a certain unit and asking them to create 2 truths and 1 lie for that particular unit. I would then use this list as part of their overall review in some way.

A selection of student answers:

 

Questions????

Recently, I read a post by Jon Orr where he used "2 Truths and 1 Lie" as a review tool for the exam in grade 10 math and it made me think about how I could use it in my classroom. Here's my account on using "2 truths and 1 lie" in Grade 12 Calculus - what worked, what I would modify, what it showed me about my students' understanding.

I posted the function  on the board and asked my class to create 2 truths and 1 lie about the derivative of that function. My goal was to see the depth of their understanding of derivatives as well as the product rule, quotient rule and chain rule. (We has spent the previous two classes on the product rule and the quotient rule and we had covered the chain rule in this class).

I had them write their 2 truths and a lie on post-it notes and hand them in (I asked for their names on the back of the post-it and to also identify their lie on the back). My hope was to then, at the beginning of the next class, display all their post-it notes and have them as a class sort through them into truths and lies.

As I read through their responses, I didn't think the activity went that well (I did this activity in 2 classes on the same day and got the same type of results in both sections). Here are some of their answers:

One of the reasons that I thought it didn't go well was that I was not getting very detailed responses or their responses were very surface level. Most of their responses were also very similar. After debriefing the activity with a colleague, I realize that perhaps their responses showed their understanding of derivatives at that moment was only surface level. It made me reflect upon my practice - had I only focused on the algorithm and not so much on the meaning of a derivative? Was what they were producing a reflection of what I had shown them in class? If I repeated this activity as exam review at the end of the course, would I get more detailed responses?

I then questioned if this was the right type of question to ask and where might I use it to get more detailed responses. As we start the curve sketching unit in calculus, this activity may be more appropriate if when shown the equation or graph of a functions, they were asked "Determine 2 truths and 1 lie about the graph of a given function." Students would have more aspects to reflect upon and be able to create more detailed truths and lies. Would I get better responses if students were shown the equation? the graph? or both?

I also pondered using this activity as part of a conversation with students. What if I created a list of truths and lies and asked each student to identify, from the list, 2 truths and 1 lie about a function or graph and describe why they know it is a truth or a lie.

Though I was not completely satisfied with the results of the first attempt at this activity, I definitely plan on using "2 Truths and 1 Lie" again in the upcoming weeks in Calculus. The students were engaged and each student did have an opportunity to participate and demonstrate their current level of understanding.

My favourite

What is one thing that I couldn't live without in my classroom or that greatly improves student learning in my classroom? Though there are many things that I think I could choose, the one that comes to mind is Vertical Non-Permanent Surfaces (VNPS).

I discovered VNPS through Twitter near the end of the last school year.  After hearing about them from a colleague again at the beginning of this school year, I decided to try them in my Advanced Functions course and quickly discovered their usefulness and power they have on learning. I'll admit that I am a fan of reading about something (usually on Twitter) in the morning and attempting it in my classroom later that day. Some of these attempts are successful, some require some thought and reconsideration before attempting again. Using VNPS was one of these events that was extremely successful.

I usually use VNPS during a lesson that is very skill based. I have used them with polynomial long division, factoring polynomials, and basic derivatives. What normally happens is that I introduce the concept, work through some examples with the student and then have them show me their understanding using the VNPS (or I have also used them during review periods).

I begin by randomly organizing my students into small groups (usually 2 - 3) and have them move to a VNPS. I ask that each group have one white board marker for the group. The first person takes the marker and answers the question posed to the class. The group works together to solve the problem but the person with the marker is considered the lead for that question. Once each group has answered the question, the white board marker is passed to another student and another question is revealed. This continues until each member of the group has had a chance to write a solution on the board. I usually have as many questions prepared as there are members of the group.

Why do I like VNPS?
- They offer another opportunity for me to formatively see their work as well as their form in questions in a less formal setting.
- They encourages conversations between students in small groups on the concepts that we are learning.
- They allow each student to demonstrate their knowledge to the group and get peer feedback on their knowledge.
- They allow me to showcase student work within the class.

Positive thoughts and positive results.

Teaching can be pretty chaotic and stressful. Even if I have taught a course before, I'm always looking to improve a lesson and adapt it to the students in my class. There is always something that needs to be done (planning a lesson; creating an assessment; developing an activity; meeting with colleagues to plan an activity, assessment, lesson; giving extra-help; coaching; etc.) And each day is different - you never know what might come across my desk on a given day. This busy work day can be draining and very tiring but being able to stay positive can make even the busiest day a good one.

Keeping a positive mind frame can be challenging on some days but by focusing on the small things can make a huge difference. Here is a list of some of my small good moments that can occur any day:

1. Getting out for a morning walk.
2. Getting a parking spot in the closest lot.
3. Finishing my coffee while it's still hot.
4. Having a student ask a really good question during the lesson - especially one that makes me have to really think about the answer.
5. Finding a new gem of a lesson or an activity to try out in my classroom (usually from someone I follow on Twitter). My latest one is probably "Polygraphs" from Desmos.
6. Having my team already have set up the volleyball nets before I arrive at practice.
7. Laughing at least once (usually at myself for something that I said or did).
8. Hearing a student say "Wow, math was fun today!"
9. Getting out for a walk after dinner for some fresh air and thinking time.
10. Getting to bed before 11 pm!

Always looking for the positive ++++++++