How To Incorporate Number Talks in Distance Learning

Number talks have so many positive benefits, and they became a regular part of my classroom practice last year. I loved listening to my students' reasoning. The class time spent talking about and comparing strategies was well worth it.

However, when we shifted to distance learning, number talks were left by the wayside. As I consider next year, I know that I want number talks to be part of my teaching....whether online or (hopefully) in person.  Here are some ideas of how I plan to use number talks if we go back to distance learning next year.

Ways to Use Number Talks with Distance Learning

How To Get Your Students Excited About Similarity

       Our current unit is Stretching and Shrinking, a unit about mathematical similarity.  This is one of my favorite units to teach every year, and this year I found a great way to launch it.  I decided that to launch our unit, we would have some class "grow beasts".  I am so excited about how this class activity turned out.  There were lots of great benefits for my student's learning, and it was pretty easy to implement.

       First, I'll talk about the implementation.  I had 7 different grow beasts:  2 frogs, 2 lizards, 2 skeletons and 1 shark.  Each table was assigned one of the grow beasts.  On day 1, we defined what measurements we would be taking each day.  I had a simple chart that defined measurement 1, 2, 3, 4 and 5 for each grow beast.  Students took initial measurements of their beasts.  I had students submit their measurements through a very simple Google form.  I also made initial measurements of each of the different types of grow beasts.  We also examined the claims on the package (I hung them on the wall as reference), which included the following claims:

       The next day, we organized all of the data and looked for outliers.  For each measurement that didn't make sense, we decided what to do about it.  In a few cases, we threw the data out that clearly seemed incorrect.  In a few cases, the decision of the class was that the measurement had accidentally been taken in centimeters instead of millimeters, in which case we fixed it.  We also made predictions of what we thought would happen.  This was a great chance for me to begin to understand my students thinking before we started our unit.  I tried to focus our discussion on key issues relating to similarity, such as whether all parts of the body would grow equally, and whether the angles in the figures (such as the angles of the legs in the frog) would stay the same or change as the beast grew. Student response was mixed on these issues, so the discussion was very eye-opening for me.  It gave me a good window into my students' initial thinking about similarity.  I also had students make predictions about what would be the size of each of the measurements if they all doubled, or if they all tripled.  At the end of the day, I put the grow beasts in water.

      I also made a graph with all of the initial measurements, as well as a line showing the measurements for growing 3 x bigger and growing 6 x bigger.  I had original measurements as my x variable and growing measurements as my y variable.  I started by putting all of the original measurements on the x-axis.  I made it on a big sheet of chart paper.  This graph became a central place for me record all of the results and became the guide for many of our discussions.  I was so glad to have this graph to refer to throughout our grow beast project.  I also really liked that it was a great way to introduce the concept of proportionality, although we didn't formally use that language.  For someone teaching 8th grade or Algebra, this might also be an interesting way to look at trend lines.  I definitely found that each day's data made a somewhat linear pattern on the graph, although measurement accuracy (or lack of) might be a problem.  You could also use Desmos to graph your data.

       Our warm-up each day was to measure the grow beasts.  I usually gave them the first 5 minutes of class to get their measurements submitted.  If you're going to do this, you'll want to make sure that you have a towel around to dry off the beasts before you measure them.  The kids got really excited to see how they grew.  Each day I would put up the chart that listed the different measurements, and the kids continued to complete the same Google Form over and over with the measurements for the grow beast.  This project definitely reminded me that having students measure is so important, and they REALLY need the practice!  I also liked having students give the data via the Google Form.  It allowed me to easily sort the data by the animal type, and then by date so we could really see the trends in growth.

       At the end of each day, I also measured the grow beasts and add the new data to the class graph.  I had each figure color coded on the graph, but it was really cool to see how you could see a trendline of growth for all of the figures.  I loved the fact that I had already put the tripling line and 6 x bigger line on the graph, so when I showed it to the kids we could compare it to those benchmarks.


       One interesting (and fortunate) thing that happened to me was that one of my frog grow beasts did not grow for some reason. That group of students was kind of bummed out, but as a teacher it was terrific!  We had initially debated whether the angles of the legs would stay the same, and this way we actually were able to check it out.  If you plan to do this project, if possible, you might get one grow beast that doesn't go in the water to compare back to.

           

       We let our grow beasts grow until they stopped.  For us, that was about 7 days.  They were close to triple in size, but none of them quite made it.  This did make a great impression on my class.  It also sparked a fantastic discussion about mathematical similarity.  By the time we finished the project, I had introduced 2-D similarity (Wumps, for you all CMP users!) and we had a great discussion of what the grow beasts had to do with Wumps and everything that we had learned about the characteristics of similar figures.

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Why Desmos Is My Favorite Technology Tool

      Earlier this year, I started writing about some of my favorite technology tools, and I've even written about Desmos as one of those tools.  But I'm here today to tell you even more about how much I love Desmos!  Previously I wrote about how much I love the teaching activities part of Desmos, but my love of Desmos started with the graphing calculator part of Desmos. Here are a few of the features that I love on the graphing calculator side of Desmos.

1.  Linear regression
I can't even begin to tell you how much easier it is to use Desmos than a regular graphing calculator.  I teach one section of Integrated Math 1, and I always used to completely DREAD the lessons on linear regression.  You know, like, give me a stomachache and make me not want to get out of bed that morning kind of dread. I would spend forever the day before making sure that every graphing calculator was working and had all the right settings....and it never mattered.  I would still spend the entire hour running around, fixing settings and putting aside the calculators that I couldn't figure out what was wrong.

But not with Desmos.  With Desmos it's soooooo easy!  No settings to worry about, much easier to enter data.  It's just EASY!  Plus, they have a super easy tutorial.  You just need to click on the question mark in the upper right-hand corner, and the tour to show you how to do regression is right there.

2.  Create tables and graphs I can screenshot to use in other things
When I'm making a test or activity for class, I often find myself using Desmos and taking a screenshot.  Easy peasy!

3.  Label axes and zoom in and out on your graph
This is kind of a subpoint from above, but I have recently discovered that you can label the axes on your Desmos graph. Plus you can easily zoom in or out depending on the view you want to see of your graph.   Makes it even better!

4.  Export and share your graphs 
This is also useful in addition to taking a screenshot.  I find that the lines are a little bit darker than a screenshot, so sometimes I prefer this method.

In addition, you can also share a graph this way.  For example, if you had created a graph that you already set up some of the labels or axes for your students, you could share it with them.

5.  You can add photos to the background of your graph.
This is a great way to incorporate real-life situations and help students connect them to the real world.

6.  Students can easily graph systems of equations and see the point of intersection with accuracy to the nearest tenth.
Last year, I was teaching a lesson from one of my favorite units, Moving Straight Ahead.  This is a unit about linear relationships, and we were working on the Henri and Emile problem.  This is a problem about two brothers who are racing.  They have different speeds, and the older brother gives the younger brother a head start.  Students are supposed to decide how long the race will be so that the younger brother wins a close race.
Most of my students solve this using a table or guess and check.  I was so excited to watch one group of students decide to use Desmos to solve the problem.  They graphed the equation of each brother and could clearly see the point of intersection (where the brothers are tied).  I thought it was absolutely amazing that students saw the technology of Desmos as a tool to help solve the problem, and understood how to use it and how to interpret the answer.  The rest of the class was so impressed with this solution....and wished they had thought to use it! 

7.   Add sliders to let students play around with concepts of slope and y-intercept.
This is such an easy way for students to play around with what happens to your line as you change elements of the equation, without having to graph everything by hand.  It helps students to see the big picture and analyze patterns.

If you haven't yet, it's time to give Desmos a try!  Leave a comment telling me your favorite feature of Desmos!

Favorite Techology Part 1: Desmos Teaching Activites

         I'm going to spend a few blog posts talking about some of my favorite technology sites and applications to use in my classroom, and my favorite features of each.

       Usually, I'm a "save the best for last" kind of gal, but for this I've decided to dive in with my favorite....Desmos!  I have written before about how much I love Desmos, but today I want to tell you about a few of my favorite features (and a little bit about how to use them!).

     There are tons of cool activities that have already been created to use on Desmos.  If you've never tried any, poke around and find one to try.  My favorites are MarblesSlides and Inequalities on the Number Line, but there are tons of great ones to pick from!

     Here are a few reasons that I love creating my own Desmos teaching activities:

1.  One of the things that I love about Desmos activities that I create myself is all of the different options that I have for what I can add to them.  Here is what you can add to your Desmos activities:

• graph
• table
• sketch
• media (picture or video)
• note  (you give information to students)
• input  (students input text or math equations)
• choice questions (choose between multiple choice, multiple mark, or explanation)
• card sort or marble slides activity

You can even combine more than one of these things on to a single screen.  For example, you could have a graph on the screen and an input box where students have to respond to a question about the graph.  

2.  Another thing that I love is that I can see all of my students work on one screen.  I can quickly see who has finished which slides.  I can click on a student's slide to get a closer view of their work.  This really allows me to give students real-time feedback!  Having everyone's answers in one place also allows for terrific use of student work to guide discussion, as I can quickly scroll through answers to find ones that I want to highlight in class.

Notice that Brahmagupta has made a mistake on the purple line.  This makes is so easy to find mistakes and talk with students.

3.  My class really likes to have me use the "Anonymize" feature, which assigns each student the name of a famous mathematician.  It's fun to hear, "Cool, I'm Katherine Johnson!" or to see kids looking up mathematicians to find out who they are.  I also like this feature because when I notice a problem, I can call out "Pythagoras, it looks like the second point on your graph is off.   Please double check it" and give kids feedback without embarrassing anyone.

4.  I like to set up my activities so that students get to see the answers of other students after they complete a slide.  I think it's really helpful for students to see what other classmates were thinking, and how they justified their thinking.

5.  Another feature that I really love about Desmos is the fact that you can carry forward a graph from one screen to another and kids could continue to work on a graph from one screen to another. 


If you're ready to create your own Desmos activity, go to https://teacher.desmos.com/ and choose  choose "Custom" on the menu on the left side of the page.  Then click "New Activity" in the upper right hand corner.    Then click "New Activity" in the upper right hand corner of the next screen.

This short video should give you some ideas of how to add things to your Desmos activity.

Good luck, and I hope you love creating these activities as much as I do!

My NCTM Experience Part 1: Dan Meyer

I feel so lucky to have been sent to the NCTM Annual Conference in San Antonio last week by the KATM Board.  I am going to do a series of blog posts about my favorite sessions and biggest conference take-aways.  I'm starting with one of the last sessions that I went to.  

 Last week I sent an email to the generic Desmos email.  Imagine my excitement when I not only got the answer I wanted, but the email came from Dan Meyer!  Yes, I in my own geeky kind of way was soooo excited.  Fast forward to the NCTM conference when I was talking to another math teacher who starts to tell me about someone (can't remember who!?!) and said, "She's my math crush.  Who's yours?"  And while I may not have thought of it in those exact words, I had to admit that it was Dan Meyer.  Now fast forward to the 8 am session on Saturday morning of the NCTM conference....what a way to start my day!

Dan with Kira and me.  He has perfect long arms to take a selfie!

The title of the session was "Math is Power, not Punishment".  The big idea of the session was based around the idea that we need to create intellectual need in our students for what we are teaching them.  As Dan said, "Math is the aspirin, but what is the headache?"  He had some really great, quick activities to illustrate this point.  The most powerful one involved the coordinate plane.  Dan started with a slide of a bunch of dots on the screen and told us each to choose one of them.

         

Then the screen changed....all those dots were there, but there was a bunch of others.  Dan got a couple of volunteers from the audience.  Volunteer #1 had the job of trying to describe which dot was hers to volunteer #2...and let's just say that was a tough job!

Then it was Volunteer #2's turn, and here is what happened.  I thought this was a great illustration of the idea of creating intellectual need.

The examples that Dan used in the presentation were very meaningful for me as a middle school teacher....the need for the coordinate grid, and another activity that looked at the value of combining like terms before solving equations.

This idea is not just powerful for secondary teachers however.  As I left this session with my K-2 Math Enrichment colleague, it got us talking about a lesson she had been telling me about earlier in the day.  She had done a lesson using non-standard measurement units, such as hands, feet and so on. As she and I talked, we realized that this lesson on non-standard measurement units would be a great way to create intellectual need in her students.  See what headaches can be cured by using standard measurement units.

As teachers, we want to help our kids find the easiest way to do things.  But perhaps we are taking some of the value of process away by not letting them experience some of the headaches first.
This is truly a powerful idea....that if kids see the value of what a method saves them it will be more meaningful to them.  Think of all those things you teach.....why did mathematicians invent those things?  What headaches did they help cure?

Have I Mentioned that I Love Desmos.....

I know I've said this before, but I really love Desmos.  Today was one more example of that.

We are working on proportional relationships right now.  We've been working on  proportions and unit rates, but now it's time to move on to the next level of understanding.  I'm trying to get students to see the connections between what we have learned about unit rates, and tables, graphs and equations.

The last couple of days I've been doing a series of partner activities about proportional relationships.  The first activity was a breeze for the kids....find the unit rate.  The first time around all of the kids had the same unit rate so they could check each other's work.  Then we did another activity where they kids had different unit rates and had to compare and choose the better deal.

Then I had the kids graph the unit rates.  I don't know why I'd never thought to do this before, but it was a great addition to this activity.  On one graph, I had the kids graph both of their points (such as 12 books for $48 and 6 books for $24).  Then they graph the unit rate (1 book for $4).  It was a good chance for kids to make the connection about all of the points in a proportional relationship falling on a straight line.

With the second activity, I decided to save some time and show some graphs on Desmos instead of having kids graph them.  It may have been a split second decision, but it worked out great! In one class, the Desmos graphs brought up a major misunderstanding that I wouldn't have realized otherwise....and Desmos made it so much easier to clear up.  We were looking at this graph below, and my students though that the green line was a lower unit rate because the line was shorter...that was all they could see.

Luckily, with Desmos it was so easy to take care of this misconception.  I quickly changed from the tables I was using to equations, as shown below.  

As students watched me type the equation and saw the lines become longer, it was like I could see the light bulbs come on, and within a couple of questions they could explain not only which unit rate was better but what they had misunderstood.  This was definitely a case of technology supporting understanding, in an immediate way that other tools couldn't have done...this is technology at its best.  Simple, but awesome!

Using Trash to teach Proportional Relationships

      What good is a bunch of empty boxes?  Last week, it was great to teach proportional relationships.  I had my students all bring an empty box or bottle that had unit conversion information on them.  The first thing that I asked my students to do was to enter the conversion information onto a table on Desmos table.  As you can see below, the student started by entering the information from the package:  10 oz. and 283 grams.

The next thing that I had the kids do was to enter at least 2 more points that would be true based on the conversion information that we had.  Most of my students cut the amount on the package in half, or divided it by 10.  At this point, we stopped to talk about the fact that the points the students were graphing were forming a straight line.
Next, I had students put (0, 0) on the table, and see if it fell on the line.  Then we talked about why this point would be on everyone's line, even though they had different conversions.
Finally, I had students trace along the line, and find the point where the x-value was 1.  We talked about what this point represented.
This simple activity was a great way to introduce proportional relationships, and some of their important characteristics.

Whole Class Participation Techniques

         As this school year begins, I've been excited about how my classes have been taking shape so far.  The kids are great, and the classroom is starting to feel like community.  We're all starting to get to know each other a bit, and the kids are starting to understand my expectations.
          I read a book over the summer about total participation techniques.  I was really excited about this idea of getting every kid to answer questions, rather than just one kid.  So as I started planning for building my classroom climate, this was one of my major considerations.  I really wanted to utilize technology in a way to increase whole class participation whenever possible.  These have been the things that I've been really pleased with in the first two weeks when it comes to increasing class participation.

1. Go Formative--I was introduced to GoFormative as one of the sessions at Space Camp this summer and thought it had potential.  I tried it last week and loved it!  I have used online quizzes on Google forms or quia, but what is different about Go Formative is that you are seeing data real-time (not just when kids finish).  And not only can you see it real-time, you can send kids individual feedback, and they can make corrections based on the feedback you sent them.  So here was a sample question:

The answer I was wanting was that students disagreed, because the numbers were not in order.    

Here is an example interaction between one student and myself.

The student's initial response (which you can't see because he fixed it) was "agree", so I sent him the first message about "What do you do first?"  This helped him realize the numbers weren't ordered, and he then changed his answer simple to "disagree".  However, I also wanted students to justify their answer, so I sent the second message about "Why do you disagree?", leading to his final response.
            I was impressed with how powerful these interactions were, and look forward to using this tool more in my classroom!

2.  Desmos--I have written before about being a big fan of Desmos, but this week as I used it as a way to encourage whole class participation and formative assessment.  We were doing one of my favorite lessons, one about students who are going into business together.  My class has to analyze the business plans in terms of how much money each partner initially has, and how much they save each week.  It's a great activity to teach distributive property, combining like terms,  and the appearance of linear functions with positive/negative slope.  I have always loved this lesson, but Desmos added something to it.  We did most of the lesson on paper, and then I used just a few Desmos questions for real-time formative assessment.

I love being able to see everyone's response to a question on one screen.  I also like being able to quickly show several different strategies for determining whose graph it was, while highlighting student work.  It was fantastic!

3.  Response Cards--This was my low-tech whole class participation technique...but still effective.  We were having a debate about a question I had on the board, so I had students take out the "agree, disgree, or not sure" cards. I was able to get a quick feel for the mood of the class, as well as who might present arguments for each side to further our discussion.  Quick, easy method for formative assessment!  (You can get these response cards free at my Teachers Pay Teacher store, by the way)

4.  Google Questions on classroom--I really enjoyed using the question function on Google Classroom.  I played a video which showed a visual representation of mean, and then asked students to respond as a Google Question whether they thought the video represented mean, median or mode.  Again, quick, easy formative assessment.  Some classes got it right away, and that was great.  Other classes were split down the middle at first, but after viewing the responses in the question, every class quickly concluded that is was mean

The whole interaction probably only took 10 minutes, and it was extremely effective!

5.  Kahoot--This is certainly not a new one to me, as I used Kahoot frequently last year as well.  This year, I'm trying to refine a few things about Kahoot.  A couple of things I'm planning to incorporate in my Kahoots this year are spiraling review and trivia. I want to add just a couple of spiral review questions to every Kahoot, just to keep skills sharp.  And the trivia is just for fun...the kids love knowing that every now and then I'll throw in something random just for fun!
       I also love to use Kahoot to teach error analysis, and trying to get the kids to figure out what error had to be made to get the incorrect answers.