Box and Whisker Plots Headbandz

This year, I learned about How to Start a Math Fight by Jon Orr. One of the ways he suggests to start a fight is an activity called Headbandz. It is taking the idea of the popular game Headbandz and using it for a mathematical purpose. I was able to purchase 8 headbands for a dollar at my dollar store so that every student had their own headband. Very cost effective way to bring forth a worthwhile strategy. I wanted to try something different with box and whisker plots for my sixth graders. A couple weeks ago, I did the activity by just telling the kids that “before we put our cards on our heads, we are doing a challenge today. You may not have seen anything like this, but remember you must only use yes or no questions to figure out what was on your head. Once we raise our cards, you will have an idea of the figure that is on your head.” I was very curious as to the language they would use. It was absolutely amazing too. I gave them 10 minute to ask questions instead of the usual 5 and no one was able to figure out what was on their head. One of the truly amazing pieces of headbandz is that in 5 to 10 minutes the students see 16 different representation of a box and whisker along with different types of number lines. See below:

Through this initial activity, the students used this type of questioning:

1. Does my number line count by 5’s?

2. Do I have a box (amazed they used this word instead of a rectangle) with lines coming out from the sides? (this lead into great discussion about the difference between a line and a line segment.)

3. Is my left segment shorter than my right segment

4. Is the line in the middle of my box between 10 and 20.

5. Is the dot on the line to the right at 30?

These are the ones I could hear them saying, but the struggle was awesome and every student could ask questions. Even after 10 minutes I had to bring them back and discuss the activity. They wanted to keep going because some were getting close to the answer. We talked about what made it hard and why? For most of them, they couldn’t figure out which question to ask and it was difficult for their partner to know exactly what they were talking about at the same time. I told them that they would be learning about box and whiskers over the next couple of weeks. Again, I can’t state enough that they saw 16 examples in a matter of 10 minutes.

Fast forward now ten days. Yesterday I gave them the same activity and only 7 minutes. I wanted to give 5, but lost track of time because the students were having so much fun. We had over half of them solve what was on their head and that was cool. I gave them another one to solve to see if they could get two in as some were done in 3 minutes. WOW!

I asked “What made it easier to figure out what was on your head?” It was the vocabulary they had built and that everyone in the room had an understanding of that vocabulary. The questions they asked this time were:

1. Is my Q1 less than 20?

2. Is my Q3 between 40 and 50?

3. Is by maximum 40?

4. Is the median between 20 and 25?

5. Does my number line start at zero?

What a difference in the questions. Is vocabulary important? YOU BET IT IS! The students were able to pin point the exact locations of the parts of a box and whisker plot based on the vocabulary they were using and that is what impressed me the most. They also were able to connect the vocabulary they said to the parts they saw ten days ago because they had to struggle through an activity where they did not have the vocabulary, but they had the visual. I believe this experiment was a success and I plan to try this again next year with them if I have the opportunity. Starting your class period with a math fight definitely gets the conversations going and the brain moving. I have used this activity thus far with expressions, equations, and exponents. Expressions was excellent to build the vocabulary of coefficient, constant, and terms. I even went as far to include like terms that the students had to simplify their expressions once they found out what was on their head. We even did this with visual patters from the website visualpatterns.org by Fawn Nguyen.

If you want to download the document I created, feel free. The vocabulary the students learn from this activity is awesome.

head-banz-activity.docx

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Notice and Wonder with Surface Area

I wish I would have learned about notice and wonder 10 years ago. It has definitely changed the way I teach and has helped me understand so much more about how my students are thinking. I put this problem to my class of 6th graders. While they have not learned about surface area yet, they have been exposed to volume in 5th grade. When I showed this problem, the students made their T charts and wrote down what they noticed and what they wondered. One student wrote as a notice that:

Volume = 44h and

Surface Area = 30h + 88

I was blown away by this. What was even more special was how she explained the way she solved it. She made so many connections to her previous mathematics and the expressions that they learned this year. It was also an amazing teachable moment for everyone in the class. She explained why she doubled each answer for the surface area and combined like terms.

A lot of students wondered what the height was. So I gave them some more information.

Most students were able to solve that the height was 17 inches by dividing 748 by 44. I then asked if they could set up an equation to solve for the height.

This challenged some students but that was great, knowing that we can use formulas to find missing measurements through substitution.

Finally, I asked them to solve for the surface area. The girl who noticed that the surface area was 30h + 88 was able to solve for the surface area very quickly because of the expression she solved in her notice. Some students were multiplying to find the area of one side and then doubling it which is great to because that is how they made sense of the problem.

At the end of the lesson, we then had a discussion connecting the idea that expressions are repeated patterns. Using the student’s expression from the notice we can quickly find the surface area regardless of what they height is based on the connection she made. This was a very powerful lesson for me and my students.

I plan to extend this lesson by asking: “the volume of a rectangular prism is 36h. What could be the dimensions of the prism be?” I am hoping to tie this into factors and that there are several prisms. I wonder if they will use decimals or fractional lengths? I also plan to give them a surface area expression to see if they can come up with the prism dimensions.

I ended the lesson showing them how this would look as a standardized test question.

Trapezoidal Understanding

Last week I had the opportunity to teach a lesson to a group of 6th graders about trapezoids. I took a great class in the fall called Making Math Moments Matter by Jon Orr and Kyle Pierce. In one of the modules they challenged us to take regular test questions and use them to spark curiosity, fuel sense making, and ignite our next moves. The question I chose was the one below.

I decided to spark curiosity by witholding information and giving a slow release of the information from the problem. I started with the slide below because it gave a low floor for kids to start and to get the conversations going with justifications.

I was able to really get at their misconceptions about trapezoids and I was blown away over the fact that some students thought trapezoids could not have a right angle, that it must have two “slanted” sides. Here are some of the trapezoids they made. I also like opening these questions so that kids have a chance to show different ways that they think about and see the problem.

The one below was created by a group because they thought that trapezoids must have two slanted sides. They then asked if the two sides were parallel. They were having trouble justifying their answer. At this point they have not heard about slope, but I was able to teach them the basics of it and show them that the two lines may look parallel, but we can prove mathematically that they are not. This was a very neat moment for the two students and I. What I found is that they were very receptive to learning about slope because they needed the math at that moment based on something they created.

After showing their many representations, we were able to solidify that a trapezoid is a quadrilateral with “exactly” one pair of parallel sides. I had a student ask, “you mean it is not “at least” it is “exactly” one pair of parallel sides?”. What would it mean if the definition was atleast? A student said “then a rectangle would also be a trapezoid.” What a neat moment for him to put that together. This type of discussion from the first slide was exactly what I had hoped for. The kids also taught me to see the problem different ways. I like opening up the questions because it also allows the kids to think on their own and you can differentiate the lesson within groups.

After the kids created their trapezoids, I had them find the area of their own trapezoids.

This was neat to see, especially when finding the height. The work we did in the beginning really helped the kids understand what base 1 and base 2 meant because they are the parallel that make up the trapezoid. The first slide really laid some good ground work for us. Now I limited it to the area must be 18 square units.

By this point, I actually had one group of students who placed point correctly so that their area was 18. They felt very validated by their work when I asked this last question which was neat so I asked if there was another spot they could place a point and get an area of 18 square units. They were kind of stumped at first but they got to work on it. This was another way we could differentiate the lesson and have kids find multiple solutions for one problem. I then show the kids the PSSA problem we were working on so that they could see what they might encounter.

As I reflect on this lesson I really like the idea of withholding the information because I can more easily differentiate, but also get at the thinking of students. The multiple choice question is too closed and does not allow for multiple representations. I also could not see what the kids know about trapezoids. Knowing your learning goals before you start a lesson and letting the kids guide the lesson with their questions really helps you and them dive into some deeper mathematics.

Why notice and wonder is important!

Okay, Graham Fletcher, Kyle Pierce, and Jon Orr have blown me away with how I can make the biggest impact on students with my teaching and helping others.  They have been big on notice and wonder and I think I am starting to understand why.  I just completed module 5 of Making Math Moments that Matter class and I think know how I can capitalize more on my math moments.  While it is in the modeling it is also in where our students are coming from and where they are going.  How can I make the most of where they are coming from and learning and where they need to go in my instruction.  Knowing our curriculum two years ahead and two years behind is extremely important.  Kyle and Jon challenged us to take questions from our textbook, but I am looking at released items as I am finding the test questions that are multi-step are also great classwork questions as they connect a lot of learning.  Here is another attempt at a question but with a planned focus on previous learning and future learning.  I hope that while I am withholding information, I can also hang on a slide, have productive struggle, and connect their previous and future learning.

I would present this notice and wonder here:

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Here I can listen to what the students are thinking and really talk about what a right angle is.  How do you know it is a right angle?  What pieces make up the right angle?  Is it two lines?  Why not?  Could they be two lines? Is it two segments?  If they kids say “no they are rays”, then could they be two segments?  That would lead into new learning as kids aways think it is two rays that make angle, but no, it is two segments that could also form an angle.  We are only on one slide and immediately, we have reviewed lesson 1 from our practice workbook, their 3rd grade knowledge, and build understanding for future learning.  Now we move on to this slide.

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This is where I see the slow release being highly effective.  The kids would do a quick Thing Pair Share (TPS) and then hopefully they would say the right angle was divided into two angles.  Now, I can have them estimate the angle and have them attend to precision by telling me exactly what angle they are estimating.  That is powerful.  We are connecting previous learning of how angles are named and now I am free to have them explore complementary angles in a variety of ways.  A student may estimate that angle TXU is 30 degrees, then I can say what would be the measure of angle RXT.  Next, a student may estimate angle TXU as 24 degrees.  What does that mean for the measure of angle RXT?  Wow, we could solve several complementary angles and we are only on slide 2. A student may say and angle is 100 degrees.  I could then help clear up a misconception about acute angles.  Just a thought. I could also have them prove their answers through an equation which would meet one of the 4 common core standards for fourth grade all because of notice and wonder.  The possibilities are endless here. So cool.

After all of the learning that went on from the last slide, we could then release more information and give this slide.

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Now we are out of complementary angles because the have three angles, but we have built some understanding in that the angle is still 90 degrees.  Complementary angles were not our learning goal, but because of future learning, I would decide to include that on the last slide without actually using the word complementary.  Great time for an TPS.  Hopefully students will notice that angle TXU and angle RXS are close to the same size.  That is what I am looking for, but you never know what they will say.  Might be able to clean up some misconceptions based on what they say.  Who knows, but that is why notice and wonder is awesome.

Now we give more information to give some productive struggle with this slide.

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Have the student quickly TPS.  Then move to this slide.

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At this point, the students would work together to solve for one of the angles.  They would then show all of their work and justify their answers.

I would then show them this last slide so that they could see this was just a sample PSSA problem.

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I am amazed at how we can develop mathematicians by using textbook or sample test questions.  I used to look at this question in an negative light, but if we open up the question, we can gain valuable insight, review previous learning, and ignite our moves for future learning.  This is an amazing piece that can really help my instruction as a teacher.

Concreteness in Expressions

The 6th grade teachers and I had a discussion about how we model expressions with our sixth graders.  Kyle Pierce showed a great way to spark curiosity in Growth Patterns which made me as the questions “what is an expression for?  Why do we have them?  Expressions are a to show repeated growth.  This year, we decided to make math visual with our expressions and used the website visualpatterns.org.  We selected the ones that were linear only.  We started out with this pattern because we felt the kids would see the pattern.

Pattern 4

After watching Kyle’s video, I decided to do a slow release like he did.  So I created this slide with this image.  I created the picture upside down from the original by accident, but it worked out in the end.

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Kids said they noticed 3 squares.  I asked how they knew they were squares.  They said it had 4 sides.  So I drew a rectangle on the board.  They said no, but you said it has 4 sides.  So then they said the sides have to “even”.  Oh, so I labeled the sides 4 and 6.  Again, they were not “Attending to precision” in their description so I asked them what they were looking for and they said 4 equal sides.  So then I drew a rhombus.  At this point the kids got a little frustrated, but I really was able to drive home what a square was because then they said it has to have 4 right angles.  What a magical path to go down with them that was all from a notice and wonder.  Hopefully we helped to clarify what makes a square, a square.

Next we moved on to this slide:

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After talking to the 6th grade teachers about this lesson, we probably should have used the word picture to start this lesson and then moved to the word term in visual pattern 3 and beyond.  The word picture would help them make sense of the problem more to start as we are focusing on representing what an expression is for.  It was so much fun to see what they came up with.  Since they drew what they thought and there was no right answer at this point, kids were very engaged and creative.  I showed the next screen which is below, of the pattern and the room erupted with excitement if they were close to the picture or and instant ah, it just “repeated”.  This was a very powerful moment of why withholding information gets everyone involved and since they are engaged, they then make sense of the problem.  WOW!

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Here is where the magic happened.  I asked the kids how they saw the pattern growing.  They actually used the word “repeated”.  NICE.  But then a student came up and drew this and my mind was blown.  He showed me a way to see the pattern differently.  See below.

IMG_2210

The student visualized the piece in picture 2 “hinging” down to form a 2 x 3 rectangle.  At that instant, I changed my whole plan of teaching the lesson as this tied right to their 4th and 5th grade work of area multiplication.  I never saw it that way until this student pointed it out.  From there, the lesson started to flow because they could see the multiplication as repeated addition.  It was unbelievable.  Again, I never intended to show that but that is how informal assessments following the curiosity pathway can lead to memorable math moments.  That student was so proud of himself.  He gave a double fist bump in the air as he walked away.

Once that connection was made, we flowed to the next term.

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Now, I asked them if we could look at this as an area multiplication and they saw the two squares on the top filling in the missing two and then the row of three filling in below.  It was so neat to see them all back to area multiplication.  They connected the term number being the number of column and 3 as the constant number of rows.  WOW!  See below

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Now it was time to make sense of the pattern so we went to the next slide.

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I thought we would provide some productive struggle, but the kids made so much sense of the problem that they came up with 21 right away.  Now we were ready to write an expression.

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This took some time and provided some great productive struggle.  6th graders struggle with what a variable is and what it is for.  I had to some modeling here.  I love the double number line that Kyle showed us in the course and chose that here too.  See below.  I should have modeled area multiplication and put “n” on the bottom as the kids connected the term number with the number of columns for the shape.  I will do that next year.

IMG_2208

We then were able to understand that the term number represented the variable number and that we substituted that in for n.  What a great connection that n is a number that can changed based on what term number you are finding.  This was just our opening lesson.

Now, we wanted to introduce the vocabulary of terms, coefficients, variable, and constants.  I did that after this lesson and I probably should have held off.  That is a change I would make for next year.  This was a great pattern for them to learn what an expression is what it is for, but it is not right pattern to show what a constant is and how it works since the constant is zero.  I would probably start out with this pattern instead:

Pattern 2

Since the pattern is 4x + 1, we could talk about how the center square is the constant.  It is present from the beginning and in every term.  It is the 4 we add in the corners that is our growth and our “rate of change”.  I would also show a ratio table now and try to have them see the constant rate of change.  I would not talk about slope, but I would start to develop that visually for 7th and 8th grade.

I like the idea of not giving the vocabulary while developing a concept.  Let the kids discover the vocabulary and give it to them afterwards so that they can connect it to something.  Julie Dixon is very big on that which makes sense.

UPDATED!

After watching module 5 I realized two things about why this is important and that I must how a ratio table.  Understanding the constant and the pattern of growth is important in a later lesson of looking at two way tables using expressions.  We could come back to this lesson during unit 7 for this lesson.

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By creating a math moment that mattered, we will save time later on because we can reuse the task.  AH, starting to understand this now, I think.  In the example above we are showing direct variation and assuming we have no money, but we can add the constant in there.  I believe we will need to bring some type of context with the 4x + ! to connect it to the graph like the eating of the popcorn.  Maybe you have $1 in your pocket and you make $4 every week cleaning your room.  Seems lame, but by putting context to the problem, it helps the kids make sense!  I might need to improve on which pattern we choose to bring back.  Maybe use this patter, teach the ratio table and tie that into the x and y coordinate, and then give it context.  They kids would even see the constant as the y intercept which will help them in 7th and 8th grade.  If we look at the sample below as 19x + 31 we could give context to you have $31 in your piggy bank and you earn $19 per hour mowing lawns?  Just a thought.  Any other suggestions?

Pattern 7

Sparking curiosity with word problems

In the course “Making Math Moments that Matter” we have been challenged to transform textbook questions and make them into curiosity questions through withholding information, estimating, anticipating, or notice and wonder.  Here is my attempt to take a released item and turn it into a meaningful task.

 

To start, I would do a notice and wonder with the picture below.

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This created good rich discussions about the octagon.  The students noticed that all of the angles were obtuse, there were 8 sides, and that it looked like a stop sign.  I extended the conversation and asked if it had any parallel sides.  While some could see one set, after sharing they realized that this octagon has 4 pairs of parallel sides.  Great intro from a simple question.  You could give a “what do you wonder” in here and see where it goes.

Next, I showed the slide below:

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Something unexpected happened here.  A student struggled with the idea of what “side” meant.  I asked them to explain what they thought it meant.  The person responded with half of the octagon. WOW!  I am not sure I would ever have learned that a student would think that way if I would not have opened this question like this. Since there are no right or wrong answers, we valued their thinking and they drew what they thought.  This lead to a great discussion about what a side of a polygon is after I showed this next slide.

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Then we went to this slide and the students were starting to get the pattern.

 

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Now they were much quicker

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Now that they had the pattern, I decided to change it up and ask a different question so I showed this as the next slide.

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It was at this point I was blown away when the kids were working to hear a student make sense of this by saying “wait, if I remove one side and replace it with 2 sides then I am only adding one side each time!” WOW, just wow.  This student was then able to share her discovery with the class. It was like a light went off for everyone.  After this discovery, I showed the next slide.

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The students by now all had 12.

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At this moment I showed the students this slide.

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I told them that they just did a PSSA question.  This is a grade 4 geometry released item.  Now, this question by itself is alright, but the kids really just have to count the sides.  Is there sense making or curiosity sparked by reading the question and answering it?  By restructuring this question, I believe we sparked curiosity and created a journey of productive struggle.  We also have the kids making sense of the problem with a slow release of information.  I also was able to learn more about the students thinking which helped guide my instruction and also cleared up misconceptions not just for one student but for everyone in the room critiquing the work of others while listening and participating.

 

If you are interested in taking your teaching to the next level, consider taking Kyle Pierce and Jon Orr’s class.  This lesson was inspired by how Kyle showed us how to deliver a visual pattern lesson to our classes.  Why can’t we use the same technique for word problems if it fits?

 

 

How does the thousandths effect numbers

I have been taking an awesome class called “Making math moments that matter” by Kyle Pierce and Jon Orr.  I highly recommend it!  During module 2, we were challenged to create a task that follows the curiosity pathway.  Here is my attempt.

First, I would start out with a notice and wonder like below.  I would give the kids one minute to write down their notices and wonders and then do a think, pair, share.  We would have a discussion as to whether 2.999 is actually $3 or not.

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After our discussion I would then show this as a quick video.

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After the kids see this, we would have a discussion about how much a gallon would cost and then I would then show the gallons video/picture below:

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With this new information, in pairs, I would have the students figure out how much 5 gallons cost, then 8 gallons.  They will need to show all of their work and justify their answers.  We would model these answers through a double number line and a ratio table.  I would then have them predict what 10 gallons would cost.  Most kids should get $30.  Then I would play the next video which will reveal this.

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What just happened?  I have the kids discuss why it wasn’t $30.  After a discussion, I would have them show work for how much would 100 gallons cost?  How much would 1,000 gallons cost?  This would hit a misconception right on the head of just adding a zero.

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I would end the class with the original picture and ask the kids, “How did the .009 or 9/10 of a cent effect the price of gas? When did the price change and why?”.

The keynote for this lesson is here:

here is a video of how I would possibly deliver this lesson.