What’s for Dessert?

If you haven’t already purchased The Classroom Chef: Sharpen Your Lessons, Season Your Classes & Make Math Meaningful I highly recommend you get it. John Stevens (@Jstevens009) and Matt Vaudrey (@MrVaudrey) have written a book that is humorous, relevant and helpful to any math teacher, from the beginners to the veterans.

While I was reading the book, I came across “Choose Your Own Assessment”in Chapter 15 on re-thinking assessment (If I could I would quote this entire section):

“We teachers often judge the cooks of our classroom solely on the product of their recipe card – without asking them to pick up a spatula.  We are relying on one simple form of assessment to gauge the level of proficiency our students have reached on any given standard or objective.  To be blunt, it’s easier – easier for the teacher to create and monitor, easier to grade, and easier for the kids to prepare for.  Or so we think.” p.157

It was the beginning of May and the last thing on my mind was another test, especially after we just finished state testing.  I was burnt out, couldn’t find any problems worthy enough to create my own test  on Rigid Transformations and to be honest, I wasn’t looking forward to grading anything.  Instead of me creating something that would have tasted like a dry piece of cake (you know you like moist cake!),  I chose to have the kids demonstrate 5 learning goals in any format of their choice.  I sent the idea out to John Stevens and Ryan Dent (@4ryandent), fearful of this being extremely messy.  This was the end result CYOA – Rigid Transformations. Because I didn’t want to add to my sanity and have all the kids choosing their own problems, lets be honest that would take forever to grade, I gave them a Rough Draft/Project Outline with some options of problems for them to choose from.  This would make the grading easier on me, but still offer enough of a variety for the students to choose.   They had two days in class to work on it, a Friday and a Monday, and then it was due the following Monday.  The only way kids were able to work in a group was if they were going to create a video.  Students submitted their rough drafts to me at the end of the day Monday, I looked at them to offer feedback, and returned them Tuesday so they could get started on their final product. In reality, I kind of graded their project before the final draft, because most of the final drafts reflected my suggestions which made grading them fast.

Here are some of my absolute favorites for the videos.  I was looking forward to grading them and found myself watching some of the videos multiple times and on the weekends.  If I had to rank my top 3 videos to watch it would be

  1. Dasia p5 (for those of you on Twitter, this is the full video of the one I posted)
  2. Raven and Jaelyn p5
  3. Devin Brody Damanza p5

The video created by Devin, Brody and Damanza does have some language in there that is typical of middle school boys, like calling each other stupid.  However, they made some great connections between Rigid Transformations and activities they do every day.

Other Projects:

Allison: A triangle wanting to bake cookies…

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Emma: Comic about Toni and his math homework…

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Kaitlyn: Informational book…

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Marquez: Informational book…

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Now I don’t know about you, but these were way better than any test I would have created.  Thanks John and Matt for the idea to have the kids Choose Their Own Assessment!


Until Next time,


NCTM 2016 – “Take Aways” and Questions

“Take Aways”

  1.  Multiplication IS NOT repeated addition.  Yes ladies and gentlemen, this goes against everything we feel in our bodies but accept it. See Kevin Devlin’s Article from Mathematics Association of America.
  2. Trying to teach students to pick out “key words” in a word problem to help them solve does not develop sense making structures for advanced learning.  Not all key words appear in all problems and they don’t always imply the same meaning.  How the words are used depends on the context.
  3. Learning Goals versus Performance Goals.  (Huinker)
    • Learning goals are what students will understand as a result of the lesson (Promotes Growth Mindset).
    • Performance goals are what students will DO to demonstrate understanding (promotes Fixed Mindsets)
    • Sentence frame that combines both:
      • I can ________ to show that I understand _______.
  4. To increase rigor for Which One Doesn’t Belong, have students come up with 2 reasons why one wouldn’t belong. (Stevens)
  5. Go buy the book The Classroom Chef by John Stevens and Matt Vaudrey. My favorites from the book:
    • “Preparing a lesson takes research and new ideas to push the learning towards the objective.  Lesson planning is the step-by-step nature of a given set of objectives.” (p.18)
    • Don’t say an answer is wrong, say it’s not correct yet. (p.42)
    • Bringing in the math when you need it will allow the math to serve the conversation.(after comparing mullets, they eventually used math to be more accurate with the mullets). (p. 79)
    • “Problem solving is what you do when you don’t know what to do.” (p.110)
    • Instead of trying to entertain our students, make them curious with the tasks you select to implement.
    • Allow kids to choose their own assessment.
  6. Teach the child, not the standard. (M. Burns)
  7. To pose purposeful questions, we need to know which type of questions we are asking: assessing or advancing?
  8. A high quality task with low implementation will still create a better learning experience than a low quality task with high implementation.
  9. School isn’t something that should be done “to” us.
  10. Collaborate around student understanding, the mathematics and the next instructional steps to keep the PLC focused and on the right track.
  11. The problems in the book won’t look like the problems in real life.  We need to “delete” the textbook (D.Meyer) by eliminating the question, values, and labels.  Make students curious and turn the math dial up slowly.

Questions for Professional Development:

  1. What should students learn first: concepts or skills? (MTSS pre-conference)
  2. How do we define learning versus doing when establishing math goals? (Huinker)
  3. How can I use the problem as a vehicle to teach math? (Seeley)

Questions I Still Have:

I attended many sessions where presenters spoke about completing interviews with elementary aged children.  I am intrigued by this idea.

  1. What would it look like in the middle school?
    • Since I have 138 students.
  2. How do you know which questions to ask in the interview?
  3. How do you know which students to choose?
    • May be a dumb question but I was thinking those with Ds and Fs?
    • Separating the can’ts and the won’ts
  4. What do interviews look like at the middle school level with in a 52 minute period?


12 math rules that expire in Middle Grades

13 Rules that Expire



Clothesline Math

My first experience with Clothesline Math was at CMC 2015 in Palm Springs where I attended a session with Andrew Stadel (@mr_stadel) where we were solving multi-step equations on a clothesline.  I remember hanging in there at first with the basics, but when he got to the multi-step equations, the session moved faster than my brain could process.  This only intrigued me.  Then I starting following Chris Shore (@MathProjects) who I think created the clothesline. I was in search of some resources on how I might do this in my class. Thankfully my colleague Lisa (@mathgeeksrock) went to a training and received more information on how to implement the clothesline.  So naturally I went to her to figure this whole thing out.

It’s the week before spring break so I definitely didn’t want to start a brand new unit.  We are approaching a unit on solving and writing equations.  I couldn’t think of a better time to introduce the clothesline than right now, and Lisa agreed. I wrote the values from the cards on the whiteboard so the kids could start placing them on their whiteboards (a piece of card stock in a page protector).  After about 30 seconds, I passed out the cards and had students go up and place them on the clothesline. Students could make adjustments on their whiteboards as needed.  When we agreed on placing and spacing, they copied the final product onto their worksheet which can be found here.

We started off easy with whole numbers to get them used to the number line.  Then we moved to fractions and decimals to connect to equivalency.  Their second shot at fractions looked something like this…(Their first attempt was with benchmark fractions and they were pretty accurate)

During the discussion of placement on the number line, they referenced percents and decimals, so as that came up we clipped them together to allow them to make comparisons.

After many partner talks, and people coming up to change things they felt were wrong, they came to the picture below…

Note: The 8/15 and the 15/29 are in the wrong order, which led to the discussion below.  I didn’t snap a picture of the correct number line :/

The 8/15 and 15/29 created a great discussion/debate on proper placement.  The kids were talking about equivalent fractions, the difference between the numerator and denominator, the size of the denominator and what that meant in terms of the size of the pieces and how all of this related to 1/2.   There were many partner and table talks that happen with  this.  Although this may sound like a script between me and one or two students, it was a class discussion:

Me: So how do we know if the 8/15 and the 15/29 are in the right order?
Student: Half of 15 is 7.5 so 8/15 is 0.5 over what the half way would be.
Me: So how does that compare to the 15/29?
Student: Well, half of 29 is 14.5, so 15/29 is also 0.5 over.
Me: So what does that mean?
Student:  That both fractions are 0.5 over the half way point.
Me: So how do you know which is bigger then?
Student: What happen if we changed 8/15 to 16/30 by using a giant one?
Me: How does that help?
Student: Well then you can see that half of 30 is 15, and 16 is one over half, and therefore would be larger than the 15/29.
Student: But what if we do the same thing to 15/29?
Student: We got 30/58.
Me: How does that help us?
Student: It doesn’t really. The giant one makes 15/29 equal to 30/58, which is one more than half since half of 58 is 29.
Me: So now what?

The students seemed frustrated at this point because they thought they had an effective method of determining which was larger.  The period was coming to a close so I asked them, “How can the denominator help us figure this out?”  In a table talk, students realized that fifteenths are larger pieces than twenty-ninths so therefore, the 8/15 would have to be larger than 15/29.  I was satisfied with that response, but sad I didn’t have time to talk about this more. The students begged to check with a calculator using division.  They felt relief when they found out they were right.

When we moved to variables the next day, they were fine with the variable placement, until I made x equal to 4. I gave out the whole number cards of 2 and 1 which threw them for a loop (see middle picture orange cards to the right).  They weren’t connecting the whole numbers to what would be the equivalent x term.  They didn’t see a relationship between 1/2x and x.  Through some partner and table talks, they eventually figured out the ½x and ¼x were related to the x.

Then I gave them some expressions to place.  As you can see below, this was clearly confusing to them…NL5

Thankfully as I was walking around I saw that this student (and some others) knew exactly where to place each one.NL6

While students were talking and collaborating I heard a lot of, “If x were _____ then _____.”  They were using the same ideas from  placing ½x and ¼x  to place the expressions.NL7

The last day we tried 3x + 5 = 5x – 3 and 4x – 3 = x + 6.  For the first equation we placed 0, 3x, 5x, 3x + 5 and 5x – 3.  They did it right the first time.  So then I asked them, what if 3x + 5 was equal to 5x – 3?  I clipped the expressions together at this point.  This created excitement in the room.  We talked about distance on the number line and how we could use that to determine equivalency between other terms.  For example, the distance between the 3x and 3x + 5 was 5.  The distance between 5x – 3 and 5x was 3.  So the total distance between 3x and 5x was 8.  But it was also 2x.  This told them that 2x was now equivalent to 8 and they could now work backwards to figure out x. A sketch of what the number line would look like is below (from my notes).  The other problem looked similar.  The students applied the same idea of distance between terms and how we could use whole numbers and variable terms to represent the same distance.  When I wrote the actual equation they solved on the board, they were amazed.  They didn’t actually feel like they were solving multi-step equations.

NL8The kids really enjoyed this week of clothesline math.  It’s like I was doing number talks all week. One of my students said, “I wish we could do this everyday, this is fun,” which is always nice to hear.

Thanks to everyone for the continued inspiration and support as we try new things in our classroom. Now off to Spring Break…..



Comment Codes

Recently on Twitter, in the Principles to Actions (#NCTMp2a) slow math chat, the topic of Comment Codes came up.  The first week of March, Question #3 was, “What are some ways to response to student thinking in your classes?”

I remember reading on Fawn Nguyen’s (@fawnpnguyen) website Finding Ways that she grades using a highlighter.  During the Principles to Actions math chat, Frank McGowan (@frankmcgowa) talked about using comment codes.  Instead of writing the same questions and comments on EVERY SINGLE PAPER, Frank attaches a code to each comment or question. I believe he collaborated with his English Department on this.  Then when the assignment is returned, he gives them a reflection sheet which includes the codes.

So here is how I applied the same idea in my class this weekend as I was grading.  Frank, maybe you can offer your insights as well.

I gave a quiz on proportionality.  Every time I wanted to write down a comment or question, I wrote it one time on a separate sheet of paper and numbered them.  It looked like this:

  1. Where would the decimal be in $4?
  2. Is there a more efficient way?
  3. How could you label the picture?
  4. Why divide these numbers?
  5. If it’s 12 ft 6 inches, how would you represent that as a decimal?
  6. what would it look like if you used ratios?
  7. What properties must proportional relationships have?
  8. No sure I understand your strategy.
  9. How could you use the diagram to help you?
  10. What are you multiplying 4 and 7?
  11. What does the 2.5 represent?
  12. How is this like Growing Rectangles?
  13. Can the same situation be both True and False?
  14. Why multiply by 16 instead of 15?
  15. Does your answer represent seconds, hours or minutes?

These questions were all specific to one quiz.  So next to the question they got wrong, I put the number of the question(s) next to the problem so they would know how to address their mistake, or reflect on their work.



It might be better to label them a., b., c., …etc. as when they are numbered it can be confused with point value.  This took less time and I found it to be way more efficient.  You can also make the comment codes key more of a reflection worksheet in which the students have to interact with the questions you asked them to understand their mistakes.  There are lots of opportunities here for students to develop their learning.

Thanks Frank, Andrew and Robin for chatting about this.


Connections with Proportionality Part 2

I have been spending a lot of time with my students on what it means for two quantities to be in a direct proportion.  Instead of working out of the textbook for the unit, I have done the following tasks:

  • Growing Rectangles
  • Vegetable Garden
  • Skittles
  • Track Meet (I didn’t blog about this one because it was a hot mess and I totally anticipated student responses incorrectly.  Although it’s a great task it won’t be in this sequence next time)
  • MARS Proportion or Not a Proportion

There are already many things I would change about how I implemented these  tasks, especially the Skittles Task (which probably could be an entire unit within itself). After the Skittles task, I gave a quiz.  Most of the kids totally bombed it.  I was so confused.  I felt like the Skittles task was so rich that they should have done better. They lacked the ability to make connections when the context changed. Then when I started grading their Skittles packets (they held on to each piece daily and turned it all in at the end), I realized that even though they were participating in the discussions, they were not writing down other student strategies presented in class.  They didn’t record how the ideas and strategies from class related to their own.  And sadly, the implementation of my new homework (HW) reflections has been mildly successful because they are just not doing their HW.  The kids that do however, have a greater depth of knowledge than those that didn’t take it seriously.  There are many reasons the kids bombed the quiz, and it’s not totally their fault either.  Because of this, I decided to do the MARS activity Proportion or Not a Proportion.

I gave them a pre-test to see how they were thinking about proportional relationships.  I graded them but didn’t assign a score as this was strictly formative. I only wrote questions for each problem they got wrong (yes this took FOREVER).  Then we did the MARS activity, which really helped them work through what it means for quantities to be in direct proportion.  After the activity, I gave them a similar quiz and saw a lot of improvement.

Even with the improvement, the bad quiz scores from before still weighed heavily in my mind because I keep thinking, “Man, I must have done something wrong.”  So on a Friday I gave a short response quiz.  I didn’t want the students to wonder, “Why are we doing these isolated activities?”  I knew they weren’t isolated tasks but did they? So I asked them these questions and requested no less than a 5 sentence paragraph for each one:

  1. How are the activities Growing Rectangles, Vegetable Garden and Skittles all similar?
  2. Does Growing Rectangles represent a proportional relationship? Explain how you know.
  3. Does Vegetable Garden represent a proportional relationship? Explain how you know.

This was also a open note quiz because I wanted them to take out all of their tasks, make generalizations and look for similar structure within the problems.  I was thoroughly please with the majority of the responses.




However, not all of the responses were as good as the ones above.  I’d be lying if I said they were.  I realized that my class that struggles with discourse during class has more trouble connecting the activities.  Their responses went something like these:

“The similarity between the Growing Rectangles, Vegetable Garden and Skittles activities are pretty much they same.  The reason why they’re pretty much the same is because the math that’s involved in it.  You have to find out what goes where and for all of them we made a graph that was also the same as the rest of the activities.”

“The Growing Rectangles, Vegetable Garden and Skittles activity all include multiplication, addition and not proportional relationships. The Vegetable Garden and Growing Rectangles only included (multiplying, addition and areas of ft. or inches).  The Skittles activity included multiplying, addition, doubling, division, number lines.”

“All include graphs, numbers, and answers. Also involve lots of thinking. These activities work without a graph but if you didn’t use a graph, you probably got something wrong.  The activities we did use addition, subtraction, multiplication and division.  Growing Rectangles, Skittle activity and Vegetable Garden all have missing numbers and you have to figure out those numbers by divide, multiply, add, subtract, graph even a number line.”

These are somewhat comical.  They are a great example of writing a whole lot about nothing.  However, they show what they don’t know.  So clearly there is more work to be done on my part.

Next Steps:

I’m really struggling with discourse in one of my classes.  I ask them to partner talk and either they don’t at all, or they talk about something else, even when I’m standing right there!  And even when I ask them a low level question that everyone should have an answer to.  Looks like I need to go back into Principles to Actions for some answers. Do you have advise?

Homework will always be an ongoing battle that I am not willing to fight.  I just know personally that the kids that do spend time on their reflections and do them thoroughly have more depth than the other students and will be more successful later.  I guess I will have to be happy with that.

So what now?  I’m thinking of creating a HW assignment where students actually graph on a coordinate plane some Growing Rectangles.  Then they will see that the rectangles that are proportional to each other fall on the same line, which may clear up some misunderstandings. Then they can make more connections with this new graph to the graphs they did in the Skittles activity.

Until next time…S

Skittles in 60 Seconds

I’d like to begin by thanking those involved with Emergent Math for organizing and compiling (and creating?) great tasks.  This task originates from John Berray.  He originally designed this activity for an “Intermediate Algebra class, which is a lighter form of Algebra 2. Really, this activity is a first year algebra activity that can extend deeper depending on the course.”  I have modified his activity based on his suggestions from his blog, and made it fit 7th grade standards for Ratios and Proportions.

Day 1: Notice and Wonder

This was a minimum day strictly dedicated to noticing and wondering about the scenario cards.  What do YOU notice?  Math concepts they notice and wonder about involve slope, unit rate and y-intercept and we haven’t even started the activity yet!

Notice wonder Skittles

Day 2: Data Collection

If you are doing with Skittles, buy 3 – 2 pound bags for two classes of 38. It was a bit crazy at first.  The students realized how hard it was to actually consume the amount of Skittles they were supposed to in their given scenario.  I may try M&M’s next year.  However, this led to different strategies for collecting data.  Some shoved their faces full of Skittles to the point of being sick, while others strategically set aside the amount of Skittles they were supposed to eat to be able to count the total a little easier.

Skittles data 3

Day 3: Unit Rates

They were asked to figure out who ate the most skittles per second. I was a little disappointed with the results.  Of course I had 5 or so students in each class who knew exactly what to do and immediately had the right answers with realizing that something was wrong with scenarios A and B.  Then the typical mistake was to mix up the order of the quantities in their unit rate.  But everyone else was lost.  Blank papers, weird strategies I hadn’t even anticipated and the questions I was asking them to move them along weren’t getting them anywhere (which means I was asking the wrong questions or I needed to come up with something else).  I had anticipated incorrectly. We didn’t get to finish this as planned so I needed to re-evaluate questioning/other strategies.

Day 4: Unit Rates cont…

Skittles Per 2 data 1

Revisiting the idea of Unit Rate, I knew I had to get them to the meaning of their answers. We spent the entire period talking about these 4 strategies.  Some of the questions I asked during the sequence referred to the  last two strategies at the bottom:

  • What do you notice?
  • How are the similar?
  • How are they different?
  • Why divide?
  • What does this mean in terms of seconds and Skittles?
  • How do we know which one is right?
  • Is there another way we could write this?
  • How can we use a Giant One?

The students struggled with defining what their answers meant. We focused just on person A when we were talking about the 0.6.  Is this Skittles or seconds?  The class was split 50/50 and it led to a great debate about why it represented Skittles per second. I wanted them to write ratios do help develop the idea of rate vs. unit rate and they just weren’t going there with their strategies.  Finally after many (and I mean many) table talks, they could remember what ratios were (i.e. 3:4, 3 to 4 and 3/4).  And then we decided which form of the ratio would be most efficient to use (they chose fraction form). They did come up with the idea that if I’m eating 3 Skittles every 5 seconds, then that is less than one Skittle per second, so the answer has to be less than 1 (which cleared up the error in the representation of the division).  To link back to their prior knowledge, I asked them about the “Giant One” a concept from the CPM text.

This led to a discussion about whether or not we should multiply or divide Skittles Per 2 data 2for equivalency and what that meant in terms of our answer as it applied to the context of the problem.  This tied back to their understanding of equivalency with rational numbers and they talked about how the to two answers could be the same, which we defined through table talks.  Now they could apply this to the rest of the scenarios. Finally, Success.

Skittles Per 6 data 1

A student in my 6th period noticed that person A and B both start with a number, and that when we try to find the unit rate, we get a different answer each time.  This wasn’t the case with scenarios C and D where no matter what we divided we got the same answer.  So then another student asked, “What would happen if we re-did the chart without the Skittles that they started with?”  I was SO HAPPY that this question came from a student and not me.  So she made a new chart without the starting amount of Skittles and started to look for patterns in numbers.  Now they noticed that they were getting the same answers as when they were dividing with the rate.  This took 2 full days plus a minimum day to get here.  But I have to remind myself…

“Teach at the speed of learning.” Phil Daro

And sometimes learning is slow.

Day 6: The Blank Graph

Unfortunately I had a sub when the kids analyzed the blank graph, but she is a seasoned sub and could trust her not to just give the kids the answers. First the kids did Notice & Wonder.  I reviewed the discussion questions with them again the next day to determine their level of understanding.   Discussion questions were:

Skittles Per 6 data 3

  • How do you know you labeled the graphs correctly?
  • Who takes the longest to eat? How do you know?
  • How is that related to what we know about the rates and unit rates?

Students were able to connect that if you eat slower, your unit rate would have a smaller value.  We already realized that if we eliminated the Skittles that person A and B started with, that we get the same unit rate, so 0.6 for Person A had the smallest value, and therefore would not be a steep as the other lines.  If this were an Algebra 1class, I would have loved to go more in depth about what it means when Line 2 intersects Line 3.  The kids thought this was interesting and noticed that Line 3 was eating more than Line 2 at first, but then Line 2 ends up eating more.  They related this to the fact that Line 3 which was person A, started with Skittles which accounted for why they were eating more at first.

Day 7: Labeled Graph

One of the major goals for Day 7 is for the kids to use the graph to determine the Unit rate, and find the relationship between the Unit Rate and the starting amount of Skittles.  Again, we “noticed and wondered” about the graphs and then they had to label which person belonged to which line.

Skittles Per 6 data 4

It was nice to see that some students used the amount they started before the time, the amount of Skittles eaten total, and a connection to the Unit Rates to justify their thinking.  The student below found a coordinate from the graph and matched it to data in the table to justify his answer.

Skittles per 2 data3a

Overall, I have been pretty pleased with my students’ ability to read the graphs and make connections between the graph, data, rates and unit rates.  Monday we will formalize the unit rate from a graph as the point (1, y).  Depending on how long that takes, it might be fun to push them into using similar triangles to see the rates! A girl can dream right? Ha!

Day 8 or 9…? (I’ve lost count)

Finally we come to the equation part of the activity. Something I didn’t anticipate was students writing equations in both minutes and seconds. First the tried to figure out how many Skittles person C would eat in 5 minutes.  I used the strategies from 5 Practices on sequencing work.

Skittles per 2 data 4

Then I asked them to come up with a more general equation that would work with any amount of time…

Skittles per 2 data 5Skittles per 2 data 6

It didn’t take them very long to come up with the equations, and as you can see they did it in seconds and minutes.  Then they applied what they did for person C to person D. We had to review a little bit the different forms of the Unit Rates and what they might look like.  This seems to keep throwing them off which means they haven’t really grasped what the unit rate is, as they keep confusing it with the rate.  After realizing this, I know my next task has to deal with Unit Rates. It’s like I keep asking them, “And where did that number come from again?”





What is interesting is what came next.  I asked them whether or not these types of equations would work with Persons A and B.

Skittles per 2 data 7Skittles per 2 data 8

They knew what the format of the equation would look like and they clearly defined what each part of their equations meant.  I couldn’t take pictures of their work because they were honestly arguing about what to write down, so I just recorded the class discussion.  They knew that the equation in seconds had to contain some type of constant.  They remembered that there wasn’t a constant when we divided the values in the chart until we took out the amount that they started with.  This led them to the 0.6.  I could see now why they were confusing the eating 3 skittles every 5 seconds as a unit rate.  Originally this was the rate at which person A ate the skittles.  However when dividing the values in the chart, they didn’t get .6 until removing the amount that they started with.  So Person A ate at a constant rate of 0.6 Skittles per second AFTER they had already consumed 6 skittles.  They even noticed this on the graph by drawing slope triangles (We didn’t call them that but they noticed it).  We tested our equations to see if they worked for figuring out Skittles consumed in 5 minutes, and noticed that they answers didn’t match.  Therefore they started tweaking the equation in minutes.  They realized that the 42 already included the 6 Skittles already consumed which led them to change the equation from y=42x + 6 to y=(42 – 6)x.  When we tested it again, we still got a different answer from what he had with the equation that was in seconds.  Then someone realized we needed to add the 6 back in (which is the blue 6 I added when this came up).  So now our equation went from y=(42 – 6)x to y = (42 – 6)x + 6.  So I asked them, “Why subtract the 6 and add it again later?”  They knew that in the equation y=(42 – 6)x they were accurately figuring out how many skittles were eaten at one minute, however they still needed to add the 6 to find total amount of Skittles consumed for any given amount of time.  We checked it again with 10 minutes and it worked.  When I asked them to apply this to writing an equation for person B, it was a piece of cake.

Without giving them a formal definition of what it means to be proportional, for their HW they had to list characteristics of what it means for quantities to be proportional to each other with regards to the Scenarios they were given in the activity, knowing that if I just give them the definition this whole activity would be a waste.  They did this on the very bottom of the handout in the chart.


So when I checked the HW that they were supposed to complete (filling in the chart), only 5 out of 37 students completed the assignment in one of my classes.  I was so frustrated and linked this to lack of perseverance.  They want me to just give them the definition which isn’t meaningful to them at all. Then when collaborating with a colleague, I realized that maybe my mistake was assigning that part for HW.  My 6th period did a much better job on pulling their thoughts together (or maybe they heard how upset I was and made sure they did it before class).  They do a better job at persevering through mathematics and making connections in general.

Skittles Per 2 data 9

The 5 students that did the assignment came up with the ideas on the left.  For the most part they had the right ideas but it wasn’t as specific as I had anticipated.

Below are the responses from my other class, which are a bit more thorough.  I felt like I didn’t even need to do direct instruction at the end of this class because everything I would have said what was included in these student responses.

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Next Steps:

Because of their lack of understanding Unit Rate, I’m going to do a task from illustrative mathematics called Track Practice.  Then use the MARS activity Proportion or Not Proportion to solidify their knowledge of proportional relationships.


In a perfect world, there would be no weekends and no absences so we could have just continued the lesson straight through.  Me being out one of the days and two (or 3?) long weekends made it a bit more difficult at the end when I was trying to help the kids pull all of the pieces together. The struggle is real right?  The lesson is posted here and I am always open to suggestions on how to make it better. I made some notes in the PD column about possible changes to the activity I may make next time. So what do you think?

What are we teaching: Reflections on Phil Daro

Phil Daro: Face to Face with National Language Experts from University Washington Tacoma on Vimeo.

It’s clear that in the United States we have a culture of education in mathematics (maybe in all subjects) that is a mile wide and an inch deep, too many topics and not enough time on each topic.  We need to SLOW DOWN! Phil Daro agrees that our curriculum has too much junk which begs the question What are we teaching? He also says the CCSS was developed to be more coherent than our previous standards, yet teachers are still struggling to “fit it all in.” Phil Daro addresses many of the barriers we as teachers face that are issues larger than our classroom.

I myself have felt the struggle of presenting a meaningful task in my class worried that I am getting further behind because it is taking longer than I had anticipated.  However, the major differences between the U.S. and countries like Japan and Singapore is that in the U.S. we teach to cover, whereas the other countries teach at the speed of learning.  The Skittles task I have been working on with my students is moving extremely slowly, but I remind myself to be patient, teach at the speed of learning.  The U.S. is behind because we are moving too fast through the curriculum.  When we move too fast, the learning is lost because their is no depth.  When we slow down we will reach higher standards, allowing for Universal Access and coherent mathematics between topics and grade levels.  So how do we Slow Down with all of the pressures created by the state and districts?  The standards for mathematical practices are part of the solution.

The standards for mathematical practice (SMP) are the expertise that go with the content. SMPWhen implemented and utilized efficiently, it forces you to slow down.  Evidence of the SMP is what we need to be looking for in each other classrooms.  Daro gives an example of how we can help students construct viable arguments and critique the reasoning of others.  Students need to be saying second sentences.  To be able to respond to a strategy and then create an argument with evidence, it takes a minimum of 2-3 sentences.  As teachers, we need to be ready to increase wait time.  This is hard as it is something I still struggle with, however we need to practice patience and use open follow-up questions: Why does that make sense to you? Is that always true?  If we move too quickly to the next student because we know he/she has the answer, we are showing we value speed over understanding math.  What we end up really doing is making the math more accessible by taking the math out of the work of the student.  If we take the thinking away, we take the math away.  This reinforces the idea of needing to go slow down, we can’t learn and do math if we are in a hurry.

Daro also talks about students being precise (SMP 6).  We not only need to be precise with our numbers but also in the use of language.  Students need to start creating their own definitions for mathematical topics within a given context and use reasoning from those definitions to develop ideas and arguments.  If teachers have students memorize definitions or concepts without context, then we have again resulted to answer getting without meaning. It’s the same as being given a procedure without knowing why it works.

We also need to change the culture of our classroom.  When students respond to a question who is the audience?  Typically they think it’s the teacher. The goal should be that students explain strategies so that other students can understand.  When the audience is the teacher, we transition from understanding to answer getting.  Answer getting results in low expectations.  Obviously within the student’s explanation there will be an answer, but the answer shouldn’t be the main focus.

Changing the culture of the classroom is not easy, especially if you are trying to make changes mid-year.  How do we change it? Two-thirds of the time we should:

  • Present a problem. Maybe even conceal the question part of the problem so students can comprehend the context.
  • Students then independently work on the problem to develop their own thinking.
  • Students pair up with a partner and work in pairs (or tables), each student producing his/her own product.
  • Prepare an explanation/presentation.

The teacher needs to have already prepared a summary of the lesson in advance.  As the teacher selects student work to display, it should be sequenced from easiest to grade level.  Then the teacher facilitates conversations to build and connect student representations.  In the end, the teacher should summarize the mathematics quoting student work in the summary as examples of the teacher’s points.  When the structure of the classroom is set up this way, we can bring students to readiness by gathering prior knowledge, which will prepare them for the direct instruction at the end of the lesson.

What does this mean in terms of Professional Development?

We need to overcome the idea of teachers having to know everything up front.  We learn every day, every year. Let’s give ourselves a break!  There have been many times I am learning the math with my students as they teach me different ways to look at things and have different ideas than what I had originally anticipated because their experiences are different.  Daro says that workshops are necessary but weak,  as they will never solve our problems.  The only way to get better is in the school, on the job learning that is more closely related to what we are doing every day.  Teachers need to create a culture where we learn shoulder to shoulder with our colleagues (Daro references David Cohen).  We need to be collaborating on lessons, what happens during the lessons, and how our students responded.  We need to be in each others classrooms observing each others pedagogy so we can critique and ask questions about instructional choices.  Until teachers accept this responsibility, everything else will be weak. How effective is a workshop if we can’t see how the strategies learned in the workshop are implemented?

Making Connections with Meaningful Tasks

After reading Jo Boaler’s book Mathematical Mindsets and collaborating with other educators, I was determined to change what I do and STICK with it!  As we came back from Christmas Break, I started my students off with a task called Growing Rectangles from Boaler’s book. With the help of the Principles to Actions math chat on Twitter #NCTMp2A, I was able to really narrow down the learning goal for the lesson…

Students create rectangles and through applying a scale factor of (k), discover the area of the new rectangle is k² larger than the original.

The first thing students had to do was create 5 rectangles with an area of 20 cm².  They quickly realized there were only three options with whole numbers.  Using focused questions, we discussed other options.

GR slide 1

Once students had their rectangles, they had to enlarge them by a scale factor of 2.  Without even knowing what scale factor means, I asked them to predict what they thought it meant based on the context.  They were able to define it themselves.

Then we played a little Notice & Wonder.  Many of them noticed that the area was 4 times as large as their original.  I asked them, “Does this always happen?”  To test this theory, they proceeded to choose a rectangle with an area other than 20cm² to see if it had the same pattern, and we continued to Notice and Wonder. Two responses are below.  Students discussed their ideas and noticed in the second example to, “take the 2’s that you multiplied with you add them and get 4 which is what the area is multiplied by.”  Knowing that adding was a mistake, instead of addressing it at the moment and taking over, I asked the students to test this theory.  Do we add the scale factor from both sides or do we multiply? (my handwriting in blue came after question 5).

GR slide 4

To test the theory of adding vs. multiplying they did this…

GR slide 5

Selecting the work here was very important.  I wanted to select something that wasn’t too hard for them to make connections with.  So I chose groups who created a rectangle with an original area that was a multiple of 10 before they scaled them by 3, 4, and 5.  When I put the student work up on the left, many students immediately noticed that the area was no longer being multiplied by 4.  They realized here that when applying a scale factor of 3, the area of the enlarged rectangle was 9 times as big, a scale factor of 4 created a rectangle with an area that was 16 times as big and a scale factor of 5 created a rectangle with an area that was 25 times as big.  At this point, I drew them back to the responses to #4 and asked, “So what does that have to do with this?” They immediately said that 3×3=9, 4×4=16 and 5×5=25 so we don’t add the sides we multiply them (this is when I changed it in blue on the slide above).  Then I asked them, “Does this always work? How do we know?”  After displaying the student work on the right with rectangles that had an original area of 20cm², they realized that this always worked which led to…

GR slide 6

My writing in blue was a record of student debating in class.  They eventually came to the understanding that applying a scale factor of k will create a rectangle with an area that is k².  I was really pleased with how successful this was as I planned on linking the idea of scale factors to scale drawings and proportionality.

The next task was taken from CPM. The task was a problem I did last year for the first time.  I made the same mistake that my students did so I knew it would be a task worth doing.  First, as a class we did an entry level scale drawing problem which included Notice & Wonder, many table and partner talks and coming to a consensus about the values the kids were assigning to the problem and whether or not they made sense.

The scale drawing at right shows the first floor of a house.  The actual 4-13dimensions of the garage are 20 feet by 25 feet.  All angles are right angles. What are the actual length and width of the living room (in feet)?

Next they completed the following as a task in their groups…

Hank is planning his vegetable garden.  He has created the scale 4-15drawing at right.  The actual area for the tomatoes will be 12 feet by 9 feet.  All angles are right angles.   If the horizontal length of the zucchini plot shown in the diagram is 1  5/8, what is the area of the real vegetable garden in square feet?

We spent a lot of time Noticing and Wondering on this one.  The more I used that strategy, the more I see how the basic “notices” turn into more “math specific notices” which definitely help focus their thinking.

4-15 SW

As they were noticing, I recorded their “noticings” on the board.  Eventually, they figured out that the scale was 1 in. for every 6 ft. They also noticed that tomatoes are really a fruit, not a vegetable, so I had to re-define that part of the question to having them find the area of the entire garden. After “noticing” they wonder the questions I’m going to ask them anyway!

I gave them some time to get started.  I walked around and kept seeing blank papers.  They had done a great job noticing things so it wasn’t making any sense to me why they weren’t writing anything down. I secretly started freaking out inside because I wasn’t sure about my next instructional move.  As I was freaking out, I was reminded of a session at CMC that I attended presented by Matt Vaudrey.  I remember him asking , “What questions do you have?”  So I stopped class and said to them, “It seems like some of you are struggling.  What questions do you have?”  They asked…

  • Do we keep the fractions in fraction form?
  • How do we convert inches to feet?
  • Do we need to pair up gardens and add them together?
  • How many feet is 1.625 inches?
  • How many feet is 3.125 inches?
  • How do I know if 3.125 inches is equal to 18.75ft?

We table and partner talked with these questions above (the first three were from a different class than the last three).  I’m glad they asked how to convert inches to feet.  In response to that question I asked them, “Well how is this related to the growing rectangles problem? How are they similar? How are they different?”  They instantly saw that if 1 in. was equal to 6 ft., then the scale factor would be 6, which led them to the idea of multiplication which led them to knowing how to turn inches to feet. They were confusing the fact that 1 foot has 12 inches with the scale in the problem. I also noticed that many of them were getting hung up on computation which was getting in the way, so I allowed them to use calculators to eliminate that struggle.  Here is what they came up with and how I sequenced it…


This first table had minor computation errors.  What the students noticed here is that they took the scale factor of 1 in. = 2 ft. and changed it to .5 in. = 3 ft. so they could easily work with converting the decimal part of 6.25.  I was impressed with their method as they were already thinking about proportionality. The solution of 37.5 ft.² came next in the sequence (the typed info on the slides came after students made connections through table/partner talks).  This helped them find the error in the previous one.  We spent most of the time comparing 37.5 ft.² with 225 ft.².  Which one makes sense? How do we know? I wish I would have video taped this lesson because I can’t even remember all of the questions I was asking them to get them where they needed to be.  One student did say, “if the tomato garden is 12 ft. by 9 ft., then the area is 108 ft.² so we know that the 37.5 ft.² can’t be the right answer.”  Through asking, “What does this/that mean?” (I feel like I ask this a lot) and having the kids continue discussing to find meaning in the numbers, they realized one group found the area in inches and then tried to scale it to feet, while the other scaled the sides to feet first, then found the area.  Some seem confused because the first strategy made sense to a lot of them (which was also the mistake I initially made when I solved this problem).  So I posed the question, “What if this were a growing rectangle?  How would that change things?” To which a student replied drawing the image below.


They hadn’t really made the connections but were almost there.  So I went back to the slide with the work above and posed a new question, “How are 37.5, 225, 6.25 and 36 related?”  Sad that the bell rang, I asked that they reflect on the question at home for HW and be ready to share the next day.  I was pretty pleased with the responses (please forgive their grammar, they have the right mathematical ideas).


Because of the growing rectangle activity, they realized that the area of the vegetable garden in feet had to be 36 times as large as the original.  If the original area is 6.25 in.² then 6.25 × 36 = 225.

If 6.25 × 6 = 37.5, the area of the new rectangle was only 6 times the original, therefore 37.5 × 6 again would give them 225, an area 36 times as large.

Then to check their understanding I posed the question, “So where are the 6’s in 18.75 × 12?”  I thought I would stump them because there are no 6’s in the problem when you look at it.  I didn’t stump them at all.  They knew that the 18.75 and the 12 had already been scaled from inches to feet so (3.125 in. × 6 ft.)(2 in. × 6 ft.) = 225 ft.².

I stressed out about how long it took to complete these activities, but I knew the learning was worth it. Fewer tasks with depth are more meaningful than multiple lessons that “practice” procedures and don’t develop understanding.  I went home feeling like THIS is why I teach EVERY DAY!

Here is the handout I gave my students. 4-15 Task

Next post: Reflections on Skittles in 60 Seconds (7.RP.1 & 7.RP.2)


The need for change

I am in my 10th year of teaching and while I still enjoy it, there is an overwhelming need for change.  There have been many change agents that have moved my progress along.  Professional development at my district has been eye opening and has definitely influenced my instructional decisions on a day to day basis.  Taking the time to read has made me realize even more that I still need to make even more changes. Books I highly recommend are:

  • 5 Practices for Orchestrating Productive Mathematics Discussions, Margaret Smith & Mary Stein.
  • Principles to Actions: Ensuring Mathematical Success for All, from NCTM.
  • Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching, Jo Boaler.
  • CCSS Math Frameworks for Grades 6-8.

Not to mention the huge change from legacy standards to Common Core Standards.  For the past two years I have attended CMC and listened to Robert Kaplinsky, Dan Meyer, Andrew Stadel and Fawn Nguyen which motivated me to sign up on Twitter so I can follow  them and their ideas.

Still unhappy with the structure of my classroom, I emailed Fawn Nguyen to ask about the structure of her classroom.  Through collaborating with her, I changed everything for the second semester.

First Semester Structure:

  • Students come in, copy agenda and immediately grade HW.
  • During HW, students come to a consensus about answers and strategies used.
  • Lesson for the day from CPM textbook (instructional strategies from 5 Practices embedded within lesson as I anticipated student strategies).

Second  Semester Structure:

  • Back to Warm Ups
  • Review Homework (HW)
  • Classwork is now split between Tasks and Lessons from the text.  The goal is to do 1-2 tasks a week like Fawn, however if a task has a lot of “meat”it obviously takes more than one day.

I noticed while working with CPM (a textbook our district is piloting) that I didn’t have time to do the warm-up in addition to grade and go over HW. So first semester I eliminated warm-ups completely.  But the more I read and evaluated my students, I realized that I needed to implement warm-ups that build Number Sense, not just practice problems (especially if they practice them wrong). So I modeled my HW similar to Fawn’s model.  HW is still assigned, however I don’t stamp, grade or enter it in the grade book daily like I used to, which now gives me time to do meaningful warm-ups.  I also give them all of the answers to the HW so they can self check at home.  The catch is that they have to show their work and they can’t redo a test or quiz unless they have shown an effort to keep up with HW.  I spot check every now and again.

My new HW is modeled after Jo Boaler’s book Mathematcal Mindsets.  Students are now given reflection questions based on the task or lesson which are turned in for a grade.  I wanted HW that is more meaningful, allowing students to reflect and make mathematical connections or ask questions that can be found belowHW Reflection questions blog

Students typically answer one reflection question a night based on the task.  Most of the time I choose a list of reflection questions they should be able to answer from the lesson, and the students to choose one of those.  I do have a higher turn in rate with the reflective HW assignments than I do with HW from the text.  And when I read their reflections, they actually “get it” which is rewarding for me.  Students are making connections like I’ve never seen before.

Homework Reflection Questions