Monday, March 27, 2017

@fractiontalks using @desmos activity by @ms_hansel Part 1

I saw Alison Hansel tweet out a super cool Fraction Talks Desmos activity she used with 4th graders after she posted some student work samples. I figured it would be perfect for my 6th grade math support class.

Now in a 53 minute period, I only got through TWO slides. And that's OK. It was a rich discussion. And we will definitely be returning to finish this activity at a later date!

I believe the images are from and she did the heavy lifting of selecting thought provoking images. This activity takes advantage of the sketch tool in a perfect situation. When working with these on paper, students can get lost in what they are subdividing shapes into. The beauty here is they can label, draw lines, and erase. Then they can describe their thoughts with words, submit to class, and see small sketches and explanations of their classmates.

I also used teacher pacing so students would be thoughtful and not just rush on to the next slide. I then PAUSED (big groan from class), asked for Chromebook screens to be almost closed to an acute angle, anonymize class responses, and review students work. Students were slowing it down by claiming work and shouting out it wasn't theres, so it went slowly as I had to remind students of my expectations.

MANY students made the following mistake on the first slide:

It is the classic mistake of seeing 4 shapes, and since 1 of the shapes is red, then 1 out of 4 shapes are red, or 1/4 of the shape is red. I asked students who disagreed why they did. Eventually they said that you can't compare them if they are not the same shape.

Then students saw the couple examples that were correct. They could see that these students had divided the other colors into triangles that were of equal size to the red triangle. My colleague had an interesting thought that some students might have said 1/4 because they were color blind and couldn't see the red and orange. I don't believe many students thought this or else they may have said 2/8 is red.

As we all know, the correct answer is 1/8. One student actually said 1/7, and I wanted to honor what was right about that. That student had described the part to part ratio, or in other words, there was 1 part red to 7 parts NOT red. This thinking is clearly further along than the 1/4 thinking.

Once we had discussed it in depth, we moved on to the next slide. Students felt much more confident and were much better equipped with strategies to succeed with this image. They immediately started dividing everything into triangles and said it was 3/8. Their explanations got A LOT better too.

And I love the anecdote about the student below here. He had come up with the answer, and kept disagreeing with all of the people that had put 3/8. I saved his for last to analyze.

So, I wrote 6/16 on the board, and 3/8 next to it. I asked the class, "Who is correct?" One student bravely raised her hand and said, they are both right. 6/16 simplifies to 3/8, because you can divide both numbers by 2. Mind you, students had worked on that skill in recent weeks but this student was able to articulate a context in which there would be a NEED to know how to simplify.

The rest of the activity has a few more sketches, and I believe uses the Card Sort feature to do some flag matching, but as I said, we didn't get more than 2 slides in, but it was a very rich discussion and I'd like to thank Alison for providing the link. I will definitely bust this out with my 7th and 8th grade math support during some free time at some point near the end of the year.

Wednesday, March 22, 2017

Fishbowl... A @CPMmath Study Team Strategy

With one third of the school year left, I felt it was a good time to try out a study team strategy I had never done before: a Fishbowl. I remember doing it at the Academy of Best Practices at Seattle University and finally decided to use it. I think students needed a strong reminder of what the study team norms were.

So, I picked 1 group that seemed to work well together the day before, and a table next to them. I then asked the whole class to get their composition book and pencil and without talking surround the two tables. We were going to silently observe them use their study team roles to start and complete the first 2 problems of the lesson 8.2.2. I selected these because the first problem set is review of exponents from yesterday's lesson where they had to write the multiplying of power numbers in a simpler form. The second problem involved error analysis where the student correctly expanded the powers, but put addition signs between them and multiplied them out instead of writing them in simplified exponent form.

I asked students take notes on positive actions and actions the team could improve upon. After they finished, we would debrief right there, of course starting with the positives. Then we talked about what they could improve, and then they returned to their seats and I said now show that you can do it just as good or better!

In first period, here were my observations:

First off, the facilitator immediately asked, "who wants to read?" I then heard one student claim they read first last time, so they didn't want to read. The person that agreed to read was reading fast and mumbling. They weren't reading clearly. I think it was to draw attention to themselves...? J was saying his work aloud, saying how he was expanding the power number or writing it in factored form. T from the other group was doing the same. J asked V, "Are you on 8-60?" One member didn't write it in simplified exponent form, but went back and fixed it. When we debriefed, students noticed that two of the group members barely said anything. In the other group, I complimented I for responding to a students answer by saying, "OK. Why?" That was a great example of a student prompting another for their reasoning. We also talked about how the recorder reporter should have been looking at everyone's paper to be sure it was written correctly to move on. The next day one student asked if we were doing a fish bowl again. I said, "No. Why?" He said that if he was in the group that's the fish bowl he could get a head start on his work... (He usually works ahead, but has gotten better at it with my reinforcements of not doing so)

In second period, a student started reading right away, but skipped the introduction! In the other group, they started reading right away, but the introduction first. Group 6 moved on when they were done, and confirmed with their group that everyone was done. Group 7 moved on to the next problem, without going over the answers to 8-59. I observed some weren't finished, and didn't speak up. Group 6 moved on, but didn't write it in simplified exponent form. In the debrief, we saw how both groups started quickly. One moved on too quickly without making sure the group was ready. Also, group 6 didn't fully read the directions of writing it in a simpler form. This reinforced making sure you understood what the question was asking.

In fifth period, both groups started reading quickly. A suggested an idea. A also asked, "Are you done writing that down?" In the other group, M confirmed that they agreed with another group members idea. There also were no arguments about who wanted to read. One higher student didn't have the courage to speak up about disagreeing about the team's idea. C suggested an idea. One student said, "It'd be like..." which I mentioned in the debrief. as a good way to describe how you are going to write it, and to get confirmation from your group. When students shared, they said that A was dominating the discussion and not giving other team members a time to share their ideas, or use wait time. I talked about how as teachers we need to give students wait or think time, and students need to give each other time to think also. We also liked how there was some positive nervous clapping when they got an answer right and all agreed.

So, in conclusion, this is a great study team strategy. It's perfect to do when you feel like students aren't working together as well as they can and to remind students what the expectations are, by seeing what you should and shouldn't do.

Also, I thought it would put the other groups way behind, but they had heard the discussions before, so it made their start a little more smoother and efficient because they had listened to two different groups.

Saturday, March 18, 2017

Fraction Talks warmup @FractionTalks @mathletepearce

I can't take credit for the below image. I found it on twitter. It's located here and above under my Fav Problems. The variety of ideas and misconceptions with this problem make it a must-do warm-up in every classroom.
As you can see below, I treated it like a number talk. I had students share all their answers, without any reaction or judgment, and asked students to defend their answer.

As you can see in the top right, Juliana's idea was that there's 1 hexagon in the middle. But if there are 6 trapezoids on the outside, 2 of those trapezoids can be rearranged to make 1 hexagon. So 6 trapezoids would make 3 hexagons. Therefore she reasoned that 1/4 of the shape was yellow, because if the whole shape is 4 hexagons, only 1 of them is yellow.

Victor said that the yellow hexagon in the middle was 2 trapezoids, or 2 reds. So, if there's 6 reds on the outside, that's 8 total. Therefore, he said 2/8 are yellow.

I did want to acknowledge the wrong answer of 2/6, because that is the ratio of yellow to red, so there's something correct about that. It's just not the part/whole ratio that we are looking for when we say what fraction of the shape is yellow.

I also asked students how someone could say 1/7. They said they probably saw 1 yellow shape out of 7 total shapes. Students said you couldn't do this because they are not all the same sized shapes.

One student said 1/2 of the shape is yellow because a hexagon is half of a trapezoid... a good thought that wasn't extended though to the whole shape.

I had to take a picture of this one because all day this was the only person who thought of it this way. The student said you can make the whole shape triangles, and a hexagon is 6 y yellow triangles, and there would be 24 triangles total, so 6/24 is yellow.

Every period I mention the way I see it if no one has said it yet, which is noticing the shape is symmetrical, and if you cut it in half you have 1 yellow trapezoid out of 4 total trapezoids, for 1/4.

More images like this can be found on We have also worked in some of the flag matches from that site which are pretty awesome and cross curricular.

This blog post was inspired to by a video @mathletepearce made and the conversations on Twitter that followed.

8.SP.4 IM Task 2 way frequency table student survey samples & rubric

I wrote a blog post for NCTM about the lesson used. I also have an idea about how to score the assignment using a rubric and already want to label the columns independent variable and the rows dependent variable so students know why they are totalling the columns for the relative frequencies.

This post will have my reflection notes, the scoring rubric that I'd love feedback to improve, and some interesting student work samples of varying levels of understanding.

I have improved the survey assignment template with the following: each question is labeled. The independent and dependent variable are labeled on the table and in the hypothesis example sentence. I also added to the conclusion by adding "Therefore if ________________________, you are more likely to ___________________.
My hypothesis was...correct or incorrect? conclusive or inconclusive?"

I also discussed this lesson with some colleagues from my FAME program and one had a great suggestion! An alternative to calling out students name for yes yes, no no, yes, no, or no yes votes you could ask the class who play a musical instrument? If yes, line up in a column on left side of room. If no in a column on the other side of the room, parallel to the rest of the class in the other line.

Then when you ask the next question, do you play a sport? Those groups separate into different corners of the room. For example yes can step forward, no can step back. You now have the four quadrants of the 2 way table and you can visually see how much bigger the group of students may be that play an instrument and a sport, compared to the students who play an instrument and no sport. I want to try this next time.

To help me make a rubric, I looked at each classes results and separated them into piles: 1 pile if they didn't finish the survey or filling out the two way table (only 2 or 3 students per class), another pile if they got the hypothesis, conclusion, and relative frequency percents. Then a middle pile if they were missing the fractions & percents and/or their conclusion.

My rough rubric was students earned a 4, or A if they had their column totals, fractions, percents, hypothesis stated, questions clear, and a conclusion using the data as evidence. Students got a 3 if they were missing the percents or conclusions. A 2 was earned by not stating their hypothesis and hypothesis. A 1 was given to students who completed the survey and filled out the table, a 0.5 if they attempted, and a 0 if I didn't receive the assignment.

Some students did not complete the fractions or relative percents because they were reading the wrong row or column or didn't understand how to do it in general.

Note to self, I need to put that rubric as a small footer to the assignment sheet so students know what I am looking for and could make it easier to grade by marking the grade with comments on the rubric.

Above is my reflection notes with items I wanted to mention in this blog post.

This student nailed the concept and even had time to decorate it a bit. Her hypothesis was correct, if you watch cartoons you have an 88% chance of watching Spongebob.

This student mixed up their conclusion a bit. Great survey idea. His hypothesis was correct. What he meant to say I believe is of the 20 people who are 49ers fans, 17 are also giants fans. I also see that his joint frequencies don't add up properly to his marginal frequencies, so there's some type of miscalculation going on here.

Good work and conclusion, just missing the percents to support the conclusion.

This was interesting... The above picture is 2 students who had one of their questions be do you like to read? What's shocking is the results are different. To me, this means either some kids changed their answers as the class progressed, or they answered the question differently based on who was asking them the question. I am not sure.

A socioeconomic question that had no surprises. Stellar work. Actually, just noticed that they made a mistake with the top right joint frequency of 7, which should have been 7/7 for 100%.

Good work here.

This was interesting. 31 students were asked if they liked school. 20/31 said they didn't like school. Of those 20, 1 of those kids wanted to be a teacher!! Shocking! I'm very curious who this student is. What was surprising also was that of the 11 kids who liked school, none wanted to be a teacher.

This student had a great hypothesis and survey. Just made a mistake when calculating the relative frequencies. They used the total student surveyed as the denominator rather than the total students who were born in the U.S. of 26.

Sunday, February 19, 2017

#clotheslinemath Linear Equations

I am going to teach this lesson on Tuesday to all 3 of my Common Core 8 classes. Any feedback or suggestions prior to that are appreciated. A reflection of how it unfolded to follow. I wanted all the images in one place for my colleagues to use and for anyone else to try. All credit goes to Andrew Stadel and Chris Shores for the lesson, I just added a small extension.

UPDATE: (2/21/17)

I noticed students drew the number line cards really big after looking at the google slides. For each period after I made sure students knew they needed room on their paper for 2 clothesline and 2 graphs to be pasted on.

I had students think for 3 minutes how the 6 parameter cards could be placed on the lower clothesline. I demonstrated how if a card lined up with a number, they would be equal or equivalent. I also demonstrated how you could clip a card underneath another using the clothespin.

I then asked students to place the cards individually with 3 minutes of silent independent work time. Then I timed them for another 3 minutes of taking turns sharing out their ideas to their group, to gain confidence in sharing their idea with the class.

With 6 parameter cards, I said that we would need 6 different students put up 6 different cards and give a statement to the class about what they did and why they chose to put it there. I had students vote using their thumbs whether they agreed or disagreed with a person's statement. I asked students put their thumb sideways if they didn't whether the person was right or wrong. In first period students immediately went for the growth of line 2, or m2, was 0.

Students seemed to see that the blue line's growth had to be 2, because it was the steepest, and it was increasing from left to right so it had to be a positive number, so they said it couldn't be 1 and picked 2. Some students also reasoned that the red line must have a smaller slope but negative because it was decreasing slowly.

One student talked about how the y-intercept of the red line had to be 2, because the blue line was also the same distance from zero being at -2. I asked the students what we call distance from zero, and some remembered that was the absolute value.

After students had placed cards m1 to m3 and b1 to b3, I handed out the desmos graph. I asked them to paste it in their notebook and then see if they agreed with how the clothesline was from the last image. Students noticed that they appeared the same, and the y-intercepts were definitely the same.

I pointed out when students were drawing slope triangles between lattice points, but am not sure if all students correctly did so as I walked around the room.

Students had a bit of trouble with the scale of the graph, with each square being 0.5. This lead them to see that the vertical growth was 1.5, and the horizontal growth was 1. I asked them what the slope triangle was in grid squares, which gave 3/2. Students reasoned that must be between 1 and 2.

I did not get to the part of putting m4 under -1, and b4 under 1. If I had more time I would have. My colleague said his class also had trouble substituting the new m values into the equation y=mx+b to try to confirm that the lines would intersect algebraically.

It was nice to see that after some students noticed the lines would intersect, that they could prove it graphically. In each class a student hung x and y under the respective coordinates for the intersection point.

Some students thought there might need to be 3 sets of x and y coordinates since there were 3 different lines.

Overall I loved the activity. It had a sense of mystery. Also, having a graph with grid lines achieved my goal of practicing calculating slope. I am still concerned how many students left the class with a solid handle on it. I liked how I made sure they had some independent think time before sharing with their group because if it's straight to group it tends to be the same people sharing. Students also liked the chance of going up to the board with the whole class watching, while others were nudged towards giving their reasons for agreeing with someone from their seat.

Wednesday, February 1, 2017

Hot Seat Study team Strategy (@cpmmath Transformations)

This past summer I had the pleasure of applying for and being accepted to CPM's TRC research group in Sacramento. After brainstorming ideas, my group was formed with Aristotle and Erica. The great part was that we all taught 8th grade and were concerned about students doing their homework, specifically the Review and Preview. We decided we would focus on taking breaks from the lessons once in awhile to focus class time on individual practice as well as group work on review preview problems. We are also trying to show students the benefits of mixed spaced practice compared to a bunch of practice problems of the same type. Another concern was if students don't do the homework, they are getting very little individual practice to self-assess. It's also important to make homework worth part of the grade, but not a big part. My colleague and I settled on 15%.

Part of the work we did was work on how to check it, and with some collaboration with Erica and some ideas from Dylan Wiliam's Embedded Formative Assessment, students have a quarter sheet of paper to check 5 assignments. I can share this if anyone wants it. Most days students self check. Others they peer check. Once in awhile, a whole group switches their work with another group so they get a different group's opinion of their work. Also, if and when students lose the checking sheet, if they want credit, they have to show it to me personally, and of course I look it over a lot more closely!

Participation quizzes have been helpful, as well as pair checks, which I detailed in an NCTM blog post recently. After a recent Skype meeting, Erica reminded me and suggested I use the Hot Seat strategy. The rules are detailed below in the Google slides, as well as the problems students focused on. I had never used this strategy before as I was concerned about the competitive feel, students feeling inferior, and the timing of the activity. I will address those concerns in this post.

So, as described in the first slide above, 1 student brings their seat to the back of the room, and works individually with no help. To make this quick, I said resource manager (assigned team roles based on the alphabetical order of their first name in their group). If they get it right, they earn their team 2 points. If everyone in their group gets it right with all work shown, the team gets 1 point. I kept track of the points, and students were very honest. It was instant formative assessment.

As you can see, we are finishing up Chapter 6 which has closure problems. Instead of saying students, work on these problems. I mixed up the order and focused it on transformations.

For the first problem, students only needed 3 minutes to figure out the scale factor. Some asked which is the original and which is the new figure? I told them they could decide. I also said to not use calculators.
Students in the back row are on the hot seat.
After keeping the score, facilitators were up next. I gave them 7 to 8 minutes for this because they had to start with one transformation, and then finish the steps. I would circulate asking questions and giving advice to students who needed it.

I asked students for feedback informally in each class. A student that doodles and doesn't engage with her group much liked that it was more individual based and the timing made her focus more on finishing the problem. A struggling student said she didn't like it because she couldn't get help from her group when on the hot seat. Students were much more quickly to adjust their paper and body to point to their work while verbally explaining a concept.
From my classroom door, you can see everyone focusing on the problem, book, or glancing at the board for the problem.
Also, when creating the graph, the class got dead silent in concentration, and then you could start hearing whispers in the groups when they were comparing how they rotated or reflected a shape if their triangles looked the same.

I was also able to implement the 5 practices using my Google Drive app by taking photos of student work to share when going over the answer and awarding the points.

For the second problem for list 1, it asks for a 180 degree rotation. A student asked which direction clockwise or counterclockwise? Another student answered, "it doesn't matter it will look the same!"

Here you can see that same triangle successfully reflected over the y-axis. Then student volunteers said you could then reflect it over the x axis and then translate it up 1 and over 2.

This photo shows the same problem, but also the second one where after graphing a triangle students were asked to reflect it over the y axis. Separately they were asked to translate it, and describe what happened to the coordinates.

Here is 2nd period's scoreboard.

Numbered steps that were easy to read.

Here you can see the horizontal, then vertical translation. You can also see the correct reflection over the y-axis. I asked students that made mistakes reflecting, "how far away is the shape from the axis you are trying to reflect it over?" This moved them in the correct direction.

Here you can see how students were able to show expressions for what happens to the coordinates when moving 4 to the right and down 6. They would have to add 4 to the x coordinate, and subtract 6 from the y coordinate.

Also, for reflecting, they had to multiply the x coordinate by -1, and leave the y coordinate alone.

My colleague and his students really enjoyed the hot seat activity as well. I picked a wednesday to try it since it was a shorter period and I knew they wouldn't be able to finish the 6.2.6 lesson during a short period.

Unfortunately, we only got through 3 rounds, but a LOT of learning happened, and these were dense problems. I had a short warm-up, reviewed 2 homework problems briefly, passed back a test that I went over, and then did the Hot Seat. The resource managers didn't get a chance to be on the hot seat, but I'm definitely going to do this again with a goal of students being on the hot seat more than once.

At the beginning, I told students that the goal of this activity was to see where you are at individually with their understanding, and to also appreciate the PRIVILEGE of working cooperatively in a group. I am very confident that students will be working much harder together tomorrow when working on a problem solving task about scale factors between models and real life objects.

Tuesday, January 31, 2017

Similar Figures Academic Vocabulary @von_Oy

This post was inspired by Suzanne blogging about the precise language she wants by not naming a method "difference of squares" when it could pigeonhole students into a procedure without knowing really why it works. I can also empathize with teaching students the definition of a function through examples. It can be disheartening when the only answer you get is it passes the vertical line test. I like when students on their assessments would draw the vertical line and the points it passes through, labeling those as the input that has 2 different outputs.

I was concerned about how my students were processing the attributes of similar figures. CPM has a great lesson where there is an original quadrilateral with a bunch of other shapes that are similar except for 2 of them. One is horizontally stretched, while the other is vertically stretched. As students compare the shapes, they can see that the corresponding angles of the non-similar shapes are clearly different measurements. Also, corresponding sides that should be parallel are intersecting. They also should be able to count how many times a corresponding side can fit into another similar shapes corresponding side to figure out the scale factor.

Since I was concerned with students understanding, I gave students a post it note, 5 or 6 minutes, and the following prompt:

How do you know if two shapes are similar? When does a dilation make the shape bigger? Smaller?

A complete answer would mention all of the following words:

size, scale factor, angles, congruent, corresponding, parallel

Feel free to add a labeled diagram to show your thinking.

I got a variety of responses. Some students included the words, but not in the correct context.

Surprisingly, many students said that similar figures are congruent. To this I commented, "always?" It was hard for students to use the word corresponding and angles in the correct way. 

We don't take notes that often, but synthesized their thinking. Students realized a similar figure that was enlarged would have a scale factor greater than 1, a shape that shrunk would be a fraction between 0 and 1, and a congruent shape would have a scale factor of 1.

This work was after dilating on the coordinate plane, and this was nailing down all the academic language.