OK….so most shared work problems suck. I apologize to my students aspiring to be pipe organ re-varnishers, but we can do so much better.

This week I used Cocoa Puffs, stopwatches and Desmos to bring some engagement to my rational expressions lessons. To start, each student was provided with a plate filled with 30 grams of Cocoa Puffs (incuding the plate) . After my 3-2-1 countdown, students picked Puffs one at a time from the plate and tossed them onto an empty plate. As they completed the task, times were recorded for each student.

After students finished, I had them partner up and consider the question: “if you worked together with your partner on this task, with one plate of Cocoa Puffs, how long would it take you?”

Students asked a number of clariying questions (yes, there is one plate. yes, you can pick them off the plate together.), partnerships developed a few ideas. We debated the validity of many of them:

- Many groups took the average of the two times, then divided the result by 2. This seemed reasonable to a number of groups, and led to a discussion of the vavlidity of averaging rates.
- Some groups attempted to find a rate per gram. This was a good start, but given that groups did not know the mass of the plate (I use Chinet, so it’s bulky!), this introduced some guesswork.

To steer discussion, we focused on one student who took 80 seconds to complete the task. How much of the job did they complete after 40 seconds? After 20? Can we write a function which depends on time here? What does it mean? Crossing the bridge from the task time (80 seconds) to the job rate (1/80 per second) is a tricky transit. Using Desmos to show the “job” function lends some clarity.

From here, many partnerships felt more comfortable with establishing their own estimates. The next day, teams shared their work and estimates on OneNote, then peer-assessed the communication. Some of the work was wonderful, well-communicated, and served as a model for the class to emulate.

The next day, we listed our calculated shared work predictions on the board, and tested our estimates. Teams timed each other with cell phone stopwatches, and did not let participants see the clock until the task was complete.

Many groups were quite close to their calculated predictions! We discussed why our predictions didn’t quite meet the actual – bumping, variability in mass, general panic – and when error is acceptable. And now we have a firm background in rates and rational functions – time to conquer those pipe organs!

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For my 9th grade class last year, I wanted an activity which would cause students to think about variability in data distributions, and introduce standard deviation as a useful measure of variability. You can preview the activity here. Take a few minutes, test drive it, and see if you can suss out the problems.

So, what went wrong. Well, a number of things – but here are the two primary suspects.

- It’s too damn long! It takes way too long to get to a working definition of standard deviation, and by screen 14 students are all over the place. Using student pacing could help remedy some of this, but I found much of the class losing interest by the time we got there.
- I was stubborn! I was looking for a “cute” visual way for students to think of standard deviation as “typical distance from the mean”. In my zeal to hammer this working definition home, I tried to build slick graphs which lost many students.

How to fix it – last year, the Desmos teaching faculty developed the “Desmos Guide to Building Great Digital Activities“. It’s worth a read (and a re-read) now and then to guide activity construction. In my variability activity, this bullet point from the guide resonated with me: **Keep expository screens short, focused, and connected to existing student thinking**. In many of the screens, I over-explained things. Students don’t want to read when they are completing a digital activity, they want to investigate, create, and explore. I robbed them of that chance.

Today I tried my new, rebuilt variability activity with 2 classes (slimmed down to 12 screens from 19), and there was a vast improvement in class engagement. There were more opportunities for students to express their ideas regarding comparisons of distributions, and we had plenty of time to pause, recap, discuss and think about next steps. A number of points from the Desmos Guide drove my thinking:

**Ask for informal analysis before formal analysis. **While I kept in the “typical distance” definition of standard deviation, it was only in a small way – we’ll move on to a more formal definition next. Students were able to conceptualize standard deviation as a useful measure, and now can move on to a formal definition. My old activity felt too “sledge-hammer-ish” and I knew it.

**Incorporate a variety of verbs and nouns**. I provided a number of ways for students to think about variability and distribution comparisons in the early screens, and strove to build different-looking screens. This kept the ideas fresh, and students talked with their partners to assess these differences in different ways.

**Create activities that are easy to start and difficult to finish**. In the last 3 screens, I ask students to extend their thinking, be brave, and apply new ideas. For those students who got this far – most did, these screens elicited the loudest debates. We ran out of time at the end, but we have some good stuff to build off of tomorrow morning.

I’ve learned that it’s important to be honest about an activity. It’s easy to blame the students when something goes wrong, especially which class heads away from learning and towards frustration. But performing an activity autopsy, focusing on clear goals, and keeping the design principles in mind is helpful to move an activity forward.

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**This week my school admin team got it right. And I feel fortunate to work with them. **

For about a year, my school’s admin team had kicked around the concept of a school-wide EdCamp. To be honest, I never thought it would see the light of day…there are just too many other things loaded into the calendar. So an invitation to work with a team of teachers to organize a high school-wide EdCamp was a true surprise…then the work began.

We planned 3 morning sessions, followed by lunch and prizes. But beyond the structure of the day, we had a lot of talking-up to do. Would our teachers, many of whom had never been to an EdCamp, understand the concept? Would people propose sessions? Could we engage the curmudgeons in our teaching ranks? At our opening faculty meeting, we showed a brief video to help teachers understand the EdCamp concept, then talked it up over the next 2 days.

The morning of the conference many teachers suggested ideas, asked questions and thought about what they’s like to learn. In the end, we had a nice variety of topics and it felt like there was something for everyone!

As I walked around during the sessions, I was thrilled to see rooms filled with discussions, and teachers from different departments with an opportunity to engage. I know there is no possible way every to reach everyone, but I hope it was a day of professional learning for most.

On my end, I offered a session “Activity Builder for the Non-Math Crowd” which seemed to be of use to those assembled. Then later, a session where we just did math – problems from Open Middle, Nrich, Visual Patterns and others let math teacher talk, learn, and think about engaging problems for their classrooms. You can download the problems I shared with this link.

Thanks to the fantastic people I work with for letting me be part of this: Baker, Dennis, Kristina and Melissa.

And a big thanks to the HH admin team: Dennis, Ralph, Tracey and JZ. I appreciate the opportunity, and promise not to complain again….until the next time…..

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This week the catcher of that team, Darren Daulton, died after a battle with brain cancer. Newspapers have shared memories of “Dutch” and among the articles is one which reminds us of the surprising number of former Phillies who have passed away due to brain cancer (Tug McGraw, John Vukovich, and Johnny Oates). A revised 2013 article from the *Philadelphia Inquirer* analyzes the the unusual number of Phillies who have developed brain cancer, and contains many appropriate entry points for Statistics courses. Some highlights from the article:

- A comparison of the observed effect to random chance – here a professor of epidemiology summarizes: “You can’t rule out the possibility that it’s random bad luck.”
- A summary of plausible variables which could lead to elevated levels of exposure, such as artificial turf (which may have contained lead) or anabolic steroids.
- An analysis of the increase rate of brain cancer among Phillies – here we are told that the Phillies’ rate is “about 3.1 times as high” as the national rate. A confidence interval, along with an interpretation and associated cautions are also included.

Let’s explore that “3.1 times” statistic…time to break out the technology.

A few weeks back, I attended the BAPS (Beyond AP Statistics) workshop in Baltimore, as part of the Joint Statistics Meetings. Allan Rossman and Beth Chance shared ideas on using their applet collection to explore simulation (see my earlier post using the applets to Sample Stars) along with a “new” statistic we don’t often talk about in AP Stats – relative risk.

To start, I used the Analyzing 2-Way Tables applet and used the “sample data” feature. Here I attempted to use the same numbers quoted in the article:

The national rate was 9.8 cases per 100,000 adult males per year, while the rate in the former Phillies was 30.1 cases per 100,000 – about 3.1 times as high.

There are two issues here: first, to perform a simulation we need counts, so numbers like 9.9 and 30.1 just don’t play nice. I’ll use 10 and 30. Also, I wasn’t surprised that this site was not real happy with my using a population of 100,000 for simulation. Here, I am going with 1,000 for convenience and to make the computer processor gods happy – we can debate the appropriateness of this down the road.

The applet will then simulate the random assignment of the 2,000 subjects here to the two treatment groups (group A: being a Phillie, group B: not being a Phillie). How likely is it that we will observe 30 or more “successes” (which here represent those who develop brain cancer) in one of the two groups? In the applet, we can see how the “successes” have been randomly assigned from their original spots in the 2-way table to new groupings.

AT BAPS, Allan Rossman then explained how we can summarize these two groups using Relative Risk, which is listed under the “Statistic” menu on the applet. In general, relative risk is the proportion of success in one group divided by the proportion of success in a second group. If we have proportions in two groups which are equal, then the relative risk would be 1. We can then link to the newspaper article which claims a 3.1 “relative risk”, simulate many times with the applet (below we see the results of 10,001 simulations), and compare to the reported statistic.

According to the simulation, we should only expect to see a relative risk of 3 or above about 0.08% of the time – clearly an “unusual” result.

But the article does not claim a significant difference, and cautioned against doing so as a number of assumptions were made which could alter conditions. This would be an opportunity to discuss some of these design assumptions and how they could change the outcome.

Rest in Peace Dutch!

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Late in May, David Wees shared materials which challenge students to investigate the relationships between “standard form” and “completing the square” form (aside – does anyone agree on proper terms for these?) using area models to build representations. Given that I use area models often to introduce polynomial multiplication, I was eager to maintain consistency in the student understanding.

But before we dove into David’s lesson, I wanted students to revisit their understanding of area models. In this Desmos Activity Builder lesson I created, students shared their interpretations of area models and worked in pairs to investigate non-square models. In one of the final screens, students argued for the “correct” interpretation of a model.

Using the Desmos teacher dashboard, we could see clear visual arguments for both representations. This was valuable as we ended the lesson for the day, and tucked that nugget away for Monday, when we would begin to formalize these equivalencies.

After the weekend, students worked independently through David’s Completing the Square lesson. Not only did students quickly move through the area models and the dual representations, the debates between students to explain how to move from one representation to the other were loud and pervasive. I’m also loving how many of my students have started to use color as an effective tool in our OneNote-taking (below).

At the end of the sheet, all students completed problems which translate standard form to vertex form with no support from me (“no fuss…no muss”). It dawned on me that something amazing had happened….my students had figured out completing the square without my ever talking about completing the square.

Tomorrow we’ll tackle those pesky odd-number “b” terms, but my students own this already!

]]>What proportion of the numbers in Pascal’s Triangle are even?

Every time I talk to Jim, he’s bound to have a neat problem for me to chew on. The last time, he shared a fun task involving the harmonic series. Take a few minutes and think about this Pascal’s Triangle scenario…I’ll even leave you some spoiler space.

At the ATMOPAV Spring Conference last month, Jim shared an entertaining talk titled “Gambling, Risk, Alcohol, Poisons and Manure – an Unfinished Life Story”. The talk led the attendees on a journey through the history of statistics, starting with games of chance and the meeting of Chevelier du Mere and Blaise Pascal, through the introduction of formal inference procedures developed at the Guinness brewery, and to identifying statistical abuses in the present day.

Jim is a life-ling educator and former Executive Director of NCTM who happens to live quite close to me. It was a thrill having him share his ideas with the group.

Had enough time to think about this Pascal question? Spoiler time is up!

So, which rows are in Pascal’s Triangle are we talking about here?

In theory, we are talking about “all” of the rows in the infinite Pascal’s Triangle, which makes this a bit tricky to think about for kids (and adults as well!). But Jim shared with me slides which show the proportion of evens in increasing numbers of rows of the triangle. You will notice that as the number of rows grows, the proportion of even entries also increases, and approaches 1. What a neat result! Below is an animated gif I made using a Pascal’s coloring applet which shows the increase in the proportion of even (white space) numbers in increasing rows.

For your class, this is a fun opportunity to talk about the parallels between Pascal’s Triangle, Sierpinski’s Gasket, and fractal area.

Already looking forward to my next encounter with Jim!

]]>Today I am teaching Power.

Power: a deep statistical concept, but one which often gets moved towards the back of the AP Stats junk drawer. The only mention of power in the AP Stats course description comes under Tests of Significance:

Logic of significance testing, null and alternative hypotheses; p-values; one- and two-sided tests; concepts of Type I and Type II errors;

concept of power

So, students need to understand the *concept* of power, but not actually compute it (which is itself not an easy task). Floyd Bullard’s article “On Power” from the AP Central website provides solid starting points for teachers struggling with this concept; specifically, I appreciate his many ways of considering power:

- Power is the probability of rejecting the null hypothesis when in fact it is false.
- Power is the probability of making a correct decision (to reject the null hypothesis) when the null hypothesis is false.
- Power is the probability that a test of significance will pick up on an effect that is present.
- Power is the probability that a test of significance will detect a deviation from the null hypothesis, should such a deviation exist.
- Power is the probability of avoiding a Type II error.

This year, I tried an activity which used the third bullet above, picking up on effects, as a basis for making decisions.

**HEY KOOL-AID MAN!**

Arriving at school early, I got to work making 3 batches of Kool Aid. During class, all students would receive samples of the 3 juices to try. Students were not told about the task beforehand, or where this was headed. Up to now, we had discussed type I and type II error, so this served as a transition to the next idea.

THE BASELINE SAMPLE:

All students received cups and as they worked on a practice problem I circulated, serving tasty Kool Aid – don’t forget to tip your server! I told students to savor the juice, but to pay attention: I promised them that this first batch was made using strict Kool Aid instructions. Think about the taste of the juice.

SAMPLE A:

Next, students received a drink from “Sample A”. Their job – to assess if this new sample was made using LESS drink mix than the baseline batch. Also, I varied the amounts of juice students received: while some students were poured full cups, some received just a few dribbles. To collect responses, all students approached the board to contribute a point to a Sample A scatterplot, using the following criteria:

Sample size: how much juice you were given

Evidence: how much evidence do you feel you have to support our alternate hypothesis – that Sample A was made with LESS mix than the baseline?

As you can see, the responses were all over the place – a mixture of “we’re not quite sure” to “these are strange directions” to “I just don’t trust Lochel – something’s up”. But the table has been set for the next sample.

Sample A: it was made with *just a smidge less* mix than the baseline. So I wasn’t totally surprised to see dots all over.

SAMPLE B:

I poured drinks again from this new sample, and again varied the sample sizes. I asked all students to think about their evidence in favor of the alternate, and wait until everyone tasted their juice before submitting a dot.

And check out those results! Except for a few kids (who admitted they stink at telling apart tastes), we have universal support in favor of the alternate hypothesis.

Sample B: this was made with 1/2 the suggested amount of drink mix. Much weaker!

FOLLOW-UP DISCUSSION:

This activity made the discussion of power much more natural. In particular, what could occur during a study which would make it **more likely** to reject the null hypothesis, if it deserves rejecting?

Larger sample size: smaller samples make it tough to detect differences

Effect size: how far away from the null is the “truth”. If the “truth” as just a bit less than the null, it could be difficult to detect this effect.

In terms of AP Stats “concepts of power”, this covers much of what we need. Next, I used an applet to walk students through examples and show power as a probability. And like most years, this was met with googly eyes by many, but the foundation of conditions which would be ripe for rejecting the null was built, and I was happy with this day!

Suggested reading: Statistics Done Wrong by Alex Reinhart contains compelling, clear examples for teachers who look to lead discussions regarding P-value and Power. I recommend it highly!

]]>**SELF-CHECKING GRAPH MATCHING**

I used this often with my Pre-Calculus class in the fall, and the concept works equally well with younger students. Simply start a new Activity Builder screen, and enter the equation you’d like students to provide. Place the equation in a folder, which you can hide so students won’t see it when they encounter the screen. Finally, by making the graph with dashed lines, students can easily see if their submission matches the requested graph, and can adjust accordingly.

**GALLERY WALK**

Here’s a neat Activity Builder hack you may not know about. If you have an existing Desmos graph, copy the URL from your graph to the clipboard. Then, in an Activity Builder screen click the “Graph” button and paste the URL into the first expression line – and PRESTO, the graph is imported into an Activity Builder screen. I often collect student work by simply having them submit a Desmos URL. Consider taking samples of student works and create a virtual gallery walk. Let students view each other’s ideas, comment and make suggestions. Thanks to my colleague DJ for providing neat student graphs!

**SELF-ASSESSMENT SLIDERS…AND OVERLAY**

Have students assess their own learning with a moveable point. Provide an “I can…” prompt and let students consider where they fall in the learning progression. Hold a class-wide discussion of unit skills by anonymizing student names and using the overlay feature to take the class pulse on skills.

**MY FAVORITE DISTRACTOR**

Activity Builder allows teachers to build their own multiple-choice questions, with the option of having students provide an explanation for the choice they make. In “My Favorite Distractor”, students select an answer they KNOW is wrong, and explain how they know. This may not work for many multiple-choice type questions, but consider using this idea in situations where the distractors have clear, interesting rationales for elimination.

Have your own quick formative assessment ideas? Share it here!

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My 9th graders have seen factoring before, but it was back in 7th grade, and it was only a surface treatment. So after a brief opener where we discussed what a “factor” means (both numerically and algebraically), I dropped the bomb –

- I’ve posted your learning targets online
- I’ve posted videos, resources and practice problems if you need them
- I’ve set up online practice if you need it
- You have a timed quiz on Friday (we started on Tuesday)

And….scene!

Panic….apprehension….incredulous looks….

So, you’re not going to teach us?

Nope. Now get to work.

Here are some details of what I posted:

LEARNING TARGETS

- F1: I can identify and factor expressions which involve greatest common factors.
- F2: I can efficiently factor trinomials of the form ax
^{2}+bx+c, where a = 1. - F3: I can factor trinomials of the form ax
^{2}+bx+c, where a does not equal 1 (or zero). - F4: I can identify and factor perfect square trinomials.
- F5: I can identify and factor “difference of squares” expressions.
- F6: I can factor expressions which may represent a combination of F1 to F5.
- F7: I can factor expressions “by parts” (or “by grouping”) when necessary.
- F8: I can factor expressions which are the sum (or difference) of two cubes.

RESOURCES

Each learning target featured a video – some from Khan Academy, and some from other sources I searched for – but I attempted to provide a variety of methods. Some featured grouping as a primary means, others demonstrated the box method or the diamond. This was the most important aspect of this learning experience: **I wanted students to experience a variety of approaches, evaluate them, and make a personal decision about what worked best for them.** The students did not disappoint.

I also posted other online resources, such as worked examples and flowcharts. One of my favorite resources – Finding Factors from nrich, was also included. Finally, I created an assignment on DeltaMath for each learning target, and a final jumbled assignment. The end of each day featured an exit ticket quiz and recap, to assess progress and provide “next steps” during the week.

SO WHAT HAPPENED?

Some students latched onto factoring by grouping for every quadratic, and explained their reasoning to their peers. Many of these same students later in the week found more confidence in their number sense and chose to group only for “tricky” problems. One student was particularly insistent that the box method was the best was to go for all problems. Others found the diamond method helpful – which led to deep conversations about number sense and how to make searches more efficient. And in one fascinating conversation, a student discovered a “trick” he had found online. The group debated the merits of the method, tried some practice…but as nobody in the group could figure out why the method worked, they quickly dismissed it. Good boys!!!

In the end, the quiz scores were great. But beyond the scores, I feel confident that the students have made choices about their learning, assessed and revised their thinking, and can move forward using their new tools.

WHAT DID THE STUDENTS THINK?

Today I asked students to reflect upon their learning experience, and provide me feedback.

What was your overall feeling about last week’s learning method? (1 = “Please never do that again”, 5 = “I loved it – do it more”.)

Describe something you LIKED about last week’s classes, and why you liked it.

- I liked being able to choose what i wanted to do. I could focus on my weaknesses and do less problems on what i was good at. I also appreciated the practice problems.
- I liked that if you knew a topic you could move on and didn’t have to wait for someone else or the next day of class.
- I liked that I could learn and do problems at my own speed.

Describe something you DIDN’T LIKE about last week’s classes, and why you didn’t like it.

- I did not like that you did not explain how to factor
- I didn’t have as much instruction from the master of factoring. {note – I suppose this is me?}
- the teacher wasn’t involved

This last comment intrigues me…and I’m not sure if I should be bothered by it…I don’t think I should be. In many respects, I feel I worked harder during the classes, as students were all over the place. But I also realize students don’t see all of this going on around them. I’ve become intrigued by how I can be less of a teacher and more of a facilitator in my classes, and this was a solid step forward I feel.

Now, off to plan to not lecture tomorrow….

]]>**How many different 5-character DesmosActivity Builder codes exist?**

This problem would have likely met my intended goal of having kids think about the fundamental counting principle in a real-world context. It also would have taken about 10 minutes of class time, and have been forgotten about by the next day. It felt like I was missing an opportunity to develop a deeper discussion. A slight tweak to the question added just the right layer:

**Activity codes for Desmos Activity Builder currently have 5 characters, as shown here. When will Activity Codes need to expand to 6 characters?**

And now we have a problem which requires a bit more than a quick calculation. To start, I asked students to work in their teams to make a list of information they would need to help solve this problem. This was not easy or comfortable for them – but a preliminary list of questions emerged from group discussions:

- How many 5-character codes are there?
- Are codes used less on weeekends and summers?
- Can letters repeat in codes?
- How many codes a day are used?

This was a good start to set kids in motion to think about how to solve the problem. I’m hoping they will think about new questions or revise their questions as we go along…the class did not disappoint!

HOW MANY CODES ARE THERE?

As kids worked, clarifying questions came up – some of which I just didn’t know the answer to, and hadn’t really thought about:

Mr. L, are there any zeroes in codes? Kids might confuse them with the letter O.

Mr. L, I don’t see any L’s in the codes?

Excellent observations, and restrictions we need to think about in our calculation. A tweet to the Desmos crew lent some clarity, and added more restrictions!

Thank for the intel, Eli!

HOW MANY CODES PER DAY ARE USED?

This was tricky for my class. To help, I reminded students that when we started the semester, codes were 4 characters. When did the Desmos 5-character era begin? A quick scroll through my history (shown here) provides some info. After further interrogation from my class, I shared that Activity Builder started around July of last year with 4-character codes. Add this to our bucket of helpful info.

SHARING IS CARING

Writing a draft solution was the next task for students. But instead of turning it in to me immediately, I formed class teams of 3 where students shared their drafts and ideas. I used this opportunity to build teams of students who I observe don’t often interact or chat. From here, I gave students another day to think about their explanation – keeping in mind that there are no right answers to this question, only answers we can defend. But it still feels like we are missing a key piece in this problem……

DID WE MISS ANYTHING?

The next morning as students were mingling before the bell, I looked across the room at the laptop of Jacob – one of my more insightful, but also introverted, students:

It’s the mother lode!

The google trends graph for student.desmos. Yes! Yes! Yes! Stop everything kids, we need to talk! Jacob – tell us all about this graph. How does this new info factor into our estimates? What should we do with it? Is this going to continue? And with this, I gave the class an extra day to think about their responses, share, and dig deeper. And while many students simply estimated a growth rate by doubling or tripling their computed rate (this is fine with me), I am getting some responses which far exceed my expectations – like Jacob, who developed a growth function and evaluated integrals (did I mention this is a 9th grade class????)

Yep, this was definitely better than my originally intended problem!

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