Using to Make Confidence Intervals

Last year, I introduced confidence intervals using an applet. There are lots of them out there, but this is my favorite. I like that it simultaneously shows the proportions on a dot plot and constructs the confidence intervals off to the side for each sample.


My problem with introducing it this way is that I’m concerned that since the data is pre-loaded, and not something familiar to the students, it’s hard to connect to.

I wanted to essentially create the same experience, but let it be with data the students collected. I wasn’t aware of a program that would let me input the information for the sample proportion then also graph the lines for the confidence interval as well. Then I remembered seeing a function in called error bars!

My Plan:

Recently, my students took data on proportions of M&Ms in fun size bags of M&Ms, letting each one be a “sample” of all M&Ms. I am going to print out sets of 3 samples on individual slips, and redistribute them to students, so that once again, all students have different data to work with. They will calculate 95% and 90% confidence intervals for each sample, and then will send me the proportion and their margin of error through a Google Form. I will use to create one graph of the 95% intervals and another of the 90% intervals, so that we can compare and contrast, and look for those that “miss” the true proportion.


  1. Open up the Workspace
  2. Since I’ll have each student calculating 2 versions of the CI for each of their 3 samples, I’m using a google form on which they will enter their sample number (on the sheet I’m giving them), the proportion of Blue M&Ms in their sample, and the margin of error for the 90% CI and the margin of error for the 95% CI.
  3. Use the Make a Plot button at the top to choose scatterplot. In the left menu, select Error Bars.
  4. Copy the data from the google form responses into the spreadsheet. You will have to relabel the columns in if you just copy and paste. I also create a column that contains 0.24 over and over again, since this is the true proportion of blue M&Ms advertised by the company.
  5. I plan to make my intervals, so I will have the Sample number as the x, the proportion as the y, and the margin of error for one of the CI’s be the Ey (error in the y direction).
  6. Scroll to the bottom of the left menu and select the blue Scatter Plot button.
  7. To add a horizontal line that represents the true proportion, I went back to the grid tab, then used the Make a Plot button to select Line Plot. I turned off the variable choices in the columns, and then chose Sample number as x and the true proportion column as y. At the bottom of the left hand menu, use the Insert Into dropdown to select the graph tab 2. Then hit Line plot for it to be added to that same graph.

Here’s my practice run:


I think we may get another snow day tomorrow. If so, I may do a screen recording video. (Now I’m thinking that would have taken less time if I had just done that in the first place…)

Unit Circle – Paper Triangles

I’m sure someone else had this idea long before me, but I didn’t consciously take it from anyone in particular. This year, I wanted to help my Pre-calculus students understand the patterns inherent in the unit circle. First I used this idea from Kate and Riley.

Since this activity didn’t make it clear how to get the radical forms of the key values for sin and cos, I wanted another activity to make those patterns evident. We spent a day reviewing special right triangles, and then the next day, I gave them this blank copy of the unit circle, having them fill in the degree measures first. Then I gave them quarter sheets of orange copy paper.

I gave these directions:

  1. Line one edge of the orange paper up to sit on top of the x axis.
  2. Align the right edge of the paper with the point is formed by the radius in in the 30 degree angle.
  3. Trace the radius (through the paper) in order to make your triangle.
  4. Turn the orange paper 180 degrees to get another corner (to save paper!)
  5. Repeat steps 1-3, but using the radius from the 45 degree angle.
  6. Cut out both triangles.

After that, we labeled the angles, and they were able to tell me that the hypotenuse was 1 unit because of earlier discussions on unit circle. We derived together the lengths of the missing sides through the formulas related to special right triangles to label the sides of the triangles. Have them label both sides of the orange triangles, so they can flip them and still read the measurements in other quadrants.

To get them started, we went to quadrant one and placed the 30-60-90 triangle in place. This helps them visualize the x-coordinate and y-coordinate of 30 degrees. I took them through the first quadrant and up to 120 degrees. In each class, someone was able to predict the coordinate sign changes that the x-coordinates are negative. At that point I left them to the task of completing the unit circle. I also gave them this handout to help them process through the patterns that they found. Here are some pics for the visual folks out there.

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Grading growth

I teach a class that was created as an intervention class for students with low ACT scores. Our school has recently become part of a blended learning pilot program, meaning that I have a class set of laptops now. In discussion with my principal, we have decided to make this class much more individualized, since one of the difficulties in managing that class is the huge gap between the highest and the lowest students. I am giving them a diagnostic ACT test and trying to figure out then where each of them should start their remediation. In order to run the class this way, we have to figure out a fair way to grade them based on their growth and effort. I’m thinking a rubric may be the fairest way to do this, but am stuck on what my categories/indicators would be. Got any ideas or even guiding questions for me?

#EduRead Reflections June 4 – Homework

For a primer on what #EduRead is

This Past Week’s Article: Homework: A Math Dilemma by Patricia Deubel

My Supplemental Articles: Algebra Homework: A Sandwich by D. Bruce Jackson, and Conceptualizing Drills by Nat Banting



The article entitled “Homework: A Math Dilemma” was a revelation for me. It put a framework on some thoughts that I had been struggling to articulate for a while. Not only did the short article talk about homework, but it also addressed a theoretical framework of learning math, which is the part that I had been lacking.

Basically, it alluded to research that there are four dimensions of learning math: Mastery, Understanding, Interpersonal, and Self-Expressive. I had generally been uneasy with my department’s pedagogy, worrying that we were worrying too much about skills and not much about connections, vocabulary, and concepts. Turns out that what I was worried about was a isolated focus on the Mastery dimension of learning math, which focuses on procedural fluency. I aspire to provide more in the realm of Understanding, which necessitates my students being reflective, making connections between ideas, etc. but they are so unused to that from other classes, that I’ve struggled to make much headway.

I have aimed to grow more at having students communicate more (Interpersonal) and to create (Self-Expressive) but have not had the mental bandwidth yet to make serious commitments to these. I’m looking forward to thinking more about those this summer, but it helps to even have categories to shoot for now.

The article introduced this paradigm to help its argument that homework shouldn’t just be a problem set. Ideally, it shouldn’t even just be a few versions of a problem set where students with higher understanding take on a harder version. They argued that there should be a variety of reasons and executions of what homework should look like, and they should cater to students who are naturally gifted in different ways in the 4 dimensions of learning math.

Here are some of the homework ideas that we brainstormed on the chat. In parentheses I noted which dimensions I think these assignments target best.

  1. Students choose a way to show mastery; “menu math” (all 4 dimensions)
  2. Assign 3-5 required problems and let students choose 3-5 more from a certain number of options (Mastery)
  3. Give differentiated assignments on a technology practice platform, such as Khan Academy, MathXL, where each student can work at own pace and get a topic that is appropriately challenging (Mastery)
  4. Students respond to some sort of journal prompt (Understanding, Interpersonal, or Self-Expression)
  5. Students create something using Desmos or Geogebra, etc. (Self-expression)
  6. Assign vocabulary terms to define or create/find examples of (Understanding or Interpersonal or Self-expression)
  7. Give students a few problems and the answers, ask them to fill in the steps and justify (Understanding or Interpersonal)
  8. Write summary notes or 3 questions the teacher could assign based on the day’s work (interpersonal or self-expression)
  9. Assign backwards problems: Give answers, make students write a problem that would solve to give that answer (self-expression)
  10. Students self-assess on knowledge, using a google form or some polling app (Understanding)
  11. Students write about which problems were hard for them and which ones were easy and why (Understanding or interpersonal)
  12. Students write on a discussion board online, respond to each others’ comments (Interpersonal)
  13. Random quizzes using a couple homework problems (Mastery)

Shout out to Sarah Aldous for a solid post of more homework ideas accumulated this week!

To grow on:

How do I grade all these ideas? I toyed with the idea of a general rubric that could apply to all homework, but I think if it is to apply to so many different purposes and formats, the language would be so broad that it wouldn’t work for any single assignment.

Summer Goals

If you have any ideas on resources I should look at for any of the above goals, let me know!

Read at least one peer-reviewed article on math education weekly

See my #EduRead posts.

Organize my curriculum into series of units and lessons

Primarily, I want to be able to offer my students a better overview at the beginning of a unit as to what is coming – big questions, key vocabulary, upcoming homework assignments/assessments, etc. so they can be more proactive. I think I need a better organization of my lessons and their hypothesized order so that I can achieve this.

Find and prepare an online learning management system for my AP Statistics class

I hope to increase the level of technology used in my AP class, starting by giving them a consistent expectation of where to find information about their class and how to participate. As I see many colleges incorporating Blackboard or something similar, I would like to incorporate a comparable paradigm so they will be able to begin transitioning to that mindset ASAP.

Develop a better paradigm for teaching students how to have mathematical discussions

I think in general, my students often struggle to communicate to one another logically, but there are many cases in which my students clearly get stuck, and they don’t even know how to ask me a question. I have heard a bit about “Accountable Talk” from some folks in my district, but am generally unfamiliar with it. I think I would like to offer my students sentence starters, and start by practicing these things with simpler math problems to help them feel confident in their ability to communicate.

Re-work my policy for following up on assessments

I have offered students the chance to do test corrections to earn credit back. I like the policy, because it offers the students a reason not to despair over testing, and also gives them a chance to learn from their mistakes. However the actual format I’ve had them use has been a bit clunky and rigid. Also, I’m trying to determine if there is a way I can have students work on similar problems, but not the same problems that were on the test so that I can prevent some incentive to cheat for those who need to do makeup assessments.

Create a plan for student jobs

My coach challenged me with a story from her teaching days where she realized a particular student was much more charismatic than she was. She engaged him as the designated emcee for learning games and other activities to get the class excited and involved. Upon further conversation, she said something to the effect of “If there are students who do ____ at the same level that I do, why should I keep that to myself?” I plan to invest students in my classroom by giving them the chance to volunteer for particular jobs (paper returner, emcee, calculator collectors, new student mentor, etc). I have a hunch that having some students responsible for the classroom could even encourage some of the more distant students to get more deeply engaged.

Make assessments cumulative

Plan out the first few weeks of school

I hope to incorporate a mix of math prerequisite skills, good student habits, and introduction to the technology I will be relying on (Desmos, Geogebra,, and a few others)

Make my syllabus into an infographic

I saw this idea somewhere online, but forgot to save whose idea it was. This may be more for my fun than for my students…

#EduRead Reflections May 28 – Mental Math

I decided that one of my summer goals would be to read at least one math education research based article per week as a way to deepen my thinking for next year. I had found out about a Twitter chat called #EduRead, which meets every Wednesday at 8 pm central time, with the goal of reacting to and sharing ideas regarding a pre-selected math education research article!

This satisfied my cravings to deepen my Twitter PLN as well as meet my goal of reading more math ed research.

My first week to participate, we read over Mental Mathematics beyond the Middle School by Rheta Rubenstein, published in 2001 in the Mathematics Teacher journal from NCTM.

Reading this article helped me to realize how my students struggles to learn new ideas can potentially be attributed to a lack of real fluency and confidence with prior skills. Teaching precalculus this past year, Rubenstein’s suggested objectives a precalculus student should master with mental math were often roadblocks to new learning for my students (evaluating common exponents, negative and zero exponents, basic roots, parent functions, intuition linking graphs to situations, etc.).

Rubenstein Precalculus Objectives

I struggled to differentiate for my students, and so I generally had to take the attitude of “if you didn’t get it then… come to tutoring?” This solution did not work for most of my students, who often have complex after-school schedules. Rubenstein suggests it is critical to integrate practice time in developing mental computational fluency in these prerequisite skills.

The other big idea I took away from her writing was summed up in this quote:

I know that I have taken the stance of opposing calculator use with my students when they get lazy on arithmetic problems and some other skills that I teach in class, but I don’t often have a supplementary strategy to offer instead. In fact the only one I feel confident in offering them a different strategy is in adding and subtracting with negative numbers.

My questions going forward, which we also struggled to answer during the hour-long chat:

  1. Do we have time for this? Or maybe can we afford not to do this?
  2. What can this look like in a classroom while you are also obligated to cover grade level content?

Would you Rather: Hypothesis testing

I am teaching Type I and Type II Errors Monday and wanted to get students thinking through the problem of balancing the probability of the two errors. I want them to understand the tension and realize they will not be able to eliminate either possibility. I plan to do this before actually introducing the names for the types of errors. I just wrote up these two “Would you rather?” scenarios. I would love some feedback or some other suggestions to use as follow up.

For the two scenarios below, decide which error would be worse. Clearly state your answer and your reasoning.

Scenario 1

As a doctor, you see a large national study that 35.9% of Americans are now considered obese (H_0:p=.359). Alternatively, you think it may be possible that your patients are below the national average (H_a:p<.359). If you conduct a hypothesis test on a sample of your patients, which would be worse?

a.Rejecting the null hypothesis and stating that less than 35.9% of your patients are obese when in fact your patients are in line with the national average.
b. Failing to reject the null hypothesis that your patients match the national average, when in fact less than 35.9% of your patients are obese?

Scenario 2

You are a researcher. Patients using the current treatment for lung cancer go into remission 55% of the time (H_0:p=.55). You believe that you have found an improved treatment (H_a:p>.55). Which would be worse?

a. Rejecting the null hypothesis and stating that your treatment is better when in fact it is not.
b. Failing to reject the null hypothesis that your treatment is equal to the current treatment, when in fact it is better.