Will it fit?

I used this lesson in MTH 221 (Mathematics for Elementary Teachers) to address Common Core State Standard 7.G.B.4. It seems to have a lot of potential, but there are still some elements that I think need to be tightened up. These are written in red - along with some other thoughts. I would appreciate any feedback on how to improve this lesson.

[Schema Activation]

How many of you know your fitted hat size? For example, I wear a seven-and-seven-eighths. Today, you are going to find your hat size and what that number means. 

I like the DC cap for a couple of reasons. First, obviously, DC are my initials. Second, it foreshadows the circumference and diameter relationship we will explore in the lesson.

In the past, I have had students measure a bunch of circles to find the ratio between circumference and diameter. It has been a struggle to make it an interesting lesson. This connection to something personal (hat size) seemed like it might be an improvement.

According to LIDS.com, there are a couple of ways to determine your hat size.

We will use both a flexible tape measure and their printable ruler in order to...

[Focus]

... consider the following questions:

• How are hat sizes and head-measures related? (In other words, if we didn't have access to their table, can we determine a hat size given a head-measure?)
• We are supposed to be working on CCSSM 7.G.B.4. Is this connected in some way to circles? Too obvious?
• Why don't hat makers just use the head-measurement as the size?

[Activity]

Measure your head using both the flexible tape measure and the printable ruler.

Feel free to wear the printable ruler as a stylish headband as you work. Optional

Place both your head-measure and hat size on a sticky note and place it at the proper coordinates on our graph.

What does our graph show? Is there a relationship? If so, what do you predict the relationship to be? (If we input 22 inches for head-measurement, what hat size is the output? What if we input C inches?)

Here are a couple of tables for hat sizes from Lids.com. Let's use them to see if we can determine the input-output rule they are using to find hat size from head-measurement. 

There are other hat size tables (for example), but I like that this one makes 22 inches a hat size of 7 because 22/7 is often used as an approximation of pi.

[Reflection]

What did you find? What is the rule?

Des found the following: y = 0.333x - 0.333. Does it work for our table? If it's correct then what hat size does a person need for a head-measurement of 24?

Okay, I wanted to play with the new Desmos linear regression feature - sue me. Is it a problem that this line doesn't go through the origin? That the slope actually represent an approximation of 1/pi? During the lesson, it seemed like this portion required a lot of scaffolding.

So what does the hat size mean? Let's take our headband and place it on the table. Notice that it is nearly the same shape as a circle. Now measure the distance (diameter) across your headband (circle).

In my case, if I measure what is approximately the diameter of my headband, I find the length is close to my hat size (seven-and-seven-eighths). How about you? What does that suggest our hat size means?

The students were most impressed by this portion of the reflection. They liked that the hat size number was not some arbitrary value - that it was actually connected to something mathematical. I looked for some history of hat sizes to explain why this value is used instead of circumference, but Google failed me.

Now what fitted hat size should I buy from Lids.com if my head measure is 24 inches? Seven-and-two-thirds doesn't look like it's an option.

In an earlier unit, students struggled with the idea of independent and dependent variables and creating graphs that accurately represent a real-life situation. Because a hat maker does not make all possible diameters, we decided it didn't make sense to connect the dots. Instead, we came up with the graph shown above.

One of the reasons I like this activity is because it does connect with so many other standards, like 6.EE.C.9 and 6.SP.B.4. What do you think? Does this lesson have merit - is it worth saving? If so, how? Please add your thoughts in the comments.

Updated: As much as I loathe Pi Day, this piece on Stormy Kromers (hats made in the Upper Peninsula of Michigan) might make a nice connection.

At what level is his thinking?

Pierre van Hiele

My colleague, Jon Hasenbank, and I have been discussing the van Hiele Levels of Geometric Thinking and what they mean for teaching and learning in mathematics. I am particularly interested in finding videos of people sharing their geometric thinking so that we can apply the Levels and evaluate their thinking. If you are not familiar with the Levels, here's how Pierre van Hiele, the architect of the Levels, described the first three Levels in Developing Geometric Thinking through Activities That Begin with Play [PDF].

In my levels of geometric thinking, the "lowest" is the visual level, which begins with nonverbal thinking. At the visual level of thinking, figures are judged by their appearance. We say, "It's a square. I know that it is on because I see it is." Children might say, "It is a rectangle because it looks like a box."

At the next level, the descriptive level, figures are the bearers of their properties. A figure is no longer judged because "it looks like one" but rather because it has certain properties. For example, an equilateral triangle has such properties as three sides; all sides equal; three equal angles; and symmetry, both about a line and rotational. At this level, language is important for describing shapes. However, at the descriptive level, properties are not yet logically ordered, so a triangle with equal sides is not necessarily one with equal angles.

At the next level, the informal deduction level, properties are logically ordered. They are deduced from one another; one property precedes or follows from another property. Students use properties that they already know to formulate definitions, for example, for squares, rectangles, and equilateral triangles, and use them to justify relationships, such as explaining why all squares are rectangles or why the sum of the angle measures of the angles of any triangle must by 180. 

And now for some practice. Given these descriptions, how would you categorize the thinking of the individual in this video?

Slice a rectangular pyramid: What happens when you slice vertically into a rectangular pyramid? What kind of geometric shape results?

What evidence do you have to support your categorization of his geometric thinking? (Perhaps there are clues in the task's questions.) If more evidence is required, what questions might you ask to get a better sense of his Level of Geometric Thinking?

As always, your thoughts are welcome in the comments - as long as they are civil.

How does it fit?

At the beginning of the semester, I wrote a post about using the history of the LEGO company as a cautionary tale for innovation in education reform. The idea came from a Diane Rhem interview with the author of Brick by Brick. But there was another idea presented in book that also caught my attention - clutch power.

When a child snaps two bricks together, they stick with a satisfying click. And they stay stuck until the child uncouples them with a gratifying tug. And therein lies the LEGO brick's magic. Because bricks resists coming apart, kids could build from the bottom up, making their creations as simple or complicated as they wanted. … it is clutch power that makes LEGO such an endlessly expandable toy, one that lets kids build whatever they imagine. (page 20)

It was the last sentence that got me thinking. How might I design learning experiences that use clutch power - allowing learners to "build whatever they imagine?" In particular, I had one of our math courses for preservice teachers in mind. It explores a variety of mathematical topics that students often see as disconnected: algebra, geometry, measurement, and data.  Was there a way to treat concepts in that course as building blocks that learners could use across the domains to build new understandings instead of simply following the instructions I provide?

I am still working on that question, but I did start experimenting with simple problems that allow learners to add their own ideas while demonstrating their content competency. The following is one problem (modified) from the final I just gave. I would be interested in your feedback.

Plot the points (2, 2) and (2, 4) on the coordinate plane. Pick two other points that could combine with these first two to be used as the vertices of a trapezoid. Use this context to develop items that would align with the following targets (be sure to justify your alignment using the indicators):

• Graph points on the coordinate plane to solve real-world and mathematical problems (5GA1, 5GA2)
• Classify two-dimensional figures into categories based on their properties (5GB3, 5GB4)
• Solve real-world and mathematical problems involving area, surface area, and volume (6GA1-A4)
• Draw, construct, and describe geometrical figures and describe the relationships between them (7GA1, 7GA2, 7GA3)
• Solve real-life and mathematical problems involving angle measure, area, surface area, & volume (7GB4, 7GB5, 7GB6)

Because this was a new assessment for most of my students, this time around I offered some example items to pick from:

1. Define the term "trapezoid" and then use that definition to identify what other names apply to your shape.
2. Find the area and perimeter of your shape.
3. Extend each side (creating 16 angles) and find all the angle measures.
4. Write your own question for this context.

I still expected them to align the item with the targets, justify their alignment, and come up with a correct response.

Does this make sense?

The following guest post comes from my friend and colleague, Robert Talbert. It is thanks to the success of the University of Michigan (MS '92) in the March Madness tournament that I earned this honor. I do not know if Robert is aware that I also attended Northern Michigan University (Teaching Certification '87) and that my grandfather was the registrar there, but these connections make this post all the more special. Thanks to Robert and GO BLUE!

*****

“It makes sense when you do it, but when I try it, I have no idea where to start!” In other words, it never made sense. The sense-making was only an illusion and did not actually happen.

Anyone who has taught mathematics to a real human being for more than 30 minutes has encountered the above statement. I think the sooner we realize that sense-making is not something that you feel but something you do, and the sooner we mathematics instructors shift our focus away from providing feelings (can I say “sensations”?) of sense-making and onto the actual making of sense in the classroom, the better off we will all be.

I’m currently at the annual meeting of the Michigan section of the Mathematical Association of America, where a number of the talks so far have centered on sense-making, none moreso than a plenary talk by Peggy House of Northern Michigan University titled “Reasoning and Sense-Making”. I appreciate Peggy’s perspective on reasoning and sense-making, since she is a veteran classroom educator and math education specialist. Regular readers of my blog over at the Chronicle of Higher Education know that I spend a lot of time thinking about various instances of the flipped classroom, including the adaptation of Eric Mazur’s model of peer instruction to university mathematics. The more I work with these pedagogical models, the more I realize that sense-making is really what they are all about.

A good peer instruction question, for example, is one that is aimed squarely at the biggest misconception on a fundamental concept — which is to say, it unearths the very thing that students most urgently need to make sense of. And then we ask them to make sense of it. And these days, when I discuss the flipped classroom with someone and they ask, “What do you do with the class time once the lectures are moved out?”, I just say: sense-making activities. I want students not only to make sense of the things they are learning but also learn how to make sense of things.

In Peggy’s talk, she posed the following problem: Find the area of the octagon in the picture below, which is formed by the intersection of the various segments connecting vertices and midpoints of a square:

One of the first points Peggy made in her talk was that sense-making is in the eye of the beholder. When instructors approach mathematics as pure procedure and then focus on showing students how to perform a procedure, this not only is not sense-making, the thing we are talking about will only make sense to the students insofar as their ways of seeing the problem coincide with our ways of seeing it. Which is to say, most of the time it won’t make sense at all, even though after a clear lecture it will seem to make sense. This is a dangerous position for students. It would be better to have concepts not make sense and for students to know that they don’t make sense, than it would be for the concepts not to make sense although students think that they do.

In the octagon area problem, for instance, I could certainly lead students through a clear discussion of a solution through direct instruction. This would not be all bad. There is some value to seeing an expert learner (that’s me) explicitly model his or her decision-making and thought processes as he or she works through a problem. But the benefit ends there, because this problem is not about “the right answer”. Very few nontrivial mathematics questions are! Peggy went on to give examples of some of the ways her geometry students solved this problem:

Some of them were wildly creative:

And some of them were creative in all the wrong ways — for example, one group got an answer by assuming that the octagon is regular, which as it turns out, it isn’t. And that’s where reasoning comes in to sense-making. When a student is solving a problem, it’s not enough to be creative. There also has to be good reasoning involved as well. If one’s reasoning behind a solution or a conjecture is flawed, then the concept being thought about has not fully made sense yet.

One of the great lessons learned from this talk, something I knew already but can always use encouragement on, is that my job as a mathematics teacher is not to download by brain into the students’ brains, or even to wire their neural pathways to look like my neural pathways. Indeed this is not really even possible, because — thank goodness! — every student is a unique human being, with a unique backstory and ways of looking at the world. Sense-making is in the eye of the beholder. My job instead is to provide them with an environment that is rich in sense-making activities and in which students are held to high standards in the ways in which they argue for their solutions.

Someone in the audience asked, not cynically, how instructors in K–12 schools can be expected to teach in this sort of way while still preparing kids for the content on standardized tests. Peggy’s answer was, you just have to do it, and perhaps there are ways to design classes where you can teach content in this way and not lose much. I have nothing to add to that answer, except that I hope that people everywhere begin to realize that it doesn’t do much good for students (K–12 or university level) to know a lot of content that makes no sense to them.

Epilogue: I’m writing this guest post because I lost a bet that I never made. My wife’s alma mater Indiana University was making a good run of it in the NCAA mens’ basketball tournament back in March. Dave, John Golden, and I thought it would be fun to see whose Big Ten alma mater (or proxy thereof; I went to Vanderbilt, which posted a relatively miserable 16–17 record and didn’t even make the NIT) went the farthest. Needless to say the Hoosiers didn’t get it done. But at least some good can come out it.

Did you get anything good?

For years, I have tried to write first semester exams using a holiday theme. Even as a middle school math teacher, I set problems in the North Pole or around different traditions celebrated this time of year. Given a recent #mathchat on holiday mathematics, I thought I would share some of the problems here. Consider it my gift to you. Happy Holidays!

On the twelfth page of the test my teacher gave to me, twelve days of giving.

Use the lyrics from the traditional song, The Twelve Days of Christmas, to complete the table shown below.

Show how you could represent the gifts given each day using simple pictures, a recursive formula, and an explicit formula.

• How many gifts are given on the 12^th day?

• How many total gifts are given over all 12 days?

On the eleventh page of the test my teacher gave to me, eleven Lions playing.

A while back, WOOD TV8 meteorologist, Bill Steffen, made a claim that the Detroit Lions’ record might be related to the amount of snowfall we get in Grand Rapids.  I emailed him asking for more information and part of his response is shown at the right.

• The scatterplot below verifies that a positive correlation does seem to exist between the number of Lion wins and Grand Rapids snowfall (based on the data provided).  Draw in an approximate line of best fit and use it to predict the amount of snowfall this season if the Lions win six games this season.

• By Steffen’s own admission, he “got to pick the years”, which is a factor.  Describe another sampling method that might better represent the relationship between wins and snowfall.

On the tenth page of the test my teacher gave to me, ten snowmen chillin'.

The idea of using snowmen glyphs to represent some of our winter and holiday preferences was introduced here. Recall that if the person making the glyph wears a hat during a typical winter day, then he or she would fill in the snowman's top hat. The same goes for wearing a scarf. Given the 10 snowmen shown below, answer the following questions.

• What percent of the people represented by these glyphs wear hats on a typical winter day?
• What percent of the people represented by these glyphs wear scarves on a typical winter day?
• What percent of the people represented by these glyphs wear hats or scarves on a typical winter day?

On the ninth page of the test my teacher gave to me, nine cookies baking.

As you might suspect, Mrs. Claus is a wonderful baker, especially when it comes to cookies. Her specialty is called Santa’s Hats, which are basically isosceles triangles with red frosting and sliced marshmallows across the bottom and on top. Because so many of the cookies she bakes are for the elves, Mrs. Claus thinks of a batch of cookies in terms of three-quarters of a dozen. If each of these batches takes eleven-and-a-quarter marshmallows, then how many marshmallows are required for a dozen cookies?

On the eighth page of the test my teacher gave to me, eight reindeer playing.

You know Dasher and Dancer and Prancer and Vixen, Comet and Cupid and Donner and Blitzen: But do you recall… what you learned in math class at all?  Use these eight famous reindeer (sorry, no Rudolph) to solve the following problems.

• Only two reindeer can lead the team.  How many different leading pairs are there (left side and right side does not matter)?
• You have probably heard about the Reindeer Games. Did you know that they only give ribbons for 1^st, 2^nd, and 3^rd place? How many different ways are there for the three ribbons to be awarded the eight reindeer? (Assume that each reindeer is equally likely to win.)
• Which of the above questions is a permutation and which is a combination?  Please be sure to explain your rationale.

On the seventh page of the test my teacher gave to me, seven candles burning.

During the week of Kwanzaa, families gather in the evenings to light the candles of the kinara and discuss the Nguzo Saba or seven principles. There are seven candles – three red candles to the right, three green candles to the left, and one black candle in the center of the kinara.

A manufacturer of Kwanzaa candles did some research and found out that the candles do not all burn at the same rate. Below are box-plots representing the number of minutes it took for each candle to burn completely, separated by color (100 candles of each color were tested).

Which candle color seems to last the longest?  Describe how the box-plots support your reasoning.

On the sixth page of the test my teacher gave to me, six bulbs a blinking.

In order to save money this year, Dave decides to buy outdoor bulbs at Ben’s Bargain Bulb Bin.  The bulbs there are cheap, and there are a lot of them (some say an infinite amount), but only three out of every four bulbs work. Dave needs 6 bulbs for the end of each snowflake in a string of lights, but if even one bulb doesn't work, then the entire snowflake remains dark. What are the chances that a snowflake using bulbs from Ben's will be unlit?

On the fifth page of the test my teacher gave to me, five Pecan Puffs.

Just in time for the holidays, a new item has been added to the J. Peterman Candy Collection – Perfect Pecan Puffs. These are five pecan puffs packed in cube-shaped boxes that are arranged to look like a "P" (see ad below) held together using green holiday paper.

The paper is expensive, so Elaine suggests using a "net" to cover the packages. What might this net look like?

PS: Kramer thinks that this is the perfect gift for Festivus!

On the fourth page of the test my teacher gave to me, four lighted trees.

Clark Griswold just purchased The Growing Tree Kit to add to his Christmas light show. Each kit has four, green, foot-high triangles/trees and four strings of red lights that fit around the perimeter of each tree. What makes these trees special is that they stack to make larger trees. Then the string of lights can be put around the perimeter of the larger, combined tree (see the examples below).

Single tree       Combined tree

When Rusty puts together the combined tree, he notices that two of the strings of lights are left over. "That's not a problem," says Clark. "We'll just add them to the roof. What I'm worried about is the visibility of the tree. I think we need to make it bigger - maybe 10-feet tall."

In order to make a 10-foot tree that is mathematically similar to the smaller trees, how many kits does Clark need to buy? Also, how many strings of lights will be needed to light the 10-foot tree's perimeter? What do you notice about the area and perimeter of these similar triangles as they dilate?

On the third page of the test my teacher gave to me, three wise men.

A version of this problem can be found in
Connected Mathematics Project's
Clever Counting

In the story of the Magi, three wise men come from the east looking for the king of the Jews. On their way, they stop at King Herod's for directions. They find out that the Messiah was to be born in Bethlehem. Imagine that the network shown to the right represents the many paths that the wise men could have taken.

• How many paths were there from the East (E) to King Herod’s Temple (H)? Include only the shortest paths (south/down or west/left).
• How many paths led from King Herod’s Temple (H) to Bethlehem (B)? Include only the shortest paths.
• In all, how many shortest paths are there from the East to Bethlehem that pass through King Herod’s Temple? Explain how you arrived at your solution and why it works.

On the second page of the test my teacher gave to me, two types of stools.

From America Trek

In a shed at the North Pole are the parts for 75 stools for the elves in Santa's Workshop. Some of the stools will be three-legged and some will be four-legged (for the bigger elves like Buddy). The shed contains 75 seats and 259 legs. If Santa does not want any parts left over, then how many of each type of stool can be made with these parts? Please use two different representations (pictures, words, symbols, ...) to communicate your thinking.

On the first page of the test my teacher gave to me, a Dreidel with a counting tree.

The Dreidel is an integral part of the celebration of Hanukkah. Children play by spinning a Dreidel and depending on the side it lands on they win or loose candy. Using the modified rules shown below and this online Dreidel, investigate what happens on two successive turns.

Specifically:

• What are the chances that the combination of these two turns will result in a negative score?
• What is the mean number of points you earned as a result of spinning the Dreidel twice?

Conduct ten trials and record the cumulative points won or lost as a result.

Dave conducted 990 trials of his own (see the table below). Add his results to your own, and answer the questions of interest. Why is this experimental probability?

Points	40	25	20	18	10	5	3	0	-2	-4
Results	57	127	127	121	61	124	111	61	136	65

Check your results against the theoretical probability. Be sure to support your work with some sort of representation (organized list, tree diagram, table, area model, or network). Why is this theoretical probability?