A time problem

Wrote this problem for my spring final and fell a little too in love with it.  Has some fun algebra behind it.

Jane glanced up at the clock and noticed that when the second hand was on the 12, the three clock hands divided up the clock into a right, acute and obtuse angle.  What time was it:

• 3:30, 
• 5: 43, 
• 1:22, 
• 9:15, 
• more than one possibility or 
• none of those  

Remember the hour hand moves during the hour.

When grading, many students ignored the idea of the hour hand moving in between, so I evaluated based on their assumptions.  In general on the test, I was trying to create the possibility of seeing some problem solving, where they could demonstrate understanding of ideas without necessarily having to get a right answer.  It was partially successful.

As I was trying to write the problem, I posed myself this question: 

If it is y o'clock and x minutes past the hour, what is the angle formed by the clock hands?

If you're considering either, I'd love to hear what you think in the comments.  How do you evaluate the first?  In the second, would you expect the equation to be linear?  Why?

Cartoon from xkcd, of course.



Some of the exam problems were pretty open-ended, like:

1.    Find an L-shaped figure with an area of 84 sq.cm and a perimeter of 44 cm.  Is there more than one?
2.    What kind of triangles can be made by connecting vertices on a regular octagon?  Specify the side-angle type.  Did you find all of the types? 

And some were more closed, but hopefully with multiple ways to do them.

5.    Sort the quadrilaterals into two overlapping Venn diagram circles: one for rotational symmetry, one for reflectional symmetry.  Quadrilaterals that don’t fit either should go outside.
6.    A Hershey’s chocolate bar is 43 g.  A kiss is 4.56g.  You remember that 1 pound is 454g and 1 pound is 16 oz.  How many ounces is a Hershey bar?  A Hershey kiss?  How many kisses in a bar?  (Make a joke if you want.)

Nobody made a joke.  How many kisses in a bar?  Come on!

Pi in June

In honor of 2pi day (June 28th)...

From the awesome Dinosaur Comics, of course.

Why celebrate half the holiday?  Get the whole circle!  That old pi day is for squares.  Er, semicircles, at least.

Seriously, we did this activity in the preservice elementary class and I thought I'd share.  It culminates in the usual measure a bunch of circles activity, but tries to motivate it.  Sure it's amazing that all those ratios are close, but why would you do it in the first place?  And why are they all the same anyway?  For me the answer is similarity.  This activity was motivated by my students wanting to know more about pi, as it came up when doing volume of solids (someone remembered a formula), and that got people wondering.  In addition, our geometry class comes before our number class, and lots of students had said that fractions were what was most confusing.  And pi practically lives in ratio city.

Similarity

Two objects are similar in geometry if one is an enlargement of the other.  Mathematicians often use ratios to investigate them.

Consider a 2x3 rectangle.  A 4x6 rectangle is an enlargement.  But a 4x5 rectangle is not.  Can you tell by looking?  Describe what you see.


People discussed how the 4x5 should be similar because its 2 wider and 2 longer, but in the 4x6, which looks the same, you can see 4 or the original rectangle.


What ratios can you make with the two similar rectangles that are the same?

Students made both 2/3 to 4/6 and 2/4 to 3/6.


If you wanted to make a rectangle that was similar and 5 squares wide, how long should it be?  Can you prove your answer is right?

Some students set up a proportion and solved with cross multiplying... although they confessed that they didn't know why cross multiplication worked.  One student saw 5 as 2 + 2 + half a 2, and calculated 3 times 2.5.  We talked about how that was excellent proportional thinking because instead of 2+2+1, she relatedit back to the original as half of 2.  I showed how in the picture you could see the 7 as 2.5 of the 3's, or 3+3+half of 3.

A 24x36 poster is supposed to be an enlargement of an 18 x 27 poster.  Is that possible or not?  Explain?

Saw quickly with ratios.

Find the lengths of the diagonal of each poster using the Pythagorean Theorem. 

Refresh, connect question.

Sum up what you see about ratios and enlargements/similarity:

Jotted down and discussed at their tables.  Enlargement and proportional were heard in a lot of the conversations.

Common photograph sizes are 3x5, 4x6, 5x7, 8x10, 11x14 and 16x20 and 20x30.  Sketch, outline or shade in the ones you can on graph paper.  Which are similar?  Which are closest to similar?


The sketched them separately, so I did a nested version on the board to show another way.  (Often a student will have done it that way.)


Most cameras record in a ratio of 3:2, movie camcorders in 4:3 and hi-def in 16:9.  How do these compare to common print sizes?  What sizes would you use for your regular prints and for an enlargement?

General agreement on the 4x6.



Circular Arguments

Looking at these circles, they look pretty similar in the usual English sense.  Are they similar in the mathematical sense?  How do you know?



This was an unexpectedly interesting discussion.  Most people thought yes, they are all similar,  and then one student made them hesitate.  That led to proportional talk and someone brought ovals into it.  I asked if people knoew what made a circle a circle and they didn't.  They discussed how the only things you could measure are the radius/diameter or the circumference.


Give three different pairs of similar rectangles.  Compute the ratios of their perimeter to their longest side.  What do you notice?

Students saw the constant ratios quickly when they shared examples.  One student made the jump to the idea of scale.  Pretty cool.

What might that mean about circles?  Give your reasons.  (This is basically asking for an educated guess.)

Time was tight, so I was a bit leading here.  What ratio would the rectangle example be like for circles?  "Circumference to radius."  So if circles are all similar... "Then the ratios should be the same."


Divide and Conquer

•    Data collection:  in your group measure the circumference and diameter of at least 10 different circles. (details to follow)
•    Add your data to the class stem and leaf.
•    Which is the best choice for typical (mean, median, mode) and why?

 We discussed how to measure diameter and circumference, and agreed on precision.  (Two decimal places for the ratio.)  We got over 50 data points, and it made a pretty nice bell curve between 2.8 and 3.8.  There were two measurements of 6.00, but she realized she had divided by the radius.


The mode was 3.20 (5) followed by 3.18 (4).  The median was 3.19.  The mean about 3.22.  Curiously, the majority of students felt that the spread was too big to be explained by measurement error, so circles must not all be similar!  I did let them know that they were, but it would take a more theoretical argument to prove it to them.

All in all it felt much more contextual to start with similarity, and students made a lot of sense about why pi might have been discovered.

The activity sheets are below.  But also be sure to check the comments, where Alexander Bogomolny has a couple of very relevant links about the idea of measuring one circle as accurately as possible.
Similar to Circles

Variable and a Problem

The Geogebra post on variable was gathering resources for a middle school inservice with math teachers and special education teachers.  (I've added the geogebra files for my sketches.)  Unfortunately, everything that worked smoothly the night before caused hangups on the day.  Sigh.

I'm working on this project with Esther Billings and Pam Wells, who both have a great grasp on teaching this subtle concept and are amazing teachers.  Pam is a wonder at working from student/participant work towards her goals, and Esther has this great integration of research understanding with what it means for teaching.

Objective:  TPW understand variables can represent changing quantities or represent an unknown; experience doing algebra with meaning and understanding of underlying concepts.

The day before they had spent a lot of time working on and watching student work for finding the patterns from pictoral sequences and expressing the relationship symbolically.  Including the very nice Modeling Middle School Math video of the Beams and Rods problem, from the Math in Context curriculum Building Formulas lesson (videos 8-13).  

We wanted to keep the element of having the teachers have a chance to do mathematics instead of just talk about it, and we also wanted it to connect to the Connected Mathematics Project Bags of Gold problem from the Moving Straight Ahead unit.  Swapping problems help develop the idea of equality, which is central to the idea of variable as unknown.  The bags of gold gets at it by putting the same amount in each bag, but you don't know how much that is.  So if 2 bags and 3 coins is the same as 15 coins...

I hate contrived situations, so my first thought about swapping was Magic cards (or Pokemon or Yu-Gi-Oh) but that seemed irrelevant to the audience.  Pam had the nice Stuart Murphy book Dinosaur Deals about that sort of thing.  My next inspiration was currency trading.  Found the current rates, jiggled it around, and realized it was entirely proportional.  Save it for later.  I wanted an exchange plus.  I thought about a classroom rewards situation (these were teachers) that a friend uses, and then complicated it.  So I wrote this problem:

Classroom Rewards

For small achievements or solid whole group work, students in Ms. Smittyck’s class earn a white chip (like presenting a problem at the board).  For more significant achievements  or an action that benefits others (like raising your grade in the class or helping another student meet a standard), a student can earn a red chip.  For very notable work or effort or action on behalf of another student (like figuring out a way to increase recycling in the classroom), students can earn a blue chip.  If you can get to 6 blue chips, you get a jolly rancher on Fridays for free.

Two white chips can be traded in for a jolly rancher.
Six white chips can be traded in for three red chips and a jolly rancher.
Five red chips can be traded in for two blue chips and a jolly rancher.

What would be a fair trade involving white chips and blue chips?  At the end of the year, what would be a fair trade for blue chips in terms of jolly ranchers?  What other questions does this raise?  

Connections:  Does the situation make sense?  Do you need more information?  What does an answer to the problem look like?

Focus:  How will you answer the question?  Have you ever solved a problem like this one?  Is there a representation that would be helpful?

Activity:  Solve the problem.  Try to record your thinking.
Extension:  If you were going to let the class trade in chips as a whole group for a pizza party or doughnut day, what would make a reasonable goal?  Why?

Reflection: How would you check your work?  Now that you know the answer would you solve it another way?

Teacher Work
This was my first time with the group, so I was worried:  was it too messy? Too easy?  Too contrived?  (I hate contrived problems - especially mine!)

But they were amazing.  They dived right into the problem, asking great making sense questions.  What is this?  How does that work?  Why wouldn't they just...?

People worked with equations (because that seemed more mathy), tables (because that seemed helpful) and making pictures with blocks (which we had put out on the table beforehand without comment).  The blocks group had the most rapid progress and worked things out in multiple ways.  Nobody's answer matched mine.  But that wasn't the point.  They did want to be told the answer, but were okay with 'later.'

The free Jolly Rancher caused the most problems.  People thought you were trading the 6 blue for 1 Jolly Rancher, or that it was in addition or a one time thing.  The other confusing thing was that the red and blue chips come out surprisingly (to people there) close in value.  Given that, I reworked the problem a bit.  This retains the messiness and elements of non-proportionality.  If you want to make the problem considerably cleaner, make it 11 white chips for three red chips and a Jolly Rancher


Classroom Rewards, v2

For small achievements or solid whole group work, students in Ms. Smittyck’s class earn a white chip (like presenting a problem at the board).  For more significant achievements  or an action that benefits others (like raising your grade in the class or helping another student meet a standard), a student can earn a red chip.  For very notable work or effort or action on behalf of another student (like figuring out a way to increase recycling in the classroom), students can earn a blue chip.  If you get into the 6 blue chips club, you get a jolly rancher every Friday without having to trade anything in.

Two white chips can be traded in for a jolly rancher.
Twelve white chips can be traded in for three red chips and a jolly rancher.
Ten red chips can be traded in for two blue chips and a jolly rancher.

What would be a fair trade involving white chips and blue chips?  At the end of the year, what would be a fair trade for blue chips in terms of jolly ranchers?  What other questions does this raise? 

Questions
How would you work on this problem?  How would students?  Is it too messy for students?  If you're interested, I'd be curious about your comments.

Geogebra: variable

A brief collection of resources for starting to consider the idea of variable on Geogebra.

Geogebra is the free dynamic geometry and algebra software available either to download onto your computer or to run as a web app.  Find out more at www.geogebra.org.   Webstart installs it, and Applet start runs it in your browser.  The main place for finding pre-made sketches on specific topics is the Geogebra Wiki.

As we consider the idea of a variable, there are two main uses in secondary algebra: to represent a changing quantity, and to represent an unknown.  Both uses are an abstraction or generalization from arithmetic, and challenging concepts for learners. 

One sketch that addresses the idea of the unknown asks students to identify an x on the numberline, where some other integers are located as clues.  Give it a try.   (More sketches from this group at realmath.)

One sketch that uses time as a variable in a 6 hour car race at constant speed explores linear functions in terms of slope and intercept.  Give it a try.  The sketch can be a little fussy, but has a lot of great features.  (See their other sketches at Math 24-7.)

This is my first attempt at a Guess My Rule sketch (webpage).  It lets you get a new rule, and pick a variety of inputs, but is pretty clunky as you have to check and uncheck boxes.  (Geogebra file.)


 

If you just want to try basic linear graphing in geogebra (webpage), give this sketch a try.  It lets you adjust the y-intercept and another point on the line. You can also enter a line of your own using the input bar.  (Geogebra file.)

What's Math Got to Do With It? - Wrap Up

We had our final book group on What's Math Got to Do With It? by Jo Boaler.  See record of our previous two discussions here.  I didn't realize that she had moved on from Stanford to the University of Sussex.  There's a video of her talking about the book, and also maybe a TED talk on the way?  (At least TED has a profile of her.)


Interestingly, Keith Devlin recently talked about this book, too.  It's worth reading his thoughts about it, in the context of emphasizing thinking over skills.  I'm probably not that extreme in practice, trying for thinking along with skills.





From LoosePartsComics.com


What the students noticed:

• Parents using puzzles or games at home.  Makes it easier for parents to be and get involved.
• Learning goals: as students we never knew the learning goals.  I can or I will statements...
• Important to make the classroom open to questions, with validation for questioners. 
• Teachers can respond without giving the answer.  What did you get vs how did you get there.  Dig deeper.

Chap. 8: would like other ideas for games.  The ones in the book were good and raised questions even for us.  Manipulatives are good, especially for children who can not do it mentally.
What about time management?  Don't manipulatives take longer?

Homework that parents can't help with...  What do those students do?  And it's anxiety-causing for a parent who doesn't understand and doesn't want to admit it.  Problems that allow flexibility in method opens it up to more students and parents.

Chap. 9: what can parents do?  Parents need to be advocates for their students if the student doesn't understand.  At the same time, if the parent forms a front with the teacher, then there's two people looking and checking for understanding.

On the teacher side, promote that kind of interaction with parents.

Overall: the preservice teachers thought it was a good book.  Many talked about keeping it for when they're teaching.  Asked to give it stars (like a review), only one student gave it less than full marks, and that was a 4/5.  Clearly the best textbook reaction I've ever had in a class.