Practice Problems

I'm trying to shift towards standards based grading (SBG) this semester for the content grade in my geometry class, and so I'm paying a lot of attention to the kinds of problems I write. My tendency is to write big, open, sprawling problems that cross many standards, but I haven't yet worked out how to do that and SBG.  I know that I want problems where students can show understanding without necessarily getting to a correct answer.  Mathematicians are wrong a lot. What distinguishes us, I think, is that we often know whether we are wrong or right.

My students asked to have the time before the test to practice, instead of the book group I wanted to do. (110 min class and a 1 hour exam.) Very reasonable.  But that meant that I had to come up with practice problems! I'll post those now, then the test when all students have taken it.

Photo by scrappy annie @ Flickr

Do you give students practice?  What's the relationship between practice and the assessment problems?  This was a big topic in my student teacher observation this morning also.  

 322 Midterm Practice

The Standards

A. Analysis of characteristics and properties of
2-D geometric objects: number of edges; side length; angle size; parallel; perpendicular; convexity, etc.

1. concept: definition and recognition
2. application: use to sort and characterize, build or draw
3. combination: consider multiple properties in a single object
4. familiarity with examples: triangles, quadrilaterals, polygons

D. Distance, area, angle

1. concept: definition and key properties
2. formulas: use and derivation
3. connections among formulas

Try your choice of the following problems. Look for problems that allow you to problem solve and share your thinking.

Which standards could you demonstrate on which problems?


1) Connect each side property to an angle property and draw a different polygon to match each pair.

(at least) 3 congruent sides		no adjacent congruent angles
pair of perpendicular sides		(at least) 1 angle > 180 degrees
3 parallel sides		pair of adjacent congruent angles

Draw 3 different connections and try again!

2) Sometimes quadrilaterals are defined by their diagonals rather than by sides and angles. Determine which quadrilateral goes with which definition below, and make your argument. If no quadrilateral goes with a definition, state why.
a. Diagonals both bisect each other.
b. At least one diagonal bisects the other.
c. Diagonals are equal length.
d. Diagonals are perpendicular.
e. Diagonals do not intersect.
f. Diagonals are perpendicular bisectors.

3) On our Area on a Grid class workshop, find the areas of the shapes using formulas.

Area on a Grid


4) On graph paper, divide a square up into exactly 7 triangles with as many different triangle types as possible. Can you get all 7 types?

5) Area
a) Make an area formula for a trapezoid, or prove the one you know, using the formulas for rectangles and triangles.
b) Make an area formula for a kite.
c) Make an area formula for a chevron, using the diagonal lengths.

6) On graph paper, find squares with areas listed or argue why you can’t: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
a. Note: 5 is possible.
b. Find the edge length for each square you found using the Pythagorean Theorem. What do you notice?

7) Draw a 2 circle Venn diagram with the labels only in your mind (eg. a line of symmetry and at least one pair of parallel sides). Write in the quadrilateral types where they go, and give to a tablemate to figure out the labels.

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.

Money Problems

Excellent variation on the "how many ways to make change problem" at the New York Times today. The Freakonomics column is reporting the work of Patrick DeJarnette. I can see giving this problem from 4th grade to linear algebra!

Click on the money tag to see my previous money games. Click on the cartoon to go to the excellent Non Sequitur website.

Good Problems

Where do you get good problems for your students?

One source is that problem-of-the-day widget at the bottom of the blog. A couple times a week, I'm copying those, put them into a Word document, and then save them for a good opportunity.

But my all time favrite source is from the English (or British?) parallel to the NCTM: Nrich. Problems are sorted by content, tagged, by grade band (stage) and challenge level (number of stars). Some are unsolved, but accessible. Almost all are clever and/or interesting. Soooo nice! Give them a try. Here's an account of a teacher and how they use Nrich.

Here's one that I gave on a math for middle school final this semester:
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Take, for example, four consecutive negative numbers, say

−7, −6, −5, −4

Now place + and/or − signs between them. e.g.

−7+−6+−5+−4
−7− −6+−5− −4

There are other possibilities. Try to list all of them. Now work out the solutions to the various calculations. e.g.

−7+−6+−5+−4=−22
−7− −6+−5− −4=−2

Choose a different set of four consecutive negative numbers and repeat the process. Take a look at both sets of solutions. Notice anything? Can you explain any similarities? Can you predict some of the solutions you will get when you start with a different set of four consecutive negative numbers? Test out any conjectures you may have. Try to explain and justify your findings.