Wednesday, October 13, 2010

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.

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