Monday, September 29, 2014

Unit Test Correction Policy

Tests are a big part of my students' grades. I want all students to have a chance to go back and relearn material they have trouble with on a test, and I want their grade to reflect those improvements. I also want all students to take classwork, homework, test review, and tests seriously as they happen, not wait till disaster strikes and then try to catch up. My policy on test corrections tries to balance these goals.

If you score below 70 on a unit test, you really need to do corrections. I don’t want anyone to have below a C and I know you can catch up if you work at it!

How To Get Higher Test Scores 

The best way to get a high score on a test is to do it on test day. If you didn't take test review seriously in class or for homework, rethink that for next time and you'll probably find it helps you get a better score on the next test.

You can raise your score significantly by doing corrections, though. Corrected problems will earn you half the missing points back, OR full corrections will give you an 80% (B-), whichever is better. Another way to think of this is that your post-correction test score is the average of your old test and your corrected test (or 80%, whichever is better). 

Examples: with full corrections, a 90 would become a 95, an 86 would become a 93, an 80 would become a 90, a 72 would become an 86, and anything 60 or below would become an 80.

(Some exceptions for higher correction or retake credit may be made in the case of excused absences.)

How To Do Corrections

Your corrections must be easily identifiable (I don't want to have to reread your whole test). You can do them on the original test and mark them clearly (highlighters are great for this), do them on a new copy of the test, or do them on a separate piece of notebook paper.

WRITE YOUR NAME on any new pieces of paper with corrections. Staple them to the original test for easy reference.

Each corrected problem must include a written explanation of what was wrong and how you fixed it (for instance, explain you mixed up factors and multiples, then do the problem correctly). The explanation can be brief. I just don't want to see anyone simply copying down the right final answers. Show work!!

Don't work on extra credit problems, if there are any. Those are a one-time opportunity.

There will sometimes be two due dates for corrections: one for drafts, and one for final corrections. Basically, you only get points for corrections that are actually correct, so if you want a second chance, you need to get me your corrections before the final due date so I can help you catch any errors.

Where To Do Corrections, and Who Can Help

Work on corrections at home, or at school outside of class time (especially after school). Students often find that just working in a calmer environment helps them remember some things they felt confused about on the original test.

You can get help from ANYONE with corrections: me, your friend, your parent, another family member, Khan Academy or elsewhere on the web, ... You can use a calculator and, of course, your class notes.

Ask me for help after school if you need it!! I am generally available on Mondays, Wednesdays, and for a shorter time on Thursdays and (sometimes) Fridays. If possible, let me know you’re coming so I’m sure to be in my room. I can also arrange to work with you at lunch with a day's warning. If those times don't work, contact me and we can try to work something else out.

Where to Put Corrections

When you finish corrections, put them in your period's in-box in Room 203.

Did I leave anything out? Please let me know if something was unclear.

Thursday, September 25, 2014

Middle School Geometry Under Common Core Standards

The Common Core State Standards adopted by Oregon and most other states in the US describe what math students should understand and be able to do at various grades and in various "domains," or topics, such as Expressions and Equations (basically Algebra) or Statistics & Probability. The standards at each grade level are meant to build on the standards of the year before.

Here is my summary of what all middle school students are expected to learn about Geometry by the end of eighth grade, based on the Common Core state standards in math for sixth, seventh, and eighth grades. (Note: "e.g." means "for example".)

I. Area, perimeter, circumference, surface area, and volume

  1. Know and use formulas for area of triangles, rectangles, parallelograms, and two-dimensional shapes made from these (6.G.1), and circle area and circumference (7.G.4).
  2. Know and use formulas for the volumes of right rectangular prisms (boxes) with fractional edge lengths (6.G.2), three-dimensional objects composed of cubes and right prisms (7.G.6), and cones, cylinders, and spheres (8.G.9).
  3. Find the surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. (7.G.6)
  4. Represent 3-D figures with nets, and use nets to find surface area. (6.G.4)
  5. Describe what 2-D shapes you would get by slicing 3-D objects like boxes, cylinders or pyramids. (7.G.3)

II. Scale and Similarity

  1. Interpret scale drawings, and reproduce scale drawings with a different scale. (7.G.1)
  2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions (e.g. triangles with certain angle measures and side lengths). (7.G.2)
  3. Recognize what conditions (e.g. combinations of side lengths or angles) determine a unique triangle, more than one triangle, or no triangle. (7.G.2)
  4. Use informal arguments to establish facts about the angle-angle criterion for similarity of triangles. (8.G.5)

III. Angles

  1. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. (7.G.5)
  2. Use informal arguments to establish facts about the angle sum and exterior angle of triangles. (8.G.5)
  3. Use informal arguments to establish facts about the angles created when parallel lines are cut by a transversal. (8.G.5)

IV. Transformations

  1. Rotations, reflections, and translations: know side lengths & angle measures are preserved, and parallel lines are still parallel. (8.G.1)
  2. Describe how to get one congruent or similar figure from another with transformation(s). (8.G.2 and 8.G.4)
  3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. (8.G.3) 

V. The Pythagorean Theorem

  1. Explain a proof of the Pythagorean Theorem and its converse. (8.G.6)
  2. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in two and three dimensions. (8.G.7)
  3. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. (8.G.8)

Friday, September 19, 2014

Great Math 8 Resources for CPM Algebra

The publisher of our Algebra Connections textbook, CPM (College Preparatory Mathematics), has a terrific homework help resource on the web. For each homework problem, it has tips, sample work, suggestions, and/or a few answers to check against as you work.

There are lots of other things for families on in addition to the homework help, including extra practice worksheets, technology resources, resource pages to go with lessons (the same ones provided in class), and advice and guides for parents. Hope you find lots of useful things! If you have particular recommendations, please leave them in the comments.

Sunday, September 14, 2014

Clapping Games and Multiples

Last week was the first week of the new school year at da Vinci -- and my first year as a math teacher here. This year, I am teaching Common Core Math 6 and Math 8, as well as a small section of high school Geometry for eighth graders who took Algebra last year before the school switched completely over to the Common Core math class structure.

For sixth grade, we start with factors and multiples, so as I gathered getting-to-know-you activities for the first week, I had those topics in mind. For eighth grade, we spend the first few months on algebra, so I wanted to find patterns we could talk about -- especially non-visual patterns, since we will do plenty of those already.

I've been connecting with other teachers on Twitter, and someone retweeted a link from @iPodsibilities to this video with this clapping game:

"Aha!" I thought, "Non-visual patterns! and factors and multiples!" So in Math 8 and, especially, Math 6, we worked with this game (through about 2:02 on the video) over the past few class days in between things like learning the bathroom pass system and passing out textbooks.

The students were very focused in learning the game -- I was amazed how many could do the whole thing after only a few times through it, and even those who took longer (like me) were willing to keep at it till they got the hang of it. Perseverance, hooray!

Eventually in all the classes we described the patterns in "Sevens" something like this:

1. slap slap slap slap slap slap slap
2. slap clap slap clap slap clap slap
3. slap clap snap slap clap snap slap
4. slap cross slap clap snap clap slap

We first explored questions like these:

When we do "Sevens," what do the four patterns have in common?
What is different among the patterns?
Which one is hardest, and why?
Where do you see repetition?

Students in every class noticed that each pattern has more different moves than the one before; each pattern begins and ends with a slap; and patterns 1-3 repeat some moves in the same order. (Note: We did only the basic "Sevens" game, not the extra pattern mentioned at the end.)

Then we started exploring what other numbers besides 7 would give the same kind of behavior for these four patterns if we kept repeating the same moves in the same order, especially the beginning and ending with a slap. We said if this happened, the number "worked" for all four patterns. For instance, we tested 9:

9 with pattern 1: slap slap slap slap slap slap slap slap slap (works)
9 with pattern 2: slap clap slap clap slap clap slap clap slap (works)
9 with pattern 3: slap clap snap slap clap snap slap clap snap (doesn't work, because you didn't end with a slap)
9 with pattern 4: slap cross slap clap snap clap slap cross slap (doesn't really work, because you stopped in mid-cycle as you repeated the moves)

Students in every class found at least one other number that "works" for all four patterns. Can you? We also talked about what kinds of numbers "work" for the second pattern and why (hint: half of the natural numbers work).

Today, for the sixth graders, I reproduced a way of writing out the four patterns that students in some classes came up with:

1. slap slap slap slap slap slap slap
2. slap clap slap clap slap clap slap
3. slap clap snap slap clap snap slap
4. slap cross slap clap snap clap slap

Then I asked them why they thought I did the underlining the way I did, why I wrote the last slap in red, and what ideas they could come up with for the kinds of numbers that would "work" for each pattern.

To my delight, all three sixth grade classes explored these questions thoroughly and in every class, someone eventually mentioned the magic word MULTIPLE... as in, "Pattern 3 will work for any number that is a multiple of 3 plus 1." This led very nicely into a review of what multiples are, which sets us up well for this week's work, which was one my main goals!

We touched very briefly on why numbers that "work" for Pattern 4 also work for Patterns 2 & 3, but that part was hazier for them... which is OK, because after we study common multiples it will probably make more sense.

There was a particularly great math moment in Period 5 when Melody came up with a mind-blowing procedure for finding numbers that "work". She noticed that 7 works, and 13 works, and 25 works. Then she decided, and started proving to herself, that in general, if a number works, you can double it and subtract 1, and you will get another number that works. Therefore, for instance, 25*2 - 1 = 49 works. I could see that the numbers she was coming up with were all multiples of 6 plus 1, so I agreed that each of them worked, but it wasn't till after class that I sat down and proved her method would always succeed.

If you've had a few months of algebra, give the proof a try! (I'll probably sic my Geometry class on this one soon.) Numbers that "work" can be described as 6n + 1, where n is some natural number. Show that if you double any number that works and subtract 1, you'll get another number that works. Isn't that an awesome discovery?

Fun Puzzle Websites

This is nowhere close to a complete list! There's so much fun stuff out there!

Oregonian's Puzzle Kingdom: Lots of logic puzzles of all sorts, including Battleships, Sudoku, Pic-a-Pix, Kakuro, and Hashi.
Maths Resources (the British call it maths): Dozens of fun puzzles and games, including classic card and board games as well as newer online games like 2048.
KenKen: Includes various difficulty levels. Great practice for thinking about numbers (especially factoring) and logic!
Numbrix: another neat logic puzzle. Try the easier levels first to get the hang of it.
Brain Teasers from the National Council of Teachers of Mathematics' Illuminations website
Calculation Nation online math games (also from NCTM)
Hotmath math games are at various levels; pick one that is appropriate for you (the cockroach one is pretty funny)
NRICH Enriching Mathematics: Lower secondary is probably the most appropriate level here
Vi Hart has a lot of amazing videos on YouTube. I haven't watched all of these, and some rely on high school or college math.
Lure of the Labyrinth is a computer game designed for pre-algebra middle schoolers. It has a storyline in which you are rescuing a lost pet from monsters in a labyrinth by solving complicated math puzzles. You can set up a free account to try it. I have not investigated it much yet. If you try it, let me know what you think of it!
Lewis Carroll Puzzles: How can you go wrong? I also strongly recommend reading Alice in Wonderland and Through the Looking Glass if you have not already!

Saturday, September 13, 2014

Math Websites with Creative or Complicated Games

Here are some math websites I recommend for middle school students. This page is a work in progress; I am reviewing, sorting, and updating math links I originally collected on my former website, so you may want to look there too.

Calculation Nation: All free, no ads. Click on "GUEST PASS" or create a login at home with your family. I recommend READING DIRECTIONS before playing any game. Any are fine, but some are more fun or better for math than others. My favorites are:
  • Square Off: Capture spaceships with rectangles with certain perimeters. Math concept level: high. Strategy level: high. Time pressure: medium.
  • Factor Dazzle: Get points by finding factors, and keep your opponent's score low by giving them numbers without many factors. Can you figure out ahead of time how many points you will get from a certain move? Very similar to NCTM Illuminations' Factor Game but has a little extra fun. Math concept level: high. Strategy level: medium. Time pressure: low.
  • Drop Zone: "Drop" your fractions on other fractions to add up to 1. More fun than it sounds! Math concept level: high. Strategy level: medium. Time pressure: low.
  • Fraction Feud: Make a fraction (less than 1) smaller or larger than the one you’re “jousting” against. Consider using the Fraction Bar Chart. Try to figure out which cards are generally best to use or keep for later. Math concept level: high. Strategy level: high. Time pressure: low.
  • Times Square: Tic-tac-toe with times tables, basically. Math concept level: medium. Strategy level: medium. Time pressure: low.
  • Flip-n-Slide: Capture ladybugs with a triangle by translating, rotating, and reflecting it. Complicated; could turn out fun after several sessions. Math concept level: medium to high. Strategy level: high. Time pressure: low.
  • neXtu: I haven't really played this, but it basically looks like a fun board game. Math concept level: low. Strategy level: high. Time pressure: low.
Fraction Game (NCTM Illuminations): Click on '+' symbols for information. Use equivalent fractions and estimation of fraction sizes to "play" fraction cards on fraction number lines. Play several times. How few cards can you use? What are good strategies to reduce the number of cards you use?

Troy's Toys: Percent discount game. In Level 1, find a discounted price from the original price and the percent discount. In Level 2, find the mystery discount percent from the price and discount. Not super creative, but fairly realistic and thorough.