Solving and Graphing Linear Equations: Practice Games

• Model Algebraic Equations with a Balance Scale: A bit confusing to start, but fantastic for visual thinkers. Build the equation, then keep the scales balanced by doing the same thing to both sides until you have solved for x. Clicking on blocks instead of dragging can reduce the tedium.
• Interactivate Equation Solver: You provide steps for solving an equation, along with reasons ("additive inverse" if you're "undoing" an addition or subtraction or "multiplicative inverse" if you're "undoing" a multiplication or division). The program responds with the new equation that would follow from doing what you suggest to both sides of the equation. Keep going till you get x by itself and know what its value is. (Leave "Use Identity Properties" off unless you want to have to do extra steps saying adding 0 or multiplying by 1 can be ignored.)
• Solving Equations Connect Four: Loads slowly. I recommend setting the timer to a longer time, or no time at all. Play with a partner or your imaginary friend. Experiment with the different difficulty levels and settings till you find the right difficulty (except don't do "Quadratic" for Math 8).
• Solving Equations Hangman: Ignore the letters! Good equations for practice; gamewise, not terribly exciting. Solve each problem (on scratch paper) and enter the answer. Mistakes cause a part of the hangman to be drawn. 
• Manga High Algebra Meltdown: Complicated, fun, some time pressure. You have to provide the "input" (solution) that will "go through the machine" (equation) to give the desired output. You can adjust the difficulty level after you've succeeded at easier levels in previous games; I'm not sure how you can save this if you play at school, however.
• Algebra vs. the Cockroaches: Cockroaches on your graph paper! Annihilate them! Choose your weapon, which will fire in the straight line whose equation you provide, hopefully hitting the cockroaches before they breed. Read the instructions (they're quick). Remember growth/change = slope = rise/run; they all say how much y increases (or decreases) when x increases by 1. Some time pressure; try using hints if this gets frustrating.

Math Information and Practice Websites

This page lists some math websites middle school students might find useful for explanations of math topics and practice of math facts and procedures. I have mixed feelings about students using some of these sites, which you can read here, but they can also be quite helpful.

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.

General Math Websites

• Khan Academy is famous for how thorough its coverage of math topics is. As a stand-alone math program it has limitations, but as far as I know it has a good reputation for accuracy. You can explore their videos for free (just pick Math as the Subject from the top of the screen), or if you want to track your progress on their questions, you can create an account for free.
• IXL is a commercial website with lots of quiz questions for practice; it gives information on math procedures when you make a mistake. You can use it for free for a limited amount of time each day. It has a very extensive list of topics. The questions tend to be somewhat repetitive and the help is very focused on procedures, but several students have told me they found IXL helpful.
• Math Is Fun has lots of informative write-ups of math topics; their information on fractions may be especially useful. Information is followed by quizzes so you can get some practice.
• Sheppard Software has many different game options, especially for grade 6 and younger. Their games are a little more complicated than some and do require some understanding, but your success still mostly depends on how fast you can recall math facts.  
• Ask Dr. Math at the Math Forum has a lot of interesting posts in answer to people's questions. They are often more thought-provoking and far-ranging than the other math help options listed here. 

Multiplication, Division, Multiples, and Factors

• Penguin Jump: For your 12 x 12 multiplication facts, check your knowledge (and speed) in a competitive way. You can join a game or create your own. If you create a private game, you can play against the computer. You can also make custom settings to play (for instance) only up to 10 x 10.
• Sigma Prime: I think this factoring game is fun! Shoot the appropriate prime factors at the invading number ships.
• Hit the Button: "Number Bonds" is about addition & subtraction; the others are multiplication & division.
• Times Tables Quiz at Crickweb: Solve multiplication problems in a "millionaire"-style quiz.

Fractions, Decimals, and Percents

• Melvin's Make a Match: Match written fractions with equivalent pictures. Good design.
• Fraction Booster: "Enter Activities" and then select Level 4 to practice putting fractions on a number line or Level 5 to find reduced fractions.
• Ordering Fractions: Sort fractions with different denominators in order of size.
• Match Fractions, Decimals and Percentages: Like it sounds.
• Decention: Find equivalent fractions, decimals, and percentages.
• Treefrog Treasure: Sophisticated game design; math seems to be basic identification of percents and decimals on number-line-like scales.
• Sheppard Software Fraction Games: PacMan-type games and others, giving practice in improper fractions and mixed numbers, equivalent fractions, adding fractions, subtracting fractions, among others.
• Fruit Splat/Place Value Decimals: This game is great practice for thinking about place value in decimals and for adding using mental math. It's designed to have several different levels, and you can play in timed mode or "relaxed" mode.
• Flower Power: put decimals in order of size -- a nice complicated game (read the directions).
• Balloon Pop Decimals Level 1 and Balloon Pop Decimals Level 2: Pop the balloons from smallest decimal to largest. If a balloon won't pop, it's because you haven't found the smallest. Score is based on time, but you can ignore it if you want and still get the practice.
• Balloon Pop Decimal Patterns: Pop the balloons that continue a pattern.
• Hungry Puppies: add decimals (quick mental math; fun speed challenge, but problems are not terribly complicated)
• Sheppard Software decimal activities: Most of the activities on this list are mainly useful if you're having trouble understanding what decimals mean.
• Fraction/Decimal/Percent Jeopardy: quiz yourself on converting between these. Use "0.3..." for 0.3 with a bar (repeating decimal)
• Troy's Toys: prices and percents: find out amounts of discount from percent, or vice versa; you pick the level of difficulty by picking the toys
• Balloon Invaders: a good challenge for finding percents FAST! only works if you are quicker with the keyboard than I am!

Solving Algebraic Equations

• Solving Equations Connect Four: Loads slowly. I recommend setting the timer to a longer time, or no time at all. Play with a partner or your imaginary friend. Experiment with the different difficulty levels and settings till you find the right difficulty (except don't do "Quadratic" for Math 8).
• Model Algebraic Equations with a Balance Scale: A bit confusing to start, but fantastic for visual thinkers. Build the equation, then keep the scales balanced by doing the same thing to both sides until you have solved for x.
• Manga High Algebra Meltdown: Complicated, fun, some time pressure. You have to provide the "input" (solution) that will "go through the machine" (equation) to give the desired output. You can adjust the difficulty level.
• Solving Equations Hangman: Ignore the letters! Good equations for practice; gamewise, not terribly exciting. Solve each problem (on scratch paper) and enter the answer. Mistakes cause a part of the hangman to be drawn. 

Geometry: Exploration of Triangle Similarity

Conditions for triangle similarity: Try the SAS tab first, then select SSA at the top and experiment with that.

If two triangles have a proportional pair of side lengths and the angles between those sides are congruent, must they be similar?

If two triangles have a proportional pair of side lengths and the angles between one of those sides and the other side are congruent, must they be similar?

When or Why You Should or Shouldn't Use Math Practice Websites

Elsewhere on this blog, I've been making a list called Math Information and Practice Websites. This page is about my mixed feelings about this kind of practice.

The games and pages on that list have some nice features. They help you practice and memorize useful "math facts" (addition, subtraction, multiplication, or division of two numbers), and give you practice at recalling them. Many people find them fun. Some of the pages give lots of information and explanations that can help you remember or learn math topics.

Some possible disadvantages to these games are that they:

• often emphasize speed and scoring, which can be stressful for some people and aren't all that related to math understanding
• generally don't lead to deep thinking about math concepts (the games on my Math Websites with Creative or Complicated Games list tend to be better for this)
• usually do not connect different math ideas; the problems are narrowly focused on certain skills
• hardly ever require complicated problem solving strategies
• may not meet middle school Common Core standards
• often do not involve "real world" problems or make you curious

These websites could definitely make your life easier by speeding up certain calculations, and some of the math teaching information is excellent. But remember: if you find a game on that list is stressing you out, or you're just learning how to move through a Pacman-type maze fast but not actually improving your recall of any new math facts, the game is not making your life easier and maybe you should go do something more fun or thought-provoking! Just make sure to stop and think every once in a while about whether the game is helping with your learning goals or not.

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.

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)

Geometry Exploration: Pythagorean Theorem and Pantographs

Explored in class today:

Pythagorean Theorem (click on the Desmos link):

Pantograph

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 cpm.org 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.

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!

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.