Building Enduring Mathematical Understanding, one lesson at a time.

Archive for the category “Mental Math”

msSunFun: Musical Math Partners

One of the “math games” we play in my classes that I would like to share for

is “Musical Math Partners.”

It is quite flexible, gets kids out of their seats, and gives students an opportunity to use mental math skills.


Each student receives some sort of “card” depending on the topic of the day.

Some sort of “path” is designed for students to travel around the room without too many “log jams.”
Here is a diagram of how I make it work in my room:

Students follow a path around a “row” of four pairs of desks, but at the end of the row they may choose to turn either left or right and continue around that “row.” (Even though there are two arrows in the diagram, students are single file when walking between the desks.)

As the teacher plays their choice of music, students “travel” around the path – usually only 10-15 seconds. Once the music stops, students take a seat in the nearest desk and become “partners” with the person in the adjacent seat. (If you have more desks than students, you might want to identify some of the desks as “out of the game.” As it is, you will often have students that must continue to “travel” after the music stops in order to find a partner. If you have an odd number of students, the “leftover” person becomes partners with the teacher.)

Students “do the math” (more details below) – usually trying to finish more quickly than the other person – followed by a short debriefing regarding how they solved the problem, and then they trade cards so that they have different experiences with each new partner.

Possibilities. . .
The possibilities are only limited by your imagination / ingenuity. ๐Ÿ™‚

Integer Operations
Each person holds a card with an integer. (A deck of playing cards work well for this: black = positive and red = negative.) When the music stops, the teacher randomly flips an operation card and students perform the operation from “left card” to “right card.” (I would only recommend division if you are interested in also focusing on fractions and mixed numbers.)

Fraction/Decimal Operations
Similar to the Integer Operations, only cards have fractions (or decimals.) I am fairly certain I would not include multiplication if decimals were involved, unless they were single digits. ๐Ÿ™‚ For subtraction, if students are not yet familiar with negative numbers, students can subtract the lesser value from the greater one.

Fraction/Decimal/Percent Comparisons
Each student has a card containing a number. (You could do all fractions, all decimals, or a mix.) When the music stops, students race to decide which value is greater.

Evaluating Algebraic Expressions
Each student has a card containing two pieces of information: an algebraic expression (as simple or complex as you wish) and a numerical value (also as simple or complex as you wish – including negatives, decimals, or fractions – but remember the goal is for students to evaluate fairly quickly.) When the music stops, students must evaluate the expression of the left card by the number on the right card and then vice versa.

Operations on Algebraic Expressions
Each student has a card with an algebraic expressions. When the music stops, teacher randomly selects add or subtract and students perform the operation from left to right. If the expressions are binomials, multiplication could also be included. (I am not sure I want students multiplying trinomials using mental math.)

Solving Linear Equations
Two options using cards throw the previous two “games” if the expressions are fairly simple.
Option 1: For the cards that have the expression and the number, students set the left expression equal to the number on the right card and solve, then switch to number on the left equal to expression on the right.
Option 2: For the cards that all have algebraic expressions, students set the two expressions equal and solve. (This would work best if the expressions were fairly simple, but again, it is up to the teacher depending on the level of students and the current unit of study.)

Measures of Center and Spread
Each student has a card containing a fairly small data set. When the music stops, the two data sets are combined, and students are to find median, mode, and range. Depending on the size of the sets and the values, you could also have students find mean or the quartiles and interquartile range.

Plotting Points
Each student has a card with an integer and each desk pair contains a coordinate grid. When the music stops, students point to the location on the coordinate grid (of course, the left card is x coordinate, and the right card is y coordinate.)

Pythagorean Theorem
Each student has a card with a value representing a side of a right triangle. When the music stops, the teacher either chooses “two legs” or “hypotenuse and leg.” Students find the remaining side. (Since relatively small values should be chosen in order for students to compute mentally, there would be some duplication. If students with equal values became partners, the second option could provide some interesting discussion!)

Distance on a Coordinate Plane
Each student has a card containing the coordinates of a point. When the music stops, student locate the two points and find the distance between them.

That’s about all for now. Feel free to add more ideas in the Comments. This is certainly an activity to use only after students have a fairly strong understanding of the concepts used in the tasks. Teachers need to be careful when choosing the values / expressions on the cards so that mental calculation is fairly accessible.


New Blogger Initiative: Mystery Number Puzzles

For the third prompt of the New Blogger Initiative, I have decided to choose Option 1, if not purely for the sake of wanting to learn more about how to use LaTeX, especially within a blog post.

My Algebra students will be diving into solving linear equations within the first week or so of the start of school. Since they have already had experience with 1-step and 2-step equations in sixth and seventh grade, I wish to quickly expand their repertoire into solving multi-step equations by first taking on those involving just a single “x” term. (Ok, seriously, I’m not going to use LaTeX for that! Would it just be x? Soooo frustrating not to know until it’s published!) (Aha! If I publish it as “Private” I can look at the preview of the actual online version – success!)

Initially, the tasks will not involve equations at all, but those goofy “Mystery Number Puzzles” such as:

I am thinking of a number. . . .
When I multiply it by 2 and then add three, multiply the result by 4 and divide by 6, then subtract 5, my answer is 1. WHAT is my number?!

Of course, if this task is just given verbally, it would be EXTREMELY challenging to solve – who can remember back that many steps?!

My lesson plan involves posting pages all over the classroom (in those lovely page protectors of course.) Students working in pairs would all start at a different spot in the room and complete a Scavenger Hunt. (I know I read something about this idea somewhere, but I don’t have the link. Edit: Found the link to Math-In-Spire thanks to @msrubinteach.) The bottom of the page will have a “Mystery Number Puzzle.” Once the pair has “solved the puzzle” they search around the classroom for that “solution” on the top half of another page. There will also be a “symbol” on the page for them to record. On the bottom will be a new puzzle to solve, and so on, and so on. There will be three different “sets” of six puzzles, so after six problems they SHOULD be back where they started. (Unless, of course, they solved a problem incorrectly and moved onto another “track” – uh oh!) The “symbols” they record will also be in the proper order if the problems were solved correctly. A class discussion involving strategies used to solve the puzzles would ensue.

After this opening activity, we will take a look at how to record this information mathematically. The first step would be to determine how to actually write down the original puzzle in mathematical form. After a brief partner brainstorming session the following “should” be agreed upon. (It will be interesting to see whether the use of parentheses will be remembered and/or emphasized.)

I am thinking of a number. . .
When I multiply it by 2: 2x
and then add three: 2x+3
multiply the result by 4: 4(2x+3)
and divide by 6: (here’s the tricky LaTeX part) \dfrac{4(2x+3)}{6}
then subtract 5: \dfrac{4(2x+3)}{6}-5
my answer is 1: \dfrac{4(2x+3)}{6}-5=1
WHAT is my number?!

It is important to note here that some students may alternatively use: (4(2x+3))\div6-5=1 instead of the fraction bar, and that is totally acceptable.

Next phase: pairs return to their “starting page,” flip up the page, and write out the mathematical representation on the top half of the BACK of the page using a dry erase marker (or dry erase crayon.) After returning their page to its original position, students do a Gallery Walk around the room and mentally conjure up the proper representation before peeking on the back side to see what other students wrote.

The last step involves representing the solution process mathematically. I am not “picky” when it comes to solving these types of linear equations. As long as there is only one term involving x (wow, it just starts to roll right out of your fingers) the entire process can be done using inverse operations. In fact, my goal would be for students to be able to write the solution for THIS equation in the following way: x=\dfrac{6(1+5)/4-3}{2}=3. I certainly do not expect that right off the bat, but I am very comfortable with something along the lines of:
I would hope that students would begin to complete at least a few steps at a time while still showing the computation appropriately, such as:
As students begin to describe their steps/explain their thinking, I would introduce the term “inverse” to move along their mathematical vocabulary.

Finally, students return to their “page” and write their mathematical solution process using as few “lines” as they feel comfortable using, as long as ALL of the steps in the process are shown. Gallery Walk THIS time involves making sure each pair DID show each operation in the process!

(Ok, it’s not really over yet, but probably for that day. Given a multi-step equation, students will write – and solve – the Mystery Number Puzzle associated with it. Students will create their own puzzles and challenge others to solve them. THEN, what to do when the puzzle involves a step like: subtract your original number. . .? JUST when they’re feeling all proud and confident you set the bar a bit higher and off they go again!)

Final note: found a link on (someone’s blog – I think it was a new blogger – yeah, that really narrows it down- I was just so awestruck when I clicked the link that I never went back) post to an AWESOME WebApp called WebEquation, at least for iOs. (Maybe Android?) You can literally just WRITE a mathematical expression/equation/whatever and it will turn it into LaTeX and “Math Script”

While I totally appreciate Sam’s link to more info on LaTeX syntax, I seriously would just go to WebEquation and handwrite the expression to find the proper way to “type” it. Unfortunately, you can’t just copy and paste the LaTeX script – it comes out in HTML or something. However, you CAN press on the LaTeX until it opens a new window showing the “Math Type” version and then click to save that as an image:

(Ok, technically I cheated. It would not actually paste the image of the equation, but I pasted it onto a Pages document and then took a picture to insert here. Not sure it’s worth the effort here. It does make a nice work around for Pages and Keynote though!)

So. . . depending on the complexity of the expression, I might just skip the LaTeX altogether!

Made4Math: Math Cards

When I was taking Math Ed. classes in college we had to choose one manipulative and describe all of the different ways they could be used in the classroom. I chose Legos, and came up with quite a list of activities. Many of them involved using Legos as replacements for other manipulatives, therefore allowing you to spend less by “just” buying Legos. I guess I was frugal even back then ๐Ÿ™‚ (Of course, that was before I knew just how EXPENSIVE Legos are!) Almost all of the Legos at our house are “claimed” by my son, and those that are left belong to his sister, so I have yet to use them in the classroom. (I did “sneak” some Duplos to school when they were younger, but those are all now in storage.)

Anyways, I originally developed the idea of “Math Cards” just for the “Find Your Match” activity listed below, but it has morphed into much more. See what other ideas you can think of!

Find Your Match
The first incarnation of Math Cards involved having students pick a slip from a bucket when they arrive in class and find their partner for the day based on their card. Now, the cards are not identical (this is middle school math, after all) but “match” in some way. Maybe they are equivalent. Maybe one card is the solution to the equation on another. Maybe they are different representations of the same linear relationship. The sky’s the limit on what you can create. Here are some I have made for this year.



I also have “Find Your Match Trios” cards where groups of three will “find each other.”


Each year, I would print out the sheet and cut it up for the class to use. I would create new “cards” for different concepts to add to my collection. (I have quite an assortment for 6th Grade, where we have a two period block with the students.) The next year I would find the file, print it out, cut it up . . .
I found that students would “cheat” (in my opinion) by putting their “answer” on the back of their slip and THEN finding their match, but in reality the “game” involves NO talking, and students are holding up their card and looking at other cards (analyzing them mentally) to find their match. The idea is that they will have to “do the math” for every other student’s card until they find their match.

Sooo, my brilliant idea was, hey, I can have these laminated and reuse them for years and they also won’t write on them! I took it up a notch and glued them on construction papers before having them laminated. Color coding made them easier to keep in sets. I had a TA first semester last year and it kept her busy quite often. Here are some of the finished products:



THEN, other uses started flowing into my brain. Below are “two more ways” that I have been using the cards, but within those two categories are MANY more uses ๐Ÿ™‚

Math Greetings
I have mentioned this briefly in other posts: First Day and Magnets, and I plan to do a more in depth post on it in the future. As students enter the room, they are required to “do math” in some way. I can put a set (or part of a set) in a “bucket,” have a student draw a card and “do it.” Sometimes it means solving the equation, sometimes it means changing the fraction to a decimal, sometimes it means finding the slope for the near relationship, sometimes it means just doing the arithmetic on the card. Again, the possibilities are endless.

Some cards are just numerical values (fractions, decimals, percents, square roots, cube roots) so I combine the cards with student magnets and they plot the values on a number line.

I have also combined the cards with dice.
-Draw a card with an algebraic expression, roll a die, evaluate the expression for that value of the variable.
-Draw a card with an algebraic expression, roll a die, set the expression equal to the value and solve.
-Draw a card with a fraction, roll a die, multiply (or divide!)

This year I plan to add some more “twists.”
-Set out a 4 by 4 grid of cards. Students find a match, I take the cards and place out two new ends.
-Pick two “ax + b” cards: set equal and solve, just add them together, or multiply together.

I really value the brief 1-on-1 with each student. It allows me a quick assessment of where they are at on a particular skill or concept. Often 2-4 students are “answering” their Math Greeting at the same time, so it really keeps me on my toes! Sometimes it really opens my eyes as to the general level of understanding that remained after 23 hours, and this may alter my plans for the day ๐Ÿ™‚

Review Games
At the end of each Unit, I often have a variety of different activities that rotate among groups of students. Many of these end up involving the cards in some way.
-Math Race: Flip a card from a stack, race to answer it, keep the card if you “win” OR everyone writes down their answer, earn one point for getting it right and one for being first (IF you are correct.)
-Match Game: Lay out cards upside down. Flip pairs and try to find a match.
-Go Fish: Students play the game while “fishing” for cards that match the ends in their hands.
ANY of the Math Greetings activities can be modified slightly as well.

Final Thoughts
Most of the time I am quite intentional in the problems I choose. For instance, for the scientific notation set in the first image, there are only two different mantissas (had to look that up) so plenty of opportunities to demonstrate conceptual errors. The Exponent Rules set provides for the same “opportunity for error.” Does 2^20 รท 2^5 = 2^4? Students who think so will “find the wrong match.” However, in the factoring/multiplying set, I could have done a better job just switching two values so that the “a and c” in the quadratic are the same. I once turned my brain into mush coming up with sets of five values where each card matched with two other cards having the same mean, but also matched with two different cards having the same median. It made for a good discussion of outliers!

Once I figure out the “tech side of things” I do plan to post links to them on a page here on this blog, but for now, feel free to come with your own ideas!

Part of Enduring Mathematical Understanding comes through practice, and Math Cards certainly provide that opportunity. However, deeper understanding comes from dialogue. “Find Your Match” is often followed up by questions regarding “why” two particular cards are a match. It’s always interesting to hear the conclusions!

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