Building Enduring Mathematical Understanding, one lesson at a time.

Archive for the month “September, 2012”

180 Days: Day 1-3

Day 1
I wrote my First Day plan about a month before school actually started, and surprisingly, it didn’t change much. (I was right on target when I said we would probably run out to time and not be able to even start the “My Math Stuff.”)

My Algebra students tried to count “Squares on a Chessboard” and became slightly overwhelmed by the enormity of the task after initially being sure it was just 64!


My Math 8 students investigated “Rice on a Chessboard” that will lead nicely into our first unit on Exponents and Scientific Notation.


Day 2
We spent a good chunk of this day making magnets, math portfolios to store their assessments and tracking forms, and Stars for Standards, but also did a “Get to Know You” activity (modified from an idea by MathMamaWrites) requiring students to think about positive and negative in the coordinate plane. Topics were randomly chosen from their “Know Me Notecards” and groups placed their magnets accordingly.


Day 3

In Algebra, we went back to the chessboard squares and made some more progress. Some groups studied the relationship between the size of the squares and the number of squares. Other groups looked at the size of the board and the number of squares. Patterns were emerging!


For the daily Math Greeting, students placed their magnets on the Venn Diagram regarding their thoughts on their relationship to math. This is an Algebra class. Some for the other periods were more interesting, but I neglected to take pictures 😦

The glare (and an old pen) made it hard to read. Top circle says “enjoy,” left one says “understand,” and right one is “work hard.”


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. 🙂

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. 🙂 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.

Made4Math Monday: Formative Assessment Forms

I definitely left some things “hanging” on my last SBG post, so I thought I would “kill two birds with one stone,” so to speak, and write about some of the ways I incorporate formative assessment into my daily routine.

A lot of people have written about “Exit Tickets,” and I guess, in a way, that’s what this is. I LOVE Sarah’s recent post at Everybody is a Genius (I’m not sure about everybody, but she IS a genius!) about the laminated dry erase ones 🙂 I think I might try that for my more “reflective” tasks to end the period, but I like the “permanence” of the system that I have set up.

Here is my:


Question of the Day
After we have spent a bit of time on a concept (but before it is time for a Mini-Assessment (Quiz)) I use Question of the Day to assess where students are at on a concept and us it to guide my instruction for the next few days. I will put up a slide such as this:


at the end of class and given them about five minutes to work on it. (Depends on the task.) Rather than just using quarter sheets of paper (or anything dry erase) I wanted to have a tool that I could use to provide feedback and also to have students keep as a record of their growth. I came up with the following simple “form”:

I print out four copies and then use the “booklet” app on our copy machine so that it shrinks it down and prints all four onto an 8.5 x 11 sheet. The original copy has more “space” on the top section because when it shrinks down there is an extra “gap” at the bottom of the page. This provides opportunities for 8 QOD’s before the students need a new sheet. They will store them in the “pocket” of their Math Notebook once they receive them back the following day. I use colored markers to write their number at the top so it’s easier to sort and hand back to color or number groups. The specific standard (CCSS) is written on the form, and we update scores on their Tracking Sheets (see SBG post) fairly regularly. When the forms are “full” students store them in their Math Portfolio – a hanging file folder in a crate that contains their tracking sheet as well as other “larger” formative assessments. Since there is only one question (ok, there’s usually two to three parts though) they don’t take long to grade and record. I can also “sort” the forms to create “Just for Today” seats (also on Sticks and Seats post) to either differentiate instruction or provide support for students who have not yet mastered the concept.

Writing About Math
I have students write in math class quite often,

20120903-130039.jpg but I haven’t always spent the time reading and commenting on their responses as I should. This year we are implementing the Common Core State Standards, and there are more than a few that employ the use of the verb “explain” or “describe.” I am looking forward to challenging students to meet standard on those in addition to the more skill-based ones. I decided to modify my Question of the Day form to use it for Writing About Math prompts as well.

Sometimes the prompt will be tied to a particular standard and graded/recorded as such, but other times I am just looking into their thinking about a concept and wanting a way to provide feedback.

Recording System
Since I record multiple scores for each standard (and multiple standards on each assessment) I needed a way to organize that information so that I can see the progressions of scores in each area. I created a “grade book” form that allows for up to four scores (more if I sneak in a re-assessment score next to the original) for each standard with room for three standards on each sheet:

Once I have my class lists I will enter them on a blank copy, make four copies of that, and again use the booklet feature on the copy machine. This time I transfer them to the 8.5 x 14 size, otherwise it shrinks down more than I want. When folded they create a nice size for up to twelve standards. I usually don’t have more than that in any one unit, so I create a new “grade sheet” for each unit. When I am entering grades into the online Gradebook, I just have to look at the most recent column and update scores that have changed from previous assessments.

Final Note: QOD and WAM are not always relegated to “end of period tasks.” I also use them at the beginning of the period. After I collect them we discuss possible responses. I am excited to use “My Favorite No” either when we do them at the beginning of the period, or the following day as a re-cap / remediation activity!

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!

msSunFun: Homework Hassles

Somewhere along the clogged up Google Reader, I missed the topic for this week’s:


It didn’t take more than opening up Flipboard to find out that the dreaded “Homework” is what’s on the menu today.

I have taught Middle School Math for the past four years. Until this year I have always taught sixth along with “something else.” Beginning Wednesday I will have three classes of Algebra (8th graders) and three classes of Math 8. I must admit that I am not exactly looking forward to the “Homework Hassles.”

I use Standards Based Grading in all of my classes, and one of the tenets that I feel strongly about is that a student’s grade is based purely on their understanding of math concepts and not on “participation,” “effort,” “behavior,” or “homework.” Therefore, even though I assign homework fairly regularly in my classes, it does not factor into their overall grade. Let me rephrase that: There are no “points” from homework involved in their grade, but I do feel there is a fairly strong correlation between a student’s homework completion rate and their overall grade. Unfortunately, (surprisingly enough,) not all middle school students necessarily have that same philosophy.

From my (albeit brief) observation of student behavior, sixth graders are much more “regular” in regards to homework completion. I think that whatever habit they developed in Elementary School tends to stay with them as the begin their life as a middle schooler. At some point, for some of the students, “grades” start to get in the way. If an assignment for another class counts as a part of their “grade” then it surely takes precedence over their math homework that “doesn’t really count” for their grade. By the time they are eighth graders, the percentage of the student population who find “more important things to do with their time” has definitely increased. Instead of completing a set of practice problems that your teacher has assigned in order to help you learn and retain the math concepts, school has become a series of earning “points” (or not earning them) to reach a desired “grade.”

Enough ranting, here is my “multi-pronged” approach for this year.

Math@Home Logs
For at least the first month of school, all students will be required to fill out a log, tracking the date, the assignment, time spent, whether it was completed, student initial, parent initial, and teacher initial.

This will be kept in the front “pocket” of their math notebook. I will stamp (or not stamp) daily and collect the log on Fridays. A “bulk” email will be sent home to any students who were negligent in completing the assignments and/or having their parent/guardian sign off on their log.

If a student has been successful at completing their homework for the first month of school they will be excused from completing the log unless their habits begin to change.

No Homework Notice
As I circulate and check off (stamp) homework, student who did not complete the assignment (didn’t start, didn’t finish, or didn’t show process) will be given the following reminder:

and will then be required to fill out a GoogleDocs form detailing the reason why they didn’t complete the homework along with a plan for completion.

I can print off the spreadsheet whenever necessary. As students do complete the assignment I can delete that row from the sheet.

Morning Math / Learn @ Lunch / Afternoon Academy
These are just my fancy ways of saying before school / during lunch / after school. Each Monday, students with assignments still missing will need to choose 15-60 minutes (depending on how much is missing) of time in which they will come to class and work on their missing work. (I have a cool grid on my board all ready to go for this. Students will each have a magnet with their name that can be placed in the location of their “first appointment.” No picture today – maybe later this week when I am at school.)

Making Homework Accessible and Appropriate.
Why have homework? What is my goal for my students when I assign it? These are important questions to consider when making decisions about a lesson each day. One goal I have for this year is to make sure I don’t assign problems “too early” in the concept development cycle. Just because a topic was introduced in class that day, doesn’t mean that students are really ready to “practice” on their own. Instead an assignment might focus on previous skills that will be helpful in solidifying the current concepts. This will take careful consideration, since I am not always sure how much progress will be made in class each day.

For some units I design quite a bit of the homework assignments myself, trying to incorporate some “puzzle/problem” solving activities that require students to come up with strategies that will, in the long run, help develop their understanding of math concepts. (See the example in this “Tricky Tables” post.)

In addition, I hope to make textbook homework less “rote” and more “reflective” at least SOME of the time. A few posts by David Coffey that I read recently share how to “flip homework” so that students are really analyzing a set of problems instead of just “cranking out answers.”

I certainly don’t have the “magic bullet.” I forced myself to refrain from reading today’s posts before composing this one, but I am looking forward to finding some ideas that will be useful. 🙂

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