The Algebra Toolbox

If the only tool you have is a hammer, you tend to see every problem as a nail. – Abraham Maslow


My Best Number

73 The Best Number in the World

73 The Best Number in the World

One of my many goals this year is to increase number awareness and number sense in my students.  This open-ended get-to-know-you assignment was a good start.

First, I showed the students this 1:00 minute video clip from youtube to introduce the idea of the “best” number. (Use caution if you just search for Sheldon Coopers best number on youtube. One version explains Raj’s favorite number 5,318,008 because it spells boobies when you turn a calculator upside down!)  After watching the video, we talked about the types of numbers that Sheldon mentioned (palindrome, prime,  binary, product). I asked what other “types” of numbers they knew. Descriptions such as odd, even, perfect squares, whole numbers, etc were brought up.

I gave each student this  My Best Number recording sheet. (I left room for more “best” numbers so students could add other cool numbers to their recording sheet as they year goes on).  They had to come up with one  Best Number for them and at least 2 reasons why it was the best. The reasons could be mathematical, historical,  or personal to them.  I gave them a couple days to think about it. On Friday of that week, each student shared his/her best number and the reasons behind it. It was a great way to get to know the students. One girl said 10 was her best number because that was how old she was the first time she met her real dad and that’s how many fingers she had to count how much she loves him.  WOW – deep for a high school freshman.  Many students chose their birth date, sports jersey number, etc. but many were very personal.

As we worked through our unit on solving equations, the students substitute their best numbers into equations that had “all real number” solutions to check the validity of the solution really being all reals. They loved that their best number was a solution. As they check their solutions to linear inequalities in the next unit, they will do the same thing.

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Super Bowl Math 2013

SB pic

In preparation for Super Bowl weekend, I updated a lesson I used a couple years ago where my Algebra I students calculated the quarterback passer rating for their favorite quarterbacks.  They really enjoyed it!  For a little history on the QB Rating, this is a good article.

I provide the data they need since most NFL sites include the rating in their usual stats for each player.

Quarterback Passer Rating Worksheets guide the students step-by-step through the calculations

Quarterback Statistics for 2012 season gives the answers

Quarterback Stats for 2012 seasion WITHOUT the QB Passer Rating is the data sheet to give to students

I am a new fan of which also has had some great Super Bowl prep activities on their site, including How Have the Super Bowl Ads changed (graphing points, recognizing an exponential function, predicting future costs of ads), Super Bowl Scores 2013 (predicting Sunday’s score, finding mean, median, mode, and range and determining which measure best represents the data), 4th Down (using data to decide whether to go for it, attempt a field goal, or punt on 4th down), Super Bowl Numerals (Roman numerals, number sense), Losing Teams in the Playoffs (ratios, percents, and proportionality) and NFL Home Team Advantage (using an inforgraphic to compare NFL home and away wins).  This site provides FREE pdfs of their activities which are linked to the Common Core State Standards, or you can purchase a $15/yr membership and have access to editable documents and solutions. Well worth the $15 a year to be able to personalize the already great activities.

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Christmas Price Index


With three days of school left before break and an impending snow storm on Thursday evening, it’s time to do some holiday activities.

Today, we used PNC Bank’s Christmas Price Index website (interesting FAQs). I gave the students a recording sheet. First, I had the students estimate the cost of each of the gifts in “The Twelve Days of Christmas,” while I told my story about how upset I would have been if my “true love” had sent me all those pooping birds, dancing ladies, men in tights, and loud drummers while I was trying to get my house decorated and cookies made for Christmas.  I would have kept all those golden rings and sent him packing.

After they make their estimates, we head to the computer lab and they go to the PNC site. This year it’s really cool. Each gift is “hidden” somewhere in the world and they have to follow clues and do little puzzles to find them before each price is revealed. Great geography tie in. Looks like google earth images may have used in the creation.  I had them record the location and the price of each gift as it is revealed. There is a link at the bottom of the screen where they can click and immediately get the prices of the gifts without the puzzles, but I wanted them to see the sights.  The website also gives the percent increase or decrease in price since last year.  Great activity for working with percents…they can figure out last year’s prices.  We talked about which costs surprised them the most, why some prices went up while others stayed the same, and some of the places they “visited” on the journey.  They got really excited when their estimate was close.  I only wanted to spend one day on this, so that’s all we did. So many options for additional activities though.  CAUTION: I don’t know if it was the site or our computers, but it got really slow sometimes. “Patience” was our word of the day.

Tomorrow, I’m trying the Auction from Math=Love. I’ve never done it before, but will give it a go. I wrote 12 auction questions with 9 correct answers and 3 incorrect. I kept score using an excel spreadsheet which I displayed on the board.


Activities for Solving Equations

I use lots of practice activities for solving one-, two-, and multi-step equations with my Algebra I class. Solving equations is part of their middle school curriculum but most still need much practice to achieve “fluency.”

Here are some of the activities I use:

By the Shapes Activity that I got at an NCTM conference years ago, but don’t know who the presenter was. If these are your creation, please let me know so I can credit you! 

Summary of Activity: Students work in groups of 4 to determine the value of each of 3 shapes, using 4 clue cards (equations).  The By the Shapes Teacher Notessay to use counters, but I don’t. I put the By the Shapes Recording Sheet in a plastic sleeve and students record the value of each shape on the sleeve. When they have the solution, they raise their hands and show me. If they are correct (By the Shapes Answers), they mark off that clue set number at the bottom of the recording sheet (to keep track of which sets they have completed) and I give them another set.  The best part of this activity is the conversation that happens within the groups. Students engage in some really good discussions about the value of each shape, verifying solutions by substituting the values into each of the 4 clues (equations) and eliminating possible values of the shapes based on the clues.

Preparation:  Copy the By the Shapes cards onto card stock. There is page of icons that can be copied onto the back of each clue page, but this isn’t really necessary…just pretty. It’s nice if each clue set is a different color. I made 2 sets of clues (16 sets total) so that as groups finished I could be sure to give them one they hadn’t done yet.  Give each group 1 By the Shapes Recording sheet and a dry erase marker/eraser. It helps to do the Teacher demo so they have an idea of what they are supposed to do.  I told them to use all the clues at the same time to figure out the value of each shape.

Solving Equations Song– even my high school boys will clap and sing along to this one.

2 step equations model for students who struggle with getting started solving two-step equations.

Gallery Walk Variables on Both Sides– I post each sheet of 3 problems around the room and give each student an answer sheet. Students work with a partner and rotate among the 10 sheets of problems, choosing one problem from each sheet to complete. I ask them to challenge themselves to complete the “hardest” problem on each sheet if they are able.

Crumple and Shoot (also known as Trashketball) – powerpoint of equations with variables on both sides of the equals sign. Students work in teams of 4, each studnet has a number from 1-4.  Every student does every problem on white board. I use a random number generator to call a number from 1-4. That student holds up their board – if the answer is correct, their team gets one point and they have the opportunity to get extra points by shooting a piece of scrap paper into our recycle box.  I have a “one point line,”  “two point line” and “three point line” from which to shoot to make extra points.  Students are encouraged to work together with their team to complete the problems since they never know who will be called to show the answer. Great communication encourager.

Pass the Probem – Equations– in groups of 4, students pass one problem around the group, each student doing one step of the problem. Directions are on the first page of this document.

Placemat Equations– I put one “placemat” in a plastic sleeve. Students work in groups 4, each student completes one of the problems on the placemat. They find the sum of their solutions and write the sum in the center of the placemat. I only have to check one number to know if they have all solved their problems correctly.  After using this activity, I found that it was better for me to write the equations onto the placemats so that the problems were oriented so that the students could see the problems easily without the problem being upside down. I also used markers so it was more colorful.

Hot Seat Solving Equations– powerpoint of equations. Students have to determine what the first step would be in solving the given equations (includes literal equations).  Students are in 4 teams. Each team has four cards:  “Add/Subtract,”  “Multiply/Divide”  “Combine like terms” and “Use Distributive Property to simplify.”  Hot Seat directions can be found here in this post.

Study Guide– prior to taking the unit assessment, we prepare a study guide.

I also played a game of Risk using equations.  I gave the students a blank recording sheet. I put the problems on the board and showed them one problem at a time. They did the work on the back of the recording sheet and wrote the answer only on the front along with how many points they wanted to risk on that answer. I revealed the answer, we discussed it, and they calculated their points (working in pairs to keep them honest).

**Update: I just read a great post by Julie at I Speak Math about solving equations.  I love this math community!


My Favorite Sub Plan

My favorite plan to leave for a sub is the FACEing Math  activity.  It takes most of my students two 42-minute periods to complete, is self-explanatory and easy for classroom management, provides review for students of concepts we’ve previously studied, and gives me great pictures to put on my walls!

Students complete 27 problems. Each problem is multiple choice with 2 answer choices so it’s moderately self-checking.  The answer choice tells students what feature to draw on the face template or what color to make each feature.  There are numerous FACEing Math books for all different levels, but my favorite is the “Create” books which are blank templates so you can put in the problems you want your students to do. 

I know we are only supposed to put free resources on here, but this is money well spent.  The author allows people to download the first lesson from each book for free, so I’m hoping not to violate any copyright laws by posting one of the templates from Create Book #2 for view.  In honor of World Series Season, here is a FACE template that makes a Baseball player ( page 2)  with the answer key (Lesson 16).  You just need to look at the answer key to see which letter (A or B) the correct answer choice needs to be for the face to come out correctly. 

My students really love this activity. They can talk quietly as they work.  I require them to show work either on the worksheet or on another sheet.  Even though all the students’ drawings should have the same facial features, each face will look unique.  We have contests to see who has the silliest, neatest, most unique, etc face when finished.  Students will need access to colored pencils or crayons and a pencil. I keep copies for emergency sub plans of content we’ve already studied so students can complete it independently.


Math Bloggers Initiation #3 – Math Quote

“On two occasions I have been asked [by members of Parliament], ‘Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?’ I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.”  – Charles Babbage

This quote reminds me so much of many of my students who just can’t understand how they got an answer wrong because, “That’s what the calculator said the answer is, so it has to be right,” with no consideration whatsoever about whether they entered the information correctly in the first place.  Especially when we are working with real-world data and applications, I ask them to estimate the solution first, even if it’s as simple as, “Should the solution be positive or negative?” or “Will be solution be greater than or less than 100?”  Knowing where mistakes will occur helps guide my questions.

One of the most frequent errors occurs when students have substituted negative numbers that are raised to a power. They don’t remember that parentheses are needed to obtain a correct calculation.  For example, recognizing the difference between (-2)^2 and -2^2 is tricky for them.  Most often this occurs when substituting values into an expression.  To combat this problem, I tell my students to always recopy any problem and put parentheses in FIRST before substituting.  For example:

b^2 - 4ac   would look like this ( )^2  – 4(   )(   )  THEN substitute the given values for a, b, and c.  This has helped eliminate at least some of the errors that occur when raising a negative number to a power.

Calculator use is so inconsistent in my district.  The elementary schools don’t use them much, the middle schools use them some (the middle school state assessment has a “calculator section” and a “no calculator” section), Algebra I students are required to use a graphing calculator for the high school assessment. The students go on to Geometry, Algebra II, Trig, and Calculus and use a graphing calculator all the time.   We offer college classes to our seniors, but to qualify they have to pass the college’s placement test, which does not allow the use of a calculator.  We have trig students who can’t pass the basic math portion of the placement test and therefore don’t qualify to take College Algebra or Prob & Stats because they’ve come to rely on the calculator and forget how to do basic math.

I’m all for calculator use in the classroom, but I believe that basic skills needs to be reviewed throughout high school and that estimation must be a huge part of our instruction as well.


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Lessons aligned to Common Core math

This website, Learnzillion,  has been recommended by our state Coordinator for Math Education as being a good resource for elementary through high school teachers of math and English/language arts.

I’ve just started exploring it, but have been impressed so far with the number of lessons available at the high school level. There are 176 for math. You can choose a lesson by grade level, domain (but conceptual categories are there) and standard.  The lesson includes a 3-5 minute video which includes lesson slides that can be downloaded and edited and a guided practice video.  The lessons are a little dry, but would be good for a student who needs a quick review or for one who was absent. What I like about it is the alignment to common core and ideas for problems for some of the topics that I don’t currently teach in Algebra I, but will be when we go full-implentation of the new curriculum (like exponential functions).

screen shot of learnzillion website

Everything on this website is FREE and available to teachers and students.