Friday, May 27, 2011

Constructing the Idea of Absolute Value

Using the NROC absolute value game, “Absolutely,” students construct the meaning of the absolute value symbol. They then collaborate to create a set of class notes and hypothesize about how the symbol effects solving for X in an equation.

Learning Objective(s)
• Find the absolute value of numbers.

Assessment Type
This 55minute high school level lesson introduces absolute value using “Absolutely,” a game from NROC's Algebra 1--An Open Course, Unit 2, Lesson 2, Topic 1: Absolute Value. The students act as collaborative investigators to solve the mystery of absolute value. The class activity is followed up with a written assignment to assess understanding.

Assignment Details
10min— Warm up your class with a reminder about negative numbers.
  • Give them a few problems to solve as bell work, or see if you can use Socratic Questioning to lure them into explaining negatives to you.

10min— Introduce the new symbol | |, but not what it means. Tell the students it’s up to them to figure out how it works.

  • As a framing device, the lesson can be built up as an archeological investigation in which the symbol, |x|, has been found as part of a machine left by a vanished civilization. The students must discover what this key symbol means!
  • If using the framing device, you can claim that there’s a number code that needs to be figured out and tested to unlock a tomb, but you’ll only have one chance to get it right! Put on the board a few absolute value problems, the answers to which will be the code.
15min— Now that students know their goal is to find them meaning of |x|, have most of your students work in pairs at computers playing the "Absolutely" game and noting what they find.
  • They are your “Field Researchers.”
  • “Absolutely” has students order integers (with or without absolute value symbols) from least to greatest by flipping around pairs of numbers mounted on gears. It provides deeper understanding than a standard introductory lecture on absolute value because of the higher level of cognitive engagement and the immediacy of feedback in the game situation.
15min continued— Meanwhile, have a few students collecting information to make a set of class notes on the white board.
  • They are the “Senior Investigators” creating “Research Notes.”
  • Clipboards and white lab coats can be fun to use.
  • Encourage revision and neatening of the notes. This is an opportunity to teach good note organization skills.
5min— When ready, the note takers should present the class notes to the people who were at the computers. Everyone copies the notes into their own notebooks.
  • Present this as a consultation between the field teams and the senior researchers to ensure consistency before attempting the key code.
  • Encourage more revision and feedback from the field teams.
5min—Name the symbol.
  • The class should propose a name for the symbol they now understand so well.
  • Reveal the symbol’s established name is absolute value and discuss its use.
5min— As a class they should now discuss the answers to the key code problems. It’s up to you whether you give them some sort of prize when they get it right, but use their answers as a wrap up for the class being sure to summarize the absolute value findings.

  • If time: Invite students to hypothesis about how |x| + 7 = 10 will be solved.

5min— Homework – Students must write a paragraph explaining what the |x| means and including a few examples of their own making.
  • Prompt: Your research has gained national attention and you need to write a one paragraph summary (at least 7 sentences) for newspaper publication explaining your techniques and findings. An original example must be included.
  • Extra Credit Extension: |x| has been found in expressions and equations in nearby tombs. In a second paragraph, develop a hypothesis about how |x| + 7 = 10 will be solved OR how a negative number placed in |2x| + x would be evaluated.
Instructor Notes
  • For the negative number review, the distance you walked to school can be a good example. Have +1 mile represent the distance to school and -1 mile represent the distance home. Overall, if you’re trying to figure out where you are at the end of the day, you’d add the positive mile and negative mile together to get 0 miles away from home. For tomorrow’s warm up or as part of class wrap up, you can introduce the idea that if you wanted to find the total distance walked (not just where you ended up), you’d do 1 + |-1| since direction wouldn’t matter.
  • Keep things fun! Props like clip boards or a few lab coats can help your researchers get into their roles.
  • If short on time, omit the “Key Code” questions or move them to homework.
  • To encourage time on task, you can have each senior investigator working on separate boards with their teams of field researchers. Class notes are then moved to a single white board as part of a peer review process. The most successful research group will receive National Science Foundation funding (or be first dismissed from class)!
  • Choose students who are not shy and who might have trouble with sitting still as your senior researchers. Advanced students also often like the Senior Researcher roll.
  • You can also have two folks working at the boards as “Research Recorders” that your senior researchers report to if you need to have roles for more students.
  • Pair a student who is likely to struggle with a stronger, helpful student.


If notes are kept in a binder by each student, just mark them with a plus, check, or minus in a brightly colored pen as you circle the room. Later, a TA or you can count up the total number of pluses, checks, or minuses to calculate an overall grade for the notebook. Commonly, plus is 100%, check is 75%, minus is 50%, and 0 (earned if nothing is there or the students significantly disrupted) is 0%.

Alternately, if using a 10 point scale and collecting the notes and the homework paragraph the next day as a single assignment the following rubric can be used:

Notes and Paragraph Rubric
  • 5pts – Class notes are neat, complete and clear. Points can be lost from this section if student was disruptive the previous day.
  • 3pts – The paragraph explaining absolute value uses complete sentences that successfully communicate the meaning of |x|.
  • 2pts – An original example is included and is correct.

Total= 10pts

Sunday, May 15, 2011

Writing Real Life Algebra Equations

After viewing the creation of an equation describing a real life situation, students create and illustrate their own algebra equation.

Learning Objective(s)
  • Translate real life situations into word problems.
  • Translate word problems into algebraic expressions and equations.
Assessment Type
This introductory lesson uses the presentation from NROC's Algebra 1--An Open Course, Unit 2 Topic 1: Solving Equations. It includes a low stakes formative assessment in which high school students create and illustrate a problem of their choice.
Assignment Details
  1. Let your class warm to the topic of making real life algebra equations. You can ask them to give examples of when they use math in real life or give them an example of your own. Have they ever had to write their problems down to figure them out? Tell them their objective for the day is to create a real life algebra problem of their own.
  2. View Unit 2, Topic 1: Solving Equations (4min). In this presentation, an equation is made to solve for the number of batches of cookies a person can make if they only have so much flour. In the presentation, "equation" is defined and properties of equality are also discussed. If you wish, you can have your students take notes.
  3. Lead the class through making a few example cookie batch equations—have fun with letting them choose ingredients and see how much they can do on their own (Socratic questioning techniques can help with this). Be sure to use correct algebra terminology as you go.
  4. When comfortable, break students into small groups (I suggest heterogeneous grouping) encourage them to discuss their favorite cookies amongst themselves to create a fun, excited atmosphere and lower apprehension about errors.
  5. Have students take about 30 min total to make and illustrate an equation for a cooking situation of their own. They should choose what they are going to make and what variable to solve for. (How many batches, X, can they make if they have 12 cups sugar to use up and the recipe calls for 4 cups of sugar in each choco-caramel cookie batch?)
  6. Have each group show you and each other what they've come up with after about 5-10min. This allows you to check their progress and give feedback as needed.
  7. Circulate the room checking on whether students have successfully made a real life equation. Once you approve a group's equation, give them a large piece of paper and marker to make their idea and equation into a large poster for your classroom.
  8. Students do not need to present a solution on their poster, unless they are done early and you need them to keep working on something! I prefer for the students to make the equation illustration with a large blank solution space. Then, the student-generated problems can be used as class warm ups in the future or student groups can trade and solve later.
Instructor Notes
  • Fun and innovation should be encouraged. If students would rather make up an equation about car parts and cars, have them go for it!
  • For a student who is feeling really stumped, the Unit 2 Tutor Sim, "Building a Swimming Pool,” can help with moving into representing word problems with symbols.
  • For students needing a challenge: Can they make up a real life problem that involves more than one step, a problem involving addition and multiplication for example?
  • For older students, a more grown up topic, could be used to make equations. Budgeting for a shopping trip or a vacation for example.
  • This is a very scalable activity. If you don't have much class time, individuals can just set up their equation problems in class, then illustrate at home.

As this is a formative assessment, emphasis should be on participation and an honest attempt at completing the task assigned.

If in a rush, you can use a basic, "Plus, check, minus," scale to represent 100%, 85%, 75%. Odds are low a group will earn less than that.

Alternately, if using a 10 point scale:

5pts -- Time on task. Students stayed engaged and on topic working to create their problem and present it. All students in the group supported one another and invited contributions from each other.
3pts -- Students created a neat and clear equation representing their problem and incorporating feedback you gave them. A word problem is written out, variables are defined, and the equation is shown.
2pts -- The equation is completely correct.

Total= 10pts

Hippocampus Correlation

Algebra 1--An Open Course, Unit 2 Topic 1: Solving Equations