Objectives
The student will be able to design a method to find the number of possible winning numbers per game.

The student will be able to explain how to find the number of possibilities.

The student will be able to defend why their method works.

The student will be able to differentiate between permutation and combination.

The student will be able to create a new lottery game.

The student will be able to evaluate a classmates lottery game.

Anticipatory Set
Keynote slide show posing the problem will be shown.

Guided Practice
As a class the brainstorming session will take place first. The format for brainstorming will be brain writing. Groups of 4 students all write all the problems they can think of on a paper for 5 minutes then the paper is passed and they write more ideas for 5 minutes. This continues until each group has written on every paper. These group papers will then be compiled on a list on the SmartBoard for groups to reference throughout the project. Examples: how many numbers to choose, how many numbers get chosen, can numbers repeat.

Independent Practice
Students will work in groups to investigate the problems the class brainstormed (problem-solving step 2) and possibly come up with new problems they need to solve. Students will then research solutions to the problems using websites, parents, or experts (problem-solving step 3). During independent group work time the teacher will circulate the each group to answer questions and to ask questions to guide inquiry and encourage deep thinking by having students explain why they need to answer the problems and why they solved it they way they did.

Check for Understanding
The class will reconvene and selected groups will share their progress so far. Groups will be selected by the teacher. At least one group will have a very clear understanding of what problems need to be solved and how to solve them. One group will be on track but have questions on how to complete one part or another. And one group may have many problems to solve without clear understanding of how to solve them. All students can learn from these groups successes or failures. Through these preliminary presentations, off track students can gain insight to a direction to solve the problems, and students who are on track may find a problem they have not thought of. Although groups are presenting, this is a time for students to ask questions, think aloud, and talk through their understanding.

Independent Practice
Students will continue in their groups to solve the problems identified earlier during brainstorming and anything they learned from the preliminary presentations.

Guided Practice
Class will come together one more time to brainstorm what information needs to be included in the persuasive presentation. This brainstorming session will be a whole class session where the class may make a mind map or a numbered list of items that need to be included. Examples: number of balls to be chosen, if numbers can repeat, example of permutation and combination.

Independent Practice
Students will continue in groups to create their persuasive presentation (problem-solving step 4). Groups will complete a persuasive presentation.

Closure
Students will share their findings with the class through a presentation (problem-solving step 5). Students will also act as critics, helping me choose which group gave the most complete, accurate, persuasive, and interesting presentation (problem-solving step 6).

Duration
5 days
Day 1: Intro to problem, class brainstorming, group work time to generate possible solutions
Day 2: Informal presentations of progress so far, groups create presentation
Day 3: Groups finalize presentations
Day 4: Presentation day
Day 5: Presentation day

Teacher Notes: Research Info General Question: What are the observed behaviors and reported experiences of 11th grade Algebra II students when implementing problem-based learning? Specific Questions:

What are the effects of problem-based learning on students' ability to think for themselves?

What are the effects of problem-based learning on students' understanding math's purpose?

What are the effects of problem-based learning on students' written communication skills?

Class: Algebra II

Topic: Permutations and Combinations

Teacher: Ms. Hill

ObjectivesThe student will be able to design a method to find the number of possible winning numbers per game.

The student will be able to explain how to find the number of possibilities.

The student will be able to defend why their method works.

The student will be able to differentiate between permutation and combination.

The student will be able to create a new lottery game.

The student will be able to evaluate a classmates lottery game.

Anticipatory SetKeynote slide show posing the problem will be shown.

Guided PracticeAs a class the brainstorming session will take place first. The format for brainstorming will be brain writing. Groups of 4 students all write all the problems they can think of on a paper for 5 minutes then the paper is passed and they write more ideas for 5 minutes. This continues until each group has written on every paper. These group papers will then be compiled on a list on the SmartBoard for groups to reference throughout the project. Examples: how many numbers to choose, how many numbers get chosen, can numbers repeat.

Independent PracticeStudents will work in groups to investigate the problems the class brainstormed (problem-solving step 2) and possibly come up with new problems they need to solve. Students will then research solutions to the problems using websites, parents, or experts (problem-solving step 3). During independent group work time the teacher will circulate the each group to answer questions and to ask questions to guide inquiry and encourage deep thinking by having students explain why they need to answer the problems and why they solved it they way they did.

Check for UnderstandingThe class will reconvene and selected groups will share their progress so far. Groups will be selected by the teacher. At least one group will have a very clear understanding of what problems need to be solved and how to solve them. One group will be on track but have questions on how to complete one part or another. And one group may have many problems to solve without clear understanding of how to solve them. All students can learn from these groups successes or failures. Through these preliminary presentations, off track students can gain insight to a direction to solve the problems, and students who are on track may find a problem they have not thought of. Although groups are presenting, this is a time for students to ask questions, think aloud, and talk through their understanding.

Independent PracticeStudents will continue in their groups to solve the problems identified earlier during brainstorming and anything they learned from the preliminary presentations.

Guided PracticeClass will come together one more time to brainstorm what information needs to be included in the persuasive presentation. This brainstorming session will be a whole class session where the class may make a mind map or a numbered list of items that need to be included. Examples: number of balls to be chosen, if numbers can repeat, example of permutation and combination.

Independent PracticeStudents will continue in groups to create their persuasive presentation (problem-solving step 4). Groups will complete a persuasive presentation.

ClosureStudents will share their findings with the class through a presentation (problem-solving step 5). Students will also act as critics, helping me choose which group gave the most complete, accurate, persuasive, and interesting presentation (problem-solving step 6).

MaterialsIntro slideshow, computers connected to internet, Wikispace, presentation tools (PowerPoint, Vuvox, Prezi, Keynote), SmartBoard, graphing calculators

Duration5 days

Day 1: Intro to problem, class brainstorming, group work time to generate possible solutions

Day 2: Informal presentations of progress so far, groups create presentation

Day 3: Groups finalize presentations

Day 4: Presentation day

Day 5: Presentation day

Teacher Notes: Research InfoGeneral Question:What are the observed behaviors and reported experiences of 11th grade Algebra II students when implementing problem-based learning?Specific Questions: