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Mathematics Lesson Plan for Algebra II
For the lesson on July 19, 2011
At Winter Sports School, Mr. Wojo's class
Instructor: Bushra Makiya
Lesson plan developed by: pcmi lesson study working group
1. Introduction to Systems: What is a Solution?
2. Goals of the Lesson:
a. Students will develop strategies to solve problems with 2 unknowns.
b. Students will develop the concept of a solution to a system of equations.
Objectives:
1) Determine the students' prior knowledge of solving systems of equations.
2) Students will explain at least one way to solve a problem with 2 unknowns systematically.
3) Students will identify solutions and non-solutions to a system of equations using graphs, tables and equations and be able to justify why a particular set of values is a solution.
3. Relationship of the Lesson to the Common Core State Standards
Related prior learning standards (topics/objectives):
8.EE.8. Analyze and solve pairs of simultaneous linear equations.
Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
Solve real-world and mathematical problems leading to two linear equations in two variables.For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
Related post learning standards (topics/objectives):
A-CED.3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
A-REI.6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
A-REI.10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
A-REI.11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) andy = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.&
4. Unit Plan
5. Instruction of the Lesson
The students just finished and were tested on a chapter on linear equations and inequalities.
The students will work on three activities in this lesson. First, they will work in pairs to write out multiple representations for solutions of a single linear equation. They then regroup to find solutions to two constraints simultaneously. The teacher will lead a discussion for students to share their work and methods. The next activity requires students to work individually with TI N-spire calculator and application in which they do a similar activity distinguishing solutions from non-solutions of linear equations one at a time before examining a system of equations. At the end of the lesson students will work on an exit task to demonstrate their understanding of the lesson goals.
In future lessons, students will learn to solve systems of equations algebraically, solve systems of linear inequalities, study linear programming and solve systems with three unknowns.
Materials:
3 Skier/Snowboarder tasks
Chart paper (graph)
Markers, tape
N-spire calculators
Student Activity Sheet (N-spire activity)
Projector, computer, and N-spire software for demonstration.
Exit task sheet
Steps, Learning Activities
Teachers Support
Points of Evaluation
Part 1: Introduction
Students work with individual constraints
Students will work in pairs on one of the 3 problems below, first individually. The teacher observes to see what they know and understand. They will write up representations of their solutions on chart paper.
A) There are 100 athletes at Jupiter Bowl. There are only skiers and snowboarders.
What are some possibilities for the numbers of skiers and snowboarders? Find several ways to represent all the possibilities.
B) There are three times as many skiers as snowboarders at Jupiter
Bowl. What are some possibilities for the numbers of skiers and snowboarders? Find several ways to represent all the possibilities.
C) There are 20 fewer snowboarders than skiers at Jupiter Bowl. What are some possibilities for the numbers of skiers and snowboarders? Find several ways to represent all the possibilities.
Next, each pair of students will briefly present their solution to the class.
It is important to push students towards more than one representation of their problem. Let students know the teacher is going to just observe the work to see what they already know how to do.
Not working together, Can you explain to me what [your partner] did?
Quickly done: Can you think of another way to represent this solution? Can you make up another problem related to this?
Group not working, or floundering: Can you explain the situation to me? Can you think of one possibility? Can you think of more possibilities?
- solutions off a table
- solutions off a graph
- solutions with equationBoth parts of solutions must be stated (number of skiers and number of snowboarders)
Students should understand each equation has many solutions.Part 1: Anticipated Student Responses
Students might not consider 0 as a possible number of skiers/snowboarders. Students may only consider multiples of 5 or 10. Students may make graphing mistakes increasing instead of decreasing. Students may confuse the role of snowboarders and skiers in the equation, table, or graph. Students may not explicitly express ideas with variables. They may have axes or variables switched relative to other groups.
Equations:
A) x+y=100
B) y=3x
C) x=y-20
x = snowboarders
y = skiers
Ask if students have considered all of the possibilities. Question students about non-integer values and whether or not they would make sense. Ask students to use additional representations and attend to their labeling. Allow students to have variation in their responses.
Part 1: Comparing and Discussing
Did your group have enough information to confidently determine the number of skiers and snowboarders?
Teacher charts list of methods: table, graph, etc..
How many solutions are there?
Whats similar/different among the posters?
Point out defining variables, labels. In the share-out, bring out the differences and allow students to defend the reasons. Ask for similarities and differences.
Part 2: Main Lesson Task
Students work with two constraints simultaneously.
Posing the Problem
Jigsaw activity: Re-pair students so that you have a pair who worked on equations A/B, B/C, and C/A. In new pairs, students will find solutions to the problem that combines both of their constraints. If we put both of your situations together, can you find the number of skiers and snowboarders?
Part 2: Anticipated Student Responses
For all pairings:
Students may obtain a wrong graphical solution by incorrect graph scaling or overlay, comparing tables directly or taking one constraint (adding skiing and snowboarders) and applying it to the second constraint.
Students may mistakenly find multiple solutions.
Students may struggle to solve the system graphically.
Point out inconsistencies in scaling or axes. Ask, Does this matter?
Ask students to why their solutions satisfy both of the equations.
Ask students to point out what they understand from the table solution on the graph.
Part 2: Comparing and Discussing
Students will do their work for solving the system of equations on the original poster with the single constraint. When the students rotate to a different group, the students who move need to make sure that the information (graph, table and equation) from their original constraint is on the poster. After reasoning through and solving their system, students will share their results with the group.
Error in work: Is the scale the same? Is there anything different between the two graphs
Done early: Find another way to represent your solution. Is this the only possible solution? Make up another problem that goes with this situation.Part 2: Comparing and Discussing
Did your group have enough information to confidently determine the number of skiers and snowboarders?
Teacher charts list of methods: table, graph, etc..
How many solutions are there?
Whats similar/different among the posters?
Point out defining variables, labels. In the share-out, bring out the differences and allow students to defend the reasons. Ask for similarities and differences.
Part 3: TI-Nspire Activity "What is a Solution to A System?"
Teacher will need to determine the entry point to this lesson and emphasize accordingly.Part 3: Posing the Problem
Students will be presented with a calculator and the student activity sheet and will be guided through the lesson.
Read introductory page.
Guide students through the procedure for dragging and dropping points.
Ask students to fill in 1-4 on student task sheet.
Discuss responses. Get at what makes the equations true, intercepts and slope. How many solutions? Between points (0,10) and (10,0) how many points are on the line?
Show students how to pull up a new equation, and have them answer question 5. Ask the students, How can you help [another student] find more points?
Ask students to move on to items 6-8 on the tasksheet. Have a whole class discussion about their answers. Bring out and share the algebraic methods for verifying the solution.
If time permits
Prompt students to make tables for the problem on the N-spire sheet. Use a separate sheet of paper divided into four for displaying four representations of one system of equations. What is the solution in each representation?
Use the software to demonstrate the procedure. If technology is really difficult for students, work through 1-4 together.
Press students for their observations and justifications.
If the students dont come up with infinitely many solutions ask them, What if x is 3.4?
Put an additional equation of a line on the board if students are not displaying comprehension of graphs of lines.
Questions for pairs, bold for class discussion:
Off of 2,b, Could you do this for any value of x? What if x is 2.7?
#3 What pattern did you find?
#4 How many answers are there? (Make sure to get multiple student answers. Are there only integers?)
#5 (Can go through quickly if short on time.)
Starting from a given point, how would I move to get to another solution? Could I do this starting from any point?
Connect to previous activity:
What part of the poster was this like? (When there was just one constraint or two?)
How many solutions were there when you only had one constraint?
#6 What did you get for/How many possible answers are there for 6a? b? c? d?
#8 How did you check/verify your solution in 7b?
Connection to previous activity:
What part of the pairs activity was this like? (one or two constraints?)
How many solutions were there when you had two constraints?
Will there always be one solution when you have two constraints?
How is the pacing of this part of the lesson helping to accomplish our lesson goal? Do the students need to stay together?
Does this activity help the students deepen their understanding? Is the N-spire technology a distraction?Anticipated Student Responses
Comparing and Discussing
Part 5: Summing up
Question:
How does this activity we did with the calculators relate to the skiers and snowboarders problem?
Chart student answers like, Solutions to two constraints can be thought of as intersection points of two lines.
A single equation has infinite solutions, but two equations has one solutions.
Exit Ticket
Hand out a worksheet with the following prompts and collect.
Debby says that (3, 5) is the only solution to the system QUOTE . Do you agree? Why or why not?
A field can hold 50 animals. For every cow there must be 4 sheep. What is the maximum number of cows that the field can hold? (Assume there are only cows and sheep in the field). Justify your answer.
Extension: Show many possible ways to solve number 1.
Ending question: Could you ever have two equations without exactly one solution?
7. Evaluation
This section often includes questions that the planning team hopes to explore through this lesson and the post-lesson discussion.
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