1. Functions are to have no more than two variables with rational coefficients.
2. Linear equations will be given in the form: Ax + By = C, Ax + By + C = 0, or y = mx + b.
   
3. Vertical lines are included.
   
4. The majority of these items should be in real-world context.
   

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The students will:

1. construct a graph.
   
2. construct a best fit line.
   
3. construct a mathematical model.
   
4. make predictions.
   

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1. Draw a scatter plot.
   
2. Draw a line of best-fit (trend line).
   
3. Select two points from the trend line and find the slope using the slope formula.
   
4. Write the equation of the line.
   
5. Predict the number of hours needed to make $30 using your graph. Verify your prediction using your equation.
   

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Teaching for Understanding in Mathematics

Activity 1: The teacher reviews yesterday's lesson.
(Estimated time: 10 minutes)

Step 1: Find the slope of the line using M = (y2-y1)/(x2-x1)

Step 2: Find 'b' using y = mx + b

Step 3: Write the equation of the line in slope-intercept form

Technology: None
Materials: Overhead projector, paper, and pencil.
Student product or performance: Written work in their notebook.

Activity 2: The teacher presents the task for the day and asks the students to work on it independently.
NOTE: It is typical to present the task for the day and allow students to solve it in their own way. Often, the task can be solved using a method the students have learned recently.
(Estimated time: 35 minutes)

Step 1: Each student receives a copy of the 'ECR for lesson I'.

Step 2: Students follow steps in Activity 1.

Step 3: Each student will work independently to complete the entire task.

Technology: Ti:83 calculator
Materials: Overhead projector and graph paper.
Student product or performance: Graph and written work.
Scoring: Students will be scored on completion of total task.

Activity 4: The teacher suggests that students continue their work in small groups. Leaders of groups share their problems with the teacher.
NOTE: It is unusual for students to work this long without a class discussion. Also, it is typical for students to struggle with the task before the teacher intervenes.
(Estimated time: 20 minutes)

Step 1: As students complete Activity 2, they are instructed to share their work with each other.

Step 2: Questions the group cannot answer are presented to the teacher.

Step 3: Teacher reviews with whole group to insure successful completion by all students.

Technology: Ti:83 calculator
Materials: Graph paper
Student product or performance: Graph and written work.
Scoring: Students will be scored on completion of total task.

Activity 5: The teacher reviews use of the graphing calculator, and instructs students to type data into calculator and find the line of best-fit.
(Estimated Time: 15 minutes)

Technology: Ti:83 calculator
Materials: Calculators and overhead projector.
Student product or performance: Calculator Graph
Scoring: Students will be scored on completion of total task.

Note: No homework is typical in this teaching model adapted from the 1996 Third International Mathematics and Science Study (TIMSS).

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Maryland High School Mathematics Rubric: Extended Constructed Response Items

LEVEL 4
The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are correct. The explanation and/or justification is logically sound, clearly presented, fully developed, supports the solution, and does not contain significant mathematical errors. The response demonstrates a complete understanding and analysis of the problem.

LEVEL 3
The response indicates application of a reasonable strategy that may or may not lead to a correct solution. The representations are essentially correct. The explanation and/or justification is generally well developed,
feasible, and supports the solution. The response demonstrates a clear understanding and analysis of the problem.

LEVEL 2
The response indicates an incomplete application of a reasonable strategy that may or may not lead to a correct solution. The representations are fundamentally correct. The explanation and/or justification supports the solution and is plausible, although it may not be well developed or complete. The response demonstrates a conceptual understanding and analysis of the problem.

LEVEL 1
The response indicates little or no application of a reasonable strategy. It may or may not have the correct answer. The representations are incomplete or missing. The explanation and/or justification reveals
serious flaws in reasoning. The explanation and/or justification may be incomplete or missing. The response demonstrates a minimal understanding and analysis of the problem.

LEVEL 0
The response is completely incorrect or irrelevant. There may be no response, or the response may state, "I don't know."

Explanation refers to the student using the language of mathematics to communicate how the student arrived at the solution.

Justification refers to the student using mathematical principles to support the reasoning used to solve the problem or to demonstrate that the solution is correct. This could include the appropriate definitions, postulates and theorems.

Essentially correct representations may contain a few minor errors such as missing labels, reversed axes, or scales that are not uniform.

Fundamentally correct representations may contain several minor errors such as missing labels, reversed axes, or scales that are not uniform.

Source:
http://www.mdk12.org/mspp/high_school/structure/algebra/index.html

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Organize: I will organize the following data each time that I assess and score the learning standard in this task

• # of students who performed at or above the proficiency level: 25
  
• # of students who performed below the proficiency level: 13
  
• 66% of students performed at or above the proficient level
  on my scoring tool.
  

Analyze: I will examine the data above to look for trends, contributing factors, and implications of student performance over a series of assessments of the same learning standard.

Reflect: I noticed that students enjoyed the activity for the mere fact that they could see how completing the whole activity helped them make the predictions. As I look at my practice, I see that my students got a better understanding of the concept of using data to make predictions. I didn't realize that so many students would have to be taught the y = mx + b form of the linear equation. However, the students that remembered were willing to share their expertise with the other students.

Summarize: Sixty-five percent of the students reached the proficient level on the scoring tool. Many of the students who did not reach the proficient level were on the right track.

Action Plan: I will complete the following TaskBuilder Figure 8 Strategy Action Plan to prepare for my next standards-based task.

1. Plan - My next standards-based task will focus on:

Title: Solving Quadratic Equations
Content Area: Algebra II

Learning Standard(s):Indicator 1.2.4 The student will describe how the graphical model of a non-linear function represents a given problem and will estimate the solution.

Assessment Limits:

1. The problem is to be in a real-world context.
   
2. The function will be represented by a graph.
   
3. The equation of the function may be given.
   
4. The features of the graph may include maxima/minima, zeros (roots), rate of change over a given interval (increasing/decreasing), continuity, or domain and range.
   
5. “Zeros” refers to the x-intercepts of a graph, “roots” refers to the solution of an equation in the form p(x) = 0.
   
6. Functions may include step, absolute value, or piece-wise functions.
   

Intent: Proficient: Students at this level demonstrate an understanding of fundamental grade level skills and concepts and can generally solve entry-level problems in mathematics.

2. Teach - I will teach the standards-based instruction task or administer the assessment task identified in Number 1 on the following date:

Target date for teaching or assessing: February 6, 2006

3. Check - I will score the standards-based instruction or assessment task identified in Number 2 and collect samples of student work for each score point on the following date:

Target date for scoring: February 7, 2006

4. Act - I will place the results of scoring the standards-based instruction or assessment task in Number 3 into a summary table and analyze the results in a written paragraph of approximately 75-100 words on the following date:

Target date to organize and summarize data: February 8, 2006

5. Training and Coaching: I will schedule the following dates in my planning calendar:
Tuesdays, Room 3319, 3:05