southern regional education board - robeson.k12.nc.us web viewcte teacher introduces cte lesson. ......

Southern Regional Education Board592 Tenth Street NWAtlanta, GA 30318

Phone: (404) 875-9211Fax: (404) 872-1477http://www.sreb.org

Building Academic Skills in Context: Enhancing Mathematics

Achievement through CTE Instruction

(Textbook)Workshop Resources

and Activities

Leslie [email protected]

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mailto:[email protected]

http://www.sreb.org/

High Schools That WorkBuilding Academic Skills in Context: Enhancing Academic Achievement

through CTE Curriculum and Instruction

Workshop Leader: Leslie Carson

The reauthorization of the Perkins Act calls for a new emphasis on integrating CTE education with rigorous academic content. Yet many high school programs continue to move CTE students through a system of basic level mathematics barely qualifying them for graduation, let alone a high-wage career. The new direction calls for CTE education that enhances mathematics achievement in CTE courses and programs. This workshop will place teams of CTE and mathematics teachers into professional learning communities focused on building authentic CTE anchor project units of study which are fully integrated with rigorous mathematics while also enhancing students’ mathematics skills in the mathematics classroom. Participating teams will produce a unit of study and will anticipate training additional CTE and mathematics teacher teams in their home district in the process they model.

Objectives: As a result of this workshop, participants will:

Learn how to use an instructional design that enhances mathematics achievement through CTE instruction;

Learn how to assess students’ mathematics awareness as it relates to CTE lessons;

Develop teacher assignments and instructional materials, and identify related study materials that enhance mathematical learning related to projects and assignments in career/technical courses;

Develop activities and assignments that require students to work through mathematics examples embedded in the CTE curriculum, planned projects and assignments;

Learn how to use available materials to enable students to work through traditional mathematics examples related to the mathematics embedded in a career/technical project;

Determine what will be required for students to demonstrate their understanding of mathematical concepts and skills;

Develop formal assessments to determine students’ mastery of mathematical skills embedded in their CTE assignments;

Learn how to work with a learning community of academic and CTE teachers to plan integrated lessons, give each other support and feedback, and collect data on student achievement; and

HSTW Key Practices addressed by this workshop:

High expectations — Motivate more students to meet high expectations by integrating high expectations into classroom practices and giving students frequent feedback.Academic studies —Teach more students the essential concepts of the college-preparatory curriculum by encouraging them to apply academic content and skills to real-world problems and projects. Career/technical studies — Provide more students access to intellectually challenging career/technical studies in high-demand fields that emphasize the higher-level mathematics, science, literacy and problem-solving skills needed in the workplace and in further education.

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Teachers working together — Provide teams of teachers from several disciplines the time and support to work together to help students succeed in challenging academic and career/technical studies. Integrate reading, writing and speaking as strategies for learning into all parts of the curriculum and integrate mathematics into science and career/technical classrooms. Students actively engaged — Engage students in academic and career/technical classrooms in rigorous and challenging proficient-level assignments3 using research-based instructional strategies and technology.

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Building a Free-Standing Tower

Objective: To design and build the tallest free-standing structure

The team is made up of three (3) or more participants.The team should select one member of the team to be an

observer.The observer notes the work of the team and reports on the team.

Planning Time: 20 minutes

* Practice assembling the tower* You may see how the parts fit* Design total height of structure* Write down the projected height* Place all pieces back into the container

Assembly Time: 90 seconds

Evaluation: Which group built the tallest tower?

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Analysis: Observer reports on team project* Design process* Correlation between planning and construction* Why it did or did not work

The observer reports answers to the following:

Who is the real leader of the group?

How did the team use its time?

Were assignments delegated?

Were there conflicts?

Did good planning exist?

Was each member of the group involved?

Did the team assign tasks to all team members during the planning time?

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Preparing our Students to be Successful Mathematically

Previously it was enough for our students to just be able to solve a given math problem such as:

What is 45 divided by 7?

Reading off of a calculator, the answer is 6.428571429…

Today, problem solving is a key component of today’s state mandated tests. Students may see the following types of questions:

You just received $45 for your birthday and have decided to take some of your friends to the movies. If movie tickets cost $7 each, how many tickets can you purchase?

45 students are going on a school field trip and you are responsible for ordering vans to transport the students. If each van holds 7 people, how many vans must you order?

You are planning on going on a cruise this summer that lasts for 45 days so you need to schedule off of work. How many weeks will you be gone? Remember there are seven days in one week.

You just received your first paycheck in the amount of $45. If you worked 7 hours, how much do you earn per hour?

Even though each of these problems involves dividing 45 by 7, they all yield different answers.

You can purchase 6 tickets. You will need to order 7 vans. You will be gone 6 weeks and 3 days. You earn $6.43 per hour.

Today’s students are required not only to have a repertoire of mathematical skills but they must also be expert problem solvers.

Studies show that students who work in groups to solve problems, use technology, and solve problems in context tend to score higher on the new state mandated

(problem solving based) tests.

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SREB’s Criteria for Authentic Anchor Project Units

Large enough to cause students to acquire the major technical, academic and personal skills implied in course goals.

Require completion of learning activities that result in work that would be done in a real workplace.

Help students understand and experience major technology used in the field Challenge students intellectually Allow students to address problems, projects and career activities they will encounter

on the job. Require students to organize information, consider alternatives and use higher-order

thinking skills Present problems and open-ended situations Require students to apply academic skills that are most needed to advance in the

career field. Require students to learn from the teacher, other teachers and experts outside the

school Involve both individual effort and teamwork Engage students in interacting and sharing ideas about ways to address a problem or

situation and the lesson learned through the project Allow students to present their projects to an audience of educators, students and

representatives from the career field. Require students to work with authentic tools and materials representative of a

career major Have clearly-defined quality standards that students can use to evaluate their work

and take corrective action.Source: Designing Challenging Vocational Courses by Gene Bottoms, David J. Pucel and Ione Phillips, 1997.

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How do new pieces fit?

Eight Steps to Developing Authentic Anchor Projects

Step One: Identify a major career/technical project that is rich with embedded mathematics content that career/technical faculty will have students complete before our next meeting.

Step Two: Identify the embedded mathematics, technical content and technology tools that can be taught through the authentic integrated project unit.

Step Three: Identify literacy skills and habits of success that students will be expected to apply in advancing their mastery of academic and technical content and skills.

Step Four: Develop a summative unit exam to assess students’ understanding of mathematics concepts, skills and procedures used in the project.

Step Five: Develop a process to pre-assess students’ current knowledge and skills as it pertains to mathematics, technical content, technology and tools embedded in the authentic integrated project unit.

Step Six: Develop how career/technical faculty will engage students with mathematics and technical content and the use of technology and tools embedded in the authentic integrated project unit.

Step Seven: Develop how mathematics faculty will engage students with mathematics and technical content and the use of technology and tools embedded in the authentic integrated project unit.

Step Eight: Describe how students will demonstrate their understanding of mathematics knowledge and skills by completing the project as well as completing assignments designed to provide additional practice.

Seven Elements for Teaching Mathematics Through Authentic Integrated Project Units

1. CTE teacher introduces CTE lesson.

2. CTE teacher assesses students’ math awareness.

3. CTE teacher works through embedded examples.

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4. Math teacher works through traditional math examples.

5. Both CTE & math teachers work through related, contextual examples.

6. Students demonstrate understanding in CTE and math classes.

7. CTE and math teachers formally assess students.

Step One: Identify a major career/technical project that is rich with embedded mathematics content that career/technical faculty will have students complete.

For the project selection, keep in mind: The project should last at least two weeks or more.

Step2: Identify the embedded mathematics, technical content and technology tools that can be taught through the authentic anchor project unit.What Does Embedded Math in CTE Look Like? The math in CTE will vary widely depending upon the CTE area and course level. It may look like arithmetic, algebra, geometry, or trigonometry. The following list will provide some hints as to what to look for while searching for math in a CTE curriculum. Keep in mind this is a list of mathematics skills sometimes found in CTE courses. Use it to jumpstart your thinking.

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Problem solving and organizational skills

Arithmetic skills, including fractions, decimals, and percents

Rounding and ordering values

Estimation Skills

Number sets

Scientific and exponential notation

Writing equations

Solving equations

Functions

Slope

Inequalities

Charts, tables, and graphs

Measurement skills

Unit conversions

Scale drawings

Graphing of relationships

Maximum and minimum values

Accuracy and precision

Square roots

Formulas

Ratios and proportions

Area, volume and surface area

Parallel and perpendicular

Trig ratios (sin, cos, tan)

Communicating mathematically

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Sample: AUTHENTIC INTEGRATED PROJECT UNITSKILL COMPETENCY CHECKLIST

Steps One, Two & Three: Use the following table to identify competencies students will need to acquire to complete the authentic integrated project. This table allows you to break the project down into logical steps and list the skills needed for each activity. Work with your mathematics colleague to identify specific mathematics understandings, procedures and methods of inquiry students will be required to master in order to complete the project.

Project Title: ___________Introduction to Electricity__________________________________________________________________________STEP 1 Identify, in order, the activities that make up

the project

STEP 1 CTE Concepts to be Covered

STEP 2 Mathematics Concepts to be Covered

STEP 2 Tools Needed STEP 3 Habits of Success/Literacy

Activity 1(Learning Basic

Characteristics of Electricity using Low

voltage circuits)

Safety requirements.Reading and recording values at designated points on circuits using a volt meter.Ohm’s law.Characteristics of magnetism, electrical resistance, amperage, voltage, batteries, conductor sizes.

Linear functions: writing, graphing, analyzing.Real number classification.Solving linear equations and inequalities.Laws of exponents.Ohm’s Law.

Use of volt meter.Use of CBL/graphing calculator.Internet to view film.

Communicating report results through writing and oral presentations.Use of anticipation guides to read about electricity (Chapter two, Power, Speed and Form by David P. Billington and David P. Billington, Jr.) and to view beginning of film (Annenberg film on electrical circuits: . www.learner.org/resources/series42.html #33)

Activity 2(Learning basic

characteristics of electricity utilizing residential 20 volt

circuits)

Construction of circuit board.Energizing circuit under supervision of teacher.Reading and recording values at designated points on circuits.Characteristics of electrical resistance, amperage, voltage, batteries, conductor sizes, power, capacitance, inductance, magnetism.Ohm’s Law

Above Use of volt meter.Use of CBL/graphing calculator.Internet for research.Presentation software.Internet to view film.

View remainder of Annenberg film.Prepare a presentation to report on contributions made by the following: Georg Simon Ohm, Andre Marie Ampere, Allessandra Volta, Charles Coulomb, James P. Joule or BTU—British Thermal Unit.

Activity 3

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This is not a complete listing of all competencies intended for this unit of study and is meant to provide an example for training purposes.

http://www.learner.org/resources/series42.html#33

http://www.learner.org/resources/series42.html#33

Step 3: Identify literacy skills and habits of success that students will be expected to apply in advancing their mastery of academic and technical content and skills.

1. Summarizing2. Paraphrasing3. Categorizing4. Inferring5. Predicting6. Recognizing Academic/Technical Vocabulary

1. Admit Slips—Students respond to one question at the beginning of class, such as “Which problem was hard for you?” or “What did you learn from your homework?” (predicting/summarizing)

2. Exit Slips—As students leave class, they give their teacher a slip on which they have responded to questions such as: “What did I learn?” and “What am I confused about?”(summarizing/inferring)

3. Double-entry or Two-column Note Journals—Each page has a line drawn down the middle. On the left side are the main ideas from reading or a class lecture; on the right are the details. It can also be used as an explanation process. On the left is a sample problem; on the right side are the steps to solve the problem. (summarizing/paraphrasing)

4. Weekly Reflections—At the end of the week, students write for three to five minutes reflecting on what they did and learned that week. Possible topics include “how I solved a problem,” “how I used reading skills to learn this week,” “the most valuable thing I learned” and “how I will use what I learned on a real job.” (summarizing/inferring)

5. Open-response Questions—On each test, students should have at least one open-response question that asks them to explain a process to solve a problem, compare different processes or ideas, analyze the importance of certain ideas or apply learning. Questions should be scored by a rubric. (inferring/vocabulary/paraphrasing)

6. KWL Charts—Used as a pre-reading and notetaking strategy, KWL charts have three columns, “What I Know (before reading),” “What I Want to Learn” and “What I Have Learned (answers to the questions).” Class discussion focuses on the columns. (predicting/summarizing)

7. Jigsaw Reading—Students are divided into groups of four. They number from one to four. All “number ones” get the same article to read. After reading their article, all those who read article one, for instance, group together and discuss the main points. They return to their home groups and share the main ideas from all articles. Each group then makes a one-minute presentation to the whole class on the common ideas. (summarizing/inferring)

8. Graphic Organizers—As students read a passage, they outline the main ideas according to the organizational pattern of the text. Venn diagrams can be used, for example, for a passage that is organized by comparison/contrast. Cause and effect matrices can be used when nonfiction is organized that way. As students gain more experience, they select the organizer that matches the organizational pattern. They are also known as mind maps or thinking maps. (categorizing)

9. Re-telling—Pairs of students have the same passage. Student one reads aloud the first section (one or two paragraphs). Student two, without looking at the text, summarizes what the first student read aloud. They both look at the text and compare it to their understanding. They switch turns until the passage is finished. (summarizing/paraphrasing)

10. RAFT—Students learn to focus their writing by defining their Role, Audience, Format and Topic, such as “As a graphic arts student, I am writing a letter to an editorial cartoonist to ask him how he designs his cartoons.” (inferring)

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Top Ten Literacy Strategies That Every Teacher Can Implement

Big Six Reading Skills

HABITS OF SUCCESSCreate Relationships

What are the HABITS that help students Create Relationships? Working in teams – developing the ability to work in teams; working in a collaborative team setting. Being responsible – developing a sense of responsibility to those around you; building trust among

peers and adults; building rapport. Communicating effectively –listening actively, forming an opinion, expressing in a non-

confrontational manner; resolving conflict.

Study, Manage Time and Get Organized

What are the HABITS that help students Study, Manage Time and Get Organized? Managing time – knowing how to spend your time and getting value out of your time. Keeping up with materials – supplies, notebooks, backpacks, homework. Using effective study skills - preparing for tests, note-taking, outlining, questioning, study teams.

Improve Reading and Writing Skills

What are the HABITS that help students Improve Reading and Writing Skills? Revising writing to meet standards. Using writing to learn the content in every class. Using reading strategies to learn the content in every class.

Improve Mathematics Skills

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What are the HABITS that help students Improve Mathematics Skills? Solving problems – being able to deal with non-routine problems and problems with insufficient or

extraneous information. Estimating/predicting – having a sense of whether solutions are reasonable. Synthesizing information – actively looking for connections between different mathematical ideas. Computing – developing an understanding of computation processes rather than memorizing

algorithms.

Set Goals and Plan for the Future

What are the HABITS that help students Set Goals and Plan for the Future? Being accountable – to yourself and others- to own learning and the future; to self-motivate. Setting goals and Planning – setting short- and long-term goals and being able to plan how to reach

them. Making real-world connections – understanding that success and failure in school affects success and

failure beyond high school.

Access ResourcesWhat are the HABITS that help students Access Resources?

Identifying appropriate questions to find information. Negotiating – locating appropriate information from a variety of sources, including technology. Researching – reporting appropriate information and drawing conclusions, giving credit to others for

their ideas. Analyzing – evaluating the quality and relevance of materials.

Step Four: Develop a summative unit exam to assess students’ understanding of mathematics and technical concepts, skills and procedures used in the anchor project.

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As your team develops assessment items to determine students’ knowledge and skills as it pertains to mathematics, consider the following proficiency levels.

Step 5: Develop a process to pre-assess students’ current knowledge and skills as it pertains to mathematics, technical content, technology and tools embedded in the authentic anchor project

Working Definitions for Proficiency Levels

Proficiency Level

A question or assignment may be deemed at this level if:

Basic Question cues, such as the following, are used: recall facts; make simple inferences or

interpretations; and demonstrate a rudimentary understanding of terminology, principles, and concepts that underlie the field.

It requires students to identify some parts of physical and biological systems. It requires students to recognize relationships presented in verbal, algebraic, tabular

and graphic forms. It requires students to answer who, what, where and when types of questions.

Simply stated, questions and assignments that require students to remember information and make simple explanations are at the Basic Level.

Proficient Question cues, such as the following, are used: use analytical skills, draw reasonable

conclusions, or make appropriate conjectures or inferences by applying logical reasoning on the basis of partial or incomplete information.

It requires student to defend ideas and to give supporting examples. It requires the understanding of algebraic, statistical and geometric and spatial

reasoning that is relevant to the field. It requires algebraic operations involving polynomials; justifying geometric

relationships. It requires the application of scientific and technical principles to everyday situations. It requires judging and defending the reasonableness of answers or solutions to

problems that routinely occur in the real world or chosen technical field.

Simply stated, Proficient Level questions and assignments require students to apply and analyze information learned.

Advanced It requires the formulation of generalizations, the synthesis of ideas and the creation

of models through probing examples and counterexamples. It requires students to communicate their ideas and reasoning through the use of

concepts, symbolism and logical thinking. It requires the design and application of procedures to test or solve complex, real-

world problems. It requires written responses that are thorough, thoughtful and extensive.

Simply stated, Advanced Level questions and assignments require students to evaluate and create work.

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Bloom’s Taxonomy with Sample Question Stems and Sample AssignmentsUSEFUL VERBS

SAMPLE QUESTION STEMS FOR ASSESSMENTS

POTENTIAL ASSIGNMENTS AND PRODUCTS

B

A

S

I

C

K N O W L E D G E

COMPREHENSION

REMEMBERING

EXPLAINING

telllistdescriberelatelocatewritefindstatename

explaininterpretoutlinediscussdistinguishpredictrestatetranslatecomparedescribe

What happened after…?How many…?Who was it that…?Name the…?Describe what happened at…Who spoke to…?Tell me why…?Find the meaning of…?What is it…? Which is true or false…?

Write in your own words…?Write a brief outline…What do you think could have happenednext…?Who do you think…?What was the main idea?Who was the main character?Distinguish between…?What differences exist between…?Provide an example of what you mean by…?Provide a definition for…?

List the story’s main events Make timeline of events. Make a facts chart. List any pieces of information you can

remember. Recite a poem. List all the animals in the story. Make a chart showing… Remember an idea or fact Question and answer sessions Workbooks and worksheets Remember things read, heard, seen Information searches Reading Assignments Drill and practice Finding definitions Memory games Quizzes Forming relationships (analogies, similes) Predicting effects of changes Dramatization Peer teaching Show and tell Estimating Story problems Cut out or draw pictures to show a

particular event Illustrate the main idea. Make a cartoon strip showing the sequence

of events. Write and perform a play based on the

story. Retell the story in your own words. Paint a picture of some aspect of the story

you like. Write a summary of the event. Prepare a flow chart to illustrate the

sequence of events.

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USEFUL VERBS



P

R

O

F

I

C

I

E

N

T

APPLICATION

ANALYSIS

APPLYING

ANALYZING

solveshowuseillustratecalculateconstructcompleteexamineclassify

analyzedistinguishexaminecomparecontrastinvestigatecategorizeidentifyexplainseparateadvertise

Do you know another instance where…?Could this have happened in…?Group by characteristics such as…?What factors would change if…?Apply the method used to some experience of your own…?What questions would you ask of…?From the information given, develop a set of instructions about…?Would this information be useful if you had a…?

Which event could not have happened if…?If…happened, what might the ending have been?How was this similar to…?What was the underlying theme of…?What do you see as other possible outcomes?Why did…changes occur?Compare your…with that presented in…?What must have happened when…?How is…similar to…?What are some of the problems of…?What was the turning point in the story?What was the problem with…?

Construct a model to demonstrate how it will work.

Make a diorama to illustrate an important event.

Compose a book about… Make a scrapbook about the areas of

study. Make a paper-mache map showing

information Make a puzzle game using ideas from the

study area. Paint a mural. Design a market strategy for your product. Design an ethnic costume. Use knowledge from various areas to find

solutions Role playing/role reversal Producing a newspaper, stories, etc. Interviews Experiments Solving problems by use of known

information Practical applications of learned

knowledge Design a questionnaire to gather

information. Make a flow chart to show critical stages. Write a commercial for a new / familiar

product. Review a work of art in terms of form,

color, and texture. Construct a graph to illustrate selected

information. Uncover unique characteristics Distinguish between facts and inferences Evaluate the relevancy of data Recognize logical fallacies in reasoning Recognize unstated assumptions Analyze the structure of a work of art,

music or writing Compare and contrast Construct a jigsaw puzzle. Analyze a family tree showing

relationships.

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USEFUL VERBS



A

D

V

A

N

C

E

D

SYNTHESIS

EVALUATION

CREATING

EVALUATING

createinventcomposepredictplanconstructdesignimagineimproveproposedeviseformulate

judgeselectchoosedecidejustifydebate verifyarguediscussdetermineprioritize

Design a…to…?What is a possible solution to…?What would happen if…?If you had access to all resources, how would you deal with…?How would you design your own way to…?How many ways can you…?Create new and unusual uses for…?Develop a proposal which would…?How would you compose a song about…?Write a new recipe for a tasty dish?

Is there a better solution to…?Judge the value of…Defend your position about…Do you think…is a good or bad thing? Explain.How would you have handled…?What changes to…would you recommend?Are you a…person? Why?How would you feel if…?How effective are…?

Invent a machine to do a specific task. Design a building. Create a new product. Give it a name and

plan a marketing campaign. Write your feelings in relation to… Write a TV show, play, puppet show, role-

play, song, or pantomime about… Design a record, book, or magazine cover

for… Create a language code. Sell an idea to a billionaire. Compose a rhythm or put new words to a

known melody. Hypothesize Write a creative story, poem or song Propose a plan for an experiment Integrate the learning from different areas

into a plan for solving a problem Formulate the new scheme for classifying

objects Show how an idea or product might be

changed Prepare a list of criteria to judge a…show. Conduct a debate about an area of special

interest. Make a booklet about 5 rules you value. Make judgments about data or ideas based

on either internal or external conditions or criteria

Judge the logical consistency of written material

Judge the adequacy with which conclusions are supported with data

Judge the value of a work or art, music, writing, by using internal criteria or external standards of excellence

Generate criteria for evaluation Evaluating one’s own products and ideas Form a panel to discuss a topic. State

criteria. Write a letter to…advising changes

needed.

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Where can we find good examples of assessment items? Released NAEP items: http://nces.ed.gov/nationsreportcard/nde andhttp://nces.ed.gov/nationsreportcard/itmrls/startsearch.asp State accountability tests-released items SkillsUSA test items : http://skillsusa.org/compete/math.shtml Textbooks (enrichment sections) Sample mathematics found in industry www.micron.com/k12

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http://www.micron.com/k12

http://skillsusa.org/compete/math.shtml

http://nces.ed.gov/nationsreportcard/itmrls/startsearch.asp

http://nces.ed.gov/nationsreportcard/nde

Comparison of Contextual and Standardized Test Items

Problem Solving• Authentic Assessment-Mechanics are sometimes paid by the “salary plus

commission” method. Suppose a mechanic is paid a base salary of $260 a week plus 15% of all sales over twice his base salary. How much in sales must the mechanic generate in order to receive a gross salary of $300?

• ACT Sample Assessment-Ms. Lewis plans to drive 900 miles to her vacation destination, driving an average of 50 miles per hour. How many miles per hour faster must she average, while driving, to reduce her total driving time by 3 hours?

Vocabulary• Authentic Assessment-A hill on a highway has a grade of 3:2. What is the rise

for ½ mile of highway?

• ACT Assessment Sample-A circular coin has a radius of 3/8 inch. When lying flat, how much area does the coin cover, in square inches?

Evaluating Results• Authentic Assessment-Uptown Cable, a cable TV provider, charges each

customer $120 for installation, plus $25 per month for cable programming. Is the following equation for calculating the cost for the first 6 months correct? If not, make corrections.6($120+$25)=cost for first 6 months

• ACT Sample Assessment-If x is a real number greater than 1,000,000, which of the following fractions is the smallest in value? F. G. H. J. K.

Reasoning• Authentic Assessment-The figure below is the path that a robot will travel

from point A to B. Write directions, using ordered pair coordinates, for shortest distance the robot must travel.

• ACT Sample Assessment -If n is an odd positive integer, which of the following is an even integer?

A. (n +1)2 -1 B. 2n2 C. 2n2 -1 D. 2n -1 E. 3n

Ratio and Proportion• Authentic Assessment-A scale drawing of a rectangular room is 5 inches by 3

inches. If 1 inch on this scale drawing represents 3 feet, what are the dimensions of the room?

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http://www.actstudent.org/sampletest/math/math_05.html#math_10k

http://www.actstudent.org/sampletest/math/math_05.html#math_10j

http://www.actstudent.org/sampletest/math/math_05.html#math_10h

http://www.actstudent.org/sampletest/math/math_05.html#math_10g

http://www.actstudent.org/sampletest/math/math_05.html#math_10f

• ACT Sample Assessment-The measures of the angles of a triangle are in the ratio of 2x : 3x : 5x as illustrated below. What is the measure of the smallest angle in the triangle?

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Measurement• Authentic Assessment-The youth center has installed a swimming pool on

level ground. The pool is a right circular cylinder with a diameter of 24 feet and a height of 6 feet. A diagram of the pool and its entry ladder is shown below. To the nearest cubic foot, what is the volume of water that will be in the pool when it is filled with water to a depth of 5 feet? (Note: The volume of a cylinder is given by pr2h, where r is the radius and h is the height.)

• ACT Sample Assessment-The edges of a cube are each 3 inches long. What is the surface area, in square inches, of this cube?

Using Data• Authentic Assessment-Tom’s long-distance service charges $0.10 per minute

from 7:00 P.M. to 7:00 A.M. on weekdays, all day on Saturdays, and all day on holidays; $0.05 per minute all day on Sundays; and $0.25 per minute at all other times. The table below gives his long-distance calls for 1 week, including the date and day of each call, the time it was placed, and the number of minutes it lasted.

• ACT Sample Assessment-What is the median of the following 7 scores? 42, 67, 33, 79, 33, 89, 21

Linear Equations and Inequalities• Authentic Assessment-Two enterprising college students decide to start a

business. They will make up and deliver helium balloon bouquets for special occasions. It will cost them $39.99 to buy a machine to fill the balloons with helium. They estimate that it will cost them $2.00 to buy the balloons, helium, and ribbons needed to make each balloon bouquet. Write an expression that could be used to model the total cost for producing b balloon bouquets?

• ACT Sample Assessment-The inequality 3(x + 2) > 4(x – 3) is equivalent to which of the following inequalities?

F. x < –6 G. x < 5 H. x < 9 J. x < 14 K. x < 18

Nonlinear Functions• Authentic Assessment-A company that manufactures cardboard boxes is

using stock material that is 18 inches wide. Write an equation that can be 24

used to calculate the area of the end of the box when different length sides are turned up from the 18 inches. Use a graphing calculator to determine the amount to turn in order to get the maximum area for a label at the end of the box.

• ACT Sample Assessment-If one leg of a right triangle is 8 inches long, and the other leg is 12 inches long, how many inches long is the triangle's hypotenuse?

Representations of Relationships• Authentic Assessment-The town of Mayville taxes property at a rate of $42 for

each $1,000 of estimated value. Create a tax chart that can be used by citizens to calculate the amount of tax they owe. What is the estimated value of a property on which the owner owes $5,250 in property tax?

• ACT Sample Assessment-If f(x) = x2 – 2, then f(a + 2) = ?

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Guidelines for Developing Authentic Problems

1. Apply desired math content. The primary goal is to flex the student’s thinking concerning one or more particular mathematics concepts. At some point in the solution to the problem, students must apply the desired math concept. Ideally, other recently learned topics should also be used to reach the solution.

2. Use a non-contrived scenario. Always ask yourself, “Would anyone ever need to actually do this? And if so, in what sort of circumstance?” The contextual problem should then briefly present that circumstance in support of the problem. Avoid manipulations that produce an opportunity to apply a mathematical technique to retrieve a solution, but in reality, would never occur in real life. For example, while an obvious algebra application, no one would ever really confront the following contrived problem: “The total of two employees’ salaries is $135,000. If the first earned twice as much as the latter, what is each of the salaries?”

3. Include real-world numbers with appropriate units of measure. Whenever possible, use real-world numbers with real-world units of measure. Avoid always using integers and values that result in integers, especially in applications for which integers are atypical. Real-world numbers frequently come from measurements, and so the units of measure should be a meaningful part of the given data as well as the answer being sought. Use measures that are typical for the scenario being described. For example, a pipe size would generally be given in inches of outside diameter, not centimeters of inside radius.

4. Remain faithful to the selected occupational area. Refer to workers, occupations, and situations that would inhabit your problem scenario. Ethnic and gender diversity will help more students to identify with the problem. Be careful that the problem represents situations that could really occur in that workplace scenario.

5. Include some extraneous data. To solve most real-world problems, we must sift through all the facts surrounding a situation to distill the pertinent data. Good contextual problems often include at least one piece of irrelevant, but meaningful, information.

6. Avoid hand-holding or step-by-step guidance. While helpful for elementary contextual problems, better problems should not include step-by-step guidance to achieve the desired solution. Rather, the given information should be presented in a way similar to how it would be encountered in real life, and the question or goal clearly stated. The student should be able to use the various supplied data and facts, along with the appropriate math concepts to achieve a solution.

©CORD2002-John Chamberlain

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Mathematics for Information TechnologySample Exercise Revisions

Example Exercises Concerns

Original version:Carlos has an annual budget of $7500 to purchase special software packages for the Accounting, Sales, and Purchasing Departments. He allocates his funds evenly among all three departments. If the Sales Department has already spent $723.87, how much more can they expend without exceeding the budget?

Revised version:Carlos is the Network Administrator and must allocate maximum disk space values to the various workgroups in the company. The total space used by all the groups can only reach 90% of the drive’s total capacity of 750 gigabytes, to allow a 10% pad for file compression space. Based on past experience, Carlos will allocate an equal share to the Engineering, Research, and Human Resources groups. But, the Accounting and Payroll groups each require twice the normal share, and the Marketing group with all their graphic design work requires five times the normal share. Write an equation that expresses these relationships and determine how much disk space allocation each group should receive.

Article I. Solution:Let x be the smallest share, for example, given to Engineering.

48.2 GB for Engineering, Research, and Human Resources

96.4 GB for Accounting and Payroll 241 GB for Marketing

1) This is more an Accounting problem than an IT problem.

2) This is a contrived situation. If anyone knows the annual budget for the three departments, they’d also know the annual budget for the Sales Dept, and hence, would immediately know how much is left after spending $723.87.

Suggested changes: Include job title. Mention computer

hardware familiar to students.

Use values typical to real world.

Require writing an equation from a problem situation.

Require answers that would actually be required of Carlos.

CORD, 2002

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Example Exercises Concerns

Original version:Bart is the line supervisor for a company that produces silicon chips. The line can produce 300 chips in 50 minutes. At this rate, how many can be produced in 85 minutes?

Revised version:Bart is the line supervisor for a company that produces silicon chips. A new automatic tester just installed on Line 5 was able to test 180 chips in the first 30 minutes of clean operation. The old testers typically tested about 2950 chips in an 8-hour shift. How does the rate of the new tester compare to the old tester? How many chips should Bart expect the new tester to produce in an 8-hour shift?

Solution:Since 8 hours = 8 hours 60 minhr = 480 minutes,

The new tester is performing a little slower than the old, at a rate of 2880 chips per 8-hour shift.

1) This problem needs a “bit more” to give it credibility with students. As written is sounds contrived.

2) Little association with IT beyond the words “silicon chips.”

3) Why would Bart need to know how many chips can be produced in 85 minutes?

Suggested changes: Include a piece of high-

tech equipment by name: a tester (I’m guessing at the testing rate—anybody know some real numbers?)

Use a time period that might really be used as a first measurement: a half hour.

Ask for a comparison that would be meaningful to Bart: is it working faster than the old machine?

Ask for a value that would be meaningful to Bart: the production for an 8-hour shift.

CORD, 2002

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Steps Documentation1. Identify a Standard for a Career and Technology Education (CTE) area.

2. Describe a “real world” (on the job) scenario where this standard would be used.

3. Identify the math concepts used in the scenario above.

4. Choose one or more math concept from step 3 and identify the math standard. (State or NCTM)

5. Write a work example math problem based on the scenario presented in step 2 that focuses on the math standard(s) in step 4.

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CORD, 2002

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Correctly scale dimensions that are not explicitly included in a blueprint.


Contractors must use the scale provided to build walls and correctly position windows and doors.


Measurement Ratio and Proportion


Make decisions about units and scales that are appropriate for problem situations involving measurement.


The location of a window is not explicitly dimensioned on a blueprint (see drawing below). The blueprint depicts the position of a window on a wall. Calculate the position of the window, in feet and inches, from the top, bottom, left, and right edges of the wall for the given scale.

⅛ inch= 1foot

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CORD, 2002

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Properly test and diagnose general electrical systems.


An automotive technician must be able to troubleshoot electrical circuits and find the causes of problems.


Use function notation. Represent relationships as ordered pairs Use formulas to solve problems.


Understand relations and functions and select, convert flexibly among, and use various representations for them.

Solve problems that arise in mathematics and in other contexts.


One can determine the wattage of an automotive 12-V light bulb by measuring its resistance. The power output of the bulb is related to its hot operating resistance by the formula

P = VR

2

where P = power output of bulb (in watts)V = operating voltage (12 volts in this case)R = resistance of bulb (in ohms)

A. Rewrite the above equation, substituting the voltage value of “12” given for this problem. Express the variables in function notation, identifying the dependent variable and independent variables.

B. Use the formula to determine the wattage of light bulbs that have a resistance of 2.6 ohms, 6.0 ohms, and 13.1 ohms. Round the wattages to the nearer 1-watt.

C. Express the results of Part b as ordered pairs.

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CORD, 2002

R.A.P. Sheet

READRead the problem carefully, all the way through. What key words are used, what are you asked to find and what information is given? Write each down.

_____ pointsFind–

_____ pointsKey Words–

_____ pointsInformation Given–

ASK Ask yourself how you will solve the problem. Write out you plan. Then write a verbal equation to reflect your plan.

_____ pointsVerbal Plan–

_____ pointsVerbal Equation–

PUTPut mathematical symbols in your word equation. Solve the resulting problem and check to see if the result is a reasonable answer for the question that was asked.

_____ pointsVariable Equation–

_____ pointsSolving–

_____ pointsAnswer–

_____ pointsReasonable? Why or Why Not?–

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RAP Example and Template. (10 point problem)Mechanics are sometimes paid by the “salary plus commission” method. Suppose a mechanic is paid a base salary of $260 a week plus 15% of all sales over twice his base salary. How much in sales must the mechanic generate in order to receive a gross salary of $300?

R.A.P.

READRead the problem carefully, all the way through. What key words are used, what are you asked to find and what information is given? Write each down.

__1___ points Find– amount of sales in order to gross $300

__1___ points Key Words– plus, gross, percent of, twice base, over

__1 __ points Information Given– $260 base salary, 15 percent of sales over twice salary, gross salary wanted is $300

ASK Ask yourself how you will solve the problem. Write out you plan. Then write a verbal equation to reflect your plan.

__1___ points Verbal Plan– Gross salary equals base salary plus 15% of sales over twice the base salary.

__1___ points Verbal Equation– Gross = Base + .15( sales – 2 x Base )

PUTPut mathematical symbols in your word equation. Solve the resulting Problem and check to see if the result I a reasonable answer for the question that was asked.

__2___ points Variable Equation– Let x = amount of sales in $$300 = $260 + .15[ x – 2( $260 ) ]

__1___ points

Solving– $300 = $260 + .15[ x – 2($260) ]$300 = $260 + .15( x – $520 )$300 = $260 + .15x - $78$300 = $182 + .15x$786.67 = x

__1___ points Answer– The mechanic must generate $786.67 in sales.

__1___ points

Reasonable? Why or Why Not?– Sales of $786.67 are reasonable for one week. The amount of this figure over $520 (twice the base salary) is about $250 and 15% of that is about $36, very close to the $40 needed to increase a salary of $260 to $300. The answer is reasonable.

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Step Six: Identify the strategies CTE faculty will use to engage students with mathematics and CTE content and the use of technology and tools embedded in the project.

Step Seven: Identify the strategies mathematics faculty will use to engage students with mathematics and make connections to CTE content and the use of technology and tools embedded in the project.

CTE Engaging Instructional Strategies

Cooperative learning Project-based learning Socratic method Anticipation guides

o Videoso Readingso Demonstrations

Technology integrationo Blogso YouTube

Literacy Strategies Use of manipulatives Multi-intelligences approach Others?

Mathematics Engaging Instructional Strategies

Cooperative learning Project-based learning Socratic method Anticipation guides

o Videoso Readingso Demonstrations

Technology integrationo Graphing calculatorso CBL’s and CBR’so Excel

Literacy Strategies Use of manipulatives Multi-intelligences approach Others?

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THE CORANDICAdapted by Dr. Kathy Oropallo

Corandic is an emurient frof with many fribs: granks from corite, and olg

which cargs like a lange. Corite grinkles several tarances, which garkers excarp by

glarking the corite and starping it in tranker-clarped storbs. The tarances starp a

chark which is exparged with worters, branking a slorp. This slorp is garped through

several other corusces, finally frasting a pragety blickant crankle: coranda. Coranda

is a cargurt, grinkling corandic and borigan. The corandic is nacerated from the by

means of loracity. Thus garkers finally thrap a glick, bracht, glupous grapant,

corandic, which granks in many starps.

Study Questions:

1. What is a corandic?

2. What does corandic grank from?

3. How do garkers excarp the tarances from the corite?

4. What does the slorp finally frast?

5. What is coranda?

6. How is the corandic nacerated from the borigen?

7. What do the garkers finally thrap?

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Sample: PROJECT ACTIVITY OUTLINESteps Six and Seven: Begin authentic anchor project units by identifying a career/technical project students will complete. Use this curriculum mapping guide to help identify a project rich with embedded mathematics and to consider necessary adjustments to pacing of curriculum and instruction.Note: Each lesson is designed for 90 minutes.

Dates StepSix: Career/Technical Course Activities

Step Seven: Mathematics Course

Activities

Steps Six and Seven:

Field Trips

Steps Six and Seven:

Guest SpeakersDec. 4 Vocabulary.

Brief demo of culminating project (teacher’s finished model).View part of Annenberg video with anticipation guide.Lecture on various types of wires/gauges.

Habits of Success: Create Relationships Improve Reading/Writing

Pre-Assessment

Dec. 5 Safety: Show part of video or part of reading that focuses on safety. Anticipation guide.Intro to project I with low voltage circuits.Students brainstorm possible safety issues related to project I.Discuss mathematical relationships between voltage, amperage and resistance (Ohm’s law).Solve sample problems.Quiz on safety rules.

The idea of inverse functions through linear relationships.

Battery Power Experiment

Homework: Linear Data Analysis Problem modeling Ohm’s Law (#2)

Dec 6 Parts inventory/learn how to use schematic symbols for parts.Demonstrate finished model for project I.Build and test circuit, take readings, complete report on findings including drawing diagram of simple circuit.Journal entry.

Exponential Growth and Decay Models: Connecting tables, equations, graphs

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Step Eight: Describe how students will demonstrate their understanding of mathematics knowledge and skills by completing a project (culminating task) as well as completing assignments designed to provide additional practice in the CTE classroom and in the mathematics classroom.

This involves preparing a project outline that will be given to students at the start of the anchor project unit.

Sample Project Outline: Manufacturing—A “Lifting Device” ProjectAdapted from: Designing Challenging Vocational Courses by Gene Bottoms, David J. Pucel and Ione Phillips, 1997.

Situation: In building and construction as well as in industry, machinery is used to lift and move heavy loads over relatively short distances.

Design Brief: Design and construct a device that can lift and lower “heavy” objects and place them at specified locations.

Technical Competencies to be Demonstrated: Weld steel Finish welds Rivet steel members Design a steel girder Drill steel Assemble pulley systems Calculate structural strength of members Operate equipment safely Assemble

Mathematics Competencies to be Demonstrated: Similarities in geometric shapes, including area and volume Trig ratios Properties of polygons Angle relationships Describe geometric relationships Ratios and proportions Representations Applying patterns from right triangles to solve meaningful problems Pythagorean theorem Congruence

Literacy and Habits of Success Skills to be Demonstrated: Technical Writing and Goal Setting Students will complete group and individual goal charts describing group and individual deadlines,

description of each phase of the project and persons responsible for completion of each component of the project.

Students will read a selection of technical manuals related to various lifting devices and will write a technical manual describing how to build the group’s device, including a description of all mathematics used in planning and development of the lifting device.

Students will complete daily journal entries describing work accomplished during the class period, including progress toward completion of goals set by the group.

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Specifications: The device must be able to lift up to three pounds (the “load”). Each “load” must be lifted vertically at least five inches before being moved rotationally. A hook must be manually attached to—and removed from—the load. All controls for vertical movement of the device or its extensions must be locatd on the same side of

the device. The part of the device that touches the “ground” may not exceed beyond an 11-inch square area. The device may be free-standing. It cannot be kept from lifting off the ground or from tilting by any

attachment to the “ground,” wall, ceiling or anything external to the device, including a person. The device must have a mechanical advantage in lifting of at least 3. All movements of the device must be accomplished with levers, linkages and/or pulleys and belts or

simple wheels. Only one person may touch the device during its operation. All group members must be able to operate the device. All group members must be able to describe and demonstrate mathematics used in the creation of the

lifting device.

Rules: Each “load” will have a hook by which it can be lifted. The “ground” will be at the same elevation as the lifting and release locations. You will be required to lift and move three “loads” in two minutes. The “loads” will be of varying sizes and weights. The locations for picking up and dropping off the “loads” will be specified at the time of the project

presentations. The “loads” will be placed for pick-up in locations between six inches and 20 inches from the center

of the device’s base area. The locations for the release of the “loads” will be within 360 degrees of the original pick-up location

and from six inches to 20 inches from the center of the device’s base area. No electrical or pneumatic devices may be used to operate the device. A maximum of $10 may be spent by each group to purchase materials. Any materials found or supplied by the teacher may be used. No prefabricated units or building members may be used to build the device. Presentations will be on (date).

Scoring:1. Hook was in position over “load” 20 points each “load” 60 points

2. “Loads” were placed accurately 20 points each “load” 60 points

3. Fastest time for completing operations 30 points(within limits)

Second fastest time (within limit) 28 pointsThird fastest time (within limit) 26 pointsFourth fastest time (within limit) 24 pointsFifth fastest time (within limit) 23 pointsSixth fastest time (within limit) 22 points

4. Economy of design (minimal amount of material by volume, needed to complete the project) 5 to 50 points

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1. Technical Competencies 5 to 50 points2. Aesthetics 5 to 50 points3. Controllability 5 to 50 points4. Creativity (in the use of materials, mechanisms and the design) 5 to 50 points5. Individual team member ability to solve related mathematics problems 5 to 50 points

Maximum points……………………………………………….450

Penalties:1. Assistance (per intervention) Minus 5 to 100 points2. Controls not within specifications Minus 5 to 100 points3. Device moves off ground (per violation) Minus 5 to 100 points4. Completion of three operations over specified time Minus 5 to 100 points

Group Grades will be based on:1. Design and performance (as specified above);2. Oral presentations;3. Written report (including group journal entries)4. Development process (including the group’s goal chart, the serious consideration of at least three

alternative solutions, testing and modification in an organized and considered manner and the way the group functioned as a unit).

Individual Grades will be based on:Individual summary evaluationsIndividual journalsPerformance on mathematics assessmentParticipation in the group

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The Cycle of Learning

Sequence of Instruction

Get Started

Engage

Explore

Explain

Practice Together

Practice in Teams/groups/buddy-pairs

Practice Alone

Evaluate understanding (Daily/Weekly/Post-Assessment)

Close

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Week _________Project Title: ____ Sample daily lesson plan for three days! ________________________________________________________________

Course:____________________________________ Week beginning:________________________ Days ________________ of ______-day unit

Resources/Instructional Materials Needed

Cycle of Learning Monday Tuesday Wednesday Thursday Friday Possible ActivitiesGetting Started

Vocabulary list: start writing in definitions and diagrams(those that are used in the first project)

Safety:

Show part of video or part of reading that focuses on safety

Anticipation Guide for reading

Take parts inventory: place each part needed for this experiment on its corresponding drawing. Notice the schematic symbol for each part. Make sure you have all parts ready..

Admit SlipPost/Discuss/Copy ObjectivesWrite in journalSolve problemsAnswer questionsPre-assessmentOther___________________

Engage/Motivation

Brief demo of the culminating project (teacher’s finished model)

Introduce Project 1 Demonstrate the finished model for project 1: Low Voltage Circuits

Hand out procedure:Connect a 9 volt battery to the battery

Display object/pictureDemonstrate reactionModel/demonstrate labDiscuss previous experiencesActivate prior knowledgeOther____________________

ExploreView part of Annenburg Video with anticipation guide

What are electrical circuitsHow does a resistor work

Students propose possible safety issues involved in the project

Build the CircuitUsing the schematic diagram, build the experiment with the electronic parts. Confirm that all wiring is correct.

Brainstorm Create listsInvestigate Build ModelWork problem Analyze dataLab activity Evaluate stepsOther__________

ExplainPresent (lecture) the various types of wires and how to identify them

Wire gagesWire materialsWire types

Discuss the mathematical relationship between voltage, amperage, and resistance (Ohm’s Law)

(Demonstrate these components on the demo project)

Connect a 9-volt battery snap and begin touching trhe probe to each resistor lead, one at a time, observing the brightness of the LEDs. You should see the LED’s glow diminish as you go from Ri to R 5. Take voltmeter readings across the batgtery, each resistor and the OEDs as each resistor is chosen as the circuit. Chart the voltage, resistance and current flow with each and show trends.

Lecture with guided notesStudent presentationsMedia presentationInteractive discussionOther___________________

Practice as a classOral questioning from info on the video and the lecture

Use the readings from the demo to verify mathematically…..substitute readings for V and I, solve for R, etc.

Following the practice in teams (see below) conduct whole class discussions, making generalizations about the findings.

Complete practice problems/labsUse manipulativesConstruct graph/timelinesMake predictionsCollaborative writingWhole group graphic organizersOther_________________

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Cycle of Learning Monday Tuesday Wednesday Thursday FridayPractice in teams/groups/pairs

Partner quizzing each other

Groups solve a few sample problems together (see group 1 problems)

Students complete the readings and charting described above.

Solve similar problemsPractice active reading strategiesAnswer questionsPeer review/editDesign problems/questions/labsResearch informationOther_________________

Practice Alone

worksheet

Practiced problems using Ohm’s Law

Individual worksheet requiring Drawing schematic for the simple circuit, diagram labeling, content facts, and written summary of today’s experiment.

Draft writingAnswer questions/problemsDesign/construct problems/labsRevise workDesign individual investigations Other____________________

Evaluate Understanding Collect the worksheet

Quiz on the safety rules Journal Entry DiscussionOpen response questionsQuiz/test (academic/authentic)Writing sampleIndividual project/investigation or PresentationOther______________________

Closing Activity Assign/explain homeworkReview major pointsAnswer questionsStudent reflection activityExit slipOther___________________

Enrichment, Refining knowledge, Extension, Re-teaching, Accommodations

ReviewPracticeReadingTutoring (peer, teacher, group)Individual AssignmentOther interventions___________

Notes: Expected learningStudent accommodationsProgress (+ or -)Next Steps

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Ticket Out the Door

Directions: Consider the activities you participated in and complete the following. Give your “ticket” to your presenter as you leave at the end of the day.

Praise: Ideas we heard that we think are promising; analysis that seems especially on target and insightful.

Question: Questions you have; things to be clarified or expanded upon.

Polish: Suggestions and ideas for consideration. (May be posed in the form of questions.)

* Praise, Question, Polish. The “praise, question, polish” format is presented by Gloria A. Neubert in PDK Fastback 277, Improving Teaching Through Coaching, 1988. The form presented by Neubert has been adapted by AEL, P.O. Box 1348, Charleston, WV 25325, www.ael.org, 800-624-9120.Adapted from: Southern Regional Education Board, Leadership Curriculum Modules, Creating a High Performance Learning Culture, November 2003.

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southern regional education board - robeson.k12.nc.us web viewcte teacher introduces cte lesson. ......

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