MathOrMagic.pdf

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UVA-SCPS Office of Mathematics Outreach with support from VADOE Mathematics and Science 1 Partnership Grant Program NCLB Title II Part B

Mathematics Senior Level Capstone Course

Unit Overview

Title of Unit: Math or Magic? Unit Designers:

(Name and School Division) Steve Gregorin

York County Public Schools Michelle Rittenhouse and Marianne Veitch

Hanover County Public Schools Diane Leighty, Editor, UVA-SCPS Office of Mathematics Outreach Context:

Summary of the issue, challenge, investigation, or problem.

People often think there is no relationship between the math taught in school and real-world scenarios. In this task, students will translate realistic written expressions into both algebraic expressions and graphical representations. Students will essentially remove the perceived “magic” component and unveil these relationships.

Number of Class Hours:

5 hours Unit

Design: _X_Task Based ___Project Based Other Subject

Areas/Disciplines Addressed:

Physical Fitness, Science

Driving Question: Is there magic in math, or is it algebra? Can we create our own graphs that represent real-life scenarios? Mathematics Content

Addressed:

Represent practical problems; linear relationships; paired data analysis and regression analysis; make predictions; transfer between and analyze multiple representations of functions; use technology to create graphical displays and to simplify algebraic expressions.

MPE Addressed:

Problem Solving, Decision Making, and Integration Understanding and Applying Function Procedure and Calculation Assumption of Prior

Knowledge:

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College and Career Readiness/21st Century Skills to be (T) during this unit or expectation (E) for student use during this unit and assessed (A): BIE Page 35-37

Collaboration – students work in pairs for some of the tasks.

T & A Research

Communication (Oral and/or Written) – written to share data analysis

T & A Technology – use of a graphing calculator to analyze data

T & A

Critical Thinking/Decision Making – Find patterns, determine the correct algebraic expression and create a graph that connects to the written expression(s).

T & A Other: (Describe)

Major Products and/or Performances:

Group – formal written answers to questions posed in each task; posters

Presentation Audience: Class School Individual – mathematician’s journal entries, individual written

answers to some of the tasks

Expert Community Other:

Launch: Event or experience used to engage the students interest and inquiry:

Jewel Ayich in the “World’s Best Magician” clip:

http://www.youtube.com/watch?v=hqiNL4Hn04A Lance Armstrong Tour de France clip: http://www.youtube.com/watch?v=-ykwOkcyFzc

Evaluation: Formative Assessments (During the Unit)

Interview Practice Presentations

Mathematicians Journal X Notes

Preliminary

Plans/Outlines/Prototypes

Checklists

Rough Drafts Concept maps

Field Tests Other:

Summative Assessment (End of Project)

Written Products, with a rubric X Peer Evaluation, with a rubric

X Oral Presentation with a rubric Self Evaluation, with a

rubric

X Other Product(s) or

Performance(s), with a rubric

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Resources Needed: On-site people, facilities: Facilitator / Teacher Equipment/Technology: Graphing calculators

Materials: Index cards and poster board for each group Community Resources: None

Reflection Methods: Individual, Group, and/or Whole Class

Mathematician’s Journal X Small/Focus Groups Whole Class Discussions X Fishbowl Discussions

Survey Other:

Material Adapted From: Phillips Exeter Academy Math Department, www.exeter.edu/academics/72_6539.aspx

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Virginia’s Senior Level Capstone Course

Instructional Plan

Unit Title: Math or Magic?

Driving Question: Is there magic in math, or is it algebra? Can we create our own graphs that

represent real-life scenarios?

Task: People often think there is no relationship between the math taught in school and

real-world scenarios. In this task, students will translate realistic written expressions into both

algebraic expressions and graphical representations. Students will essentially remove the

perceived “magic” component and unveil these relationships.

Note: Prior to this unit, the Self-Assessment Prerequisite Quiz –HO #1 should be administered

to students. For those students whose results indicate they need additional review, a list of

tutorials is included on HO #2.

ENGAGE

How will students’ interests be peaked so they want to engage in the inquiry in this unit?

Number of hours:

Independent intro

task – 5 min

Discussion –10 min

Introduction – show all or part of Jewel Ayich in the

“World’s Best Magician” clip:

http://www.youtube.com/watch?v=hqiNL4Hn04A.

Post Fabulous Five Math Trick (HO#3) on the

board.

Students work independently trying at least three

different numbers. Encourage students to choose any

real number between 1 and 10.

Whole class discussion of results – what happens

when a rational or irrational number is used? Must the

initial value be between 1 and 10? What questions are

raised? Is this math or magic?

Note: A Teacher Guide-Chronology of Lesson and

Answer Keys is found on HO #4A.

Mathematician

Journal

Prompts –

How did the

result come

about? Is it

magic or is it

mathematics?

EXPLORE

Teacher provides guidance for the explorations to prepare students with the knowledge and skills to engage in the task.

Exploration 1: Fabulous Five Math Trick

(HO#3)

Teacher guides the students through the proof of the

“Fabulous Five Math Trick.” Students recall how to

create equations and simplify rational expressions.

Exploration 2: The Next Number Problem

(HO#3)

Mathematician

Journal

Prompts –

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Students will self-assess on the prior knowledge and skills assumed for the unit

Number of hours

Approximately 30

minutes

The goal of this activity is for the students to identify

a pattern from a student-generated set of data, create

an algebraic expression, prove an algebraic equation,

and develop the linear function through graphing

sample data.

Description of activity and student directions sheet is

attached.

Materials needed – graphing calculator for each

student

Directions for instructor – students work

independently on The Next Number Problem. Class

discussion of results when finished.

Students are put into small groups and begin HO #3A

Elaboration 1: M

3

: More Math Magic

.

your

subsequent

analysis of the

data – what do

the graphs

indicate?

EXPLAIN

Teacher introduces the main task of the unit and prepares students to in small group independent work...

Number of Hours

Approximately 45

min

Introduce The Bicycle Problem with a short video

about Lance Armstrong and the Tour de France.

http://www.youtube.com/watch?v=-ykwOkcyFzc.

Discuss with students the components of the bicycle

race, including speed of the riders, the different terrain

of the course, etc.

Put students in groups of 2 or 3. Student task sheet,

The Bicycle Problem, is attached as HO#4.

As a small group, students are expected to work

collaboratively on analyzing, solving the problem, as

well as recording, and presenting their findings. A

graphing calculator can be used to graph the data and

find the trend line. Students record the answers to the

posed questions in their small group.

After students have completed the questions with the

task, move on to Exploration 2: The Bicycle

Problem, Alternate Scenarios.

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ELABORATE

The student groups are working independently with teacher

consultations.

Number of Hours

Elaboration 1 –

approximately 1 hr

Elaboration 2 –

Approximately 45

min

Elaboration 1: M

3

: More Math Magic (HO#3A)

The goal of this activity is to allow pairs of students to

create their own written expression that will result in

linear equations.

Place students in groups of 2 or 3. Each group has a

note card and a graphing calculator. Student task

sheet is attached-

Elaboration 1: M

3

: More Math

Magic.

Once the students have completed their own problem,

the students place their note card, written side up, in

the center of the table.

When all of the groups have completed the initial

task, the groups rotate to a different table and create a

simplified algebraic expression that represents the

new problem. Students may rotate groups as often as

time permits.

To conclude the exercise, the teacher leads a

class-wide discussion of their findings and the correct

expressions for each problem. Students have the

opportunity to discuss the different methods used to

arrive at the solutions.

The activity continues with The Bicycle Problem

under the “Explain” category.

Elaboration 2: The Bicycle Problem, Alternate

Scenarios (HO#4)

The goal of this activity is to allow groups of students

to create graphical representation from a given written

scenario.

Once students have answered questions a-d, the

teacher passes out one of the four alternate scenarios

to each group (on HO#4A, page 2) These scenarios

replace the previously given data and start with

Marianne at home. Each group explores the new

Mathematician

Journal

Prompts

M

3

: Which task

was more

difficult,

creating the

written

expression or

creating the

algebraic

expression?

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scenario and creates a graphical representation on

poster board with labeled axes but without titles that

identify the scenarios. The posters are then displayed

around the room.

The teacher engages the class in a discussion to see if

the students can determine which graphs came from

the same scenario. The teacher can physically move

these posters to group them together, based on the

class conclusions. The students then have the

opportunity to discuss the graphs and hypothesize on

the given scenario. Upon conclusion of the activity,

each scenario is revealed to the class and graph

correctness is discussed.

EVALUATE

Working groups submit products or make presentations

Number of Hours

Approximately 45

min

Student created algebraic expressions from the M

3

:

Student responses to Bicycle Problem question, as

well as the posters from the alternate scenarios.

Completion of Self-Assessment (HO #1

)

prior to

lesson and Peer and Self Evaluations (HO#5).

Mathematician

Journal

Prompts

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Map the Unit

What do students need to know and be able to do to complete the task/project/problem

successfully? How and when will they assess their own necessary knowledge and skills?

How will they remediate their own gaps or weaknesses in knowledge and skills? Look at

each major task for the unit and analyze the tasks necessary to produce a high-quality

product.

Task: People often think there is no relationship between the math taught in school and

real-world scenarios. In this task, students will translate realistic written expressions into both algebraic expressions and graphical representations. Students will essentially remove the perceived “magic” component and unveil these relationships.

KNOWLEDGE AND SKILLS

NEEDED

Assumed

already

learned

Students

will

self-assess

Will be

taught

during the

unit

1. Writing algebraic expressions from

written expressions

x

2. Simplifying rational and irrational

expressions

x

3. Apply knowledge of slope

x

4. Graph paired data and find line of

best fit

x

5. Create graphical representations of

written expressions

x

6. Paired data analysis and regression

analysis with technology

x

7. Make predictions from data

x

What project tools will student’s use?



Know/need to know lists



Daily goal sheet

X Mathematician’s Journals



Briefs/Memos



Task lists



Planning Calendar

X graphing calculator_______________

□ ________________________________

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HO #1

Name: ________________________________ Period:_______ Date:________

Math or Magic? Self-Assessment

Prerequisite Quiz

Convert the following written expressions to algebraic expressions: 1. Half a number x decreased by four.

2. Six less than double the quantity of a number x increased by four divided by 2.

Simplify the following rational and irrational expressions. 3.

3 18 ) 2 ( 6 x 

4.

4 5 3 x x

Slope

5. Arrange the slopes from greatest to least. a) -3x + 2y = 6

b) y = -3x + 10 c) y = 20 d) 4x = y – 7

Graphing paired data (Scatterplots) and Linear Regression (trend lines)

6. Create a scatterplot of the following data. Be sure to include title and labels. Average Daily Temperature for January 1-7 in Degrees Fahrenheit

7. Find the linear regression equation for the results of question 6. Date Temperature

1 10

2 25

3 30

4 32

5 23

6 25

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HO #1A

Math or Magic? Self-Assessment

Prerequisite Quiz-Answers

:

1. 1/2x – 4

2. 2(x+4) - 6

2

3. 2(x-1)

4. 2 x

5. d, a, c, b

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HO #2

Tutorials

If you are having difficulty with any of the above topics, check out the links below for extra practice before you get to class.

Converting the written expressions to algebraic expressions: http://www.mathgoodies.com/lessons/vol7/equations.html http://www.themathpage.com/alg/algebraic-expressions.htm

Simplifing rational and irrational expressions:

http://www.khanacademy.org/video/simplifying-rational-expressions-1?playlist=Algebra%20I%20Worked%20Examples

http://www.khanacademy.org/video/simplifying-radical-expressions1?playlist=Algebra%20I%20Worked%20Examples

Slope:

http://www.khanacademy.org/video/slope-of-a-line?playlist=Algebra%20I%20Worked%20Examples http://www.khanacademy.org/video/slope-example?playlist=Algebra%20I%20Worked%20Examples

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HO #3

Name: ________________________________ Period:_______ Date:________

Exploration 1: Fabulous Five Math Trick:

1. Choose a number from 1 to 10. 2. Double the number.

3. Add 10 to your new number. 4. Now, divide the total by 2.

5. Finally, subtract your original number that you started with.

Exploration 2: The Next Number Problem

Pick any number. Add 4 to it and then double your answer. Now subtract 6 from that result and divide your new answer by 2. Write down your answer. Repeat these steps with another number. Continue with at least three more numbers, comparing your final answer with your original number.

Number Result

a) Write an algebraic expression to represent the steps used to create this pattern. Use n as your variable.

b) Write an algebraic expression for the output when the number n is chosen. .

c) Prove that the expressions from parts a and b are equivalent.

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HO #3A

Elaboration 1: M

3

: More Math Magic

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HO #4

Name: ________________________________ Period:_______ Date:________

The Bicycle Problem

The table below shows data that Marianne collected during the first 50 minutes of a bike ride. The distance (in miles) from her home is tabled at ten-minute intervals.

a) Make a scatter plot of this data on your calculator and write the equation of best fit. Why might you expect the data points to line up? Why do they not line up exactly?

b) If Marianne collected data for 90 minutes, estimate her distance and explain your method. What if she collected data for t minutes; what would its distance be?

c) If she continues away from home, will the graph remain linear? Why or why not?

d) Will the graph always increase? Explain in context of the problem.

Elaboration 2: The Bicycle Problem, Alternate Scenarios

Your teacher will give you a new scenario for Marianne’s bike ride. Your scenario will be kept secret from the other groups in the class. On your poster board, create a graphical representation of this ride. Be sure to label the axes with time and distance, but do not give your graph a title. There should be no text or pictures to indicate what is happening on her journey. Your graph will only be a line or

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HO #4A

Math or Magic?

Teacher Guide – Chronology of Lesson and suggested answers

Exploration 1: Fabulous Five: 1. Choose a number from 1 to 10. 2. Double the number.

3. Add 10 to your new number. 4. Now, divide the total by 2.

5. Finally, subtract your original number that you started with.

Expected result: 5

2 10

2  _ _

x x

Exploration 2: The Next Number Problem

Pick any number. Add 4 to it and then double your answer. Now subtract 6 from that result and divide your new answer by 2. Write down your answer. Repeat these steps with another number. Continue with at least three more numbers, comparing your final answer with your original number.

a) Is there a pattern? If so, what is the pattern? Write an algebraic expression to represent the pattern using n as your variable.

b) Write an algebraic expression for the process when the number n is chosen. c) Prove that the expressions from parts a and b are equivalent.

d) Plot your data onto a scatter plot. Find the equation a function that represents the data.

Expected result:

1

2

6 )

4 (

2 







n

Elaboration 1: M3: More Math Magic

Create your own problem similar to the ones we just worked. Your problem must have a different result. In other words, the result may not be the starting number or one more than the starting number. Write your final “magic problem” on the front of the notecard. It must be in complete sentences and

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HO #4A continued

The Bicycle Problem

The table below shows data that Marianne collected during the first 50 minutes of a bike ride. The distance (in miles) from her home is tabled at ten-minute intervals.

a) Make a scatter plot of this data. Why might you expect the data points to line up? Why do they not line up exactly? As the tie increases, the distance increases. They do not line up exactly as the

relationship between time and distance is not the same.

b) If Marianne collected data for 90 minutes, estimate her distance and explain your method. What if she collected data for t minutes; what would the distance be? Answers will vary depending upon method. The distance is approximately 0.2 mile per minute.

c) If she continues riding away from home, will the graph remain linear? Why or why not? Yes, because she will cover some amount of distance as time increases.

d) Will the graph always increase? Explain your reasoning in the context of the problem. No. At some point she will stop riding and the distance will be the same. When she returns home, the distance will decrease.

Elaboration 2: The Bicycle Problem, Alternate Scenarios:

1) Marianne takes 50 minutes to ride her bike to the store. Unfortunately, the store is closed, so she turns right around & heads home. Her return trip takes 100 minutes.

2) Marianne rides toward her friend’s house. After 50 minutes she stops for a 20 minute picnic and then continues on to her friend’s house. It takes her 30 more minutes to arrive.

3) Marianne takes 50 minutes to ride her bike to the store. She shops for about 10 minutes. Her return trip takes 100 minutes.

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HO #5

Name: ________________________________ Period:_______ Date:________

Math or Magic?

Peer and Self Evaluations

Peer Evaluation

The following is a list of statements to be answered by you about your partner. Think carefully about assigning values for each of the following statements.

Directions: Put an ‘X’ in the box that applies.

My partner… Strongly

Agree Agree Neutral Disagree

Strongly Disagree Contributed positively to discussions

Did an equal portion of the workload Helped to keep me focused on the task Was respectful of my ideas and opinions Is someone I would work with again

Self Evaluation

The following is a list of statements to be answered by you about yourself. Think carefully about assigning values for each of the following statements.

Directions: Put an ‘X’ in the box that applies.

I ,___________________ ,(insert name here)… Strongly

Agree Agree Neutral Disagree

Strongly Disagree Contributed positively to discussions