Radical Relationships
Schedule and Assignments
Week of 3/16
Monday
Tuesday
Wednesday
Thursday
Friday
Schedule
Initial Content Assessment
Unit Vocabulary Quest
Ratios
Equivalent Ratios
Rates
Assignments
Week of 3/23
Monday
Tuesday
Wednesday
Thursday
Friday
Schedule
Rhythmic Ratios
Proportions
Solve Proportions
Write Proportions
Trail Mix – group activity
Assignments
p 381 14-32 even
p 383 17-31 odd
p 385 16-34 even
p 387 8-17
Week of 3/30
Monday
Tuesday
Wednesday
Thursday
Friday
Schedule
Proportion & Similar Figures
Use Proportions
Scale Drawings and Maps
Ratio Rendezvous Project
Excel Exercises
Assignments
p 388-389 2-14 even
p 391 3-11
p 392-393 1-12
Review/select project options
Week of 4/6
Monday
Tuesday
Wednesday
Thursday
Friday
Schedule
Relate Percents to Fractions
Relate Percents to Decimals
Decimals, Fractions and Percents
Percents Greater / Less Than 100%
Ratio Mind Map & Ratio Rendezvous
Assignments
p 395 4-32 every 4th & 39-40
p 397 11-29 odd
p 399 1-35 odd
p 401 & 403 37-47 odd
Week of 4/13
Monday
Tuesday
Wednesday
Thursday
Friday
Schedule
Ratio Rendezvous work day
Ratio Rendezvous Peer Reviews
Problem Solving - Review
Unit Test
Ratio Rendezvous Presentations
Assignments
Begin Vocabulary Quest
p 377 15-33 odd
p 379 12-32 even
p 406-407 1-19 odd
Title:
Topic:
Radical Relationships
Ratio Rendezvous
Subject:
Grade: 6
Mathematics
Designer: D Beiswanger
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Stage 1 - Desired Results
Standards Addressed: MN K-12 Academic Standards in Mathematics 6.1.2
Understand the concept of ratio and its relationship to fractions and to the multiplication and division of whole
numbers. Use ratios to solve real-world and mathematical problems.
Benchmarks:
6.1.2.1 - Identify and use ratios to compare quantities; understand that comparing quantities using ratios is not
the same as comparing quantities using subtraction.
6.1.2.2 - Apply the relationship between ratios, equivalent fractions and percents to solve problems in various
contexts, including those involving mixtures and concentrations.
6.1.2.3 - Determine the rate for ratios of quantities with different units.
6.1.2.4 - Use reasoning about multiplication and division to solve ratio and rate problems.
Understandings:
Essential Questions:
Students will understand….
Ratios express relationships.
Ratios compare quantities.
Ratios relate multiplication and division of whole
numbers.
Ratios, fractions, decimals and percents are related
Ratios have many real-world applications.
Relationships exist in all areas of work and leisure.
What is the relationship of ratios to fractions,
decimals and percents?
What is the role of multiplication and division in the
use of ratios?
How do ratios compare quantities?
How do rates evolve from ratios?
How do rates compare quantities?
How can ratios help us solve problems?
What real-world applications use ratios and rates?
Students will know….
Students will be able to…
How to express relationships as ratios.
How ratios, fractions, decimals and percents relate.
How multiplication and division relate to ratios.
Rates are ratios comparing quantities with different
units of measure.
Percents compare a quantity to 100.
Proportions are equivalent ratios.
A scale is a proportion.
Scale drawings are proportional to real objects.
How to use ratios to solve real world problems.
Indentify and describe ratios.
Translate ratios to fractions, decimals and percents.
Interpret the relationship of multiplication and
division of whole numbers to ratios.
Solve rate problems.
Use ratios to compare quantities.
Apply ratios to real world problem solving situations.
Formulate indirect measurements.
Construct ratios to solve real-world problems.
Select an appropriate scale.
Create scale a drawing.
D Beiswanger
Radical Relationships UBD
Stage 2 – Assessment Evidence
Performance Task
The learner will create a scale drawing (goal) using skills learned and understandings gained during the unit on
ratios, rates, fractions and percents. The learner will choose the role of a professional (role) from a list of career
scenarios that regularly use ratios and rates to develop work products. The customer, client or boss (audience)
in each scenario will expect the learner to present a drawing to scale based on actual measurements of the
work product identified with the professional role (situation) from the list of career choices. The learner will need
to develop a system of measuring and recording (performance) so they can successfully scale the drawing
(product), and demonstrate their understanding of ratios, rates and fractions (purpose). The 24”x30” scaled
drawing (standard) needs to be accurate, with measurements and conversions documented in an excel
spreadsheet posted to the class wiki, https://stf116.wikispaces.com/ (criteria for success). The drawing will be
presented (criteria for success) to the class and assessed by the teacher. There will also be a peer review of
the presentation and aesthetics of the drawing.
Options: the metric system may be used for measuring and scaling if preferred. GPS measuring devices may
be used for projects with a large area, but all conversions must be documented.
Key Criteria
- Distances must be accurately measured and scaled.
- A minimum of 10 objects within the drawing must be accurately measured and scaled.
- Documentation of original measurement, scaled measurement, scale, ratio, fraction and percent must be
completed in an excel spreadsheet and posted to the class wiki, https://stf116.wikispaces.com/.
- Drawing components must include a title, an accurate scale, a legend or key and a compass.
- Aesthetics will be judged for labels and lettering, effective use of color, shape and size and neatness,
- Presentation of the project (3-5 minutes) must include an introduction, what was drawn, the scale used,
answers to reflection questions and a conclusion.
Other Evidence:
- Daily Assignments - observation checklist of key proportion concepts
- Informal hand signals and homework checks
- Trail Mix Group Activity
- Graphic Organizer – Ratio Web of real-life applications
- Self Assessment
- Vocabulary Quest
- Unit Test
Stage 3 – Learning Plan
Learning Activities
- Current Content Assessment will determine existing content understanding.
- Vocabulary Quest will introduce pertinent vocabulary for the unit.
- Learning content will be sequenced to develop understanding, beginning with simple ratios and expanding to
real-world applications.
- Rhythmic Ratios will apply ratio concepts to music.
- Trail Mix will apply ratio concepts to mixtures and allow for group activity.
- Ratio Rendezvous performance task will accomplish the following:
- Student selected, real-world, scale drawing application (conversion of measurements using ratio/proportion)
- Poster-sized graphic of drawing (may include use of graphic technology to create)
- Technology exposure through use of spreadsheet tools and class wiki space (additional options available for
use of graphic software and GPS measurements)
- Self-reflection with regard to performance task and unit learning
- Self-assessment of performance task
- Peer-review of performance task
- Presentation opportunity (including practice session)
D Beiswanger
Radical Relationships UBD
Radical Relationships
WHERE TO
W
Discuss what will be learned in the unit (ratios, fractions, decimals and percents).
Provide the vocabulary list.
Anticipate the primary performance task and provide rubric.
Develop a bulletin board to provide examples of performance task.
H
Discuss unit content in real-life applications.
Build a web of real-life applications for ratios.
Initiate Vocabulary Quest.
E
Equip students with an understanding of the concepts through direct instruction.
Instruct students on the use of the Excel spreadsheet tool.
Supply daily assignments to allow opportunity for practice.
Provide experiential learning activities illustrating real-life applications of ratios.
R
Assign student made web of real-life applications for ratios.
Include Ratio rendezvous Reflection in performance task.
E
Appraise daily assignments using observation checklist of key concepts.
Use hand signals and homework checks for informal evaluation.
Assess Trail Mix Group Activity.
Review graphic organizer – Ratio Web of real-life applications.
Look at Self Assessment.
Check Vocabulary Quest.
Examine Unit Test.
Evaluate Ratio Rendezvous performance task.
T
Assess initial understanding of content.
Provide a variety of working modalities.
Allow opportunity for rehearsal of presentations.
Make available ‘Teacher and Me Time’ for additional help.
Afford options and scalability in performance task choices.
O
Typical Daily Sequence
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Begin class with a Problem of the Day. (challenge level question or brain teaser of
interest to the students which pulls from previously learned mathematical concepts).
Exchange and correct homework papers from the previous assignment to provide
feedback. Assignments are corrected in class to reinforce understanding, establish a
feeling of success and learn from mistakes.
Invite learners to share discoveries from the day’s assignment. This is an opportunity for
reflection, additional understanding and growth and recognition (reward) of success.
Challenge learners to apply understanding learned to real-world problems. Application
invokes higher level thinking and provides validity to the concepts learned.
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Radical Relationships UBD
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Brief learners on content and objectives of the day’s lesson. Introduction prepares
students for the day’s concepts and raises a level of concern.
Instruct learners on the concept of the day. Use verbal instruction, written notes and
physical manipulatives where applicable.
Elaborate Vocabulary Quest definitions applicable to the day’s lesson.
Discuss the connection the day’s concept to previously learned concepts.
Model sample problems on the board.
Solve several examples as a class checking for understanding (key questions).
Ask questions calling for a response (hand signal) when students get the answer.
Invite students to attempt sample problems on their own allowing for questions and
sharing.
Assign homework problems allowing a few minutes to begin the assignment.
Circulate the room while students are working independently to answer any questions or
redirect misconceptions.
Provide additional modeling and possibly simpler assignments for learners below-level.
Provide enrichment assignments of greater complexity for learners above-level.
Conclude lesson by inviting students to apply understandings learned through the day’s
lesson and activities to real-world problems.
Day 1
• Hook learners by playing a whiteboard version of “Fraction Tracker” to direct thinking.
• Introduce Radical Relationships and ratios.
• Complete Initial Content Assessment to determine current understanding of Minnesota
K-12 Academic Standards in Mathematics 6.1.2 for Number and Operation.
Day 2
• Introduce unit vocabulary by sending learners on an internet Vocabulary Quest.
• Play “Fly Swatter” to reinforce vocabulary.
Day 3
• Based on Initial Content Assessment, learners may be placed in groups according to
their learning levels allowing for differentiated instruction leveraging the appropriate level
of content complexity for students.
• Introduce Ratios following the typical daily sequence.
Day 4
• Introduce Equivalent Ratios following the typical daily sequence.
Day 5
• Introduce Rates following the typical daily sequence.
Day 6
• Expose learners to mathematical relationships in music through “Rhythmic Ratios”.
Day 7
• Introduce Proportions following the typical daily sequence.
Day 8
• Teach learners to Solve Proportions following the typical daily sequence.
D Beiswanger
Radical Relationships UBD
Day 9
• Teach learners to Write Proportions following the typical daily sequence.
Day 10
• Apply ratio relationships to mixtures using Trail Mix – group activity.
Day 11
• Teach Proportion & Similar Figures following the typical daily sequence.
Day 12
• Use Proportions following the typical daily sequence.
• Perform indirect measurement of objects on school grounds.
Day 13
• Introduce Scale Drawings and Maps following the typical daily sequence.
Day 14
• Introduce, discuss and assign Ratio Rendezvous Project.
• Allow learners time to briefly research “career” options of interest.
• Select career options from those reviewed.
Day 15
• Teach spreadsheet tool skills using Excel Exercises.
• Model step-by-step procedures for specific skill required.
• Provide exemplars.
Day 16
• Relate Percents to Fractions following the typical daily sequence.
Day 17
• Relate Percents to Decimals following the typical daily sequence.
Day 18
• Relate Decimals, Fractions and Percents following the typical daily sequence.
Day 19
• Explore Percents Greater and Less Than 100% following the typical daily sequence.
Day 20
• Assess students understanding of ratio relationships and application to real-world
problems using Ratio Mind Map.
• Provide Ratio Rendezvous work time.
Day 21
• Provide Ratio Rendezvous work day.
Day 22
• Form groups of to perform Ratio Rendezvous Peer Reviews using observation checklist.
• Polish Ratio Rendezvous projects as time allows.
D Beiswanger
Radical Relationships UBD
Day 23
• Review Problem Solving following the typical daily sequence.
Day 24
• Administer Unit Test to assess content understanding of MN K-12 Academic Standards
in Mathematics 6.1.2.
Day 25
• Present Ratio Rendezvous Projects.
o Poster size scale drawing
o 3-5 minutes presentation
Introduction
Description of drawing
Scale used
Oral self reflection
Closing
o Documentation posted to class wiki space
D Beiswanger
Radical Relationships UBD
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Radical Relationships
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Radical Relationships UBD
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Student Reflection
The student reflection for this unit is imbedded in the performance task. As part of the
performance task, each student will be asked to reflect on the following questions. Their
reflections will then be incorporate into the presentation of their projects.
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Self Assessment
The self assessment for this unit is imbedded in the performance task. As part of the
performance task, each student will perform the following self assessment.
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Ratios
Using what you know about ratio relationships, create a 3-deep mind map of real-life applications and uses.
Ratio Mind Map
Ratio Mind Map (Key)
Using what you know about ratio relationships, create a 3-deep mind map of real-life applications and uses.
baseball
population
radio
weather
body temp
Decimals
baking
Percents
Ratios
Fractions
sharing
mileage
Rates
energy
Proportions
money
speed
cooking
D Beiswanger
statistics
maps
Radical Relationships UBD
mixtures
Vocabulary Quest (Day 2)
Standard addressed:
• Minnesota K-12 Academic Standards in Mathematics 6.1.2
• Grade: 6
• Strand: Number and Operation
• Standard: Understand the concept of ratio and its relationship to fractions and to the
multiplication and division of whole numbers. Use ratios to solve real-world and
mathematical problems.
Environment:
• This lesson is taught in a classroom setting at computer stations. Alternately, it may be
taught in a computer lab if there are not enough computers available.
• The learner will work independently at computer stations.
• The last ten minutes of the period they will work on one of two teams.
• The teacher should be circulating the room and then leading the game.
Learners:
• This lesson is intended for sixth grade mathematics classes of learners of all levels.
• The ages range from eleven to thirteen.
Context:
• This lesson meets the Minnesota Academic Standards because it pre-teaches the basic
vocabulary of a unit on ratios.
• Students will demonstrate their ability to perform internet searches, document
vocabulary terms and build retention of those terms.
Goals:
• Teach sixth graders of any level mathematics vocabulary terms they will be using in a
unit on ratios and to reinforce those terms with a fun game.
Objectives:
• Given a set of vocabulary terms, the learners will use the internet to explain definitions of
terms in an attempt to become familiar with them for a unit on ratios.
• When definitions have been documented, the learners will participate in a team
competition to illustrate their understanding and reinforce the meaning of the vocabulary.
Estimated time:
• Thirty minutes for internet research
• Ten minutes for game
Materials needed:
• Vocabulary Quest worksheet
• Pencils
• Computer workstation with internet access
• Whiteboard with markers to create a matrix of selected vocabulary terms. Not all terms
need to be used.
• Two fly swatters
D Beiswanger
Radical Relationships UBD
Prerequisite skills:
• Ability to work independently
• Familiarity with internet searches and use
• Ability to participate competitively on a team
Introduction:
• Class begins with a Problem of the Day as students enter.
• Pass back Initial Content Assessment and talk about focus areas for the unit.
• Invite learners to think about new terms and ideas they will be using in this unit.
• Challenge learners to a Vocabulary Quest using the internet to become familiar with
vocabulary pertinent to the unit on ratios.
• Inform them of the importance of getting a good idea of the meaning of the terms to
enhance their performance later in the competition.
Instruction / Student Activity:
• Pass out Vocabulary Quest worksheet.
• Provide learners with referenced websites.
o http://www.amathsdictionaryforkids.com/
o http://www.mathwords.com
o http://harcourtschool.com/glossary/math_advantage/glossary6.html
• Inform the learners of the time allotment and invite them to begin.
• Prepare the board for the game and circulate the room to help answer questions.
• When time is up, ask students to review their Vocabulary Quests and put them in their
notebooks or folders.
• Invite them to the board to play “Fly Swatter,” forming two teams by counting off.
• The teams line up facing the board, each team with one person in front as the player.
• A matrix of selected vocabulary terms has been prepared during the Vocabulary Quest.
• Read the definition of a vocabulary term, the first player to find the term on the matrix
and “swat” it earns a point for their team.
• Rotate player for each round to allow the most involvement by everyone.
• When all the terms have been defined, the game ends.
• The winner is the team with the most points
Closure:
• Discuss the importance of leaning the vocabulary terms for the unit on Radical
Relationships.
• Assure students there will more opportunities to further refine and use the terms in the
context of Radical Relationships as the unit progresses.
Assessment:
• As the learners are working through this activity the teacher will circulate the room to
ensure understanding of the process and progress toward completion.
• This is not a time for assessment of the terms.
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Radical Relationships UBD
Vocabulary Quest
1. Ratio ______________________________________________________________
2. Terms _____________________________________________________________
3. Equivalent ratio ______________________________________________________
4. Rate _______________________________________________________________
5. Unit rate ____________________________________________________________
6. Unit price ___________________________________________________________
7. Proportion __________________________________________________________
8. Cross-product rule ____________________________________________________
9. Extremes ___________________________________________________________
10. Means _____________________________________________________________
11. Corresponding sides of similar figures ____________________________________
12. Indirect measurement _________________________________________________
13. Scale drawing _______________________________________________________
14. Scale ______________________________________________________________
15. Percent ____________________________________________________________
16. Decimal percent______________________________________________________
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Radical Relationships UBD
Rates (Day 5)
Standard addressed:
• Minnesota K-12 Academic Standards in Mathematics 6.1.2.3 and 6.1.2.4
• Grade: 6
• Strand: Number and Operation
• Standard: Understand the concept of ratio and its relationship to fractions and to the
multiplication and division of whole numbers. Use ratios to solve real-world and
mathematical problems.
• Benchmarks:
o Determine the rate for ratios of quantities with different units.
o Use reasoning about multiplication and division to solve ratio and rate problems.
Objectives:
• The learner will relate rates to ratios, understanding the comparison includes quantities with
different units of measure.
• Given specific instruction, modeling and guided practice, the learner will solve rates for
ratios of quantities with different units.
• The learner will construct real-world ratio and rate problems using reasoning about
multiplication and division.
Motivation:
• The typical daily sequence provides motivation in the following ways:
o The Problem of the Day and applications to real-world problems addresses interest.
o Correcting homework papers provides knowledge of results (feedback).
o Correcting assignments reinforces understanding, establishing a feeling of success.
o Sharing discoveries from the assignment is an opportunity for recognition (reward).
o Briefing learners on content and objectives of the lesson raises the level of concern.
o The combination of these motivators establishes a positive feeling tone.
Materials needed:
• Math text
• Whiteboard with markers for modeling
Prerequisite skills:
• Ability to focus and pay attention to instruction
• Basic understanding of ratios
Environment:
• This lesson is taught in a classroom setting at tables or desks.
• Some time will be allowing for independent practice.
• The teacher will be instructing from the board and circulating the room to check for
understanding.
Estimated time:
• Forty minutes
Learners:
• This lesson is intended for sixth grade mathematics classes of learners of all levels.
• The ages range from eleven to thirteen.
• Differentiation will be addressed through the typical daily sequence.
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Radical Relationships UBD
Context:
• This lesson meets the Minnesota Academic Standards because it allows the learners to
determine rates for ratios of quantities with different units.
• Students will demonstrate their ability to use unit rate and unit price for making conversions
to different units.
Goals:
• Teach sixth graders of any level mathematics to understand rates using ratios of quantities
with different units.
Anticipatory Set:
• Begin class with a Problem of the Day as students enter.
Review:
• Exchange homework papers from the previous assignment.
• Invite learners to share discoveries from the day’s assignment
• Challenge learners apply understanding learned to real-world problems.
Objective:
• Brief learners on content of the day’s lesson:
o Rates are ratios comparing quantities with different units of measure.
o Methods to solve rate problems will be taught and practiced.
o Real-world ratio and rate problems will be constructed using new knowledge.
Input and Modeling:
• Instruct learners on the concept of rates, unit rate and unit price.
• Embellish Vocabulary Quest definitions as needed.
• Discuss the connection of rates to ratios.
• Model sample problems on the board.
Checking for Understanding during instruction
• Solve several examples as a class checking for understanding (key questions).
o How do you know which quantity is the numerator and which is the denominator?
o What is the value of labels on rate problems?
• Ask questions calling for a response (hand signal) when students get the answer.
Guided and Independent Practice
• Invite students to try sample problems on their own.
• Allow for questions and sharing.
• Assign homework problems allowing a few minutes to begin the lesson.
Evaluation / Assessment
• Circulate the room while students are working independently to answer any questions or
redirect misconceptions.
• Provide additional modeling or enrichment assignments for students in need.
• Review homework assignment the next day.
Closure:
• Discuss the value of understanding rates and rate conversions.
• Apply understanding to real-world scenarios.
• Collect prior day’s assignment to review and record.
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Radical Relationships UBD
Rhythmic Ratios (Day 6)
Standard addressed:
• Minnesota K-12 Academic Standards in Mathematics 6.1.2.4
• Grade: 6
• Strand: Number and Operation
• Standard: Understand the concept of ratio and its relationship to fractions and to the
multiplication and division of whole numbers. Use ratios to solve real-world
mathematical problems.
• Benchmark: Use reasoning about multiplication and division to solve ratio and rate
problems.
Environment:
• This lesson is taught in a classroom setting at tables or desks.
• The learners will be working individually at their desks listening and participating.
• The teacher should be leading the class and circulating the room.
Learners:
• This lesson is intended for sixth grade mathematics classes of learners of all levels.
• The ages range from eleven to thirteen.
• Learners above grade level will have an opportunity for extension activities.
Context:
• This lesson meets the Minnesota Academic Standards because it allows the learners to
solve rate and ratio problems using multiplication and division.
• They will demonstrate the ability to express ratios and rates of musical rhythms.
Goals:
• Teach sixth grader mathematics students to solve ratio and rate problems using
multiplication and division.
Objectives:
• Given the musical context of the interval of a measure, the learner will classify rhythms
within the measure and express them as fractions, ratios and rates.
• Using multiplication and division, the learner will solve ratio and rate problems within the
context of music rhythms.
Estimated time:
• Forty minutes
Materials needed:
• Pop music selections appropriate for presentation to sixth graders
• Electronic metronome
• White board marker
• Rhythmic Ratios worksheet
• Rhythm instruments (optional)
• You Tube “Music and Math?” video (optional)
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Radical Relationships UBD
Prerequisite skills:
• Understand basic ratio and rate concepts of relationship between two things.
• Understand the relationship and expression of ratios, rates and fractions.
• Strong listening skills
• Effective use of individual work time
Introduction:
• Hook students with a heavily rhythmic pop music song appropriate for a sixth grade
audience.
• Ask students if they like music. If so, then they like math. Music is math. More
specifically, music is relationships of rhythm and tone all of which can be expressed as
ratios or rates.
Instruction / Student Activity:
• Inquire as to the musical involvement and proficiency of the class by asking if anyone
takes piano, voice or other music lessons. Complement their commitment and
encourage them to share their knowledge.
• Invite learners to listen more closely to the music feeling for the beat. Practice fraction
patterns. Clap a certain number of times within a time-interval, noting it cuts the timeinterval into equal parts of the whole, and thus creates fractions of the interval.
• Describe the interval of time as the unit of a ‘measure’ representing a musical segment.
• Provide the time signature of the piece describing how it identifies the number of beats in
each measure (rate) and the note that represents one beat. It is simplest to find
samples written in 4/4 time. In 4/4 time there are four beats in a measure and the
quarter note gets one beat.
• Identify the rate of four beats per measure and the ratio of 4:1.
• Engage students while explaining the following ideas with regard to a quarter note: (you
may want to call on those with musical experience and use wait time)
o The ratio of notes to measures is expressed as 4:1.
o The rate of notes to measures is expressed as 4 beats / measure.
o The quarter note consumes ¼ of the time in the measure.
• Extrapolate the ratios, rates and fractions of whole notes and half notes.
• Expand the idea to ratios, rates and fractions of multiple measures.
• Listen again to the heavily rhythmic pop music selection, clapping (or tapping rhythm
instruments) to the rhythm of:
o the natural beat of the quarter note
o the one and three counts of the half note
o the down beat of the whole note
o the eight beat pattern of the eighth note
o the sixteen beat pattern of the sixteenth note (ensure your sample includes
sixteenth notes)
• Settle learners into assessment of their learning as they complete the Rhythmic Ratios
Worksheet.
• Allow 15 minutes to hand out and complete the worksheet. Have learners switch papers
and perform a peer review of the assessment.
• Open a class discussion to ensure understanding and disputed responses to the
worksheet assessment.
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Radical Relationships UBD
Closure:
• Remind learners that music is math.
• Realize mathematics is all around us in their every day lives and in some of our favorite
recreational activities.
Assessment:
• To ensure learners have discovered how rates and ratios relate to the rhythms in music
and are able to use reasoning about multiplication and division to solve ratio and rate
problems related to music, the assessment worksheet will include the following activities:
o Expression of notes as ratios and rates of one measure
o Multiplication of rates expressed as ratios for multiple measures
o Division of notes into measures expressed as rates
Extensions:
• Rhythmic Ratio Patterns Activity
• Tone Ratios Exercise
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Radical Relationships UBD
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Radical Relationships UBD
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Trail Mix Activity (Day 10)
Standard addressed:
• Minnesota K-12 Academic Standards in Mathematics 6.1.2.2
• Grade: 6
• Strand: Number and Operation
• Standard: Understand the concept of ratio and its relationship to fractions and to the
multiplication and division of whole numbers. Use ratios to solve real-world and
mathematical problems.
• Benchmark: Apply the relationship between ratios, equivalent fractions and percents to
solve problems in various contexts, including those involving mixtures and
concentrations.
Environment:
• This lesson is taught in a classroom setting at tables or desks.
• The learners will be in groups of 5-6 working as a team.
• The teacher should be circulating the room.
Learners:
• This lesson is intended for sixth grade mathematics classes of learners of all levels.
• The ages range from eleven to thirteen.
Context:
• This lesson meets the Minnesota Academic Standards because it allows the learners to
apply the relationship between ratios, fractions and percents to a mixture.
• Learners will demonstrate the ability to express the ingredients of a trail mix as ratios,
fractions and percents.
Goals:
• Teach sixth graders of any level mathematics ability to express the relationship of
ingredients in a trail mix as ratios, fractions and percents.
Objectives:
• Given a set of known ingredients in a trail mix, the learner will identify the relationship of
each ingredient to the whole as a ratio, fraction and percent of the whole.
• When expressing the relationship of the ingredients in the mixture, the learner will apply
mathematical reasoning to a real-world problem.
• Given a variety of ingredients from which to choose, the learner will create a trail mix
intended to be tasty.
• Given mixtures with varying proportions of ingredients, the learner will decide which
mixture is most flavorful.
Estimated time:
• Forty minutes
Materials needed:
• A large bowl, measuring cup and mixing spoon for each group
• A variety of trail mix ingredients such as raisins, nuts, dried fruit, granola, cocoanut, etc.
• Trail Mix Activity worksheet to record ingredients and document understanding
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Radical Relationships UBD
Prerequisite skills:
• Effectively work in groups
• Understand there exists a relationship between ratios to fractions and percents
Introduction:
• Class begins with a Problem of the Day as students enter.
• Remind students of the learning that has taken place during the prior lessons with regard
to ratios, fractions and percents.
• Invite learners to think about real-world situations where those relationships are
important. (cooking, drafting, etc.)
• Challenge learners to understand the relationship between ratios, fractions and percents
and apply that understanding to the real-world application of a trail mix.
Instruction / Student Activity:
• Teach the relationship between ratios, fractions and percents using examples on the
board.
• Invite students to come forward to solve problems on the board. Re-teach any
misunderstandings.
• Explain the Trail Mix Activity per the worksheet.
• Divide students into groups.
• Allow time to distribute ingredients, create mixtures and record relationships.
Closure:
• Allow students to eat their trail mix and sample the mixtures created by other groups.
• Take a class poll on most flavorful trail mix.
• Consider any problems the groups had with developing the mixture.
• Discuss the relationships recorded and discoveries made in the activity.
• Apply the understanding of the relationship of ratios, fractions and percents to the realworld applications discussed in the introduction.
Assessment:
• As the learners are working through this activity the teacher will circulate the room with
an observation checklist to ensure the learners are able to represent ingredients of a
mixture as ratios, fractions and percents.
D Beiswanger
Radical Relationships UBD
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Observation Checklist
Names
Group 1
Amount
Ratio
Fraction
Percent
Group 2
Group 3
Group 4
Group 5
D Beiswanger
Radical Relationships UBD
Use Proportions (Day 12)
Standard addressed:
• Minnesota K-12 Academic Standards in Mathematics 6.1.2.1
• Grade: 6
• Strand: Number and Operation
• Standard: Understand the concept of ratio and its relationship to fractions and to the
multiplication and division of whole numbers. Use ratios to solve real-world and
mathematical problems.
• Benchmark: Identify and use rations to compare quantities; understand that comparing
quantities using ratios is not the same as comparing quantities using subtraction.
Objectives:
• Given specific instruction, modeling and guided practice, the learner will use proportions to
compare quantities and objects.
• Using experience with proportions and the cross product rule, the learner will synthesize
understanding and knowledge to develop indirect measurements.
• Using indirect measurement techniques, the learner will determine the size of real-world
objects.
Motivation:
• The typical daily sequence provides motivation in the following ways:
o The Problem of the Day and applications to real-world problems addresses interest.
o Correcting homework papers provides knowledge of results (feedback).
o Correcting assignments reinforces understanding, establishing a feeling of success.
o Sharing discoveries from the assignment is an opportunity for recognition (reward).
o Briefing learners on content and objectives of the lesson raises the level of concern.
o The combination of these motivators establishes a positive feeling tone.
Materials needed:
• Math text
• Whiteboard with markers for modeling
Prerequisite skills:
• Ability to focus and pay attention to instruction
• Basic understanding of ratios and proportions
Environment:
• This lesson is taught in a classroom setting at tables or desks.
• Some time will be allowing for independent practice.
• The teacher will be instructing from the board and circulating the room to check for
understanding.
Estimated time:
• Forty minutes
Learners:
• This lesson is intended for sixth grade mathematics classes of learners of all levels.
• The ages range from eleven to thirteen.
• Differentiation will be addressed through the typical daily sequence.
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Radical Relationships UBD
Context:
• This lesson meets the Minnesota Academic Standards because it allows the learners to
compare quantities and objects using proportions.
• Students will demonstrate their ability to use proportions to perform indirect measurements.
Goals:
• Teach sixth graders of any level mathematics to understand using proportions to compare
quantities and objects.
Anticipatory Set:
• Begin class with a Problem of the Day as students enter.
Review:
• Exchange homework papers from the previous assignment.
• Invite learners to share discoveries from the day’s assignment.
• Challenge learners to apply understanding learned to real-world problems.
Objective:
• Brief learners on content of the day’s lesson:
o Proportions will be used to compare quantities and objects.
o Indirect measurement techniques will be developed.
o Indirect measurements will be determined.
Input and Modeling:
• Instruct learners on the concept of using proportions to perform indirect measurements.
• Embellish Vocabulary Quest definitions as needed.
• Perform indirect measurement of objects on school grounds.
• Discuss the connection of indirect measurement to proportions.
• Model sample problems on the board.
Checking for Understanding during instruction
• Solve several examples as a class checking for understanding (key questions).
o How do you know which quantity is the numerator and which is the denominator?
o What is the value of labels on rate problems?
• Ask questions calling for a response (hand signal) when students get the answer.
Guided and Independent Practice
• Invite students to try sample problems on their own.
• Allow for questions and sharing.
• Assign homework problems allowing a few minutes to begin the lesson.
Evaluation / Assessment
• Circulate the room while students are working independently to answer any questions or
redirect misconceptions.
• Provide additional modeling or enrichment assignments for students in need.
• Review homework assignment the next day.
Closure:
• Discuss the value of understanding rates and rate conversions.
• Apply understanding to real-world scenarios.
• Collect prior day’s assignment to review and record.
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Radical Relationships UBD
Technology
There are a number of ways technology is utilized in this UBD unit. A moderately equipped
classroom is assumed. Several forms of technology are used as teaching tools in this sixth grade
mathematics unit. Daily lessons are presented via smartboards. This allows understanding of the
lesson to be developed sequentially and still provide an artifact to post to the class wiki space for
students who may have missed the lesson. The class wiki space is used in several capacities. As
a teaching tool it allows for dissemination of information to the entire class in a timely manner,
without wasting paper. It keeps students up to date with schedules, assignments, assessments,
class notes and any other information valuable to the group. It also provides a vehicle for
collecting input from students. Assuming the resources are available, a digital projection system is
used in the classroom for enrichment and extension lessons. The capability to evaluate students
using a student response system has also been considered when developing some of the
assessments, and would be another opportunity to integrate technology. While the nature of
teaching mathematics does not demand all of these tools, they enhance the learning experience
and allow further exposure to and integration of technology and learning.
Even as these tools use technology to teach, they also help students learn. They not only learn
mathematics content, but also technology integration. As learners, they are exposed to wiki
spaces where they can communicate and collaborate. Students are expected to access the class
wiki space to inquire about schedules and assignments. If they have missed a lesson, this is
where they are expected to get the class notes for the day. Students are also expected to post to
the wkikspace when they submit their work for assessment. Part of the performance task for the
UBD unit requires the use of a spreadsheet tool to perform critical thinking, problem solving and
decision making tasks. The students are expected to gather data and record it using a
spreadsheet tool. They are expected to use the power of the tool to perform simple calculations
which they will use to develop an end product. They are expected to use basic functions of the tool
to enhance the aesthetics of their spreadsheet. The spreadsheet is part of the assessment of the
performance task. The learners also have the opportunity to use GPS technology and PowerPoint
for their performance task if they have the availability, expertise and desire to do so. These
options enable creativity and innovation.
Technology integration in this UBD unit is only limited by the imagination of the person leading it
and the resources available. It is imperative that every existing resource be used to effectively
engage students in learning content. The content alone is exciting and worth effort to understand.
But in today’s world the delivery of the content has to fit a new paradigm. Any technology available
should be leveraged as part of the daily learning activities of the students.
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Radical Relationships UBD
Diversity
Diversity is addressed in this UBD unit in a variety of ways. First the Initial Content Assessment is
intended to identify learners of varying levels of understanding with regard to the content of this
unit. The intention is to group learners by level of understanding. Students below grade level will
be grouped to focus on basic understandings which allow them to meet the academic standard.
Students at grade level will move a little faster and have the opportunity to explore more complex
problems and content extensions. Students above grade level who may already meet the
understandings of the academic standard will be allowed to explore more complex, real-world
problems and extensions.
For the group of students below grade level, assignments with simpler content examples may be
made available. The length of assignments may also be modified for these learners. On the other
end of the spectrum, content of greater complexity may be provided for students needing additional
challenge. Enrichment assignments may also be made available making connections to higher
level content such as algebra.
In addition to grade level groupings, the diversity of learners will be addressed within each group
through the amount of scaffolding done for individual students in a particular group. Each day, as
the students begin their independent practice in class, additional modeling will be provided for
those in need. Additional challenge assignments and opportunities for class leadership may be
made available for those easily understanding the day’s content.
The performance task in this unit can also be modified to accommodate the diversity of learners.
The content of the activity can be scaled from very simple objects with very clear and direct scaling
to multifaceted, detailed drawings with complex scaling. In this way, learners can experience the
authentic learning experience of this activity while challenging themselves at their personal level of
understanding.
Cultural diversity has not been addressed specifically. An amazing thing about mathematics is that
it transcends culture. The most important facet of diversity needing to be addressed is the
principle of equity. This is where there seems to be a cultural divide. Women compared to men.
Middle class compared to lower class. There are groups of people who have typically expected to
perform less successfully in mathematics. It is the expectation in this UBD that all students are
capable and have the need to learn these mathematics skills.
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Radical Relationships UBD
Student Resources:
"Alice B. Landrum Middle School - Home." Alice B. Landrum Middle School. 23 Mar. 2009. 23 Apr.
2009 <http://www.lms.stjohns.k12.fl.us/math/Math%20Games%202>.
Eather, Jenny. "Welcome to A Maths Dictionary for Kids 2009 by Jenny Eather." Welcome to A
Maths Dictionary for Kids 2009 by Jenny Eather. 19 Apr. 2009
<http://www.amathsdictionaryforkids.com/>.
"Harcourt Math Glossary." Harcourt School Publishers. 19 Apr. 2009
<http://harcourtschool.com/glossary/math_advantage/glossary6.html>.
Sommons, Bruce. "Mathwords." Mathwords. 29 July 2008. 19 Apr. 2009
<http://www.mathwords.com/>.
Tourneau, Catherine D. Le. Progress in Mathematics - Grade 6. New York: William H Sadlier,
2006.
Walser, Hans, and Peter Hilton. "Welcome." The Fun Works. 19 Apr. 2009
<www.thefunworks.org/>.
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Radical Relationships UBD
References:
"Academic Standards - Mathematics." Minnesota Department of Education. 19 Apr. 2009
<http://education.state.mn.us/MDE/Academic_Excellence/Academic_Standards/Mathemati
cs/index.html>.
Chard, Dr. David. "Vocabulary Strategies for the Mathematics Classroom." Education Place. 19
Apr. 2009 <http://www.eduplace.com/state/pdf/author/chard_hmm05.pdf>.
"E-Example 5.1.1: Communicating about Mathematics Using Games." NCTM Standards. 23 Apr.
2009 <http://standards.nctm.org/document/eexamples/chap5/5.1/index.htm>.
"Frequency of Middle C." hypertextbook.com. 19 Apr. 2009
<http://hypertextbook.com/facts/2003/DanielleDaly.shtml>.
Gann, Kyle. "Just Intonation Explained." Kyle Gann'
s Home Page. 19 Apr. 2009
<http://www.kylegann.com/tuning.html#tune1>.
LeTourneau, Catherine D.. Progress in Math Teachers Edition Grade 6 (Sadlier-Oxford). New
York, NY: William H Sadlier, Inc, 2006.
"Peter Appelbaum: Rhythm Patterns/Ratios Thematic Unit." The Math Forum @ Drexel University.
19 Apr. 2009 <http://mathforum.org/workshops/sum96/paths/rhythm.html>.
"Teaching Ideas - Math." I Love That Teaching Idea!. 19 Apr. 2009
<http://www.ilovethatteachingidea.com/ideas/subj_math.htm>.
"Two Against Three." Ancient-Future.Com: World Music Online. 19 Apr. 2009 <http://www.ancientfuture.com/2X3.html>.
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