Mathematics Seventh, Eighth & Ninth Grades
Elementary Algebra - The Importance of 1
Description
Students will use the property of one to obtain equivalent rational expressions so that they may add and subtract
rational numbers and expressions and simplify rational expressions. Using the theme, the importance of one, a
connection between the number one and the importance of one person or individual will be made using information
from Winter Olympic Games web sites.
Themes
Respect and Striving for Personal Best
Core Curriculum
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5250-01
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Mathematics as Problem Solving
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5250-02
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Mathematics as Communication
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5250-03
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Mathematics as Reasoning
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5250-04
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Mathematical Connections
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5250-05
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Algebra, Skills and Strategies
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Learning Outcomes
Students Will:
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Use the multiplicative property of one, students will be able to add,
subtract and simplify rational expressions
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Practice computation involving rational expressions and explain
the thought processes and logic used
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Develop a greater understanding of their self-worth and appreciation of
other people they may encounter in their life
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Understand that they and others may make positive contributions to society
even when working as individuals
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Activity 1: Mathematical Explorations with the number 1
Preparation
Students should have prerequisite knowledge of the greatest common factor and least
common denominator (least common multiple). Review addition and subtraction for
rational numbers. Create a list of web sites relating to past Olympics. Seek out
information about numbers of medals, names and countries that have received medals
during the Olympics. List the names and countries of participants by event.
Tools and Resources
Access to computers connected to the Internet
The following Internet sites:
Medal Data for the 1998 Olympic Winter Games - Listings by sport and by country
Olympic Almanac
AAF Olympic Primer
Instruction
Introduce the concept that the multiplicative property of one is what allows to add,
subtract and simplify rational expressions.
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a * 1 = 1 * a = a and ac/bc = a/b * c/c = a * 1 = a
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Review
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What is a rational expression? Students will give examples such as 2/3,
5/7 or other fractions. Questions from the teacher should direct students
to construct a basic definition for rational expressions.
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a/b is a rational number or fraction when b is not zero and a and b are integers.
a/b is a rational expressions when a and b are polynomials and the value of b is
not zero.
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Review
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1/3 + 1/5 = ? Have a student explain the process needed to add these two rational
numbers or fractions.
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Summary
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Students need to summarize important components of the problem. Their summarization
should cover the following points.
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Identify the needed common denominator. Use multiplication by a convenient form of one,
5/5 and 3/3 to obtain the equivalent fractions with the needed common denominator.
Display all steps needed for the completion of the problem with students justifying or
explaining their thought processes.
1/3 + 1/5 = 1/3 * 1 + 1/5 *1 = 1/3 * 5/5 + 1/5 * 3/3 = 5/15 + 3/15 = 8/15.
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The application of the multiplicative property of 1. 1 * a/b = a/b * 1 = a/b when b is
not equal to zero, further c/c * a/b = a/b * c/c = ac/bc when b and c are not equal to zero.
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Review
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2/3 - 1/6 = ? Have students complete the subtraction problem and a summarization process as before.
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Note:
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If students are having trouble with rational numbers, the teacher should schedule
time to work with these students on an individual basis using pattern blocks or
other similar manipulatives to help them master addition and subtractions of simple
fractions.
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Exploration
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How would rational expressions be different from rational numbers? Questions from the
teacher should direct students toward the involvement of variables in rational expressions
and toward a more formal definition. a/b is a rational expression when a and b are
polynomials. The expression is defined only when b is not equal to zero.
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Would rational expressions operate under a different set of properties than fractions? No,
remind students of the field properties that have been developed and that must continue to
be used. Common denominators must still be used to perform the operations of addition and
subtraction and these common denominators are obtained by using the multiplicative property of one.
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Exploration
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Have students suggest a solution that could be used to perform the following subtraction.
x/4 - x/9 = ?
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Summary
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Students should summarize the important components of the problem. Their summarization
should cover the following points.
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Identifying the needed common denominator.
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The use of multiplication by a convenient form of one, 9/9 and 4/4 to obtain the
equivalent fractions having the needed common denominator.
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Display all steps needed for the completion of the problem with students justifying
or explaining their thought processes.
x/4 - x/9 = x/4 * 1 - x/9 * 1 = x/4 * 9/9 - x/9 * 4/4 = 9x/36 - 4x/36 = 5x/36.
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The application of the multiplicative property of 1. 1 * a/b = a/b * 1 = a/b
when b is not equal to zero, further c/c * a/b = a/b * c/c = ac/bc when b and c
are not equal to zero.
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Exploration
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2x/3 + 5x/9 = ? Have students complete the addition problem and a summarization process.
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Review
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Have students create a problem similar to x/4 - x/9 = ? or 2x/3 + 5x/9 = ? and exchange problems with a
classmate or partner. Students check each other's work.
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Summary
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Samples of student generated problems and answers are recorded on the board. Questions from the teacher
direct the students to recognize that similar problems would need to involve rational expressions. During
the summarization the whole class checks the answers to the problems written on the board.
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Exploration
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During the review session the teacher should be looking for answers in the form ac/bc. The teacher begins a
discussion reviewing the simplification of fractions or rational numbers and the reasons for writing fractions
in simplest form. The discussion should be directed to rational expressions and the need to have these written
in simplest form. Have students identify answers that could be simplified. Invite students to simplify those
answers expressing their thought process aloud for the class. The general statement ac/bc = a/b * c/c = a/b *
1 = a/b should be developed in the summary of this section of the lesson.
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Exploration
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Have students suggest a process that could be used to perform the following addition problem. 2/y + 5/x = ?
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Summary
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Students should summarize the important components of the problem.
Their summarization should cover the following points.
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Identifying the needed common denominator xy.
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The use of multiplication by a convenient form of one, x/x and y/y to obtain the
equivalent fractions with the needed common denominator.
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Display all steps needed for the completion of the problem with students justifying
or explaining their thought processes.
2/y + 5/x = 2/y * 1 + 5/x * 1 = 2/y * x/x + 5/x * y/y = 2x/xy + 5y/xy or (2x + 5y)/xy
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The application of the multiplicative property of 1. 1 * a/b = a/b * 1 = a/b, further
c/c * a/b = a/b * c/c = ac/bc
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Recognition that 2x + 5y is the numerator and that the terms may not be combined together.
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x or y must not be equal to zero for the rational expressions to be defined.
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Review
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Have students create a problem similar to the one just solved and exchange it with a
classmate or partner. As a sample of problems and answers are written on the board,
the rest of the class is checking to make sure the answers are correct and written
in simplest form.
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Activity 2: The Importance of the Number 1 at the Olympic Winter Games
Exploration
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A teacher directed discussion about the number one and its importance should now
take place. The discussion should summarize the properties of the number one used
in the lesson of the day. The discussion should be directed to the Winter Olympic
Games and the importance of 1 in the Winter Games.
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The following are some suggested focus questions for the discussion
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Is there any individual who has made an important contribution to the Winter Olympics
this year or in past years? Who? What did they do?
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Do the people who make important contributions to the Winter Olympics always need to be athletes?
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Is there is any country that has sent only one person or team to the Winter Olympics?
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Is there a country that has won only one medal?
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Is there a person who has not won a medal or an award in the Winter Olympics and yet became
well known or famous because of what they did during the Olympic games?
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By random selection create teams composed of three students. Each team will review information
from web sites, books or news articles or reports and then select an individual or country for
their report. The teams are encouraged to include pictures, posters or other items along with
a written report. The reports with the related materials can then be displayed in the classroom
before or after the teams give an oral report on their person.
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Assessment
Students will:
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Add, subtract, and simplify rational expressions whether given as simple problems or in context
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Create scenarios involving rational expressions to share with their classmates
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Justify solutions to problems in both written and oral formats
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Make entries in their mathematics journals or notebooks summarizing the lesson content
and identifying important topics from the lesson
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Complete oral or written reports on their selected individual or country as student teams
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