Web Links:
http://www.purplemath.com/modules/fcns.htm
http://www.mathwarehouse.com/algebra/relation/mathfunction.php
http://en.wikipedia.org/wiki/Function_(mathematics)
http://www.nointrigue.com/docs/notes/maths/maths_relfn.pdf
Standard 2
Students will use patterns, relations, and algebraic expressions to represent and analyze mathematical problems and number relationships.
Objective 1
Analyze algebraic expressions, tables, and graphs to determine patterns, relations, and rules.
Describe simple relationships by creating and analyzing tables, equations, and expressions.
Draw a graph and write an equation from a table of values.
Draw a graph and create a table of values from an equation.
Activity:
http://illuminations.nctm.org/LessonDetail.aspx?ID=L293
3rd Grade Standards:
Represent numerical relationships as expressions, equations, and inequalities.
Order and compare whole numbers on a number line and use the inequality symbols <, >, ≠, and = when comparing whole numbers.
Represent numerical relationships as expressions, equations, and inequalities.
4th Grade Standards:
Objective 2
Use algebraic expressions, symbols, and properties of the operations to represent, simplify, and solve mathematical equations and inequalities.
5th Grade:
Use algebraic expressions, inequalities, or equations to represent and solve simple realworld problems.
6th Grade:
Describe simple relationships by creating and analyzing tables, equations, and expressions.
Write, interpret, and use mathematical expressions, equations, and formulas to represent and solve problems that correspond to given situations.
Properties of Equality
Properties of Equality
• If the same real number is added or subtracted to both sides of an equation, equality is maintained.
• If both sides of an equation are multiplied or divided by the same real number (not dividing by 0), equality is maintained.
• Two quantities equal to the same third quantity are equal to each other.
EQUATIONS & INEQUALITIES: Rules of arithmetic and algebra can be used together with notions of equivalence to transform equations and inequalities so solutions can be found.
Examples of Mathematical Understandings:
• A solution to an equation is a value of the unknown or unknowns that makes the equation true.
• Properties of equality and reversible operations can be used to generate equivalent equations and find solutions.
• Techniques for solving equations start by transforming the equation into an equivalent one.
• A solution or solutions to a linear or quadratic equation can be found in the table of ordered pairs or from the graph of the related function.
• Techniques for solving equations can be applied to solving inequalities, but the direction of the inequality sign needs to be considered when negative numbers are involved.
More Information:
An equation is simply an assertion that two expressions are related by equality.
In mathematics, an inequality is a statement about the relative size or order of two objects, or about whether they are the same or not.
An understanding of equalities and inequalities is so important. Students at a young age should be exposed to the importance of equality and the true meaning of an equal sign. Children can also be exposed to inequalities at a young age. It’s important for students to develop their own understanding of equalities and inequalities so they can apply future mathematics principles to these big ideas. A lot of mathematics rely on using equations and that is why it is such a big idea that needs to be used inside the classroom.
An equation is a sentence involving numbers, or symbols representing numbers where the verb is equals (=). There are various types of equations:
3+4=7 True Equation
3+4=9 False Equation
2x+5x=7x Identity Equation
x+4= 9 Conditional Equation
Equations are used to state the equality of two expressions.
Inequalities are demonstrated through the following signs:
In all these cases, a is not equal to b, hence, “inequality”.
These relations are known as strict inequality
Lesson Plans:
http://www.uen.org/Lessonplan/preview?LPid=11051 Commutative Cookie activity for 3rd graders where students create a commutative equation.
http://www.instructorweb.com/lesson/numbersequal.asp Lesson plan for kindergartners or 1st graders on understanding what the equal sign means.
http://www.uen.org/Lessonplan/preview?LPid=16330 Lesson Plan for 4th graders on understanding what is an equation and what is not an equation.
http://www.uen.org/Lessonplan/preview?LPid=16326 Lesson plan addressing different strategies to solve equations for 4th graders.
http://www.uen.org/Lessonplan/preview?LPid=6103 Lesson plan for 4th graders on symbols in inequalities.
http://www.uen.org/Lessonplan/preview?LPid=11207 Lesson plan for 4th graders on relationships between equations. Helps students understand equal values.
http://www.uen.org/core/lessonList.do?courseNum=5050&itemId=3022
A list of 5th grade lesson plans on equations.
Games:
http://www.mathplay.com/Equation/EquationGameOnline.htmlMatching equations with answers. For upper Level elementary grades.
http://www.mathplay.com/OneStepEquationGame.html Another equation game finding answers to equations for upper level grades.
http://education.jlab.org/sminequality/question.php?7466566 Speed math inequality game for 56 graders.
Literature can be a great resource when teaching mathematics. Integrating literature and math can both teach students literacy and motivate them to stay engaged during math time.
“Some suggest that the literature connection motivates students (Usnick & McCarthy, 1998), provokes interest (WelchmanTischler, 1992), helps students connect mathematical ideas to their personal experiences (Murphy, 2000), accommodates children with different learning styles (Murphy, 2000), promotes critical thinking (Murphy, 2000), or provides a context for using mathematics to solve problems (Jacobs & Rak, 1997; Melser & Leitze, 1999).”
http://www.apples4theteacher.com/resources/modules.php?op=modload&name=News&file=article&sid=65
List by WelchmanTischler (1992)
http://www.apples4theteacher.com/resources/modules.php?op=modload&name=News&file=article&sid=65
Many types of literature can be used to integrate with math such as picture books, short stories, novels, folktales, poetry, songs, news articles, and more! Some books explicitly present problems in the context of the reading but others are implicit and will require some creativity. Here are some compiled lists:
· Anno’s Counting Book by Mitsumasa Anno
· Anno’s Counting House by Mitsumasa Anno
· Anno’s Mysterious Multiplying Jar by Mitsumasa Anno
· The Napping House by Audrey and Don Wood
· One Monkey Too Many by Jackie French Koller and Lynn Munsinger
· How Much is A Million? by David M. Schwartz
· A Million Dots by Andrew Clements
· The Hershey’s Milk Chocolate Bar Fractions Book by Jerry Pallotta
· The Greedy Triangle by Marilyn Burns
· Sir Cumference and the Great Knight of Angleland by Cindy Neuschwander
· Sir Cumference and the Sword in the Crone by Cindy Neuschwander
· Sir Cumference and the Dragon of Pi by Cindy Neuschwander
· The Grapes of Math by Gregory Tang – and other books by Greg Tang
· Round Trip by Ann Jonas
· Eight Hands Round by Ann Whitford Paul
Compilations:
http://www.edselect.com/mathbooksbowen.htm
http://childrenspicturebooks.info/articles/picture_books_for_math.htm
http://teacher.scholastic.com/reading/bestpractices/pdfs/mbmath_TitleList.pdf
· Goldilocks and the Three Bears
· The Three Little Pigs
· Three Billy Goats Gruff
· Rapunzel (hair length, time for hair growth)
· Marvelous Math: A Book of Poems by Lee Bennett Hopkins
· Math Poetry Book: Linking Language and Math in a Fresh Way by Betsy Franco
http://literacyconnections.com/Tang.html
A list of songs that can be integrated with math can be found at the following website: http://www.songsforteaching.com/
http://www.apples4theteacher.com/resources/modules.php?op=modload&name=News&file=article&sid=65
Bahr L. Damon & de Garcia A. Lisa, Elementary Mathematics Is Anything but Elementary 2010
In order for students to understand surface area, they first need to understand concepts such as base, height, diameter, radius, pi, etc.
Surface area can be a challenging concept for students because of the different formulas that are used when dealing with different geometric shapes.
http://www.math.com/tables/geometry/surfareas.htm This is a link to a website that lists the formulas to find the surface area of specific shapes.
Standards
Here are the links to the Utah State Core Curriculum for teaching surface area.
http://www.uen.org/core/core.do?courseNum=5060 5^{th} grade core for surface area. It is found in Standard 4.
http://www.uen.org/core/core.do?courseNum=5050 6^{th} grade core for surface area. It is also found in Standard 4.
The standard from the NCTM for 5^{th} grade is listed like this:
Geometry and Measurement and Algebra: Describing threedimensional shapes and analyzing their properties, including volume and surface area.
Students relate twodimensional shapes to threedimensional shapes and analyze properties of polyhedral solids, describing them by the number of edges, faces, or vertices as well as the types of faces. Students recognize volume as an attribute of threedimensional space. They understand that they can quantify volume by finding the total number of samesized units of volume that they need to fill the space without gaps or overlaps. They understand that a cube that is 1 unit on an edge is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating or measuring volume. They decompose threedimensional shapes and find surface areas and volumes of prisms. As they work with surface area, they find and justify relationships among the formulas for the areas of different polygons. They measure necessary attributes of shapes to use area formulas to solve problems. (http://www.nctm.org/standards/focalpoints.aspx?id=334)
Lesson PlansHere are some links to some websites that have great ideas for surface area lesson plans:
http://www.uen.org/Lessonplan/preview.cgi?LPid=21572
http://io.uwinnipeg.ca/~jameis/PAGES/MYR64.html
http://www.shodor.org/interactivate/lessons/SurfaceAreaAndVolume/
http://www.iit.edu/~smile/ma8808.html
ActivitiesHere are some links to some websites that have great activities that can be used to teach surface area. The first two links are especially good ideas for teaching surface area of cylinders.
http://www.uen.org/Lessonplan/downloadFile.cgi?file=21572227998Net_of_Cylinder.pdf&filename=Net_of_Cylinder.pdf –This link is a printable net of a cylinder that students can cut out and then fold in order to see how the different shapes make up a cylinder.
http://mathteachermambo.blogspot.com/2007/08/surfaceareaactivity.html This is an idea of finding the surface area of spheres by using oranges.
http://www.teachingideas.co.uk/maths/files/areapractical.doc This link is a worksheet that students can complete in the classroom. It has to do with finding the surface area of certain things in the classroom.
Standards – UEN is a great resource for finding out the grade level standards in Utah. 5th grade and 6th grade both have standards which cover volume.
Lesson Plans for Teaching Volume
Getting Serious About CylindersSurface Area and Volume
Interactive Websites for Teaching Volume
3D Model Maker – Demonstrates Rectangular and Triangular Prisms
Interactive Overview of 3D Shapes – Reviews the volume of various 3D shapes
Interactive Cube Volume Simulation – demonstrates how to find the capacity of a rectangular prism
http://standards.nctm.org/document/appendix/meas.htm NCTM standards and expectations for k12
UEN Core Curriculum
http://www.uen.org/core/core.do?courseNum=5040– Utah Core curriculum for 4^{th} grade. Area is Standard 4 Objective 2. There are lesson plans and links included for the standard.
http://www.uen.org/core/core.do?courseNum=5050Utah Core curriculum for 5^{th} grade. Area is Standard 4. There are lesson plans and links included for the standard.
http://www.uen.org/core/core.do?courseNum=5060– Utah Core curriculum for 6^{th} grade. Area is Standard 4. There are lesson plans and links included for the standard.
Definitions/Formulas
Area describes how much surface a shape take sup in square units.
Perimeter is the measurement in units of length of the outer edge of an area.
The terms width and length can be compared and connected to the words rows and columns.
Width: Width is the short dimension of any object.
Length: Length is the long dimension of any object.
Square A polygon with four equal side and all angles = 90 degrees. A = S^2
Triangle: A polygon with three corners or vertices and three sides or edges, which are line segments. A = 1/2bh
Parallelograms: A parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size. A = L x W
Circle: A circle is a simple shape consisting of those points in a plane which are the same distance from a given point called the center.
Misconceptions
Perimeter and Area are always the same
Perimeter and Area are interchangeable
Square units means the object is a square
Student examples of misconceptions
How to best teach
Inquiry: Studies have proven that inquiry is one of the best ways to teach mathematics. Inquiry implies that students explore and solve mathematical problems through trial, error, and personal experience rather than simply being handed equations or given answers. If concepts are derived for self, this information is much more likely to become solidified in the students mind and aid in concrete understanding. The three parts of inquiry include launch, explore, and summarize.
Building figures with square tiles
Drawing models with 11 corresponding pictures using graph paper
Using open arrays or sketches and labeling them with numbers
Real life connection: Students learn area and perimeter best when they can make connections to real life examples. Ideas include tiling a room, surface area for pool covers, fencing, and room layouts.
Lesson Plans
–Using Inquiry to Teach Measurement
Deriving the area of a quadrilateral  
Launch Cycle 1 
Teacher shows picture of a quadrilateral made from graph paper on board or overhead. She asks students to discuss with their partner how many square units is inside the quadrilateral. 
Explore Cycle 1 
Students discuss with a partner. 
Summarize cycle 1 
As a class students share answers and strategies of how they figured it out. Some will have counted, some using repeated addition, and some using multiplication. Teacher makes sure to introduce the vocabulary of length and width. 


Launch Cycle 2 
Teacher states that the students are going to figure out a strategy that they can use with any rectangle, even if they don’t know what the measurement of the sides are. She hands the small groups with several quadrilaterals they can explore. 
Explore Cycle 2 
Students work in small groups to find the areas of their quadrilaterals and try to generalize a way to find the area of any quadrilateral. 
Summarize Cycle 2 
As a class, students discuss that they notice that are multiplying one side times the other side, or length times width, in each figure. When asked to write an equation to show this, they write: length x width = area. The teacher further asks them to write the equation using letters to stand for the words, so students derive l x w = a. 
Supplements
http://college.cengage.com/education/bahr/elementary_math/1e/resources.html This site includes information for purchase as well as a few free useful links for teachers.
http://www.shodor.org/interactivate/activities/PerimeterExplorer/?version=1.5.0_16&browser=safari&vendor=Apple_Computer,_Inc. Generate shapes to explore perimeter in an interactive and fun way!
http://www.brainpop.com/math/geometryandmeasurement/areaofpolygons/preview.weml Use this site for free information or sign up for cool videos. Free 5 day trial.
Resources
Bahr L. Damon & de Garcia A. Lisa, Elementary Mathematics Is Anything but Elementary 2010
Wikipedia. Retrieved April 15, 2009, from http://en.wikipedia.org/wiki/Main_Page
Exceptional Learners
Struggling Learners and Students with Disabilities
“Everything about our educational system is changing, including the students who are in our classrooms. Between dramatic increases in the numbers of minority students and legislation such as the Individuals with Disabilities Education Act and the No Child Left Behind Act, classroom teachers are experiencing a different population of children than in decades past. Today, it is not uncommon to have children with language, poverty, and learning issues all within the confines of the general education class” (Bahr & De Garcia, 2010).
What kinds of behaviors indicate struggling learners?
Can students with disabilities learn how to do every kind of math? In what areas do these students struggle most?
What can you do to accommodate students with specified disabilities such as ADHD, Dyscalculia, or Autism?
On this site you will find some definitions, characteristics, and accommodations that can be made for exceptional learners on the struggling side of the spectrum. Each student is a unique case, however, “If we expect all children to learn, then we have to know what kinds of additional accommodations or modifications we need to provide so that all students can achieve success. Many such accommodations require resources and supports. Studies have shown that traditionally underserved children can learn mathematics when provided with the proper support” (Bahr & De Garcia, 2010).
Struggling learners may not have been formally identified or diagnosed as having a specific learning disability although it may be obvious that they have difficulties in mathematics or other subject areas. The reality of these learners is that if they fall behind, they rarely catch up without interventions. Not all students struggle because they have a learning disability. Some students come to school less prepared than their peers, are ill prepared with basic skills such as counting, or the teaching methods are not conducive to their learning styles.
Struggling learners succeed when:
Students with specified learning disabilities and attention deficit/hyperactivity disorder must have a significant gap between intelligence and achievement in one or more of the following areas: oral expression, listening comprehension, written expression, basic reading skills, reading comprehension, math calculations, and math reasoning. The lack of achievement cannot be caused by visual, hearing, motor disability, mental retardation, emotional disturbance, or environmental, cultural, or economic factors. Barriers to learning students might have are in memory, selfregulation, visual or language processing, and motor skills.
Ways to overcome barriers for students with learning disabilities:
“… Many special needs children can engage in inquiry, can develop deep conceptualizations of fundamental mathematics, and can problem solve, communicate, reason, represent, and connect in ways tat often surprise and astound their teachers and their regular education peers” (Bahr & De Garcia, 2010).
For more information on these issues, please visit the National Center for Learning Disabilities, Learning Disabilities Worldwide, and LD Online.
Students with Dyscalculia have difficulty with calculation defined as a wide range of life long learning disabilities involving math. If students are suffering with specific language disorders, their math skills are also at risk of suffering as well. Students with dyscalculia may not engage in “internal chatter” to organize thoughts and manage problem solving strategies when attempting mathematical tasks. Students may be able to mimic procedures and language of problems, but have no conceptual understanding of the mathematics behind it.
Strategies to help students with dyscalculia:
For additional information, visit the Dyscalculia~ Math Learning Disability Resource
Students with more severe learning needs include disabilities such autism and metal retardation. When working with students that have severe learning needs, it is helpful to focus on big ideas and think of all the possible connections they could make to develop and understanding of that big idea. “Each child must be treated as an individual with his or her own set of strengths and weaknesses kept in mind, … the general principles to how children learn math can be applied to children with more sever learning needs as well” (Bahr & De Garcia, 2010). There is no “quick fix” for students with learning difficulties in mathematics. Teachers should analyze whether or not a product or tools helps support the development of a big idea.
“Children with specific learning disabilities have normal IQ’s, but a blockage in the way they learn. They need every opportunity to try to make connections and generalizations and to get information into their heads” (Bahr & De Garcia, 2010).
Resources:
Bahr, D. L., & De Garcia, L. A. (2010). Elementary mathematics is anything but elementary.Belmont: Wadsworth, Cengage Learning.
Carter, N., Prater, M. A., & Dyches, T. T. (2009). Making accommodations and adaptations for students with mild to moderate disabilities. Upper Saddle River, New Jersey: Pearson Education, Inc.
Pictures (in order of appearance):
http://www.pinellas.k12.fl.us/Schools/regionV/images/32575FCF97BA4DCD8D6AEC93B5908AA9.jpg
http://cdnwrite.demandstudios.com/upload//4000/500/10/9/64519.jpg
http://www.skybluepink.com/images/counters.jpg
•Operation Meaning and Relationships
Lesson Plans:
http://www.uen.org/Lessonplan/LPview.cgi?core=2
A site that has various math lesson plans including several on shapes and solids.
http://www.instructorweb.com/basicskills/lessons/geosolids.asp
Teach about shapes through venn diagram and the world around you.
http://www.teachnology.com/teachers/lesson_plans/math/shapes/
http://www.lessonplanet.com/search_b?keywords=solid+shapes&rating=3
Find a lesson that is right for your class from these great selections of ideas.
Website Helps:
http://illuminations.nctm.org/ActivityDetail.aspx?id=70
Manipulate different shapes to learn more about number of faces, edges, and vertices.
http://www.compasslearningodyssey.com/sample_act/math3_4/MA3CA05apackage_preloader.swf
Practice visualizing 3D shapes through a simple game of hangman.
http://www.learner.org/interactives/geometry/index.html
A site that covers all sorts fo different aspects of shapes and solids.
http://gwydir.demon.co.uk/jo/solid/index.htm
A site that talks about all the different shapes as well as a game to help visualize the shape.
References:
Randall, C.I. (2005).Big ideas and understandings as the foundation for elementary and middle school mathematics. Journal of mathematics education leadership. 7,
3rd Grade Standards:
Represent numerical relationships as expressions, equations, and inequalities.
Order and compare whole numbers on a number line and use the inequality symbols <, >, ≠, and = when comparing whole numbers.
Represent numerical relationships as expressions, equations, and inequalities.
4th Grade Standards:
Objective 2
Use algebraic expressions, symbols, and properties of the operations to represent, simplify, and solve mathematical equations and inequalities.
5th Grade:
Use algebraic expressions, inequalities, or equations to represent and solve simple realworld problems.
6th Grade:
Describe simple relationships by creating and analyzing tables, equations, and expressions.
Write, interpret, and use mathematical expressions, equations, and formulas to represent and solve problems that correspond to given situations.
Properties of Equality
Properties of Equality
• If the same real number is added or subtracted to both sides of an equation, equality is maintained.
• If both sides of an equation are multiplied or divided by the same real number (not dividing by 0), equality is maintained.
• Two quantities equal to the same third quantity are equal to each other.
EQUATIONS & INEQUALITIES: Rules of arithmetic and algebra can be used together with notions of equivalence to transform equations and inequalities so solutions can be found.
Examples of Mathematical Understandings:
• A solution to an equation is a value of the unknown or unknowns that makes the equation true.
• Properties of equality and reversible operations can be used to generate equivalent equations and find solutions.
• Techniques for solving equations start by transforming the equation into an equivalent one.
• A solution or solutions to a linear or quadratic equation can be found in the table of ordered pairs or from the graph of the related function.
• Techniques for solving equations can be applied to solving inequalities, but the direction of the inequality sign needs to be considered when negative numbers are involved.
More Information:
An equation is simply an assertion that two expressions are related by equality.
In mathematics, an inequality is a statement about the relative size or order of two objects, or about whether they are the same or not.
An understanding of equalities and inequalities is so important. Students at a young age should be exposed to the importance of equality and the true meaning of an equal sign. Children can also be exposed to inequalities at a young age. It’s important for students to develop their own understanding of equalities and inequalities so they can apply future mathematics principles to these big ideas. A lot of mathematics rely on using equations and that is why it is such a big idea that needs to be used inside the classroom.
An equation is a sentence involving numbers, or symbols representing numbers where the verb is equals (=). There are various types of equations:
3+4=7 True Equation
3+4=9 False Equation
2x+5x=7x Identity Equation
x+4= 9 Conditional Equation
Equations are used to state the equality of two expressions.
Inequalities are demonstrated through the following signs:
In all these cases, a is not equal to b, hence, “inequality”.
These relations are known as strict inequality
Lesson Plans:
http://www.uen.org/Lessonplan/preview?LPid=11051 Commutative Cookie activity for 3rd graders where students create a commutative equation.
http://www.instructorweb.com/lesson/numbersequal.asp Lesson plan for kindergartners or 1st graders on understanding what the equal sign means.
http://www.uen.org/Lessonplan/preview?LPid=16330 Lesson Plan for 4th graders on understanding what is an equation and what is not an equation.
http://www.uen.org/Lessonplan/preview?LPid=16326 Lesson plan addressing different strategies to solve equations for 4th graders.
http://www.uen.org/Lessonplan/preview?LPid=6103 Lesson plan for 4th graders on symbols in inequalities.
http://www.uen.org/Lessonplan/preview?LPid=11207 Lesson plan for 4th graders on relationships between equations. Helps students understand equal values.
http://www.uen.org/core/lessonList.do?courseNum=5050&itemId=3022
A list of 5th grade lesson plans on equations.
Games:
http://www.mathplay.com/Equation/EquationGameOnline.html Matching equations with answers. For upper Level elementary grades.
http://www.mathplay.com/OneStepEquationGame.html Another equation game finding answers to equations for upper level grades.
http://education.jlab.org/sminequality/question.php?7466566 Speed math inequality game for 56 graders.
Big Idea #11: If two quantities vary proportionally, that relationship can be represented as a linear function.
Examples of Mathematical Understandings:
• A ratio is a multiplicative comparison of quantities.
• Ratios give the relative sizes of the quantities being compared, not necessarily the actual sizes.
• Ratios can be expressed as units by finding an equivalent ratio where the second term is one.
• A proportion is a relationship between relationships.
• If two quantities vary proportionally, the ratio of corresponding terms is constant.
• If two quantities vary proportionally, the constant ratio can be expressed in lowest terms (a composite unit) or as a unit amount; the constant ratio is the slope of the related linear function.
• There are several techniques for solving proportions (e.g., finding the unit amount, cross products).
• When you graph the terms of equal ratios as ordered pairs (first term, second term) and connect the points, the graph is a straight line.
• If two quantities vary proportionally, the quantities are either directly related (as one increases the other increases) or inversely related (as one increases the other decreases).
• Scale drawings involve similar figures, and corresponding parts of similar figures are proportional.
• In any circle, the ratio of the circumference to the diameter is always the same and is represented by the number pi.
• Rates can be related using proportions as can percents and probabilities (Randall, 2005).
A proportion is a name we give to a statement that two ratios are equal. It can be written in two ways:
· two equal fractions, a/b =^{ }c/d
or,
· using a colon, a:b = c:d
When two ratios are equal, then the cross products of the ratios are equal.
That is, for the proportion, a:b = c:d , a x d = b x c
Retrieved from http://www.math.com/school/subject1/lessons/S1U2L2GL.html
Proportional thinking is developed through activities involving comparing and determining the equivalence ratios and solving proportions in a variety of problembased contexts and situations without recourse to rules or formulas (Van de Walle, 2004).
Practical Applications:
Cooking is one way that ratios and proportionality can be applied to a reallife situation. As one ingredient is increased or decreased in a recipe the other ingredients must also be increased or decreased proportionally. Another practical activity is to enlarge or reduce images or photos on the computer. If they are not altered proportionally the image will become distorted. Graph paper or dot paper can be used to draw similar shapes or images in proportion to each other. Maps drawn to scale can be used to show ratios and how the distance on a map correlates to the actual distance. Students can recreate a map to a different scale
(Bahr, D. L., and de Garcia, L. A. 2010).
Informational Web Sites:
http://www.learner.org/interactives/dailymath/cooking.html
http://www.mathleague.com/help/ratio/ratio.htm#proportion
http://www.bbc.co.uk/skillswise/numbers/wholenumbers/ratioandproportion/ratio/
http://nrich.maths.org/public/viewer.php?obj_id=4825
Lesson Plan Web Sites:
http://www.eduplace.com/math/mathsteps/6/a/index.html
http://www.moneyinstructor.com/lesson/ratio.asp#LESSON_PRINTABLE_MATERIALS__WORKSHEETS
http://www.iit.edu/~smile/ma8809.html
http://www.themathpage.com/arith/ratioandproportion_1.htm
More Practice and Games:
http://www.quia.com/rr/35675.html
http://wps.ablongman.com/ab_vandewalle_math_6/54/13941/3569084.cw/content/index.html
http://math.rice.edu/~lanius/proportions/rate.html
References:
Bahr, D. L., and de Garcia, L. A. 2010. Elementary mathematics is anything but elementary: content and methods from a developmental perspective. Belmont, CA: Wadsworth.
Randall, Charles I. 2005. Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics. NCSM Journal (Spring/Summer 2005). Retrieved on April 18, 2009 from http://www4.uwm.edu/Org/mmp/PDFs/CharlesBig%20Ideas_NCSM_Spr05v7(3)p924.pdf
Van de Walle, J. A. 2004. Elementary and middle school mathematics: Teaching developmentally, 5^{th} e. Boston: Pearson.