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 FRACTIONS
EDUC – 4274 Mathematics Education Dan Jarvis Cherie Attrill, Elaine Chan, Jaime Kawa, and Paulla Leclair Primary/Junior Section 1 1/2 = 3/6 half, quarters, … 5 ¼ + ½ = Rationale: The need for fractions stemmed from the inadequacies of whole numbers in measurement and division (Adam, Ellis & Beeson, 1977). In measurement, to determine a distant between two points using only whole or “natural” numbers, a standard unit would be used. However, the distance may not match the exact multiple of the standard unit chosen. We could say that, for example, the measurement would be either 6 or 7 and a decision would have to be made as to what the best description would be. However, if we chose smaller units, the measured distance would be more accurate. Thus, fractions would be an important concept to incorporate into the solution to the measurement problem. In division, fractions are also used to solve division problems when “natural numbers” do not divide evenly (Adam, Ellis & Beeson, 1977). For example, 5 ÷ 2 could not be solved without the understanding and use of fractions. Other mathematical concepts that require an understanding of fractions include decimals and percents, as well as ratio and proportions (Van de Walle, 2007). Learning fractions is basically about understanding the relationship between whole and parts of a whole. It extends our concepts of number sense and numeration by examining numbers between 0 and 1 (Adam, Ellis & Beeson, 1977). Fractions are important because it forces students to look at numbers in a multiple of ways because it is not dealing with just whole “natural” numbers. Since, students primarily worked with whole numbers prior to learning about fractions, the concept of fractions actually interferes with early fraction development (Van De Walle, 2007). However, students come to school with some understanding of fractions, in an incidental way. They understand the idea of fair shares through separating quantity into two or more parts to give to friends and the importance of having equal shares. It is educators aim to enrich and enhance the meaning that, “fractions deal exclusively with fair shares” (Wiebe, 2004, p i). With a clear understanding of fair shares, students will be able to recognize fractional parts in the real world (Wieve, 2004, p i). In fact, “Fractions with halves, thirds, fourths, sixths, eighths, tenths and twelfths constitute more than 95% of those the average student will encounter in real life” (Wiebe A.,2004 pii). Therefore, it is essential that education allows students to make connections to the outside world through the use of manipulatives, pictures and objects so that students could solve problems in and out of school. The ultimate goal in mathematics is to equip students with the understanding, skills and strategies to solve problems in everyday life situations (Adams, Ellis & Beeson, 1977). As educators, it is our role to give students the tools they need to examine and solve mathematical problems in a variety of ways, and teaching fractions is among one of the answers as it offers students a solid foundation for them to learn other strategic concepts (e.g. decimals, percents, ratios and proportions to solve real life problems). Research: Fractions are often considered to be among one of the most difficult topics in mathematics for students to comprehend and do. However, with appropriate teaching strategies and the use of models and manipulatives, the difficulties that students typically encounter can easily be deciphered and resolved, thus making the learning process more meaningful and successful. Common conceptual difficulties that many students face with fractions in the primary/junior level include the inability to grasp the relationship between the two numbers, also known as the numerator and the denominator. As a result, this leads to inaccuracies when describing a fraction. For example, when two pizzas are presented, each divided into fourths, and 1/4 of the first pizza is pepperoni, students will have a tendency to describe it as 1/8 as opposed to 1/4 due to the eight parts that they see. To correct this, Marian Small suggests that “students should be encouraged to circle or highlight the whole unit and use language to reinforce the whole” (2008, p. 206). In this instance, 1/4 of one pizza is pepperoni. Similarly, when it comes to comparing two fractions, students will have a tendency to look solely at the numerators without taking into account the denominators or the wholes. To help students realize the importance of the denominator, fraction strips and visual representations of fractional parts can be introduced so that students could compare and visually see that 4/10 is not actually greater than 1/2 and ultimately, come to the realization that in order to make fair comparisons, a common denominator must exist (Small, 2008). Other common errors that students make include the inability to see the need for all parts to be equal in parts of a region or measure. An effective strategy to use is sharing tasks in the form of simple story problems so that students could apply their understanding of sharing to fractions (Van de Walle, 2007). Another important aspect that students often get confused with is the fact that sets can contain different items, especially if they have only worked with regions. To help remedy this problem, opportunities to work with parts of a set, in addition to the use of pattern blocks to teach both parts of a set and region will most likely help (Small, 2008). To elaborate, it could be explained to students that among the yellow hexagon, green triangle, red trapezoid, and blue diamond, the red trapezoid is 1/2 of the yellow hexagon (fraction of a region) but 1/4 of the set of 4 blocks (fraction of a set). In addition to the common errors mentioned above, there are several other misconceptions that students have toward fractions as well. One of them is the belief that all fractions are less than one. To clarify this, the teacher can plot a series of fractions on a number line to show that fractions can be greater than one, as well as provide opportunities for students to work with mixed and improper fractions in a wide range of scenarios (Small, 2008), such as Sam ate two and a half chocolate bars for breakfast. Second is the “misconception that a fraction always increases in value if the numerator and denominator are increased” (Small, 2008, 207). In this case, the best method is to use a counter‐
example in which students are to add different amounts to the numerator and denominator to show that the value of the fraction increasing does not always hold true (Small, 2008). Generally speaking, offering students a wide variety of pictures and physical representations of fractions will definitely make the learning experience more memorable and the content more comprehensible so that a solid foundation is built by the time students enter grade four when operations are taught. This is supported by Dr. Arthur J. Wiebe who stated: Many activities involving appropriate manipulatives and objects should be used to nurture understanding so that when operations with fractions are performed, students have a mental image of what is transpiring and what constitutes a realistic answer (Wiebe, 2004, p. ii). To summarize, allowing students the opportunity to experience, play, and investigate with fractions is vital to helping them learn. Also, the use of visuals and manipulatives, such as fractions pieces, pattern blocks, square tiles, egg cartons, Cuisenaire rods, money, counters, and fraction bars are all great resources that teachers can and should use when teaching fractions. Group Member Thoughts/Reflections: Cherie Attrill: I understand that the importance of teaching mathematics is to give students the tools and strategies and the understanding of concepts in order for them to solve problems. However, in school I strongly disliked problem solving because it was challenging for me to figure out which concept to use to solve the problem. If I was given the standard notational problem, such as ½ + 2/3, I would know how to manipulate the numbers to solve the problem, but I could not solve it using manipulatives. I would not know to replace the parts with smaller parts in order to simply add those parts together. But if I was just given the equation, I would know to multiple the denominators (bottom number) with a number in order to get the same whole, multiply the top number and then simply add the numerators together. Math was never a struggle for me in elementary school, except for problem solving. I could not find meaning in fractions, aside from the simple fractions, and therefore was unable to apply my knowledge to real life situations. I still struggle with some of the problems that are presented in class and in the text when using manipulatives, but with enough time to think it through in various ways, I can solve fraction problems with manipulatives. I believe that it is important to teach students fractions with manipulatives first in order to solve problems, instead of introducing the concept with standard notation. Manipulatives give students the skill of visualizing the fractions and solving problems using mental images, instead of relying on standard notations. If I do end up teaching fractions in my next placement, I will be sure to make it meaningful by using manipulatives, pictures and real life problems. I also found our activities to be interesting, heads‐on and engaging for students. I will most likely use a problem‐based activity with my students at my next placement. Elaine Chan: I personally feel that fractions are among one of the hardest topics to understand when I was a student as it was difficult for me to form a mental representation of a fraction in order to truly understand what that number meant and is in standard notation. Part of the reason may be because I have always associated fraction with the pie, which is fine when the fraction is dealing with a low denominator such as halves, thirds and quarters, but when I encounter a fraction with a high denominator, it is difficult for me to picture a pie cut into that many pieces and then to associate the numerator relative to this denominator/whole. Fortunately, as I was doing the research for this assignment, I was glad to discover that different visuals and manipulatives of various shapes and forms are encouraged to be used to supplement the teaching of fraction. This way, students could build a more solid foundation and understanding of fractions. More importantly, this hands‐on‐approach to teaching will be very beneficial for visual and kinesthetic learners, and will definitely make the learning process more fun and engaging for all. I also appreciate how fractional names are being introduced and taught in the early grades before standard fraction notations are. I believe that this will give students the opportunity to really integrate the terminologies into their vocabulary so that they could apply those terms in their everyday life so to see the relevance of fractions in the real world, thus making fractions more worthwhile to learn. Jaime Kawa: When we began researching fractions, I felt somewhat overwhelmed by the topic. While it is an important topic for students, it can sometimes be a daunting one, as many students may have difficulty with the concept of wholes and parts. For example, in our own group, we disagreed with the grade one activity. Some individuals felt that representing whole apples as fractions as opposed to cutting them would ultimately confuse students. This situation arose again during my group presentation, when an individual in the group I presented for had the same concern. However, we felt that it was important to represent the apples in this manner in order to show students that fractions can be represented in a variety of ways and are not limited to breaking one whole into several fractions. During my personal mathematics education, I usually struggled very little with proper fractions as was able to solve the equations relatively easily. However, as it had been many years since I had dealt with fractions in a formal setting, I felt daunted by this task and at times became frustrated when I did not understand something. This experience allowed me to relate to how my students might feel when I am teaching them fractions and has opened my eyes to the importance of having a variety of strategies that I can use to teach. It has also made me realize that when you are accommodating an activity for a few students, it can be easily done. Fractions are an important part of mathematics as they teach students how to see parts of a whole and are just one tool that can be used to compare numbers or to determine how to split wholes into equal parts. This tool can also be useful when comparing numbers as some decimals can be complex and difficult to understand whereas a fraction may simplify the equation. Paulla Leclair: Through all of the reading and research I have done for this assignment, what I found most interesting was the fact that when I was doing my practicum, teachers were still following the text book as if it were the golden rule; very seldom did I see any math games being used in order to grab the students’ attention and have them want to learn about math. There are so many resources available for every different style of learner, that it is hard to believe that schools do not insist that these are tapped into. Knowing that the schools have computers, and that there are math programs available for all of the students in the computer labs, but having the excuse of booking lab time or not having enough computers is one thing, but teachers not using the internet for assistance when it is so readily available is far beyond my realm of understanding. Yes, we have to stick to the curriculum; but it does not say anywhere in the curriculum that we have to rely heavily on any one particular book. If a school is using one set of texts, then using them as a guideline while making math fun is so very vital. Learning through play, as we have been demonstrating in all of the math presentations, makes a subject that children learn to fear out of lack of understanding into something that they can’t wait to do. Being able to show children that there are other things to separate into fractions that are not round, and making them understand what fractions are so that they can go from a set, to a whole, to a fraction and into a ratio and decimal is so very important to do in the six year step that the Curriculum requires so that we know that each child has developed a good base to build upon for the intermediate and senior years of school. Using the wide assortment of manipulatives to do this is as vital to me as is the need to make the learning of not only fractions, but the entire curriculum a fun thing to do for all styles of learners. Seeing the other presentation as well as the assortment of activities available for fraction learning has opened a whole new world of learning possibilities for the upcoming years for me and hopefully made me a better teacher. Developmental Analysis: Activity 1: Memory Game For our first activity we had a Memory Game for grade one students. As this game required the students to turn cards over and match them into sets, it was very important that they were able to meet the Kindergarten expectation of being able to “sort, and classify objects into sets according to specific characteristics, and describe those characteristics” (The Kindergarten Program). What this game would hopefully lead into is the ability to understand the concept of more and less, and what it would take to divide something equally within a group, which falls under the operational sense expectation of a grade two student. Activity 2: Pizza Play Dough Problems Knowing what was taught through the previous game, we then moved to the grade two level and presented the students with a play dough problem that allowed them to “determine, through investigation using concrete materials, the relationship between the number of fractional parts of a whole and the size of fractional parts” as is stated in the curriculum expectations (The Ontario Ministry of Education, 2005, p. 43). In order to introduce this activity, students must be able to describe parts of a whole using fractional names, which is expected of them in grade one. Through this activity, students will then be prepared to “divide whole objects and sets of objects into equal parts, and identify the parts using fractional names” as is listed under the Number Sense and Numeration strand in the grade three expectations (The Ontario Ministry of Education, 2005, p. 55). Activity 3: Word Play Our third activity was a word game designed for grade four students who would have by then been exposed to fractional names and have understanding of fractional parts. In this game, students will learn to write in standard fractional notations and come to understand that the denominator is the whole/sum of its fractional parts, and that the numerator is the number of parts within the whole. This activity will help students develop a solid foundation to writing fractions and in identifying numerators and denominators so that they could be prepared to compare and order fractions, and later, to add and subtract fractions. Activity 4: Rainbow Fractions: Make it Even! Our final activity was the most difficult of the four. It is mainly designed for grade five students as it requires them to “demonstrate and explain the concept of equivalent fractions, using concrete materials” (The Ontario Ministry of Education, 2005, p. 78). Without the previous years’ scaffolding and understanding of ordering the size of fractions, this activity would have been near impossible to play with enjoyment. This activity will ultimately lead and prepare students to the grade six study of Number Sense and Numeration in which students are to “represent, compare, and order fractional amounts with unlike denominators, including proper and improper fractions and mixed numbers, using a variety of tools” (The Ontario Ministry of Education, 2005, p. 88). Bibliography: Adams, S., Ellis, L. & Beeson, B. (1977). Teaching Mathematics with emphasis on the diagnostic approach. New York: Harper & Row Publisher. Small, M. (2008). Making math meaningful to Canadian students, K‐8. Toronto, ON: Nelson. Ontario Ministry of Education. (2005). The Ontario curriculum, grades 1‐8: Mathematics (Revised). Toronto, ON: Queen's Printer for Ontario. Van de Walle, J., & Folk, S. (2007). Elementary and middle school mathematics: Teaching developmentally (6th ed.). Toronto, ON: Pearson Education, Inc. Wiebe, A. (2004). Actions with Fractions. Fresno, CA: AIMS Education Foundation. Activity 1: Memory Game Grade Level: 1 Strand: Number Sense and Numeration – Quantity Relationships Specific Expectations: 1m19 Divide whole objects into parts and identify and describe, through investigation, equal‐sized parts of the whole, using fractional names (e.g. halves; fourths or quarters). Purpose of Activity: To have students develop an early association and recognition between fractional names and its visual representations. In addition, it will also help students understand how fractions and its vocabulary apply to everyday life. Materials: • A set of 16 or more cards per student, pair, or group • Each set of cards contain fraction names (e.g. whole, halves, thirds, fourths, etc.) and three different picture representations of that fraction name. Instructions: 1. Students can work individually or in pairs or small groups. 2. Each student, pair or group will receive 16 cards – four cards containing various fraction names (e.g. whole, halves, thirds, fourths/quarters) and three corresponding picture cards (e.g. apples, pizzas, chocolate bars). 3. Students will shuffle their cards and lay them face down on their desks creating a large square. 4. Students will turn any two cards up and try to match them according to the same book of fractions (e.g. match “halves” with a pizza cut in two, or match two picture representations of “halves”). 5. If the student successfully matches them, he/she will remove the cards from the table. 6. If the student was unsuccessful, then he/she must turn both cards over and try again. 7. If students are working in pairs or groups, they can collect the cards matched, and count them once they have completed the game to determine who collected the most cards. Accommodations: • For weaker students, a sheet of reference, in which the fractional names correspond to the visuals can be supplemented. Another option is to have both the fractional names and visuals present on every card. • For advanced/gifted students, fraction numbers as opposed to words can be used instead. Selecting more challenging fractions, such as fifths, sixths, and eighths is another option. This activity can also be adapted to match equivalent fractions (eg. Matching ¼ with 2/8). Also, this ‐
activity can be played like gold fish, in which students have to match two cards to form one whole (e.g. ¼ and ¾) Idea adapted from: http://letsplaymath.wordpress.com/2007/11/20/fraction‐model‐game Activity 2: Pizza Play Dough Problems Grade Level: 2 Strand: Number Sense and Numeration – Quantity Relationships Curriculum Expectations: 2m15 Determine, through investigation using concrete materials, the relationship between the number of fractional parts of a whole and the size of the fractional parts (e.g., a paper plate divided into fourths has larger parts than a paper plate divided into eighths) (Sample problem: Use paper squares to show which is bigger, one half of a square or one fourth of a square.). Purpose of Activity: To let students see for themselves that certain fractions are bigger than others and vice versa, in addition to having them be familiarize with fractional names. And since this is a hands on activity in which students create their own pizzas, it encourages participation and exploration. Materials: •
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Play dough of various colours Rollers Circular/rectangular cookie cutters Plastic knives Instructions: 1. Describe the following scenario to the students: “You and your friend are having a play day. For lunch, the two of you have decided to eat pizza, so mommy baked a small pizza for the two of you. Both of you wanted a fair share of the pizza, so it has been decided that the pizza be cut equally into halves.” 2. Give each student one chunk of play dough. 3. Ask the students to make the pizza using the roller and cookie cutter, and then cut it into halves using the plastic knives. 4. When completed, ask the students to set aside their pizza. 5. Inform the students: “Later that day, your three cousins came over, and the three of you have decided to eat pizza for dinner. So, mommy went to bake another small pizza. Now, including you, there are four people in total, and each person will receive an equal amount.” 6. Give each student another chunk of play dough. 7. Ask the students to make another pizza using the roller and cookie cutter, and then cut it into four equal parts using the plastic knives. 8. Discussion questions: •
Looking at the slices, one of which you received half of, and the other, a quarter, which pizza slice is bigger? How do you know? •
Accommodations: • Weaker students could be given more time to work out the problems or be paired up with learners of the same learning style and speed in order to allow the exploration without the frustration. • Some students may require you holding their hand while scoring the pizza into equal parts. • For ELL students, pictures describing the scenario can be supplemented. • For advanced/gifted students, they could create two different shape pizzas, so that they could come to the conclusion that it is not the shape (circle and rectangle, for example), but the set (pizzas) that matters. • If needed, this activity can be spread into several days. * Emphasize that the shape of the pizza is not that significant, rather it is the way in which it is cut. Activity 3: Word Play Grade Level: 4 Strand: Number Sense and Numeration – Quantity Relationships Curriculum Expectations: 4m17 Represent fractions using concrete materials, words, and standard fractional notation, and explain the meaning of the denominator as the number of the fractional parts of a whole or a set, and the numerator as the number of fractional parts being considered. Purpose of Activity: To help students understand that the denominator is the whole/sum of its fractional parts, and that the numerator is the number of parts within the whole. Also, as names could be used, students will find this activity to be meaningful and relevant to them as they are able to form a connection. Materials: • Paper with three columns: word, consonants, and vowels • Pencils • Erasers • Word List Instructions: 1. Provide a word list (this can be the spelling list created in Language), and a sheet of paper with three columns for the students. 2. In the first column, students are to print the words from the word list. 3. In the second column, students are to find the fraction of the vowels in the word. 4. In the third column, students are to find the fraction of consonants in the word. Note: Explain that the total number of letters in the word will be the denominator (the number of the fractional parts of a whole or of a set), and that the number that changes is called the ‘numerator’, which is the number of fractional parts being considered. For example: using the word “determine”, vowels would be 4/9 and consonants would be 5/9. 5. Students are then asked to identify the words that would fit certain statements (e.g. half of my letters are vowels, which word am I? – Answer: neighbour) Accommodations: • Some students may require assistance in understanding the concept of making words into fractions; they may be paired up with a more advanced student or receive one‐on‐one assistance from the teacher. • For ELL students, the vowels and consonants can be written in different colours so that students can just count the numbers to determine the numerator. •
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For advance/gifted students, they may be asked to find other words that have the same fraction as the first word. And if needed, a dictionary may be provided. To increase the level of difficulty or to introduce the idea of a set, students could be asked to identify the fraction of certain vowels within the total number of vowels in the word (e.g. In the word “neighbourhood”, 6/12 letters are vowels and of the vowels, 3/6 are “o’s”). Idea adapted from: www.superteacherworksheets.com/fraction‐cont.html Activity 4: Rainbow Fractions: Make it Even! Grade Level: 5 Strand: Number Sense and Numeration – Quantity Relationships Specific Expectations: 5m17 Demonstrate and explain the concept of equivalent fractions, using concrete materials (e.g., use fraction strips to show that); is equal to 3/4 9/12. Purpose of Activity: To enable students to visually see how a combination of fractions can add up to form another fraction (e.g. ½ + ¼ +1/4 = 1 whole) and conversely, how a fraction can be composed of smaller fractions. Materials: • Fraction strips • Containers (optional) Instructions: 1.
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Students will then form small groups (3 ‐ 4). Each student will receive a strip of a whole to begin with. The rest of the fraction strips are then placed in a pile/container. Taking turns, students are to draw one fraction strip out of the pile/container at a time, without peeking, and within the quickest amount of time, they are to form one whole. 5. If needed, students can trade strips with others. 6. The game then proceeds with students trying to form a half, quarter, and so forth. Accomodations: •
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Students with dexterity problems may need fraction strips that are attached to wooden blocks for easy handling. • For weaker students, a fraction strip sheet can be supplied so that they can refer to it when creating whole pieces. • For advance/gifted students, they s can arrange the strips from greatest to least or vice versa. Also, unnumbered fraction strips can be used, and students would have to label them afterwards. Idea adapted from: Making Sense of Fractions, C. Thornton and J Wells, 1995. p19 Websites: The following websites offer examples, definitions, and links if you would like to learn more about adding, subtracting, comparing, ordering, mixed, equivalent fractions, and more: • Math Forum: Fractions: Elementary Lessons and Materials: http://mathforum.org/paths/fractions/e.fraclessons.html •
AAA Math – Fractions: http://www.aaamath.com/fra.html#topic1
The websites listed below offer fun hands‐on activities dealing with fractions for a range of grade levels, some of which are submitted by educators: • Fractions Idea Bank: http://www.mathcats.com/grownupcats/ideabankfractions.html
• Ideas for Teaching Fractions: http://www.superteacherworksheets.com/fraction‐cont.html
This website provides links to a wide range of online fractions games in which students can play to help them recognize, compare, and relate fractions to decimals and percents: • Johnnie’s Math Page: Fractions Activities – Fun Math for Kids! http://jmathpage.com/JIMSFractionspage.html Blackline Masters of Tangram puzzles, circular fraction pieces, fraction strips, and fraction worksheets can be downloaded in PDF format from these websites: • Blackline Masters: http://wps.ablongman.com/ab_vandewalle_math_5/0,7959,796754‐,00.html
• Fraction Strips and Fraction Worksheets: http://www.superteacherworksheets.com/fractions.html
• Fractions, Fraction Worksheets, Fraction Help: http://math.about.com/od/fractionsrounding1/Fraction_Worksheets_and_Rounding_Numbers.
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