B1 - Number Sense

Fractions, Decimals, and Percents

B1.3

represent equivalent fractions from halves to twelfths, including improper fractions and mixed numbers, using appropriate tools, in various contexts

B1.4

compare and order fractions from halves to twelfths, including improper fractions and mixed numbers, in various contexts

B1.5

read, represent, compare, and order decimal numbers up to hundredths, in various contexts

"We believe that students need to be given time to

understand what fractions are about (rather than moving quickly to computation) and that the ultimate goal should be to develop students who can reason proportionally (Clarke et al., 2008, p. 374).

B1.4

read, represent, compare, and order decimal numbers up to thousandths, in various contexts

B1.3

compare and order integers, decimal numbers, and fractions, separately and in combination, in various context

Learning Experience: Guess Who? Fractions, Decimals and Percents Edition

This is an engaging educational game tailored for sixth-grade students, designed to reinforce their understanding of fractions, decimals, and percentages in a fun and interactive way. Drawing inspiration from the classic game "Guess Who," players are presented with a board/mat featuring various fractions, decimals, and percentages represented visually and numerically. Through a series of strategic questions, players must deduce their opponent's hidden fraction, decimal, or percentage, narrowing down the options with each turn. With its blend of deductive reasoning and mathematical concepts, this game promotes critical thinking skills while reinforcing key mathematical principles.

What the Research Says:

"Two findings emerge consistently from these studies of teachers' knowledge and patterns of reasoning. One is that making mathematics fun and engaging is the central concern for many beginning and experienced teachers. Assuming that mathematics is not interesting to most students, they think that their role is to find ways to correct for that. In their study of eight prospective middle school teachers, for example, Borko et al. (in press) found that making mathematics class fun was central to these teachers' pedagogical reasoning. These researchers report uncovering a "pervasive belief' among the prospective teachers they studied that mathematics is inherently boring and hard to learn. In search of games that would

lighten the load for students, the prospective teachers justified their choices

most often in terms of how they would motivate or engage students rather

than on the basis of concerns for the mathematical content. (Bell , 1993, p. 187)

"Many middle school students, when given a problem to solve involving fractions, will choose to convert it to decimals or percents to make sense of it. This flexible thinking is to be encouraged, as percentages particularly seem to make sense to many students intuitively... Being able to translate among different representations may enable the students to get a clearer understanding of the different constructs of rational numbers." (Clarke et al., 2008 p. 377)

Grade 4

B1 - Number Sense

B1.4 represent fractions from halves to tenths using drawings, tools, and standard fraction notations and explain the meanings of the denominator and the numerator

B1.7 read, represent, compare, and order decimal tenths, in various contexts

B1.9 describe relationships and show equivalences among fractions and decimal tenths, in various contexts

(depending on the fraction cards you choose, this game can be adapted to any grade)

Learning Experience: Fraction Card Game (Which is Larger?)

In this learning experience, students will play with a partner using a deck of fraction cards. Within this deck, there will be different fraction representations shown on each card. The cards can include fraction strips, geometric models, word forms, decimal forms, or standard forms. Students will flip cards simultaneously, and whichever card has a higher number will take both cards; once the whole deck is finished, the game is over. In this activity, students are engaging in different representations of fractions when they look at the variation in cards and will need to understand the relationship between the two cards to determine which one is larger. 

What the Research Says:

“Fruitful representational contexts are framed clearly enough to facilitate the development of sound mathematical understandings and skill in students. Fraction bars, pie diagrams, number lines-all these can help to focus learners on certain key features of fractions, such as the meanings of fractional terms.” (Ball, 1993, p. 164).

“Much of the confusion in teaching and learning fractions appears to arise from the many different interpretations (constructs), representations (models), and coding conventions (5/4, 1 1/4, 1.25, 125 percent).” (Clarke et al., 2008, p.373).

“If students are to become flexible in moving between different constructs, they need to be familiar with different representations (and manipulative), as each model differs in its ability to reflect each construct or concept under investigation.” (Clarke et al., 2008, p.375).

Grade 5 - Number Sense

B1.3 represent equivalent fractions from halves to twelfths, including improper fractions and mixed numbers, using appropriate tools, in various contexts

B1.4 compare and order fractions from halves to twelfths, including improper fractions and mixed numbers, in various contexts 

B1.5 read, represent, compare, and order decimal numbers up to hundredths, in various contexts

B1.7 describe relationships and show equivalences among fractions, decimal numbers up to hundredths, and whole number percents, using appropriate tools and drawings, in various contexts

Learning Experience - Tug-of-War: Fraction, Decimal and Percent 

Designed to be played like the card game War, this interactive game helps students practice comparing decimals, fractions, and percents. Students play in pairs or small groups. Each student has their own deck of cards face down. They both flip the top card and must determine which player has the largest value. The player with the largest value wins the round. Players repeat until the deck of cards is complete.

What the Research Says

“Link fractions, decimals, and percents wherever possible [ to encourage] flexible thinking. A number of researchers believe that decimals and percentages should be introduced far earlier [i.e., before middle school] than many teachers typically do (see, e.g., Moss and Case 1999).” (Clarke et al., 2008, p. 377).

“Link fractions to key benchmarks, and encourage estimation, [for instance] asking them to decide, for each pair, which fraction was larger and why. [...] Students were required to do the comparison in their heads. [Students used two strategies: benchmarking and residual thinking] [...] Benchmarking: students compare the size of fractions with 0, 1/2, or 1. Residual Thinking: students refer to the amount required to build up to the whole. We believe that when students share their strategies for comparing fractions during class, other students may be convinced to use both benchmarking and residual thinking to tackle relative-size and ordering problems [because these strategies] are often more efficient and focus on meaning in their application [and] prepare[s] students for real-life encounters with fractions, where mental estimation is the key skill.” (Clarke et al., 2008, p. 376)

“The teacher must structure what Nesher called a "learning system" - in which learners can explore and test mathematical ideas. Nesher's framework reminds us that the representation

of ideas is more than just a catalog of ideas or a series of models - rather it is interactive and takes place within a larger context of ideas, individuals, and their discourse.” (Ball, 1993, p. 160).

“B1.6 count to 10 by halves, thirds, fourths, fifths, sixths, eighths, and tenths, with and without the use of tools” (MOE, 2020)

“B1.9 describe relationships and show equivalences among fractions and decimal tenths, in various contexts” (MOE, 2020)

Learning Experience: Fraction Marathon

https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Fraction-Game/

In this learning experience, students work on identifying fractions and equivalent fractions while being able to see them represented visually. The aim of the game is to try and get all cursors or pieces to the far right side of the number line in as few moves as possible. The students will start by flipping over a card from the deck, identifying the fraction, and deciding which number line and cursor to use to represent that fraction. Students can choose between 7 number lines, each representing different denominators and can choose to represent the fraction shown on the card or one that is equivalent in order to try and get all the cursors over to the right side by flipping over the least amount of cards. This learning experience can be played virtually using the simulation on NCTM’s website or the educator can make a physical version. Additionally, this game can be played individually or in a pair or group. This learning activity supports the curriculum expectations above as students practice using tools (like a number line) to represent fractions and count by different denominations (like halves, thirds, etc.). Additionally, students get to practice identifying fraction equivalences using this number line tool. These equivalences can be compared by looking both at the number line representation and the fraction numerators and denominators.

What the Research Says:

“The power of mathematics lies in part in its capacity to represent important relationships and

patterns in ways that enable the knower to generalize, abstract, analyze, understand. Learning to represent is therefore a goal of mathematics instruction, not just a means to an end. ” (Thomas et al., 1993, pp.163).

• The learning activity above allows students to explore representing fractions in different ways to complement representations they may already be familiar with (ex. with shapes or numerical representations).

“We believe that students need to be given time to understand what fractions are able (rather than moving quickly to computation) and that the ultimate goal should be to develop students who can reason proportionally” (Clarke, Roche, & Mitchell, 2008, pp.374).

• The games allow students to explore fraction relationships and develop an understanding of what equivalent fractions are and how they look in proportion to one another (i.e., ⅓ looks smaller than ½).

References

Clarke, D. M., Roche, A., & Mitchell, A. (2008). Ten Practical Tips for Making Fractions Come Alive and Make Sense. Mathematics Teaching in the Middle School, 13(7), 372–380

Halves, Pieces, and Twoths: Constructing and Using Representational Contexts in Teaching Fractions: Deborah Loewenberg Ball. (1993). In Rational Numbers (pp. 168–206). Routledge.

Ministry of Education (2020). Curriculum and Resources: Grade 4 Mathematics. Ontario. https://www.dcp.edu.gov.on.ca/en/curriculum/elementary-mathematics/grades/g4-math    

Grade 5 (B1- Number Sense)

B1.4 Compare and order fractions from halves to twelfths, including improper fractions and mixed numbers, in various contexts

B1.5 Read, represent, compare and order decimal numbers up to hundredths, in various contexts

B1.7 Describe relationships and show equivalences among fractions, decimal numbers up to hundredths, and whole number percents, using appropriate tools and drawings, in various contexts

Learning Experience: Equivalency Scavenger Hunt!

In this learning experience the class works together to find puzzle pieces that have been hidden around the classroom. Every puzzle piece will contain an image with either a written fraction, decimal, percentage, number line, or geometric models of fractions. After locating all the pieces, students will collaborate together to match all the equivalent pieces. Each puzzle will be made up of four pieces, as the example above shows.

For an additional challenge students could then arrange the puzzle pieces from smallest to largest based on the values they display. A fun extension for this activity could be students getting to make their own puzzle demonstrating equivalency, and their pieces could be used for the next scavenger hunt.

What the Research Says 

“Many middle school students, when given a problem to solve involving fractions, will choose to convert it to decimals or percents to make sense of it.This flexible thinking is to be encouraged, as percentages particularly seem to make sense to many students intuitively.” (Clarke et al., 2008, p. 377). 

Fruitful representational contexts are framed clearly enough to facilitate the development of sound mathematical understandings and skill in students. Fraction bars, pie diagrams, number lines-all these can help to focus learners on certain key features of fractions, such as the meanings of fractional terms. At the same time, the context is sufficiently open to afford students opportunities to explore-to make conjectures and follow important mathematical tangents. (Ball, 1993, p. 164). 

Two findings emerge consistently from these studies of teachers' knowledge and patterns of reasoning. One is that making mathematics fun and engaging is the central concern for many beginning and experienced teachers. (Ball, 1993, p. 187). 

• This activity allows students to enhance their understanding of the relationships between fractions, decimals and percentages in a fun way. It allows students to engage in discussions about equivalences with one another, a process that helps develop critical thinking and collaboration skills.

• The extension of having students create their own equivalency puzzle pieces encourages students to investigate their understanding of fractions in a way that they can make connections to their own lives and interests. 

Grade 4 (B1 - Number Sense)

B1.4 - represent fractions from halves to tenths using drawings, tools, and standard fractional notation, and explain the meanings of the denominator and the numerator.

B1.6 - count to 10 by halves, thirds, fourths, fifths, sixths, eights, and tenths, with and without the use of tools.

Learning Experience - Comparing Fractions

In this learning experience, students are prompted to look at the fractions (represented numerically and visually) and use a sign to make the sentence true. Students go through various levels and compete against themselves. Competing against yourself provides a less threatening approach to fractions. When I was playing this game, it took me a while to understand how 1/3 is bigger than 1/9, however having the visuals helped in my understanding. By level 4, I was zooming by the questions and felt empowered (as cheesy as that sounds). Sometimes the visual representation of the fractions are represented using different colours, making it a bit more challenging to quickly get the answer, however analyzing it helps with the understanding of fractions. This will help students with the representation of fractions, as outlined in the math curriculum for grade 4.

https://on.mathgames.com/skill/4.68-compare-fractions-same-numerator-or-denominator

What research says

"In order to help students develop mathematical understanding and power, the teacher must select and construct models, examples, stories, illustrations, and problems that can foster students' mathematical development" (Ball, 1993, p. 159)

"Fruitful representational contexts are framed clearly enough to facilitate the development of sound mathematical understandings and skill in students. Fraction bars, pie diagrams, number lines - all these can help to focus learners on certain key features of fractions, such as the meanings of fractional terms" (Ball, 1993, p. 164)

"We believe that students need to be given time to understand what fractions are about (rather than moving quickly to computation) and that the ultimate goal should be to develop students who can reason proportionally" (Clarke et al., 2008, p. 374)

"Overall, if students are to become flexible in moving between different constructs, they need to be familiar with different representations (and manipulative), as each model differs in its ability to reflect each construct or concept under investigation" (Clarke et al., 2008, p. 374)

Fractions and Decimals

B1.7 read, represent, compare, and order decimal tenths, in various contexts

B1.9 describe relationships and show equivalences among fractions and decimal tenths, in various contexts

Learning Experience: Decimal Shopping Challenge 

This fun activity is called Decimal Shopping Challenge! You are given an item found in the grocery store (ex- 2 liters of milk) and are given coins and bills at the bottom of the page and you must select the correct combination of coins and bills that adds up to the cost of the item. The game starts out quite easy and then progressively gets harder. Fractions are included as well in this game. For example, a quarter is listed as .25 and you must select the correct fraction (¼) that corresponds with the decimal in order to get a point. At the higher levels of the game, you have to add up multiple items. There is also an option to add a timer meaning you need to complete this task in under a certain amount of time. If you do not complete the task on time, you don’t get the point and drop down to the previous level.

What the Research Says

“We believe that students need to be given time to understand what fractions are about (rather than moving quickly to computation) and that the ultimate goal should be to develop students who can reason proportionally.” (Clarke et al., 2008, p. 374).

The Decimal Shopping Challenge gives students the time to understand what fractions are about and presents them as decimals as well which ultimately helps them reason proportionally.

Link fractions, decimals, and percents wherever possible “Many middle school students, when given a problem to solve involving fractions, will choose to convert it to decimals or percents to make sense of it. This flexible thinking is to be en- couraged, as percentages particularly seem to make sense to many students intuitively. A number of researchers believe that decimals and percentages should be introduced far earlier than many teachers typically do.” (Clarke et al., 2008, p. 377).

Our game is aimed for students in elementary school but can of course be played by students in middle school. Our game involves converting fractions into decimals and vice versa which encourages flexible thinking and shows student the different ways numbers can be represented.

“In order to help students develop mathematical understanding and power, the teacher must select and construct models, examples, stories, illustrations, and problems that can foster students' mathematical development. Lampert (1989) wrote of the need to select a representational domain with which the children are familiar and in which they are competent to make sense - in other words, a domain in which they can extend and develop their understandings of the ideas, as well as their capacity to reason with and about those ideas. For instance, because students are familiar with relationships among pennies, dimes, and dollars, and because they are comfortable with the notation, Lampert argued that money may provide
one helpful terrain in which they can extend their understanding of decimal numeration.” (Ball, 1993, p.160).

Decimal Shopping Challenge does just this. We incorporate money as children are familiar with it and have seen it be used in various contexts before. As this quote states, money allows children to extend their understanding of decimal numeration.

Grade 6 (Number Sense)

B1.3

compare and order integers, decimal numbers, and fractions, separately and in combination, in various contexts

Learning Experience: Fraction War

Materials: Deck of playing cards, pencils

Students will evenly split a deck of cards between two partners. Each player draws two cards at a time (simultaneously) per round. Each person places one card above their pencil and one below. Comparing the fractions, students must determine who has the higher fraction. The person with the higher fraction gets to keep all four cards. If students recognize that their cards create equivalent fractions, they go into a ‘fraction war’. At this point, each player draws two new cards to create two new fractions. Again, the player with the higher fraction keeps all the cards. The person with the most cards wins the game; however, the goal is that students are able to quickly compare fractions and determine their value (higher or lower).

What the research says:

"When students are first trying to make sense of common fractions,

teachers have typically defined a fraction as follows:

The denominator tells you how

many parts the whole has been

broken up into, and the numerator

tells you how many of these parts

to take, count, or shade in.

This explanation works reasonably

well for fractions between 0 and 1 but

not for improper fractions, which are

fractions greater than 1. We prefer

this explanation for students:

In the fraction a/b, b is the name or size of

the part (e.g., fifths have this name

because 5 equal parts can fill a whole),

and a is the number of parts ofthat

name or size. If we have 7/3, the 3

tells the name or size of the parts

(thirds) and the 7 tells us that we have

7 of those thirds (or 2 1/3).

We believe that this new rule may

help students use more appropriate

language when labeling fractions. For

example, we have noticed that some

students refer to three-quarters as

"three-fours" and "four-threes." This

use of whole-number rather than

fractional language appeared to be an

indicator that the students do not yet

understand which digit refers to the

number of parts or the size of the parts" (Clarke et al., 2008 p. 375)

• If students know what the denominator and numerator truly stand for, they will accurately determine the larger fraction, regardless of whether the fraction is proper or improper.

  
  

“ “Marta ate 2/4 of a sandwich at noon and 2/4 of a sandwich after school. How much did she eat?" Students might be able to discuss that she ate the equivalent of one whole sandwich or four-fourths of a sandwich. They also could discuss the notion that she has eaten 4/8 of two sandwiches - and thereby reach some agreement on the importance of identifying the unit - and of choosing a useful unit.” (Ball, 1993, p.166)

• In relation to my last point, if students have an accurate understanding of what the numerator and denominator stand for, (while also having a mutual understanding of what the unit is when comparing fractions), students can productively discuss why or why not a fraction may be equivalent/ higher.
  

B1.7 Fractions and Decimals read, represent, compare, and order decimal tenths, in various contexts 

B1.9 Fractions and Decimals describe relationships and show equivalences among fractions and decimal tenths, in various contexts 

Learning Experience - Fractional Equivalent Bingo

This activity allows students to work in small or large groups and play bingo to work on their ability to compare and find equivalences in fractions and decimal tenths. Students take turns rolling a fraction die. They role the die and get a fraction and they see if they can find an equivalent fraction or decimal on the bingo board. If they can, they mark it on the board then the next person goes. If they cannot find an equivalent number then the next person goes. This activity can be modified to include fractions, decimals, and pictures of fractions to allow students to see the numbers represented in various ways.

What the research says

According to Clarke et al. (2008, p 375) one of the ten practical tips is "Take opportunities early to focus on improper fractions and equivalence". This was described in the context of having students use number lines to examine improper fractions. Clarke et al. (2008 p 337) also stated educators should "Link fractions, decimals, and percents wherever possible. Many middle school students, when given a problem to solve involving fractions, will choose to convert it to decimals or percents to make sense of it. This flexible thinking is to be en-couraged, as percentages particularly seem to make sense to many students intuitively". Therefore, it is important to include both decimals and fractions on the bingo board.

"One is that making mathematics fun and

engaging is the central concern for many beginning and experienced teachers. Assuming that mathematics is not interesting to most students, they think that their role is to find ways to correct for that. In their study of eight prospective middle school teachers, for example, Borko et al. (in press) found that making mathematics class fun was central to these teachers' pedagogical reasoning." (Ball, 1993, p 187).

Grade 5: Number Sense

B1.3

represent equivalent fractions from halves to twelfths, including improper fractions and mixed numbers, using appropriate tools, in various contexts

Learning Experience: Teachers shuffle and distribute the cards amongst students. Each student can have one or more cards, all game cards need to be distributed. Teachers assign a player that goes first (or teachers start with their card). That student or the teacher reads the “Who Has” part (example “Who has 3/4”). Other students check the shape on on their cards and looks at the portion of the shape that is coloured in. The student who has three quarters of the shape coloured on their card says “I have three quarters”. They continue the game by reading the “Who Has” part on their card. The game continues until the student who started the game reads their card.

What the research says:

2. Develop a generalizable rule for explaining the numerator and denominator of a fraction. When students are first trying to make sense of common fractions, teachers have typically defined a fraction as follows: The denominator tells you how many parts the whole has been broken up into, and the numerator tells you how many of these parts to take, count, or shade in. We prefer this explanation for students: In the fraction a/b, b is the name or size of the part (e.g., fifths have this name because 5 equal parts can fill a whole), and a is the number of parts of that name or size. If we have 7/3, the 3 tells the name or size of the parts (thirds) and the 7 tells us that we have 7 of those thirds (or 2 1/3). We believe that this new rule may help students use more appropriate language when labeling fractions. For example, we have noticed that some students refer to three-quarters as "three-fours" and "four-threes." This use of whole-number rather than fractional language appeared to be an indicator that the students do not yet understand which digit refers to the number of parts or the size of the part (Clarke et al., 2008)

• In this game, students identify the fractional part shaded on the card by looking at the denominator (e.g., thirds, fourths) and the numerator (e.g., three, four).

Overall, if students are to become flexible in moving between different constructs, they need to be familiar with different representations (and manipulatives), as each model differs in its ability to reflect each construct or concept under investigation. Sowder (1988) noted that when it comes to fractions, students are "model poor," with many viewing a circular region as the only model of a fraction (Clarke et al., 2008)

• This quote emphasizes the value of using different representations of fractions. This activity uses different shapes to represent fractions, which can help students develop a more well-rounded understanding of this concept.

B1.9

describe relationships and show equivalences among fractions and decimal tenths, in various contexts

Learning Experience

• i was lucky enough to do a bit of work with fractions for the Grade 4 class i had in my 2nd placement! One of the ways that I tried to encourage an understanding of fractions was to have the kids in small groups match equivalent fractions together.

• here are the slides that I used! pretty basic i know lol

• to kind of level/scaffold depending on the understanding that the students might have and to afford more choice i also put it in numbered form (e.g. 1/4 or 1/6) and also as as fraction circles bc i felt that they would both show the same kinds of understanding.

• i also used the fraction strips website as a little way for the children to understand the whole part fraction connection - they could play with it after finishing the other work they were doing just to explore it a little bit! https://toytheater.com/fraction-strips/

• later on i tried to include decimals, showing how both can represent the same thing by making use of the spideman pointing meme - https://docs.google.com/presentation/d/18H8i_oM0nbUQTYXhP32frzPM0moBXeJnx_PlMVgTV6o/edit?usp=sharing

• i guess this isn't like one learning experience, but here were a few activities i used to try and stimulate the minds of the learners i had the opportunity to be with!

Reading Connections

• it was actually really interesting working backwards, seeing what i did in the past and seeing if it aligned with research, i think it's given me a bit more food for thought in the ways that I do it next time but still thinking on that haha

Clark et al., 2008

• this reading spoke about how it's important to really give time for the kids to gain understanding in what they were learning rather than rushing through proportional reasoning, I'm hopeful that I was able to do this, it is tough though when there are other curriculum areas that you need to get to.

• I also believe that in these lessons i was able to give some choice in using different manipulatives, which is something that the reading spoke about, though i'm more curious now about how to approach this differently in the future, having equivalent fractions that look different from one another, or having the children generate their own models, still thinking a little bit about this.

• i also think hopefully i have provided the kids a way to combine their understandings around decimals and fractions through the lessons i taught as this is also noted to encourage flexible thinking!

Carpenter et al., 1993

• This article also i believe spoke on the importance of using multiple models to engage students and encourage flexible thinking, I realyl wanted to also have a bit of fun with fractions with the children too, which is why i used the pointing spideman meme, and i wanted to provided different ways of engaging that allows the chidlren to learn in a manner most comfortable to them, using the different manipulatives and ways of expressing fractions. Still definitely have a lot to learn here too

This game allows students to individually and collaboratively practice working with relationships among fractions and ways of combining fractions. Students flip a card over and move the markers over depending on the fraction on the card they flipped.

Curriculum Expectation:

Grade 4: B1.6 count to 10 by halves, thirds, fourths, fifths, sixths, eighths, and tenths, with and without the use of tools

What the research says:

Emphasize that fractions are numbers, making extensive use of number lines in representing fractions and decimals. Kilpatrick, Swafford, and Findell (2001) commented that the fact that rational numbers are numbers is so fundamental that it is easily overlooked. Using a number line has many advantages. It helps students see how whole numbers, fractions, and decimals relate; it provides a way of understand- ing why 5/3 is the same as 1 2/3 and that 6/3 is the same as 2, and it makes it easier for students to understand the notion of the density of rational numbers (i.e., that between any two distinct fractions or decimals, there is an infinite number of fractions and decimals.

Provide a variety of models to represent fractions. A range of manipulatives and other tools have been employed during teaching experiments (Post, Wachsmuth, Lesh, and Behr 1985; Steencken and Maher 2002), such as fraction bars, Cuisenaire rods, paper folding, laminated shapes, and computer programs.

Further evidence for the importance of using multiple representations of fractions: