Fractions for Middle Years

Rational Numbers for Middle Years

Click here for pictures taken during the Middle Years Fraction workshop.

These rational number activities have been designed to meet a number of the rational number learning objectives in the Middle Years Mathematics Curriculum.

In designing these activities I was working with three other overarching and guiding principles: Multiple Intellgencies, Differentiated Instruction, and Technology Integration. I invented an acronym for this: MIDITI, but if we add middle years to the mix we can have MYMIDITI or MY(squared) DITI; then if we add rational numbers (RN) we get MY (squared) DITIRN--just a thought--play with it if you want--maybe we're on to something and can make some money here.

Before I brought into play any of the MIDITI stuff I first examined the curriculum to determine what I would like the students to know at the end of a unit of study on rational numbers.

Many concepts are 'contained' within the term "Rational Numbers." We have rate, percent, decimals, integers, and of course our common fractions. I have decided for this workshop to focus on common fractions. Within common fractions we can study ratio, part-whole, quotient, operator. The most frequently addressed area within common fractions is the part-whole set of concepts. Within part-whole we need to examine the region model, the part-of-a-set model, and the measurement model.

Let's visit the online curriculum guide and see where we're going with fractions!! Click on Numbers and Operations and go to N-42.

I decided to focus today on a grouping of concepts that I see as related and have designed activities that more or less address all of the following concepts simultaneously.

N-42 (a) and (b); N-45 (a) and (c); N-46 (a), (b) , and (c); N-47 (b); N-48; N-49; N-50; N-51; N-53; N-55; N-56 (a); N-57 (a).

You can see that we have addressed a number of concepts that are related and that can be designed into activities, such that almost all of these concepts are embedded (by design) into all of the activities. That being said, however, a teacher would have to know that all of these concepts are built into the activities and would need to be able to sort them out for assessment purposes through observing the performace of the students, unless of course we get really creative and design a new assessment tool that assesses holistically and conceptually--wouldn't that be arevolutionary idea!!!

I think that many teachers teach mathematics according to very isolated specific concepts--especially in fractions. We may decide, for example, that we are going to teach the concept of addition and subtraction of fractions with like denominators using manipulatives and pictures. We may find or design an activity or activities that address this concept. The children would work on this for a few days, we would assess their competence and then we would move on to the next concept.

I think that many teachers do not see the relationships between the concepts, teach each concept as a separate entity, assess it as an isolated curriculum component, and in so doing we pass on to our learners the seeming unconnectedness of the concepts. If we can connect the concepts for ourselves mentally--if we can SEE the relatedness of the concepts, then we can design a "road map" (an idea developed by Diane Hanson, the former Catholic Board math consultant) through the fraction mine field. This road map would group together 'like' concepts and embed these concepts into meaningful activities such that our learners can see the relatedness of the concepts. It's much more meaningful for the learners, it's much more fun to learn in this conceptually integrated way, it's easier for design and planning purposes, and I think it's also easier to assess.

The activities you will participate in today have been designed for today's workshop, drawn from existing resources, or are avilable on the Internet. I also have software that addresses the learning of fractions and have this software installed on my laptop for you to explore.

What is crucial for me is that the mathematics is substantive and visible to the learners. The resources, whether they be samples of children's literature, a virtual learning experience, a game, or manipulatives, they are simply there as tools to enable learning. If the tool, or the use of the tool, moves to the forefront and takes precedence over the mathematics then in my mind it has failed as a tool.

The activities you will work on today employ different material or perceptual bases. The fractional concepts are multiple but related. What you will go through today would not be where I would begin with children. I would start more basic, perhaps even having a whole class teacher-directed lesson, but in that environment I would still give children a hands-on experience, perhaps beginning with one single concept and then moving quickly to related ones.

You will be exposed today to a variety of different resources that can be used in rational number classroom learning stations. Children can rotate through these stations at regular intervals, or they can move "in their own time" from station to station. These stations would normally occur at the end of a unit on fractions (as a performance assessment opportunity) as the activities and materials vary from station to station. As teacher, you can observe, ask questions, mediate learning, work with the children, encourage communication of ideas, encourage sharing, take notes of how the children are "making sense" based on what you see them do and what you hear them say--to you and to each other.

There are many websites that include fraction activities/ideas/lesson plans. Some sites are like workbooks where all a student would do is insert the one right anwser. Other sites offer a more open-ended environment--a place to create. I have pulled together a few sites that I think are useful for children to use in the classroom as environments for fraction learning.

During and/or after each activity discuss the following with your partner:

What rational number concept(s) is/are being addressed in this activity?
Are there other manipulatives and/or other learning environments which would address this concept better?
What did I learn today, either about the fraction content, or about resources/activities to use to enable children to learn fractions -- or both?
Write some notes to remind yourself of the above points (lines provided after each activity)

We cannot suddenly decide we are going to use pattern blocks to teach fractions if (1) we have not played with these blocks ourselves and understand their potential, and (2) if we have not provided time for children to play without any interference of teacher direction (other than to play).

I find in my travels that some teachers desperately want to use, for example, manipulatives but have not had sufficient or appropriate inservice in how they can be used effectively. On the other hand there are some teachers who really would rather NOT use them because they prefer other resources, do not see the potential, have no idea how to use them, or have a very rigid view of the potential of these materials.

This workshop is about fractions and it includes a variety of different resources that can be used to teach fractions. If you like, for example, pattern blocks and can see how you would structure a learning environment using pattern blocks to enable fraction learning then try it. Likewise with any other material you will work with today. For example, if in your school you only had pattern blocks available to you as a resource to work with, then you could become very creative in how you would use pattern blocks for some of the following activities. There are multiple ways that each of the following material bases can be used to teach fractions and there are multiple ways that each fraction concept can be embedded in an activity.

VARIABILITY

Zoltan Dienes, many years ago outlined two major ideas that have stuck with me: Perceptual Variability and Mathematical Variability. The following table illustrates a possible way to introduce and then build on concepts. We'll pick fractions.

Perceptual Variability

Mathematical Variability

Teaching Strategy

one variable (e.g., pattern blocks) one concept--equivalence teacher-directed/whole class--everyone doing the same thing/hands-on; experiential

two variables (e.g., pattern blocks; geoboards) one concept--equivalence teacher works with one group; other group works independently

multiple variables (e.g., pattern blocks, play-doh, paper coverings) one concept--equivalence interactive--indirect teaching; small groups; stations

one variable (e.g., paper coverings) two or more concepts (e.g., equivalence; comparing fractions, addition of fractions) teacher-directed/whole class--everyone using the same material--perhaps going tthrough each concept with teacher and/or working in groups with task cards to explore the different concepts

multiple variables (e.g., paper coverings, games, software, children's literature, play-doh, geoboards, cuisenaire, etc) multiple concepts (e.g., equivalence, area concept, part of a set concept, measurement concept, comparing, addition and subtraction, etc) small groups working at stations--rotating; this is a useful strategy at the end of a unit and can be effective as performance stations

I advise my preservice teachers that they start with one perceptual variable and one mathematical variable and then build each as they and the students they are teaching are comfortable with both the materials and the mathematics.

I have seen many teachers use stations in mathematics and I have seen children do very interesting mathematics, but I have also observed that some teachers have difficulty assessing the mathematics that is occurring. If totally unrelated concepts are being worked on then I think it would be a nightmare for assessment, but if the concepts are truly related then the children can be doing one activity working on what may appear as one concept, but a teacher can use that opportunity to question and assess the other related concepts. A multi-variable approach is great for the end of a unit for assessment purposes. For concept-building purposes I feel it is best to work with one or related concepts and gradually build the perceptual elements.

CONCEPTUAL INTEGRATION AND PERCEPTUAL AND MATHEMATICAL VARIBILITY

In this workshop we will work with rational numbers--common fractions and we will concentrate on the above learning objectives. All the resources, including the Internet resources, and all the activities, have been selected to create learning environments in which the above concepts are most likely to be explored. For the purpose of this workshop I will assume that you all know about fractions and this workshop will thus serve as an opportunity for you to review your fraction understanding. I have therefore included both perceptual and mathematical variability.

I have really tried to vary the perceptual mediums and the kinds of 'activity' that children will actually do. This will hopefully address different multiple intelligences. I have not included any musical fractions, but if you are musically inclined I'm sure you can think of ways to incorporate music into how children learn fractions (e.g. West Coast Swing music is in 8 beats to a bar, but often the steps are in patterns of 6, 8, 10, or 12. It feels awkward until you get back to a multiple of 8--after the third bar a 6 step pattern coincides with an 8 step pattern). One of YOUR tasks, as you work with these activities, is to try to identify which of the multiple intelligences is most prominent in the activity you are doing.

Sometimes the activities are requiring children to work alone, sometimes with a partner, sometimes in small groups. They are actively 'doing' math and they are actively 'sharing' math. The teacher is actively assessing the children's mathematical understanding as expressed in some form of action. There is some direct instruction at the beginning of a unit on rational numbers, but even then there should be hands-on activity. There are activities below for children who need a lot of concrete manipulation and review, and there are more abstract activities. Each activity could easily have built into it the range of possibilities for children with special needs. In almost all of the activities it is critical that students share their understanding and work together to perform tasks. One of YOUR tasks, as you work with these activities, is to try to identify how the activity is differentiated or could be differentiated. Think especially of the students you are working with and what adaptations would be necessary for you to make to the activity to help all of your students experience success.

For TI, I have included some good websites, and some software. Of course, you can always use calculators.

Click here for pictures taken during the Middle Years Fraction workshop.

Play-do Fractions

This is perhaps a bit elementary, but I have done this activity with middle years students, especially with students who are having difficulty understanding basic fraction concepts. At the workshop you will receive a bag of play-doh, a small rolling pin, a plastic mat, a plastic knife, and some tracers. The following are some directions you can follow, if you wish, or you can make up your own task. What kind of fractional concepts are possible to work with/learn using a lump of play-doh. Stretch your imagination.

Make three identical rectangles out of play-doh. Cut each rectangle in half in a different way. Are the halves the same amount or put another way--if these rectangles were pizzas all cut in half would it matter which half you ate; would you be eating the same amount of pizza? How do you know?
Again make three identical rectangles out of play-doh. Cut each rectangle in fourths in different ways. Is each fourth the same amount? How do you know?
Make three identical rectangles again. This time cut one in fourths, one in eighths, and one in sixteenths. How many eighths equal one fourth? How many sixteenths equal one fourth? Try out any other questions you would ask your students. Try making a different shape (e.g., a circle) and asking fraction questions about the circle.
How would you divide (partition) a play-doh circle (pretend it's a pizza) among three people, such that each person got a "fair share?" Is this a "fair" question? Why? Reword it!!
How would you partition 5 equal-sized round pizzas so that 10 children all got the same amount?
How would you partition 3 equal-sized round pizzas so that 12 children all got the same amount?
How would you partition 3 equal-sized round pizzas so that 8 children all got the same amount?
Generate your own question and ask someone near you to solve it.

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Covering One Activities

At the workshop you will receive a fraction kit. Please use this kit to answer the following questions.

Fraction Task Sheet

Find 6 ways to make 1.
Find 5 ways to make ½. How many ¼, 1/12, 1/24, or 1/8 make a ½?
How many 1/8 slices equal a ¼ slice?
How many 1/8 slices make up '2' ¼ slices? What else is this size called?
How many 1/8 slices make up a whole?
How many ½ slices would make 2?
How many 1/12 would equal 1/3?
How many 1/12 would make up 2/3?
We do not have a ¾ square but how could we make one with 2 different squares?
How many 1/24 pieces equals 1/8?
Does 1/24 + 1/24 + 1/24 = 1/8?
How many 1/24 make a ¼?
What is larger 1/3 of ¼?
What is larger 2/3 or 2/4?
What is larger 1/3 or 3/8?
What is larger 1/3 or 8/24? Or 12/24?

Note: The above tasks can be accomplished using a commercial product such as Fraction Factory (rectangles or circles).

The following are other more directed activities that you can do with small groups of students.

ELEMENTARY COVERING ONE KIT ACTIVITIES

(the fraction kit can be home-made from cut out paper pieces or it could be a commercial kit such as Fraction Factory)

EXPLORING ONE-HALF --1

How many eighths equal one-half?
How many eighths make one-whole?
How many eighths is one-eighth less than one-half?
How many eighths is two-eighths more than one-half?
When we write 1/2 = 4/8, what kind of fractions are these? Can you record other fractions that equal 1/2?
What is 1/2 + 3/8 equal to?

EXPLORING ONE-HALF -- 2

How many fourths equal one-half?
How many fourths make one-whole?
How many fourths is one-fourth less than one-half?
How many fourths is one-fourth more than one-half?
When we write 1/2 = 2/4, what kind of fractions are these? Can you record other fractions that equal 1/2?
What is 1/2 + 1/4 equal to?

EXPLORING ONE-HALF -- 3

Make one-half with your fraction kit.

Make nine-sixteenths with your kit.

Compare your one-half with your nine-sixteenths and answer the following questions. Discuss the questions with a neighbor and then record your answers in your scribbler.

How many sixteenths equal one-half?
How many sixteenths make one-whole?
How many sixteenths is one-sixteenth less than one-half?
How many sixteenths is two-sixteenths more than one-half?
When we write 1/2 = 8/16, what kind of fractions are these? Can you record other fractions that equal 1/2?
What is 1/2 + 5/16 equal to?

MIDDLE LEVEL COVERING ONE KIT ACTIVITIES

EXPLORING ONE-HALF --1

Make one-half with your fraction kit.

Make five-twelfths with your kit.

Compare your one-half with your five-twelfths and answer the following questions. Discuss the questions with a neighbor and then record your answers in your scribbler.

1. How many twelfths equal one-half?
2. How many twelfths make one-whole?
3. How many twelfths is one-twelfth less than one-half?
4. How many twelfths is two-twelfths more than one-half?
5. When we write 1/2 = 6/12, what kind of fractions are these? Can you record other fractions that equal 1/2?
6. What is 1/2 + 5/12 equal to?

EXPLORING ONE-HALF -- 2

Make one-half with your fraction kit.

Make five-eighths with your kit.

Compare your one-half with your five-eighths and answer the following questions. Discuss the questions with a neighbor and then record your answers in your scribbler.

1. How many eighths equal one-half?
2. How many eighths make one-whole?
3. How many eighths is one-eighth less than one-half?
4. How many eighths is two-eighths more than one-half?
5. When we write 1/2 = 4/8, what kind of fractions are these? Can you record other fractions that equal 1/2?
6. What is 1/2 + 3/8 equal to?

EXPLORING ONE-HALF -- 3

Make one-half with your fraction kit.

Make three-fourths with your kit.

Compare your one-half with your three-fourths and answer the following questions. Discuss the questions with a neighbor and then record your answers in your scribbler.

1. How many fourths equal one-half?
2. How many fourths make one-whole?
3. How many fourths is one-fourth less than one-half?
4. How many fourths is one-fourth more than one-half?
5. When we write 1/2 = 2/4, what kind of fractions are these? Can you record other fractions that equal 1/2?
6. What is 1/2 + 1/4 equal to?

EXPLORING ONE-HALF -- 4

Make one-half with your fraction kit.

Make four-sixths with your kit.

Compare your one-half with your four-sixths and answer the following questions. Discuss the questions with a neighbor and then record your answers in your scribbler.

1. How many sixths equal one-half?
2. How many sixths make one-whole?
3. How many sixths is one-sixth less than one- half?
4. How many sixths is one-sixth more than one- half?
5. When we write 1/2 = 3/6, what kind of fractions are these? Can you record other fractions that equal 1/2?
6. What is 1/2 + 1/6 equal to?

EXPLORING ONE-HALF -- 5

Make one-half with your fraction kit.

Make nine twenty-fourths with your kit.

Compare your one-half with your nine twenty-fourths and answer the following questions. Discuss the questions with a neighbor and then record your answers in your scribbler.

1. How many twenty-fourths equal one-half?
2. How many twenty-fourths make one-whole?
3. How many twenty-fourths is one twenty-fourth less than one-half?
4. How many twenty-fourths is one twenty-fourth more than one-half?
5. When we write 1/2 = 12/24, what kind of fractions are these? Can you record other fractions that equal 1/2?
6. What is 1/2 + 7/24 equal to?

EXPLORING ONE-HALF -- 6

Make one-half with your fraction kit.

Make one-third with your kit.

Compare your one-half with your one-third and answer the following questions. Discuss the questions with a neighbor and then record your answers in your scribbler.

1. How many thirds equal two-halves?
2. How many thirds make one-whole?
3. How many thirds is one third less than one-whole?
4. How many thirds is one third more than one-whole?
5. What is 1/2 +1/3 equal to?

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Number Line

For this activity you need a strip of tape or adding machine paper on the floor. One end of the strip can be 0 and the other end can be 1 (or you can go from -2 to +2--whatever range you want; you can keep the numbers to simple fractions such as 1/5 or 3/4, or you can have mixed numbers such as 1 1/2). Each person gets one card and has to position himself/herself on this number line. After the first person is in place then each person in turn stands in relation to each other depending on the fraction on their card. Make sure you have some equivalent fractions!! I will bring a set of cards to the workshop. Later you can make your own by generating the fraction names you want to use given the range you wish to work with. This is an excellent easy-to-do and very active game for total involvement of the class. You can also play it as a board game with playing pieces.

Fractions with Cuisenaire Rods

1. Find All the Fractions

Find pairs of rods for which one is half as long as the other.
Investigate all the fractional relationships that exist between the rods.

2. Fractional Equivalents

If the brown rod is assigned the value of 1 find all sets of rods that equal 1. Simply lay the rods end to end such that they measure the same length as the 'one' rod. Designate a different rod to be 1--or maybe use two rods together to designate 1. Now find all the fractional equivalents for this new "1."

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Fractions with Cuisenaire Rods

Take a hundred flat. This will now become a "1."

Your orange rods will be a fraction of "1." How many orange rods cover your 1 board?

How many white rods will cover 1?
Use combinations of the orange and white rods to create the following fractions. Read them as decimal fractions, rod combinations, and common fractions:

Number

Decimal Fraction Amount

Cuisenaire Rods

Common Fraction Amount

1.

.3
= ______ orange + _____ white = ______ tenths + ______ hundredths

2.

.24

3.

.05

4.

.11

5.

.76

6.

.19

7.

.99

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Fractions with Cuisenaire Rods

Again begin with a hundred flat to designate one whole. Use the hundred flat as a template on which to construct your fractional amount.

This time use whatever combinations of rods you like to create the following fractions. Record each fraction both as a common fraction and as a decimal.

Number

Decimal Fraction Amount

Adding Common Fractions

Adding Decimal Fractions

1.

.35
(1/10 + 10/100 + 3/100 + 1/50 + 7/100 + 3/100)

= 1/10 + 23/100 + 1/50

=10/100 +23/100 + 2/100

=35/100 .1 + .10 + .03 + .02 + .07 + .03

2.

.67

3.

.51

4.

.48

5.

.83

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Color Tiles

Use color tiles to build rectangles with each of the fractional proportions given:

1. 1/2 red, 1/4 yellow, 1/4 green
2. 1/3 blue, 2/3 red
3. 1/6 red, 1/6 green, 1/3 blue, 1/3 yellow
4. 1/2 red, 1/4 green, 1/8 yellow, 1/8 blue
5. 1/8 red, 3/8 yellow, 1/2 blue
6. 3/8 blue, 1/4 red, 3/8 green
7. 1/2 red, 1/3 blue, 1/6 yellow
8. 1/4 yellow, 2/3 green 1/12 blue

For the following problems, determine what fraction is missing:
9. A rectangle is 1/2 red, 1/3 green, 1/10 blue, and the remainder is yellow. How much of the rectangle is yellow?
10. A rectangle is 3/5 red, the rest is blue and yellow, but not in equal amounts. What could the rectangle look like?

Write a riddle similar to # 9 or # 10 for someone else to solve.

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Pattern Blocks

Build a triangle that is 1/3 green and 2/3 red.
Build a triangle that is 2/3 red, 1/9 green, and 2/9 blue.
Build a parallelogram that is 3/4 blue and 1/4 green.
Build a parallelogram that is 2/3 blue and 1/3 green.
Build a trapezoid that is 1/2 red and 1/2 blue.

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Geoboards

On your geoboard make a 2-dimensional shape with an elastic. With a different colored elastic try to divide your first shape in half. Experiment with different shapes and with different fractional amounts. Make shapes such that you can find the fractions 1/2, 1/3, 1/4, and 1/5.

Work with a partner. Make half a shape. Your partner will make the other half--with a different colored elastic. Now your partner will make a third of a shape and you will complete the whole shape. Try other amounts--such as 1/4 of a shape, 1/5, 1/10 etc.

Again work with a partner. This time make 2/3 of a shape and your partner will complete--with a different colored elastic--the remaining 1/3. Try starting with 3/4 of a shape, 2/9, 3/5 etc.

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Rational Numbers Websites

The following are some interesting sites for learning about fractions. Visit these sites and determine for yourself if you think they would be effective sites for your classroom teaching/learning environment.

The following site (Math Forum: Fractions: Elementary Lessons and Materials) found at http://mathforum.org/paths/fractions/e.fraclessons.html has great resources for creating fraction learning stations. This site is a list of many sites that could have potential as learning tools for fractions.

Find one site at Math Forum that would be useful for you in teaching fractions. Write a few comments about the usefulness of the site you selected.

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K-grade 8 math lessons at AOL can be found at http://members.aol.com/_ht_a/iongoal/mathlessons.htm There seems to be lots to do at this site. It appears to be mostly interactive drill and practice, but probably a lot more fun that workbook pages.

Do one review drill. What do you think? Would this be better than or at least as good as a worksheet or page from a textbook?

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At the Internet Learning Center you'll find a variety of different learning centers, one being fractions. The activities are very basic and similar to worksheets, but they may have some value.
http://www.geocities.com/EnchantedForest/8112/learning.html

Go to the fraction part of the above site. It's quite elementary, but it might have some value for you. What do you think?

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The Fraction Shape site uses the familiar pattern block shapes to build fractions. This is a fun site that uses a java applet interactive environment.
http://math.rice.edu/~lanius/Patterns/

Find the java applet part of the above site and "play" with the interactive pattern blocks. If the yellow block = one whole, then make the following numbers with the blocks. Beside each rational number below make a small drawing of the blocks you used.

2 1/2 3 2/6 5/6

1 1/3 2 1/6 4 1/2

3 2/3 1/2 4 5/6

The following site has an interactive environment like the one above: http://www.arcytech.org/java/patterns/patterns_j.shtml

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The Math Central Resource Room database has some fraction resources. The following is an activity which can be worked on offline--following the on-line instructions.

http://mathcentral.uregina.ca/RR/database/RR.09.95/hanson4.html

Using pattern blocks (off-line), follow the directions at the above site.

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Online geoboards have potential for fraction learning. How do you think you would use this resource for helping children learn fractions?

http://standards.nctm.org/document/eexamples/chap4/4.2/

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The following site is one that teachers would use in their planning of fraction learning. It has many drill and practice types of examples, that may be quite useful in some situations.

http://www.visualfractions.com/

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E-Labs, grades 3-8 may also have useful ideas. They look great--I haven't had time to go through them yet to see if they would be good for fraction learning, but it's a great resource for math.

http://www.harcourtschool.com/elab/index.html

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Fraction Hotlist (from one of my graduate students)

http://www.kn.pacbell.com/wired/fil/pages/listfractionj.html

Go through the hotlist above and examine the sites my student found. What do you think of them? Are they useful?

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Other websites that may provide useful material for the fraction-learning environment are:

http://illuminations.nctm.org/lessonplans/6-8/rationalnumbers/

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The following site is one for grade eight and nines. It is more like a text book but the
questions do relate to rational numbers.

http://www.wcape.school.za/malati/RatNumbr.pdf

The following site has activity ideas and some links to other math sites

http://www.gettysburg.edu/~tuttwh01/Educ180/assignments_rational_numbers.html

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The following is a very elaborate site of activities and lesson plans for all
different strands of mathametics including rational numbers

http://mathforum.org/library/resource_types/lesson_plans/?start_at=651

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An awesome resource--the National Library of Virtual Manipulatives for Interactive Mathematics can be found here:

http://matti.usu.edu/nlvm/nav/vlibrary.html

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Fraction pieces can be found here:

http://matti.usu.edu/nlvm/nav/frames_asid_274_g_3_t_1.html?open=activities

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Adding fractions at this link: http://matti.usu.edu/nlvm/nav/frames_asid_106_g_3_t_1.html

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Comparing fractions at this link: http://matti.usu.edu/nlvm/nav/frames_asid_159_g_3_t_1.html

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Fraction equivalents can be found here: http://matti.usu.edu/nlvm/nav/frames_asid_105_g_3_t_1.html

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Pre-K-Grade 2 Fraction Websites

General

http://mathforum.org/library/levels/elem1/

Fractions

http://mathforum.org/varnelle/knum.html

The above site has some very simple rational number activities for very young children. Explore the site and see what you think.

Richard Skemp

Skemp created hundreds of structured activities for mathematics based on a theory on intelligent learning (his theory). His work is commercially available as SAIL (Structured Activities for Mathematics). I will bring copies of his books for you to explore and also some fraction activities. Again, most, if not all, of the fraction concepts in the curriculum can be addressed through these materials.

Make Two Dozen and Scale the Dragon

These are special games that I will not put on the Internet as they are not available anywhere in print form. I will bring gameboards to the workshop and demonstrate the games there. These are games that my Ph D supervisor, Dr. Tom Kieren (U of A) designed about 16 years ago for rational number research he was doing in Edmonton. I used them in my dissertation research with grade 3 children in a community school. The following year, these children, now in grade 4, used the same games with grade 8 children and the grade 8 children could not do the math!! The grade 4 children taught them rational number concepts through the playing of these games. The second game, "Scale the Dragon," requires that the covering one concepts be understood.

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Fraction Games From Box Cars and One-Eyed Jacks

One of the Box Cars books is called "Piece it Together with Fractions." (I will bring it to show you). There are many quite engaging activities in this book. I will bring one for us to work on--"Wholey Moley" on page 29. This activity needs dice and fraction pieces (which I will bring).

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For Fun

Draw a circle and make a pie chart of a regular day in your life. Make sure you write in the fractional amounts --either as a common fraction or a decimal fraction.

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Jot down some other ideas that you have for activities to enable students to learn fractions.

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E-Mail: maeers@uregina.ca

Phone: 585-4601

Perceptual Variability	Mathematical Variability	Teaching Strategy
one variable (e.g., pattern blocks)	one concept--equivalence	teacher-directed/whole class--everyone doing the same thing/hands-on; experiential
two variables (e.g., pattern blocks; geoboards)	one concept--equivalence	teacher works with one group; other group works independently
multiple variables (e.g., pattern blocks, play-doh, paper coverings)	one concept--equivalence	interactive--indirect teaching; small groups; stations
one variable (e.g., paper coverings)	two or more concepts (e.g., equivalence; comparing fractions, addition of fractions)	teacher-directed/whole class--everyone using the same material--perhaps going tthrough each concept with teacher and/or working in groups with task cards to explore the different concepts
multiple variables (e.g., paper coverings, games, software, children's literature, play-doh, geoboards, cuisenaire, etc)	multiple concepts (e.g., equivalence, area concept, part of a set concept, measurement concept, comparing, addition and subtraction, etc)	small groups working at stations--rotating; this is a useful strategy at the end of a unit and can be effective as performance stations

Number	Decimal Fraction Amount	Cuisenaire Rods	Common Fraction Amount
1.	.3	= ______ orange + _____ white	= ______ tenths + ______ hundredths
2.	.24
3.	.05
4.	.11
5.	.76
6.	.19
7.	.99

Number	Decimal Fraction Amount	Adding Common Fractions	Adding Decimal Fractions
1.	.35	(1/10 + 10/100 + 3/100 + 1/50 + 7/100 + 3/100) = 1/10 + 23/100 + 1/50 =10/100 +23/100 + 2/100 =35/100	.1 + .10 + .03 + .02 + .07 + .03
2.	.67
3.	.51
4.	.48
5.	.83

2 1/2	3 2/6	5/6
1 1/3	2 1/6	4 1/2
3 2/3	1/2	4 5/6