Version Acstl-csm.semo.edu/cjmcallister/mainpage/Appendixestotal.doc · Web viewVersion A. Version B. Fraction Assessment – MA318. The purpose of this test is to determine what

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Appendix AVersion A

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Version BFraction Assessment – MA318

The purpose of this test is to determine what you know about fractions. You may not use a calculator to do this test. Please show all work on the items where you need to do calculations to answer the question. You will have 45 minutes to do this test.

(1a, b) Look at each picture carefully. Circle the two pictures that show shaded in. You may

need to draw in lines to determine if are shaded.

(2) What fraction of the circle is part c? __________________

(3) If shape A is the unit, what fraction can you give to 2 triangles? __________________

Shape A

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(4) Look at this picture. The yellow rod is what fraction of the brown rod? ________________

Yellow

White White White White White White White White

Brown

(5)What fraction of the set is shaded? Name the amount in two ways: _____ = _____

(6) Name the fraction represented by the shaded part of the figure below in two ways:

_____ = ______

(7) Using pictures of circles for chips or tiles, draw a picture to show the fraction using 15

circles.

(8) The whole circle is the unit. Write 2 fraction names for the picture for the amount shaded.

________ _______

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(9) Imagine that you have a bag of chips (tiles) to work with. You are to show the fraction

with the chips in two different ways. Draw pictures of your two models in the boxes below.

(10) Write the number that should go in the box on this number line: _________________

0 1 2

(11) Write the number that should go in the box on this number line: _______________

0

(12) This rectangle below is of some unit. Extend the rectangle to show the whole unit.

Explain what you did to draw the whole unit.

For problems 13 and 14 give two names for the shaded amount:

15) Show where the number should be on this number line by placing an “x” at the

appropriate spot.

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0 1

(16) Name three fractions equal to .

For problems 17 – 24 circle the larger fraction. If equal, circle both fractions. For each problem either show work or explain your reasoning for your answer.

(17) (18)

(19) (20)

(21) (22)

(23) (24)

For each of the next three problems, use a model of the situation to illustrate the operation indicated and solve each problem.

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(25) On Monday Sally ate 1-third of a small pizza. On Tuesday she ate 3-fourths of a small pizza. How much of a pizza did she eat altogether over the two days?

(26) Bill and Jim receive the same allowance. Bill spent of his allowance on video games.

Jim spent of his allowance buying a gift for his mom. Bill spent how much more of his

allowance than Jim?

(27) Sara had a board that was meter in length. She then cut a piece of a meter. What

fraction of a meter was left?

(28) Draw a picture to solve this problem: Rodrigo owned ¾ acre of land. He planted corn on 2/5 of the land. What fraction of an acre did he plant with corn?

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(29) Pedro had 3 and ¾ pounds of jelly beans. He put them in ¼ pound bags. How many bags did he fill?

(30) Imagine a length of ribbon ½ yard long. You cut the ribbon into three equal lengths. What fraction of a yard is each piece?

Perform each of the following operations using the method you are most familiar with. Then write a short word problem that goes with the problem you just worked.

(31) Word problem:

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(32) Word problem:

(33) Word problem:

(34) Word problem:

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Appendix B

Mathematics Attitude Scale (Aiken, 1972)

Directions: Please write your name in the upper right-hand corner. Each of the statements on this

opinionnaire expresses a feeling or attitude towards mathematics. You are to indicate, on a five-

point scale, the extent of agreement between the attitude expressed in each statement and your

own personal feeling. The five points are: Strongly Disagree (SD), Disagree (D), Undecided

(U), Agree (A), Strongly Agree (SA). Draw a circle around the letter letters giving the best

indication of how closely you agree or disagree with the attitude expressed in each statement.

1. I am always under a terrible strain in a mathematics class. SD D U A SA

2. I do not like mathematics, and it scares me to have to take it. SD D U A SA

3. Mathematics is very interesting to me, and I enjoy arithmetic and

mathematics courses.

SD D U A SA

4. Mathematics is fascinating and fun. SD D U A SA

5. Mathematics make me feel secure, and at the same time it is

stimulating.

SD D U A SA

6. My mind goes blank and I am unable to think clearly when working

mathematics.

SD D U A SA

7. I feel a sense of insecurity when attempting mathematics. SD D U A SA

8. Mathematics makes me feel uncomfortable, restless, irritable, and

impatient.

SD D U A SA

9. The feeling that I have toward mathematics is a good feeling. SD D U A SA

10. Mathematics makes me feel as though I’m lost in a jungle of

numbers and can’t find my way out.

SD D U A SA

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11. Mathematics is something that I enjoy a great deal. SD D U A SA

12. When I hear the word mathematics, I have a feeling of dislike. SD D U A SA

13. I approach mathematics with a feeling of hesitation, resulting from a

fear of not being able to do mathematics.

SD D U A SA

14. I really like mathematics. SD D U A SA

15. Mathematics is a course in school that I have always enjoyed

studying.

SD D U A SA

16. It makes me nervous to even think about having to do a mathematics

problem.

SD D U A SA

17. I have never liked mathematics, and it is my most dreaded subject. SD D U A SA

18. I am happier in a mathematics class than in any other class. SD D U A SA

19. I feel at ease in mathematics, and I like it very much. SD D U A SA

20. I feel a definite positive reaction toward mathematics; it’s enjoyable. SD D U A SA

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Semantic Differential Test

Put an X on the space closest to your attitude about working with fractions, relative to each

adjective pair.

good - - - - - bad

soft - - - - - hard

red - - - - - green

small - - - - - large

quiet - - - - - lively

strong - - - - - weak

dirty - - - - - clean

ugly - - - - - beautiful

slow - - - - - fast

kind - - - - - mean

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Appendix C

Traditional Lecture/Discussion Format Lessons

Introduction to Fractions -Lesson 1Note: Lessons may take 1 – 3 class periods to cover)

I. Discuss students’ responses to “what is a fraction? Why are they needed?”

A. In this first lesson we will: define the concept of fractions in several ways, look at various concrete and pictorial representations of fractions, define equivalent or equal fractions, how to represent fractions in simplified form, determine how to order fractions, and review representations for mixed numerals.

II. There is no standard way to define a fraction

A. Some of the ways to conceptualize or define fractions include:

1. Part-whole: means 3 parts out of 4 equal parts of a unit whole

2. Operator: of something – multiply by 3 and divide by 4 OR divide by 4, then

multiply by 3.

3. Ratio and Rates: means 3 parts compared to 4 parts

4. Quotient: means 3 divided by 4. is the amount each person receives when 4

people share a 3-unit of something (concrete model)

5. Measure: means a distance of 3 one-fourth units from 0 on the number line or

3 one-fourth units of a given area

B. The Part-whole concept of fractions is traditionally presented in school mathematics

1. a and b are whole numbers (integers), b is not equal to zero, then or a/b

means ‘a’ amount of ‘b’ equivalent parts

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2. This concept sometimes leads children to think of fractions as two numbers

instead of as a single number.

C. Our text defines a fraction as a number that can be represented by an ordered pair of

whole numbers ‘a’ and ‘b’ as or a/b, where b is not equal to zero. The number ‘a’ is

called the numerator and the number ‘b’ is called the denominator.

D. Fractions can be considered a numerical representation of a relative amount (what

does relative mean? Must know what the whole or unit is to determine the actual

amount.) It is important when teaching the fraction concept to identify the unit being

divided into equal parts. Let the large rectangle below represent the unit.

1. What fractional part of the rectangle is shaded?

2. If we allow half of the rectangle to be the unit, now what fractional part

is shaded?

3. What if we redraw the rectangle to look like this and we consider that the rectangle

only has 4 equal parts instead of 12. Now what fractional part of the rectangle is shaded?

What has changed? Is the relationship ¼ the same as 3/12 for this rectangle?

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E. Fractions that represent the same relative amount are called equivalent fractions.

III. Models.

A. Set Model – demonstrate how to model 2/3 and 4/3 using 3, 6, 12 etc. discrete objects.

B. Area Model – demonstrate 2/3 and 4/3 using a circle and a rectangle

C. Number line model – demonstrate locating 2/3 and 4/3 on a number line.

IV. Equality of fractions

A. (Discuss cross products and cross multiplication)

1. Example: Determine which of , , or are equal

B. A fraction is considered simplified (reduced or in lowest terms) when the

numerator and denominator are relatively prime (have no common prime factors)

(Relate this to finding the GCF from MA118)

1. Examples: simplify 3/12 and 30/75

C. Since a/b = an/bn for n = 1, 2, 3, … every fraction has an infinite number of numeral

representations. These all represent the same relative amount.

1. 1/3 = 2/6 = 3/9 = 4/12 = … (show as an area model)

V. Ordering fractions

A. Let a/c and b/c be fractions. Then a/c < b/c if and only if a< b.

B. Let a/b and c/d be fractions. Then a/b < c/d if and only if ad < bc.

C. Arrange in order and find a fraction in-between

1. 17/23 and 8/15

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D. Faster way to find a fraction in-between 2 given fractions – add numerators and add

denominators.

1. 8/15<17/23, so 8/15 < 25/38 < 17/23. Verify this using cross multiplication.

E. The fact that you can find a fraction between any two given fractions is called the

density property of fractions.

VI. Improper fractions and mixed numbers (numerals)

A. Improper fractions: In a/b if a>b then a/b is an improper fraction.

1. 5/2 (draw shaded circles that represent this amount)

B. In the diagram we can see that the 5 halves form 2 whole units and ½ of another. This

can be represented by the mixed numeral 2 ½ or by the improper fraction 5/2.

Note: 2 ½ can be considered as 2 + ½

C. Represent the following mixed numerals as improper fractions: ,

D. Represent the following improper fractions as mixed numerals: ,

VII. If time - p. 230 problems for writing and discussion Assign #1,2,3 to different tables – talk

about responses.

=================================================================

Notes for Addition and Subtraction of Fractions (Lesson 2)

In this lesson we will review and explore the operations of addition and subtraction of fractions.

We will discuss the following: algorithms for addition and subtraction of fractions, common

denominators and how they relate to LCM, and properties of addition and subtraction of

fractions. We will also discuss how to write addition and subtraction word problems that involve

fractions.

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I. Adding Fractions

A. With a common denominator

1. Let a/b and c/b be fractions then:

2. Models (demonstrate on overhead how to use various models to add fractions

with a common denominator).

a. Area – manipulatives and shaded regions

b. Number line

B. Without a common denominator

1. Let a/b and c/d be fractions, then :

2. Models (demonstrate on overhead how to use various models to add fractions

without a common denominator).

a. Area – manipulatives and shaded regions

b. Number line

3. Discussion of why finding the lowest common denominator is a good strategy

II. Properties for fractions

A. Closure : The sum of any two fractions is also a fraction

B. Commutative : a/b + c/d = c/d + a/b

C. Associative: (a/b + c/d) + e/f = a/b + (c/d + e/f)

D. Additive Identity: a/b + 0 = 0 + a/b = a/b

III. Using Properties to simplify computation

a.

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IV. Subtraction of fractions

A. With common denominators


B. Without common denominators

1. Let a/b and c/d be fractions where a/b is greater than or equal to c/d,

then :

2. Models (demonstrate on overhead how to use various models to subtract

fractions with and without a common denominator).

a. Shaded regions or area

b. Number line

3. Missing addend definition of subtraction

a. Let a/b and c/b be fractions, then .

V. Mental Math and Estimation techniques for adding and subtracting fractions

A. Look at Examples on page 236.

B. Estimation

1. Range

2. Front – end with adjustment

3. Rounding to the nearest ½ or whole

4. Example and

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VI. Discuss creating addition and subtraction word problems involving fractions that are

meaningful and unambiguous.

VII. If time, p. 241 discuss #1-4 in groups, then report to class.

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Multiplying and Dividing Fractions (Lesson 3)

I. How do you interpret 2/3 x ½?

A. Can you come up with a word problem to go with this math fact?

B. A whole number times a fraction is the easiest to interpret – should start there with

students

1. 3 x ½

a. region model

b. repeated addition model

c. number line model

d. create a word problem for this example

C. Fraction times a whole: 1/3 x 4

1. We could use the commutative property to explain

2. We could view 1/3 x 4 as taking 1/3 of 4 or 1 of 3 equal parts of 4.

a. region model

b. number line model

c. create a word problem for this example

D. Fraction times a fraction: 2/3 x ½

1. We can interpret this as dividing ½ into 3 equal parts and taking 2 of them.

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a. regions model – note: the area that represents one of the fractions

becomes the whole or unit for the other fraction

b. number line model

c. create a word problem for this example

E. Mixed number times a mixed number: 2 1/3 x 1 ½

1. Shaded regions model

2. Discuss why the answer is NOT 2 1/6

a. How could you demonstrate that this is incorrect?

b. (2 + 1/3) x (1 + ½): FOIL

II. Definition of fraction multiplication and properties of fraction multiplication

A. Let a/b and c/d be fractions. Then

1. It is helpful to teach students to simplify first – why?

2. Example:

B. Properties

1. Closure: the product of two fractions is a fraction

2. Commutative: a/b x c/d = c/d x a/b

3. Associative: (a/b x c/d) x e/f = a/b x (c/d x e/f)

4. Multiplicative Identity: a/b x 1 = 1 x a/b = a/b

5. Multiplicative Inverse (reciprocal): for every non-zero fraction a/b there

exits a unique fraction b/a such that .

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This is a very useful property in algebra:

6. Distributive property over addition and subtraction

a/b(c/d + e/f) = ac/bd + ae/bf

III. Division of fractions – tough concept to model in a meaningful way. Invert and multiply is a

rote procedure that does not provide students with any meaningful concept of fraction division.

There are several, more meaningful ways to teach fraction division than just asking student to

memorize a rule.

A. Can you make up a word problem that would go with ?

B. Common denominator approach

1. Allows division of fractions to be viewed as an extension of whole number

division

2. can be thought of as: How many groups of 2/3 are there in 8/3?

3. Area model

4. Number line model.

5. Rule: Let a/b and c/b be fractions. Then

C. Fractions with out common denominators

1. Find a common denominator, then use rule from above.

2. Rule: Let a/b and c/d be fractions. Then

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Note: d/c is the reciprocal of c/d (this is where invert and multiply comes

from.)

3. Example:

D. Maintaining the quotient (a little know property)

a. If , then

b. So

c. Example:

E. Missing factor approach to fraction division

1. If with b, d, and f not equal to zero, then it must be true that

. Note that in the first problem, a/b is dividend, c/d is divisor, and e/f

is the quotient. Recall from the study of whole numbers, that

2. Examples using missing factor approach

a.

b.

E. Rectangular shaded regions to model fraction division

a.

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b.

F. Divide numerators, divide denominators approach

1. , becomes a complex fraction

a. complex fractions

2.

3.

4. When is this method efficient? Would you teach this method to your students?

IV. Wrapping up division of fractions

A. For all whole numbers a, b, (b not equal to zero),

B. Mental arithmetic (p. 251)

C. Estimation (p.251)

V. If time – p. 256, #1,2,3 and p. 259 # 1,2,5,7 divide into groups and discuss, report to class.

=====================================================================

Putting it all together (Lesson 4)

Fraction Manipulatives Activity

You will work in groups of 3-4 students. Each group will be given a type of manipulative used to

teach fraction concepts to students. Work through the problems using the manipulatives to

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represent your thinking and answer the questions below for your particular type of manipulative.

Be prepared to present to the entire class how you would use this type of manipulative to teach

various fraction concepts. Each group should turn in one copy of the work on all parts with

everyone’s name included.

1. On a piece of paper, trace the piece (or pieces) you decide will represent the ‘whole’ or ‘unit’,

then trace the piece (or pieces) that represent the following fractions: ½, 1/4, 1/3, 1/6, ¾, 2/3, 5/6.

Example: This would be the work for pattern blocks and 1/6

2. Use your pieces to represent ½ + 1/3. Draw that representation on a piece of paper. Develop a

word problem that goes with this math fact.

3. Use your pieces to represent 5/6 – 2/3. Draw that representation on a piece of paper. Develop

a word problem that goes with this math fact.

4. Use your pieces to represent 3 x ¾. Draw that representation on a piece of paper. Develop a


5. Use your pieces to represent 2/3 x 3/4 Draw that representation on a piece of paper. Develop

a word problem that goes with this math fact.

6. Use your pieces to represent ¾ ÷ 1/2. Draw that presentation on a piece of paper. Develop a


=====================================================================

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Common Error Patterns for Fractions (Lesson 5)

I. Present students with examples of students’ incorrect work with the definition of

fractions and operations on fractions

A. Students determine the error pattern for the various examples

References

Ashlock, R. B. (2002). Error patterns in computation: Using error patterns to improve

instruction (8th ed.). Upper Saddle river, NJ: Pearson Education, Inc.

Beckmann, S. (2005). Class activities to accompany mathematics for elementary teachers.

Boston, MA: Pearson/Addison Wesley.)

Musser, G. L., Burger, W. F., & Peterson, B. E. (2003). Mathematics for elementary teachers:

A contemporary approach (6th ed.) New York, NY: John Wiley & Sons, Inc.

Litwiller, B., & Bright, G. (Eds.). (2002). Making sense of fractions, ratios, and proportions:

2002 Yearbook. Reston, VA: NCTM.

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Appendix DHands-on Laboratory/Discussion Format Lessons

Introduction to Fractions - Objective 1Note: Objectives may take 1 – 3 class periods to cover

I. Discuss students’ responses to “what is a fraction? Why are they needed?”

A. In this first lesson we will: define the concept of fractions in several ways, look at various concrete and pictorial representations of fractions, define equivalent or equal fractions, define simplest form and review how to represent fractions in simplified form, determine how to order fractions, and review representations for mixed numerals.

II. There is no standard way to define a fraction

A. Some of the ways to conceptualize or define fractions include:

6. Part-whole: means 3 parts out of 4 equal parts of a unit whole

7. Operator: of something – multiply by 3 and divide by 4 OR divide by 4, then

multiply by 3.

8. Ratio and Rates: means 3 parts compared to 4 parts

9. Quotient: means 3 divided by 4. is the amount each person receives when 4

people share a 3-unit of something (concrete model)

10. Measure: means a distance of 3 one-fourth units from 0 on the number line or

3 one-fourth units of a given area

B. The Part-whole concept of fractions is traditionally presented in school mathematics

1. a and b are whole numbers (integers), b is not equal to zero, then or a/b

means ‘a’ amount of ‘b’ equivalent parts

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2. This concept sometimes leads children to think of fractions as two numbers

instead of as a single number.

C. Provide students with ‘Exploring the Part-whole Concept of Fractions’ (Activity adapted

from activities in National Council of Teachers of Mathematics. (2002). Making sense of

fractions, ratios, and proportions: 2002 Yearbook. Litwiller, B. and Bright, G. (Eds.) Reston,

VA: NCTM and Beckmann, S. (2005). Class Activities Accompany Mathematics for Elementary

Teachers. Boston, MA: Peasron/Addison Wesley.)

=====================================================================Exploring the Part-whole Concept of Fractions

The most common way fractions are presented to students in school mathematics is by the concept of equal parts of a whole. The following activity is designed to help you see the importance in identifying the ‘whole’ or ‘unit’ when dealing with fraction representations.

1. How does the figure above represent ? What is the whole? What do the smaller

subdivisions of the rectangle represent? In the symbolic representation , what does the 5

mean? What does the 3 mean? Could this figure represent ?

2. Can you now look at the same figure and see it representing of something? What is the whole in this case? What do the smaller subdivisions of the rectangle represent? Interpret the

symbolic representation in this situation.

3. Can you see of in this situation? Explain.

4. Can you see of in this situation? Explain.

5. Can you see in this situation? Explain.

6. (Be prepared to present to whole class). Create your own figure representing a fraction other than those here and explain how it can be viewed from different perspectives based on the way the ‘whole’ is defined.

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7. Jenny has a large piece of fabric that she has cut into two pieces: a larger piece and a smaller piece. By laying the pieces of fabric next to each other, Jenny can tell that the smaller piece of fabric is the same size as of the larger piece. What fraction of the original whole piece of fabric (before cutting) is the small piece of fabric? Use the part-whole meaning of fractions and a picture to help you explain your answer.

8. Each of the eight pairs of rectangles shown below represents what is left of two cakes, one with pink frosting (the lighter shading) and one with chocolate frosting (the darker shading). For each pair, look at what is left of the two cakes and tell which portion is more or whether they are the same.

9. Benton used cup of butter to make a batch of cookie dough. Benton rolled his cookie dough out into a rectangle, as shown below. Now Benton wants to cut off a piece of the dough so that the portion he cuts off contains cup of butter. How could Benton cut the dough? Explain.

10. Mira has a recipe that calls for 5 cups of flour. Mira wants to make of the recipe. Instead

of figuring out what number of 5 is, Mira measures of a cup of flour 5 times, and uses this

amount of flour for of the recipe. Use the part-whole meaning of fractions and the following

picture to help you explain why Mira’s strategy is valid:

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11. Write a clear definition of fractions based on the part-whole concept of fractions. =====================================================================

D. Provide students with ‘The Whole Associated with a Fraction’ (adapted from activities in

Beckmann, S. (2005). Class Activities Accompany Mathematics for Elementary Teachers.

Boston, MA: Peasron/Addison Wesley.)

=====================================================================The Whole Associated with a Fraction

1. Maurice says that the next picture shows that 1/3 is bigger than ½. The shaded pieces representing 1/3 is larger than the shaded piece representing ½, so why is Maurice not correct?

2. Kayla says that the shaded part of the next picture can’t represent ¼ because there are 3 shaded circles, and 3 is more than 1, but ¼ is supposed to be less than 1. What can you tell Kayla about fractions that might help her?

3. When Ted was asked to say what the 4 in the fraction ¾ means, Ted said that the 4 is the whole. Explain why it is not completely correct to say, “4 is the whole”. What is a better way to say what the 4 in the fraction ¾ means?

4. Explain why you can use fractions , , and to describe the shaded region in the next

figure. How can three different numbers describe the same shaded region? How must you interpret the shaded region in each case?

5. Suppose a textbook problem asks student to “name the fraction shown in the shaded region of the next figure.” Assuming there is no other information given in the problem, explain why the problem is ambiguous. What should the textbook problem clarify?

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6. At a neighborhood park, 1/3 of the area of the park is to be used for a new playground. Swings will be placed on ¼ of the area of the playground. What fraction of the neighborhood park will the swing area be? Draw a picture to help you solve the problem, and in your solution, describe the whole associated with this fraction. In other words, describe what each fraction is of. Are all the wholes the same or not?

E. Fractions that represent the same relative amount are called ‘equivalent fractions’

1. Do number line activity (Students will use three aligned number lines: one divided

into eights, one divided into sixths, and one divided into tenths to determine placement

and relative order of sevenths, ninths, and twelfths.)

2. Do equivalent fractions activity (adapted from activities in Beckmann, S. (2005). Class

Activities Accompany Mathematics for Elementary Teachers. Boston, MA:

Peasron/Addison Wesley.)

Equivalent fractions

1. Use the meaning of fractions of objects to give a detailed conceptual explanation for why

of a pie is the same amount of pie as of pie, which is of the pie. Draw pictures to

support your explanation. Discuss how your pictures show the process of multiplying both the

numerator and denominator of by 4.

2. Discuss how to modify your explanation for Problem 1 to show that of a pie is the same

amount of pie as of the pie.

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3. Suppose A and B are whole numbers and B is not 0. Also suppose that N is any natural

number. Explain why of some object is the same amount of that object as of the

object.

4. Anna says because starting with , you get by adding 4 to the top and the bottom.

If you do the same thing to the top and the bottom, the fractions must be equal. Is Anna right? If, not, why not? How might Anna have come up with her rule?

5. Don says that because both fractions are one part away from a whole. Is Don

correct? If not, what is wrong with Don’s reasoning?

III. Review Models used in class.

A. Set Model – discrete objects.

B. Area Model – circle, rectangle, or other 2-D shape

C. Number line model – previous activity

IV. Review and summarize equality of fractions

A. (Discuss cross products and cross multiplication)

1. Example: Determine which of , , or are equal

B. A fraction is considered simplified (reduced or in lowest terms) when the

numerator and denominator are relatively prime (have no common prime factors)

(Relate this to finding the GCF from MA118)

C. Have students complete ‘Simplifying Fractions’ activity (adapted from activities in

Beckmann, S. (2005). Class Activities Accompany Mathematics for Elementary


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Simplifying Fractions

1. Use the next diagram to help you explain why the following equations that put in simplest

form make sense: .

2. Use blocks or snap-together cubes in two different colors, or draw pictures to help you

demonstrate why the following equations that put in simplest form make sense:

3. Show two different ways to write equations that put in simplest form. (One way can take

several steps to simplify the fraction.) Then use blocks or snap-together cube in two different colors, or draw pictures to help you explain why your equations make sense.

4. Some people say that because you can “cancel the zeros” in . But if you can

“cancel the zeros” in , then why can’t you “cancel the 5’s in and say that ?

Can you “cancel the 5’s” in and say that ? What is a better way to explain why

than saying we “cancel the zeros?” What are you REALLY doing when you “cancel the

zeros?”

V. Ordering fractions

A. Let a/c and b/c be fractions. Then a/c < b/c if and only if a< b.

B. Let a/b and c/d be fractions. Then a/b < c/d if and only if ad < bc.

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1. Techniques for ordering fractions that are intuitive methods from a well

developed sense of fractions.

a. Same denominator method

b. Same numerator method

c. type fractions

d. type fractions

e. Comparison to ½.

C. Fast way to find a fraction in-between 2 given fractions – add numerators and add

denominators.

D. The fact that you can find a fraction between any two given fractions is called the

density property of fractions.

E. Provide students with Ordering Fractions Activity (adapted from activities in

Beckmann, S. (2005). Class Activities Accompany Mathematics for Elementary


Ordering Fractions

1. Claire says that because 4 > 3 and 9 > 8. Discuss Whether or not Claire’s reasoning is

correct. Give counterexamples.

2. Use reasoning other than finding common denominators, cross-multiplying, or converting to decimal numbers to compare the sizes ( =, <, > ) of the following pairs of fractions.

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3. In your own words, explain why the following rules for comparing fractions and are

true:

4. a, b, c represent whole numbers different from 0 and a > b > c. What can you say about:a/b b/a b/c c/b a/c c/a

5. Let a, b, c be defined as in problem 4. Which is larger: a/c or b/c? a/b or b/b? a/b or a/c?

VI. Improper fractions and mixed numbers (numerals)

A. Improper fractions: In a/b if a > b then a/b is an improper fraction.

B. Provide students with ‘Exploring Improper Fractions’ activity (adapted from activities

in Beckmann, S. (2005). Class Activities Accompany Mathematics for Elementary


Exploring Improper Fractions

1. Enrico says that it doesn’t make sense to talk about of a piece of paper because if you

divide a piece of paper into 4 equal pieces, then you only have 4 pieces and can’t show 5 pieces. What can you tell Enrico that might help him?

2. Show a plot of land that is the area shown on the map in the next figure. Is more than one correct answer possible?

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3. If you have only one pie, would it be a good thing to use this pie to talk about the fraction ?

What kinds of things are useful for discussing improper fractions such as ? Explain.

4. Suppose you use a picture like the next one to talk about the fraction . What kind of

confusion could arise about this picture? What must we do in order to interpret the shaded region

as ?

5. Now use number lines to discuss the meaning of improper fractions such as and .

6. What are the advantages and disadvantages of using objects and pictures to discuss improper fractions? What are the advantages and disadvantages of using number lines to discuss improper fractions?

C. Review by asking students to represent the following mixed numerals as improper

fractions: ,

D. Review by asking students to represent the following improper fractions as mixed

numerals: ,

VII. If time - p. 230 in Musser, Burger, & Peterson, problems for writing and discussion

Assign #1,2,3 to different tables – talk about responses.

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Notes and Activities for Addition and Subtraction of Fractions (Objective 2)

In this lesson we will review and explore the operations of addition and subtraction of fractions.

We will discuss the following: algorithms for addition and subtraction of fractions, common

denominators and how they relate to LCM, and properties of addition and subtraction of

fractions. We will also discuss how to write addition and subtraction word problems that involve

fractions.

I. Provide students with ‘Fraction Addition and Subtraction’ activity (adapted from activities in

Beckmann, S. (2005). Class Activities Accompany Mathematics for Elementary Teachers.

Boston, MA: Peasron/Addison Wesley.)

Fraction Addition and Subraction

1. Use a story problem and a picture or a number line to explain why the procedure for adding makes sense. In particular, explain why we need common denominators.

2. Use a story problem and a picture or a number line to explain why the procedure for adding makes sense. In particular, explain why we need common denominators.

3. When we add or subtract fractions, must we use the least common denominator? Why or why not? Are there any advantages or disadvantages to using the least common denominator when

adding or subtracting fractions? Illustrate your answer with the example and with other

examples.

4. Patti thinks that it would be easier to add fractions by “adding the tops and adding the

bottoms.” For example, Patti wants to add 2/3 and 4/5 this way: . Patti uses the

picture below to explain why her method makes sense. Why is Patti’s method not a valid way to add fractions, and why doesn’t Patti’s picture prove that fractions can be added in her way? Do not just state the proper way to add fractions: explain what is wrong with Patti’s reasoning.

+ =

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II. Adding and subtraction of Fractions – summary notes

A. With a common denominator



B. Without a common denominator

1. Let a/b and c/d be fractions, then :

2. Let a/b and c/d be fractions where a/b is greater than or equal to c/d,

then :

C. Missing addend definition of subtraction

1. Let a/b and c/d be fractions, then .

III. Properties for fractions

A. Closure : The sum of any two fractions is also a fraction

B. Commutative : a/b + c/d = c/d + a/b

C. Associative: (a/b + c/d) + e/f = a/b + (c/d + e/f)

D. Additive Identity: a/b + 0 = 0 + a/b = a/b

IV. Using Properties to simplify computation

a.

V. Mental Math and Estimation techniques for adding and subtracting fractions

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A. Look at Examples on page 236 of Musser, Burger, & Peterson.

B. Estimation

1. Range

2. Front – end with adjustment

3. Rounding to the nearest ½ or whole

4. Example and

VI. Discuss creating addition and subtraction word problems involving fractions that are

meaningful and unambiguous.

A. Use pages 64-65 of Beckmann book for the discussion

VII. If time, p. 241 of Musser, Burger, & Peterson. Discuss #1-4 in groups, then report to class.

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Multiplying and Dividing Fractions (Lesson 3)

I. How do you interpret 2/3 x ½?

A. Can you come up with a word problem to go with this expression?

B. A whole number times a fraction is the easiest to interpret – should start there with

students.

C. Multiplication activity (adapted from activities in Beckmann, S. (2005). Class

Activities Accompany Mathematics for Elementary Teachers. Boston, MA:

Peasron/Addison Wesley.)

Multiplication Activity

1. Write a story problem for the multiplication problem 3 x 1/3, then show how to solve the

problem using a diagram or number line.

2. Write a story problem for the multiplication problem 1/3 x 2, then show how to solve the

problem using a diagram or number line. How is problem 1 different in meaning than problem 2.

3. Write a story problem for the multiplication problem 2/3 x ½ , then show how to solve the

problem using a diagram or number line. (Hint: We can interpret this as partitioning ½ into 3

equal parts and taking 2 of them. Or we can think that the area that represents one of the

fractions becomes the whole or unit for the other fraction)

4. Write a story problem for the multiplication problem .

a. Show how to solve the problem using a diagram or number line.

b. Use the distributive property or FOIL to calculate by rewriting this product

as (2 + 1/3) x (1 + ½)

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c. Identify the four terms produced by the FOIL in part b and identify them in your work

for part a.

d. Discuss why the answer is NOT 2 1/6

e. Now write the mixed number 2 1/3 and 1 ½ as improper fractions and use the

procedure for multiplying fractions to calculate . How do you see the product of

the numerators in your picture in part a? How can you see the product of the

denominators in your picture in part a?

II. Definition of fraction multiplication and properties of fraction multiplication

A. Let a/b and c/d be fractions. Then

B. Properties

1. Closure: the product of two fractions is a fraction

2. Commutative: a/b x c/d = c/d x a/b

3. Associative: (a/b x c/d) x e/f = a/b x (c/d x e/f)

4. Multiplicative Identity: a/b x 1 = 1 x a/b = a/b

5. Multiplicative Inverse (reciprocal): for every non-zero fraction a/b there

exits a unique fraction b/a such that .

This is a very useful property in algebra:

7. Distributive property over addition and subtraction

a/b(c/d + e/f) = ac/bd + ae/bf

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7. Multiplication by zero: a/b x 0 = 0 x a/b = 0

III. Division of fractions – tough concept to model in a meaningful way. Invert and multiply is a

rote procedure that does not provide students with any meaningful concept of fraction division.

There are several, more meaningful ways to teach fraction division than just asking student to

memorize a rule.

A. Provide students with ‘Developing an understanding of fraction division’ (adapted

from activities in Beckmann, S. (2005). Class Activities Accompany Mathematics for

Elementary Teachers. Boston, MA: Pearson/Addison Wesley.)

Developing an understanding of fraction division

1. a. Write a “how many groups?” (measurement model) story problem for .

b. Write a “how many groups?” story problem for . (If it works, you may modify your

story problem from part (a).)

c. Assume that you don’t know the “invert and multiply” method for dividing fractions.

Solve your story problem for in a simple and concrete way, for example, by drawing

pictures. Explain your solution.

d. Use the “invert and multiply” procedure to calculate . Verify that the answer agrees

with your answer in part c.

2. Tonya and Chrissy are trying to understand using the following story problem: One

serving of rice is of a cup. I ate 1 cup of rice. How many servings of rice did I eat?

To solve the problem Tony and Chrissy draw a diagram of a large cup divided into three equal pieces, and the shade two of those pieces.

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Tonya says, “There is one 2/3-cup serving of rice in 1 cup and there is 1/3 cup of rice left over,

so the answer should be .”

Chrissy says, “The part left over is 1/3 cup of rice, but the answer should be . Did we do

something wrong?” Help Tonya and Chrissy.

3. a. Write a “how many groups?” story problem for . Choose your story problem so that a mixed-number answer makes sense.

b. Write a “how many groups?” story problem for . (If it works, you may modify

your story problem from part (a).)

c. Assume that you don’t know the “invert and multiply” method for dividing fractions.

Solve your story problem for in a simple and concrete way, for example, by drawing

pictures. Explain your solution.

d. Use the “invert and multiply” procedure to calculate . Verify that the answer agrees

with your answer in part c.

4. Do steps b, c, and d above for the problem .

5. Write a “how many in one group?” (sharing model) story problem for . Using this as a

model, write a “how many in one group?” story problem for and show how to solve your

problem in a simple and concrete way. Check by using “invert and multiply.”

6. Write a “how many in one group?” story problem for . Choose your story problem so that a mixed number answer (in this case a fraction answer) makes sense. Use this as a model for

writing a story problem for and show how to solve your problem in a simple and concrete

way. Check by using “invert and multiply.”

B. Common denominator approach

1. Allows division of fractions to be viewed as an extension of whole number

division

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2. can be thought of as: How many groups of 2/3 are there in 8/3?

3. Area model

4. Number line model.

5. Rule: Let a/b and c/b be fractions. Then

C. Fractions with out common denominators

1. Find a common denominator, then use rule from above.

2. Rule: Let a/b and c/d be fractions. Then

Note: d/c is the reciprocal of c/d (this is where invert and multiply comes

from.)

3. Example:

E. Maintaining the quotient (a little known property)

a. If , then

b. So

c. Example:

E. Missing factor approach to fraction division

1. If with b, d, and f not equal to zero, then it must be true that

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. Note that in the first problem, a/b is dividend, c/d is divisor, and e/f

is the quotient. Recall from the study of whole numbers, that

2. Examples using missing factor approach

a.

b.

F. Divide numerators, divide denominators approach

1. , becomes a complex fraction

a. complex fractions

2.

3.

4. When is this method efficient? Would you teach this method to your students?

IV. Wrapping up division of fractions

A. For all whole numbers a, b, (b not equal to zero),

B. Mental arithmetic (p. 251)

C. Estimation (p. 251)

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V. Common Error Patterns for Fractions

A. Present students with examples of students’ incorrect work with the definition of

fractions and operations on fractions

B. Students determine the error pattern for the various examples

References

Ashlock, R. B. (2002). Error patterns in computation: Using error patterns to improve

instruction (8th ed.). Upper Saddle River, NJ: Pearson Education, Inc.

Beckmann, S. (2005). Class activities to accompany mathematics for elementary teachers.

Boston, MA: Pearson/Addison Wesley.

Musser, G. L., Burger, W. F., & Peterson, B. E. (2003). Mathematics for elementary teachers:

A contemporary approach (6th ed.). New York, NY: John Wiley & Sons, Inc.

Litwiller, B., & Bright, G. (Eds.). (2002). Making sense of fractions, ratios, and proportions:

2002 Yearbook. Reston, VA: NCTM.

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Appendix E

Field Experience Journal

MA318 – Spring 2005

(Worth 80 points)

We will be conducting an after school mathematics program for 5th and 6th grade students at the

Cape Central Middle School on Caruthers Street on the following Wednesday dates: 2/9, 2/16,

2/23, 3/2, 3/9, 3/23, 3/30, 4/6, and 4/13. You will need to report to the Middle School library

each date by no later than 3:20 pm. We will work with the students from 3:30 until 4:15 pm. If

you need transportation to the school, let Ms. McAllister know as soon as possible. You are

required to attend eight of these sessions for full credit for this assignment. I would prefer that

everyone come every time so as not to disappoint the students or cause a hardship to the other

student teachers who are working with the middle school students. Take your professional

responsibilities very seriously. MA318 students who completes at least eight of the nine

sessions will receive a letter from the school principal recognizing their participation. This letter

may be used later for your resume as teaching experience.

I will assign two or three middle school students to each college student for work on solving

problems and playing games using fractions. You and another student may wish to combine your

groups. We want the program to be motivational, interesting, and fun, as well as educational, for

the middle school students and for you. You do not need to plan anything for the sessions. I will

provide the problems, activities, games, etc. If you do have an activity that you would like to

share, just let me know. This is a very flexible program and I’ll be happy to include material that

you want to bring in.

As soon as possible after each session, you are required to write a short reflection (see format

below) on the experience you had working with the 5th and 6th graders. You will also write a 2 –

3 page final reflection about your experience. You should turn in your reflections the next

Tuesday class meeting after the Wednesday session. For example, after the first session on

February 9, a reflection on that session is due February 15. At the end of the after school

program on Wednesday, April 13, you should complete your final session reflection as well as an

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overall reflection on your experiences with the after school program. All reflections (including

the final reflection) are due on April 21. Reflections should be typed in either 10 or 12-point

font. Be sure to spell check and use proper grammar and punctuation. You may print these off

and turn them in as a hard copy or e-mail them to me as an attachment. We may also explore the

DROP BOX feature of the class website. If you decide to use the electronic option, then the

document needs to be Saved as either a WORD.doc or as a RichText.rtf file. My computer

cannot read Microsoft Works files.

Here is the format for the session reflection:

Your Name ________________________________ Section of MA318 _____________

Date of after school session ____________________ Number of students you worked with ____

1. What activities did you and your students do today?

2. What successes did you see with your students?

3. What problems did your students have with the activities or concepts?

4. What did you learn about teaching students from today’s experience?

5. What did you learn about math from today’s experience? Did the material we covered in class

prepare you to work with the fraction problems/activities you worked with today?

6. Any other thoughts or observations? Any suggestions on how to make the program better?

For the final reflection, it might be helpful for you to go back and read your earlier reflections,

then write an essay about what you learned from the experiences. Did your experiences help you

increase your understanding of fractions? Did your experiences help you better understand how

children learn mathematics? Share whatever you feel will give me an idea of how combining a

specific subject content course (MA318) with a teaching field experience (this after school

program) designed to cover the same content might improve the experience student teachers

have as they take their block and content courses.

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Appendix F

MA318 – Outside reading assignmentYou are required to read 8 selections from the list below over the period of time beginning February 7 and ending April 15. The readings are on reserve in Kent Library or you may locate some of them through MOBIUS. There is no specified order and you may choose whichever eight selections you wish to read from the list provided. As evidence to me that you have read the selections I require that you turn in a short summary for each article in the format indicated at the end of this handout. I would prefer that you turn in one summary per week during the period of time indicated above. The purpose of the readings is to give you a broader view of teaching fractions concepts to children. You may turn in additional summaries (above the required 8 selections) for 5 extra credit homework points each.

Reading Selections1. National Research Council. (2001). Developing proficiency with other numbers, Chapter 7, pp. 231-254 in Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, & B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

2. Ma, L. (1999). Generating representations: Division by fractions, pp. 55-83 in Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates, Publishing.

3. Carraher, D. W. (1996). Learning about fractions, Chapter 14, pp. 241-266 in L. P. Steffe & P. Nesher (Eds.), Theories of mathematical learning (pp. 449-463). Mahwah, NJ: Lawrence Erlbaum Associates, Publishers.

4. Lamon, S. J. (2001). Presenting and representing: From fractions to rational numbers, Chapter 12, pp. 146-165 in National Council of Teachers of Mathematics 2001 Yearbook: The Roles of Representation in School Mathematics.

5. Huinker, D. (1998). Letting fraction algorithms emerge through problem solving, Chapter 21, pp. 170-182 in National Council of Teachers of Mathematics 1998 Yearbook: The Teaching and Learning of Algorithms in School Mathematics.

6. Flores, A. (2002). Profound understanding of division of fractions, Chapter 25, pp.237-246 in National Council of Teachers of Mathematics 2002 Yearbook: Making Sense of Fractions, Ratios and Proportions.

7. Siebert, D. (2002). Connecting informal thinking and algorithms: The case of division of fractions, Chapter 26, pp. 247-256 in National Council of Teachers of Mathematics 2002 Yearbook: Making Sense of Fractions, Ratios and Proportions.

8. Smith, J. P. III. (2002). The development of students’ knowledge of fractions and ratios, Chapter 2, pp. 3-17 in National Council of Teachers of Mathematics 2002 Yearbook: Making Sense of Fractions, Ratios and Proportions.

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9. Taber, S. B. (2002). Go ask Alice about multiplication of fractions, Chapter 7, pp. 61-71 in National Council of Teachers of Mathematics 2002 Yearbook: Making Sense of Fractions, Ratios and Proportions.

10. Sinicrope, R, Mick, H. W., & Kolb, J. R. (2002). Interpretations of fraction division, Chapter 17, pp. 153-161 in National Council of Teachers of Mathematics 2002 Yearbook: Making Sense of Fractions, Ratios and Proportions.

11. Sharp, J. M., Garofalo, J, & Adams, B. (2002). Children’s development of meaningful fractions algorithms: A kid’s cookies and a puppy’s pills, Chapter 3, pp. 18-28 in National Council of Teachers of Mathematics 2002 Yearbook: Making Sense of Fractions, Ratios and Proportions.

12. Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational number concepts (pp. 91 – 126) in R. Lesh & M. Landau (Eds.), Acquisition of mathematical concepts and processes. New York: Academic Press.

Summary FormatSummaries should be typed, double spaced in either 10 or 12 point font. Be sure to spell check and use proper grammar and punctuation. You may print these off and turn them in as a hard copy or e-mail them to me as an attachment. If you decide to use the electronic option, then the document needs to be Saved as either a WORD document (.doc) or as a Rich Text File(.rtf). My computer cannot read Microsoft Works files.

The summaries should contain the following:

Your Name: ____________________ Section or time: __________ Date: __________

Name of Reading

Author:

1. (In one or two paragraphs, summarize the main ideas of the article)

2. (Write at least one question you had about what you read)

3. (Write one idea or concept you really agreed with in the article)

4. (Write about how this article would or would not change the way to teach fractions)

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Appendix GAfter School Mathematics Program Curriculum

Session Learning Objective

Skill Builder Activity Challenge Problem

Practice Activity ExtraActivity

1 Development of part/whole definition of fractions

Lessons 3-5 from Cramer, et al., Level 1*

Benny’s Bakery

Fraction Capture Game from Everyday Mathematics

2 Equality of fractions

1) Lesson 8 from Cramer, et al., Level 1*2) Equivalent fraction chart

Fair share candy bar problem

1) Fraction Fill game2) Fraction Concentration game

3 Ordering fractions

Lessons 6 and 11 from Cramer, et al., Level 1*

Exploring shaded regions models of fractions

Fraction War game Fraction Circle Activity

4 Mixed numerals and improper fractions

Lesson 17 from Cramer, et al., Level 1*

That’s the way the cookie crumbles

Fraction Concentration game with improper and mixed fractions

Fractions with Tangrams Activity

5 Review of equivalent fractions


Break the code on equivalent fractions

Students may choose any of the games introduced so far

6 Addition and subtraction of fractions


Problem solving with Dr. Loyd’s Fraction Kit

Lesson 19 Student Page A, Cramer, et al., Level 1*

7 Addition and subtraction of fractions


Pattern Block Problem #1

Fraction Addition and Subtraction Worksheet

Fraction Squares A and B

8 Multiplication of fractions


Pattern Block Problem #2

Fraction Multiplication Worksheet

Fraction Squares C and D

9 Locating fractions on a number line

Students will use sidewalk chalk, string, and paper clips to make a fraction number line

*Full citations in reference list at the end of this Appendix.

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Central Middle School After School Mathematics Program

Session 1 Lesson Plan

Collect from Ms. McAllister before the session begins: 1. Worksheets needed for the entire session2. Fraction circle kits3. Pencils4. Paper5. A pair of dice for each pair of students

Objective: To explore the definition of fractions with fraction circles and other 2-D region models. Note the following:

1. the whole must be well-defined2. each part must be of equal value3. the denominator of the fractional representation indicates the number of equal sized

pieces the whole has been ‘cut’ into4. the numerator indicates the number of equal sized pieces you are considering

Skill Builder Activity: Level 1, Lessons 3 – 5 from Cramer, K., Behr, M., Post T., Lesh, R., ( 1997) Rational Number Project: Fraction Lessons for the Middle Grades - Level 1, Kendall/Hunt Publishing Co., Dubuque Iowa.Accessed on 2/4/05 from http://education.umn.edu/rationalnumberproject/rnp1.html

1. Review the part/whole definition of fractions with the students. Give each student a set of fraction circles and encourage them to empty their packet on the table and assemble the circles of various colors. Start with Lesson 3, student page A. Work thru the worksheet with the students.

2. Give the students Lesson 5, Student Page D. Everyone do the ¼ problem together, then have the students do the rest of the sheet on their own, helping them as needed. Have everyone compare answers and work on any answers that do not agree.

3. Give students Lesson 4, Student Page G. Discuss each figure with the students and come to an agreement on what are the correct answers.

From these three activities, assess the students’ understanding of the part/whole definition of fractions. If they still seem not to understand the definition, review the worksheets just completed. Once the students seem to understand the definition, move on to the practice activity.

Practice Activity: Play Fraction Capture game. This game calls for two players, so if you have three students, you may assign the two that seem to have the best understanding to play against each other and you play against the weaker student OR you may team up your students against the students assigned to your co-teacher (assuming you have the same number of students) OR if you only have two students, watch them play and assist if needed. Play the game once. If the students like the game and there is time, they may play again either with the same partner or against a new partner. Be sure to leave time for the Team Problem.

Team Problem of the Day: Benny’s Bakery. Give the students a copy of the Team Problem. You may help them understand the problem, give hints, suggests, etc., but the idea is for the

http://education.umn.edu/rationalnumberproject/rnp1.html

http://education.umn.edu/rationalnumberproject/rnp1.html

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students to solve the problem. Try to stay out of it if they are doing ok with out you. Ask leading questions. You job is to coach and cheerlead!!!! Turn these into Ms. McAllister at the end of the session.

Take Home Activity: Students are to be encouraged to take home a copy of the Fraction Capture game to play with their family members.

Be sure to collect the following at the end of the session:1. Extra worksheets that were not used2. Fraction circle kits3. Pencils (any that I provided)4. Paper (any that I provided)5. A pair of dice

Reference to District Mathematics Curriculum Knowledge Strands: 1, 3, 11

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Name ____________________________________ Lesson 3, Student Page A

Name Fraction Amounts Using Fraction Circles

Use fraction circles to find the names of the different fraction pieces.

I. The white circle is the unit or whole. What fraction name can you give these pieces?

1 yellow __________________ 1 orange _________________

1 blue ____________________ 1 green __________________

1 red ____________________

II. Now make 1 orange piece the unit or whole. What fraction name can you give these pieces?

1 yellow _________________ 1 green __________________

III. Change the unit to 1 blue piece. What fraction name can you give 1 green piece?

_______________________

IV. Suppose the white circle is the whole or unit. Come up with two ways you could determine what size twelfth sized pieces would be.

V. Again suppose the white circle is the whole or unit. Determine what fraction would name each of the following collection of pieces.

What fraction is represented by 2 blue pieces? ________________________

What fraction is represented by 3 red pieces? ________________________

What fraction is represented by 2 green pieces? ________________________

What fraction is represented by 3 yellow pieces? ________________________

What fraction is represented by 5 green pieces? ________________________

What fraction is represented by 7 red pieces? ________________________

What fraction is represented by 3 blue pieces? ________________________

What two fractions are represented by 6 red pieces? ________________________

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Team Challenge Problem for Session 1

List Team Leader: ________________________________________

List Team Members: __________________________________

__________________________________

__________________________________

Problem: Each day Benny makes several rectangular sheet cakes, which he cuts into eights as shown.

He sells 1/8 of a sheet cake for $1.59. As part of a new promotional campaign for his store, he wants to cut his sheet cakes into eighths a different way each day. On the rectangle below draw as many ways as your team can think of to cut the cake into 8 pieces that represent the same amount of cake. You may put additional answers on the back of this page.

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Skill 2

Objective: Understanding of the equality of fractions and fraction equivalence

Supplies needed: One set of Dr. Loyd’s fraction kit per team Equivalent Fractions chart (one per student) Pencils and scratch paper

Fraction Fill Up game boards (one per student) Take-home activity packets (one per student)

Team builder activity (one per team) Fraction numeral cards (one set per team)

Skill Builder Activity: Work with Dr. Loyd’s fraction kit to fill out an equivalent fractions chart1. Supply your group with 1 Dr. Loyd’s fraction kit. Have the students sort the pieces,

then determine the fractional value of each color. Work through covering simplified fractions such as ½ and ¼, etc. with smaller pieces to determine equivalent or equal fractions. Discuss the fact that equal areas can be covered with and equal number of different sized so that each area has many different ways to represent it. Suggest that you can start with something like a cake or candy bar and cut it up in different ways.

2. When the equivalence chart is completed, look at the relationship between the numerators and denominators of the various equivalent fractions. See if the students can notice that if you multiply the numerator and denominator of a simplified fractions by the same factor, you get an equivalent fraction to the original, just cut into more pieces.

3. Ask how many equivalent fractions ½ has altogether (stressing that the chart doesn’t show them all.) Try to get them to say that a fraction can have an infinite number of equivalent fractions.

Practice Activities: Play the Fraction Fill Up game firstRules for Fraction Fill Up1. Prepare for students to play the game by selecting the following fraction numeral cards from your deck: ½, 1/3, 2/3, ¼, 2/4, ¾, 1/6, 2/6, 3/6, 4/6, 5/6, 1/12, 2/12, 3/12, 4/12, 5/12, 6,/12, 7/12, 8/12, 9/12, 10/12, 11/12. 2. Give each student a Fraction Fill Up game board and a pencil3. Mix up the fraction numeral cards and draw one. Show it to the players. Students must try to shade in that fraction or its equivalent on one of the circles on their game card. They can only shade 1 representation for the fraction amount. For example if the ¼ card is drawn, then the students may shade in ¼ of the circle that is divided into 4 equal pieces or 3/12 on the circle cut into 12 equal pieces. Students cannot add lines to any circle to make it the right sized pieces.4. Continue drawing a numeral card and having students shade areas if possible. 5. Students may use their fraction equivalence chart to help them select an area to shade.6. The first student who has been able to shade two complete circles says “Fraction Fill Up” and is the winner.

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You may choose to have students play Fraction Fill Up again or teach them to play Fraction Concentration. Rules for Fraction Concentration

1. Materials: Select the following Fraction Cards from those provided in the deck of numeral cards (or you may choose other pairs of equivalent fractions to use, but the equivalent fractions must be chosen in pairs so that all of the cards will have a ‘match’.

2. Purpose of Activity: To help student identify equivalent fractions and improve memory skills.3. Directions: Game may be played by two to four players. Shuffle the cards and deal them face-down into four rows of five cards in each row. The player to the left of the dealer turns any two of the cards face-up in place. If the two cards show equivalent fractions, the player takes them. She then plays again. If the cards do not show equivalent fractions, they are turned, face-down, in the same places. The next player to the left then plays. The game ends when all of the cards have been paired. The winner is the player with the most pairs. Players who pay attention and remember the placements of the exposed cards have an advantage. The number of cards used can be varied.

Team Problem of the Day: Fair Share problem

Take-home Activity: Directions to make a set of fraction cards and rules for fraction matching games

Reference to District Mathematics Curriculum: 1, 3, 11

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List Team Leader: ________________________________________


__________________________________

__________________________________

Problem: Eddie, Sam, Jason, and Ben are best friends. One day at lunch Eddie, Sam and Jason are surprised to find their moms all packed them the same kind and size of Hershey’s chocolate bar. Ben’s mom forgot to send a treat. The boys with candy bars felt bad because Ben didn’t have any chocolate, so Sam said, “I know, since three of us have a candy bar, if each of us that have a candy bar cuts it into thirds and each of us give Ben 1/3 of our candy bar it will be fair.” Eddie agrees with Sam, but Jason says, “That’s not right. If we do that, Ben will have more candy than we will.” Draw a picture below to determine which of Sam or Jason is correct. If Jason is correct, suggest a way the candy bars could be divided so that each boy gets a fair and equal share of the candy bars.

What if there were 5 friends and 4 candy bars? Six friends and 5 candy bars? Suggest a general rule for how this type of situation could be solved when there is always one less candy bar than person to share it.

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Skill 3

Supplies needed: Fraction circles or Dr. Loyd’s fraction kit Worksheet on ordering fractions (one for each student) Scratch paper and pencils Fraction numeral cards Fraction Circle Take Home Activity (one for each student)

Objective: To help students identify which of two fractions is larger.

Skill Builder Activity: Ordering fractions worksheet

Practice Activity: Fraction War

Purpose of Activity: To help student identify equivalent fractions and determine which of two fractions is larger.

Materials: Use the Fraction Numeral Cards (those that name fractions between and including 0 thru 1), a pencil and paper for each player to use for computation (if needed), one set of Dr. Loyd’s fraction kit, and the fraction equivalence chart.

Directions: The game may be played by two to four players. The deck is shuffled and dealt face down to the players in turn, until all the cards have been distributed. Each player keeps his or her cards face down in front of him. Each player turns his top card face up. The players must then decide which card shows the fraction having the greatest value. The students may use the equivalence chart, the fraction manipulatives, and scrap paper to do this. The player whose card has the fraction with the highest value takes all of the cards that were turned up and places them face down at the bottom of his pack of cards.

In the event of a tie (two equivalent fractions are turned up at the same time) a “war” is declared. Each of the players involved in the war places the next three cards from his pack face down. He turns over the fourth card. The person with the fraction of highest value now showing takes all the cards involved in the “war”. Play continues in a similar manner until one player has lost all his cards, or until time is called. In this latter case, the winner is the player with the most cards at the end of the time period.

Team Problem of the Day: A Sampler of Fraction Problems.

Take-home Activity: Fraction Circle Activity


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Directions: For each pair of fractions given, use either the fraction circles or Dr. Loyd’s fraction kit to represent each fraction and determine which fraction is larger. Circle the larger fraction.

From the problems above, what can you say about comparing fractions that have the same denominator?

From the problems above, what can you say about comparing fractions that have the same numerator?

What do you notice about each of the fractions above? How does this help you come up with a rule for deciding which of the fractions is larger?

For the next set of fractions, determine if each fraction is larger or smaller than ½ and use that information to decide which fraction is larger.

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List Team Leader: ________________________________________


__________________________________

__________________________________

Problem 1: The shaded regions in the figures represent the fractions ½ and 1/3. In the blank figures provided show (1) ½ different from the one shown, (2) 1/3 different from the one shown, (3) 1/4, (4) 1/6, (5) 1/12. Be sure to label your answers.

Problem 2: Find a fraction less than 1/12. Find another fraction less than the fraction you found. Can you continue this process? Is there a “smallest” fraction greater than 0? Explain

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Skill 4

Supplies needed: Fraction circles or Dr. Loyd’s fraction kit (Optional) Worksheets A, B, and C on Mixed Numerals (one for each student) Scratch paper and pencils Fraction numeral cards TangramTake Home Activity (one for each student)

Objective: To help students work with fractions larger than 1. To write improper fractions as mixed numerals and mixed numerals as improper fractions.

Skill Builder Activity: Worksheets A, B, and C on Mixed Numerals

Practice Activity: Rules for Advanced Fraction Concentration

1. Materials: Use all of the Fraction Cards in the deck of numeral cards (or you may choose which pairs of equivalent fractions and mixed numeral/improper fraction pairs to use, but the fractions must be chosen in pairs so that all of the cards will have a ‘match’).2. Purpose of Activity: To help students identify equivalent fractions smaller and larger than 1, to help students work back and forth between mixed numerals and improper fractions, and improve memory skills.3. Directions: Game may be played by two to four players. Shuffle the cards and deal them face-down into rows and columns. The player to the left of the dealer turns any two of the cards face-up in place. If the two cards show equivalent fractions, the player takes them. She then plays again. If the cards do not show equivalent fractions, they are turned, face-down, in the same places. The next player to the left then plays. The game ends when all of the cards have been paired. The winner is the player with the most pairs. Players who pay attention and remember the placements of the exposed cards have an advantage. The number of cards used can be varied.

Team Problem of the Day: Baker’s Dozen problem.

Take-home Activity: Tangram Activity


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List Team Leader: ________________________________________


__________________________________

__________________________________

Problem: You bought a baker’s dozen (13) cookies at Benny’s Bakery. You want to share the cookies equally with your family. How many cookies will each person get if:

Number of people in the family Number of cookies for each family member

2

3

4

5

6

7

N

How many cookies would each person get if three people shared twenty cookies?

How many cookies would each person get if eight people shared twenty cookies?

How many cookies would each person get if P people shared twenty cookies?

What is a rule for finding the number of cookies each person will get if P people share C cookies?

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Skill 5

Supplies needed: One set of fraction circles. One set of Dr. Loyd’s fraction kit.Fraction numeral cardsWorksheets A1, A2, B, C for each studentScratch paper and pencils

Objective: To review fraction equivalence and develop an algorithmic ‘rule’ for finding equivalent fractions.

Skill Builder Activity: 1. Review with the students the concept of equivalent or equal fractions. Use manipulatives

to make up several examples. See if they can create a rule for how to find the equivalent fractions for ½, 1/3, 2/5 WITHOUT using the manipulatives. If they can’t, go on to #2.

2. Do Worksheets A1, A2. They may use either fraction circles or Dr. Loyd’s kit for this IF THEY NEED TO. Encourage them to find a pencil and paper or mental way to do this.

3. Now can they develop a rule for doing this with just the numbers?4. Do worksheet C

Practice Activity: Play either fraction concentration or fraction war – students choose.

Team Problem of the Day: Student worksheet B. Each student will have a copy of the worksheet. Divide up the problems so that each student is responsible for some of them. Faster students should be encouraged to work additional problems to help check their teammates work. Put all of the final work on ONE worksheet to turn in for points for this week’s challenge. Be sure they work quietly on breaking the final code. It would be easy for another team to overhear the answer and STEAL their work. This is a secret code!!!!

Take-home Activity: none this week


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Skill 6

Supplies needed: Fraction Addition and Estimation Worksheet (1 per student) Fraction Circles Dr. Loyd’s fraction kit (only one per table please) Scratch paper and pencils

Objective: To use manipulatives and story problems to introduce the concept of fraction addition. NOTE: The objective here is NOT to teach an algorithm yet. They are to develop a sense of the size of the fractions and get an answer from modeling with manipulatives or drawing pictures. The algorithm will be introduced in a future session after they have had the opportunity to explore these problems from an informal perspective.

Skill Builder Activity: A. Read the following word problems to the students. Have them estimate (or guess) the answer before you do anything else. You might guide them thru a set of questions such as: Is the sum less than ½ or bigger than ½? Is the sum less than 1 or bigger than 1? Have them model the problems with either fraction circles or Dr. Loyd’s fraction kit to find the correct answer.

1. William ate ¼ of a pizza for dinner. The next morning he ate a piece that equaled 1/8 of the pizza. How much of a whole pizza did he eat?

a. estimate the answerIs the sum less than ½ or bigger than ½? Is the sum less than 1 or bigger than 1?

b. model with manipulativesc. find the correct answer in simplest form

2. Maria received a chocolate chip cookie as big as a birthday cake for a present. She cut it into 6ths and shared the cookie with her friend LeAnna. Maria ate 3/6 of the cookie, LeAnna ate 1/3 of the cookie. Together, how much of a cookie did they eat?



3. Martin was making playdough. He added ¾ cup of flour to the bowl. Then he added another 3/6 cup. How much flour did he use?



Go to the Practice Activity when you feel the students are ready to work independently. You may make up some additional problems if you feel the students need more guidance.

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Practice Activity: Give students The Fraction Addition and Estimation worksheet to work on independently. Help where needed. If there is time left over the students may work on the Search for Wholes puzzle.

Team Problem of the Day: Find the Whole from the Part

Take-home Activity: Search for Wholes Puzzle (may be started during your session)


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Frack Jack (Adapted from an activity created by Mary Simon)

Materials: 4 sets of fractions circles Two decks of fraction numeral cards (use the 0/2, 1/2, 0/3, 1/3, 2/3, 0/4, ¼, 2/4, ¾, 0/6,

1/6, 2/6, 3/6, 4/6, 5/6, 0/8, 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8 cards from each deck to create a deck of 46 cards)

Scratch paper and pencils

Rules for Frack Jack1. The tutor is the dealer.2. The game plays in a similar manner to blackjack, except the object of the game is to get as close to 1 (or whole) as possible without going over.3. Each player should have a white whole circle in front of them. 4. Place the fraction pieces in a pile in the center of the table for everyone to reach and use.5. The dealer shuffles the cards and deals one card to each player face up.6. Each player takes fraction pieces to model the fraction on their face up card.7. Then the dealer asks each player if he/she wants to ‘Stick’ or ‘Hit me’. Stick means they don’t get another card and they are playing with the fraction they currently have showing. ‘Hit me’ means they receive another card and they must add that fraction piece to the one they already have showing. A player goes ‘Bust’ if his/her fraction pieces add up to more than a whole. If the players pieces exactly form a whole they get a ‘Frack Jack’. 8. This continues until all players are either Sticking, Bust, or have made a Frack Jack.9. If no one goes Frack Jack, the player with the fraction sum closest to 1 wins the round.10. Players start with 10 points.

If they go Bust they lose 3 points. If they get a Frack Jack, they gain 5 pointsIf they win a round they get 2 points. If there is a tie for a round, the players split the

round and get 1 point each.

(If your students have played this for a while with the fraction circles, switch to the Dr. Loyd’s fraction kits and add additional fraction numeral cards to the set for fifths, tenths, twelfths, and fifteenths.)

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List Team Leader: ________________________________________


__________________________________

__________________________________

Problem Solving and Dr. Loyd’s Fraction Kit

I. Find the unit (whole) given the following information. Explain how you solved the problem. [You may want to draw pictures.]

a) The pink piece is ¼ of some amount. Find that amount. ___________________

b) The pink piece is 1/6 of some amount. Find that amount. ____________________

c) The smallest dark green piece is 1/3 of some amount. Find that amount. _________________

II. If the pink piece is ¼ what value do these pieces have. Explain your reasoning.a) 1 orange piece b) 1 small dark green piece c) 1 smoky gray piece

Extra challenge: If the smoky gray piece is 2/3 what value does one pale green piece have?

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Skill 7

Supplies needed: Fraction Operations: Finding the Exact Answer (1 per student) Fraction Circles Dr. Loyd’s fraction kit (only one per table please) Scratch paper and pencils You will need ONE set of pattern block for the Team Challenge

Objective: To use manipulatives and story problems to introduce the concept of fraction subtraction. NOTE: The objective here is NOT to teach an algorithm yet. They are to develop a sense of the size of the fractions and get an answer from modeling with manipulatives or drawing pictures. The algorithm will be introduced in a future session after they have had the opportunity to explore these problems from an informal perspective.

Skill Builder Activity: A. Read the following word problems to the students. Have them estimate (or guess) the answer before you do anything else. You might guide them thru a set of questions such as: Is the difference less than ½ or bigger than ½? Then have them model the problems with either fraction circles or Dr. Loyd’s fraction kit to find the correct answer. Be sure to help them reduce their answers to simplest form.

4. Alice noticed that there was ¾ of a pizza left after the party. She ate a slice of pizza that was the size of 1/8 of a whole pizza. How much pizza was left after Alice ate a slice? (Note: this is a take-away context)

a. estimate the answerIs the difference less than ½ or bigger than ½?


5. Martin was making playdough. He poured 5/6 cup of flour into the bowl. Then he decided that was too much flour and took 1/3 cup out of the bowl. How much flour did he use? (Note: This is a take-away context.)



6. Joe and Renata each receive the same allowance. Joe spent 2/3 of his allowance on records. Renata spent 1/6 of her allowance repairing her bicycle. How much more did Joe spend than Renata? (Note: this is a comparison context)



7. Bobby drank 5/8 of a gallon of milk. Joe drank ¾ of a gallon of milk. Which boy drank the most milk? How much more? (Note: This is a comparison context.)

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Practice Activity: Give students The Fraction Operations: Finding the Exact Answer worksheet to work on independently. Help where needed.

If there is time left over the students may play Frack Jack for addition or the subtraction version:To play Frack Jack for subtraction, the rules are the same except that each card that is

turned up is subtracted from 1 whole. The object is to get as close to zero as possible without trying to take away more of a fraction than you have left.

Team Problem of the Day: Pattern Block Area Fraction Puzzles

Take-home Activity: Fraction Squares A and B


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Fraction Operations: Finding the Exact Answer

1. Joes lives 4/10 of a mile from school. Mary lives 1/5 of a mile away. How much farther from school does Joe live than Mary? Draw a picture to show what you did.

2. Because of a rainstorm the water level in a swimming pool rose by 9/12”. The following day it dropped by 4/6”. What was the total change in the water level? Draw a picture to show how you found the answer.

3. Veronica pent 1/3 of her allowance on a CD and ½ of her allowance on a movie. What fraction of her allowance did she have left? Draw pictures to show what you did.

4. A clerk sold three pieces of ribbon. The red pieces was 1/3 of a yard long. The blue piece was 1/6 of a yard long. The green pieces was 10/12 of a yard long.

a. How much longer was the green ribbon than the red ribbon?

b. How much longer was the green ribbon than the blue ribbon?

Page 161

c. Is the red ribbon and blue ribbon together greater than, less than or equal in length to the green ribbon?

d. If the red and blue together is greater than the green, how much greater? If shorter, how much shorter?

5. Review of addition and subtraction. With either fraction circles or Dr. Loyd’s fraction kit to help you, find the exact answers to the following problems.

a.

b.

c.

d.

e.

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List Team Leader: ________________________________________


__________________________________

__________________________________

Pattern Block Area Fraction Puzzles

Use pattern blocks to solve each of the area fraction puzzles below. Draw each solution on pattern block paper. Label each color with what fraction of the whole shape that color represents.

1. Build a parallelogram with an area that is one-third green, one-third blue, and one-third red.

2. Build a parallelogram with an area that is one-eighth green, one-half yellow, one-eighth red, and one quarter blue.

3. Rebuild each of the above puzzles in a different way.

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Fraction square

Page 164

Central Middle SchoolAfter School Mathematics Program

Skill 8

Supplies needed: Fraction Multiplication worksheet (1 per student) Fraction Circles Dr. Loyd’s fraction kit (only one per table please) Scratch paper and pencils

Objective: To use manipulatives and story problems to introduce the concept of fraction multiplication. NOTE: The objective here is NOT to teach an algorithm yet. They are to develop a sense of the size of the fractions and get an answer from modeling with manipulatives or drawing pictures. The algorithm may be used by students that already know it. We want the students to explore these problems from an informal perspective.

Skill Builder Activity: A. Read the following word problems to the students. Have them estimate (or guess) the answer before you do anything else. You might guide them thru a set of questions such as: Is the product less than 1 or bigger than 1, less than ½ or bigger than ½? Then have them model the problems with either fraction circles, Dr. Loyd’s fraction kit, or a drawing of their own to find the correct answer.

1. John has a rectangular garden. He is going to plant flowers on half of his garden. He wants 1/3 of the flowers to be roses. What fraction of the whole garden will be roses?

d. estimate the answerIs the product more than 1 or less than 1? less than ½ or bigger than ½?

e. model with manipulatives or a drawingf. find the correct answer in simplest form

2. Mark, Allie, and I ate ¾ of a pizza. My share was 1/3 of that amount. What fraction of a whole pizza did I eat?

a. estimate the answerIs the product more than 1 or less than 1? less than ½ or bigger than ½?

b. model with manipulatives or a drawingc. find the correct answer in simplest form

3. The McAllister family had 2/3 of a sheet cake left over from a birthday party. Today Jackie ate ¼ of the cake that was left over. How much of the whole cake did she eat?

a. estimate the answerIs the product more than 1 or less than 1? less than ½ or bigger than ½?

b. model with manipulatives or a drawingc. find the correct answer in simplest form


Page 165

Practice Activity: Give the students The Fraction Multiplication worksheet to work on independently. Help where needed.

If there is time left over the students may play Frack Jack for addition or the subtraction version:To play Frack Jack for subtraction, the rules are the same except that each card that is

turned up is subtracted from 1 whole. The object is to get as close to zero as possible without trying to take away more of a fraction than you have left.

Team Problem of the Day:

Take-home Activity: Fraction Squares C and D


Page 166

Fraction Multiplication

Solve each of the following by drawing a picture to represent the solution.

1. Tom earned some money babysitting for the Ellis family. He put 2/3 in the bank. He spent ½ of what was left on admission to the museum. What fraction of the babysitting money did he spend at the museum?

2. The length of a book shelf is ¾ of a yard. Maria put her tape deck on 1/3 of the book shelf. What fraction of a yard did her tape deck cover?

3. Study hall lasts 5/6 of an hour. Molly spent ½ of her time in study hall on her math lesson. What fraction of an hour did she spend on math?

4. Jane buys 4 whole pizzas for a party. Her best friend Brittany gets the chicken pox and can’t come to the party. Jane feels bad for Brittany, so she takes 1/3 of each pizza and send it home with Brittany’s sister to make Brittany feel better. How much pizza did Brittany get? How much pizza does Jane have left to feed to the rest of her guests?

5. There are 15 students in Ms. Cannon’s class. 2/3 of them are girls. ½ of the girls are on the girls’ volleyball team. What fraction of Ms. Cannon’s class is on the girls’ volleyball team? How many students are on the girls’ volleyball team?

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List Team Leader: ________________________________________


__________________________________

__________________________________

Pattern Block Area Fraction Puzzles

Use pattern blocks to solve each of the area fraction puzzles below. Draw each solution on pattern block paper. Label each color with what fraction of the whole shape that color represents.

1. Build a trapezoid with an area that is one-tenth green and nine-tenths red.

2. Explain why it is not possible to build a parallelogram with an area that is one-half yellow, one-third green and one-quarter blue.

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Fraction Square

Page 169


Skill 9

Send a student to me to pick up these needed supplies for the project: 1 pieces of string per group 4 pieces of sidewalk chalk 12 paper clips

Objective: To use what the students have learned about fractions to construct a number line from zero to two. Various fractions will be located and placed on the number line. Several addition and subtraction problems will be solved using the number line model.

Directions for the Fraction Number Line Activity.1. Depending on the weather, your group will be assigned either a section of concrete sidewalk or a large piece of paper taped to the floor.2. Once your group has all of the needed supplies do the following:

a. Using the piece of string to help guide you, draw as straight a line as possible on your ‘sidewalk’. Position the line near the top of your ‘sidewalk’. Label a point on the left end of your line as 0 and the point on the right end of your line as 2. Figure out how to locate the point that would represent 1. Be as accurate as you possibly can be with the string and paperclips to help you.

b. Now use the string to subdivide your 0 to 2 number line into halves, thirds, fourths, sixths, and eights. Label each of the following fractions

(Note: some of the points on the line will have more than one ‘label’. Why is this?)

3. Once you have your number line completed, use it to find the answer to the following problem and write them on the ‘sidewalk’ under your number line:

4. Extra Challenge:

A. Locate the following on your number as accurately as possible:

B. Which is bigger, 4/5 or 4/6? Which is bigger, 5/8 or 4/5?


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References

Cramer, K., Behr, M., Post, T., & Lesh, R. (1997). Rational Number Project: Fraction Lessons for the Middle Grades – Level 1, Kendall/Hunt Publishing Co., Dubuque, Iowa. Retrieved on 2/4/05 from http://education.umn.edu/rationalnumberproject/rnp1.html

Cramer, K., Behr, M., Post, T., & Lesh, R. (1997). Rational Number Project: Fraction Lessons for the Middle Grades – Level 2, Kendall/Hunt Publishing Co., Dubuque, Iowa. Retrieved on 2/4/05 from http://education.umn.edu/rationalnumberproject/rnp2.html

The Math Forum (1994-2005). Fraction Squares. Retrieved 3/10/2005 from http://mathforum.org/paths/fractions/fracsqrs.html

Simon, M. (2003). Frack Jack: A magnet game to teach fractions. Retrieved March 14, 2005 from www.raft.net.

Teacher’s Guide to Games: An Everyday Mathematics Supplement (Grades K-6). (2003). Chicago, IL: SRA/McGraw-Hill.

NCTM: Illuminations (n.d.) Understanding rational numbers and proportions. Retrieved 9/9/04 from http://illuminations.nctm.org/print_lesson.aspx?id=284

http://education.umn.edu/rationalnumber

http://education.umn.edu/rationalnumber

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Appendix H

Raw Data

Section ID no. Tutoring Score

1Score

2Score

3Score

4cum hour

ACT sub

math 118

math 318

2 1 1 51 60 58 58 25 27 3 42 2 0 51 57 57 56 70 20 4 32 3 0 51 60 62 59 61 19 4 32 4 1 52 57 61 60 31 29 4 42 5 0 30 58 46 46 70 20 3 32 6 0 50 58 55 51 15 22 4 32 7 0 50 58 59 51 54 21 3 42 8 1 48 61 63 63 31 24 4 42 9 0 45 55 47 54 33 24 4 42 10 1 25 43 43 50 98 18 3 32 11 0 33 45 46 41 42 18 3 22 12 0 40 47 52 49 40 18 3 32 13 0 35 50 54 47 80 19 3 32 14 1 49 57 57 57 65 20 3 42 15 0 53 56 61 62 60 27 4 42 16 0 20 26 29 29 62 18 2 22 17 0 48 49 56 50 45 17 3 32 18 1 42 51 52 53 39 17 3 32 19 0 50 57 62 61 21 25 3 42 20 1 34 55 48 41 33 18 4 32 21 0 33 49 56 48 73 3 42 22 0 37 38 45 44 110 16 2 22 23 0 39 55 61 60 41 26 4 42 24 1 46 58 61 57 15 23 4 42 25 0 52 63 62 56 24 29 4 42 26 0 43 54 61 56 45 25 4 43 27 0 47 55 59 57 66 21 2 33 28 1 52 55 60 56 24 24 3 43 29 0 46 60 62 62 34 25 4 43 30 1 34 43 50 48 81 4 33 31 0 46 62 52 59 91 21 2 33 32 0 39 56 53 43 15 21 3 13 33 1 27 51 46 50 76 17 3 23 34 0 39 49 48 51 73 20 2 13 35 1 35 47 48 57 64 17 4 33 36 0 35 48 52 55 48 19 2 33 37 1 50 58 59 58 50 23 3 43 38 0 18 37 31 39 28 3 33 39 0 45 54 57 54 15 27 4 43 40 1 49 53 55 58 23 23 4 33 42 0 44 50 60 55 43 23 3 3

Section ID Tutoring Aiken Aiken Aiken Aiken Semantic Semantic Semantic Semantic

Page 172

no. 1 2 3 4 Diff 1 Diff 2 Diff 3 Diff 42 1 1 68 67 67 67 70.00 84.00 85.00 49.002 2 0 45 40 50 41 56.00 48.00 52.00 90.002 3 0 74 81 76 83 75.00 70.00 78.00 54.002 4 1 86 72 85 85 90.00 78.00 54.00 50.002 5 0 47 52 46 63 23.00 37.00 60.50 38.002 6 0 60 59 61 69 52.00 54.00 52.00 52.002 7 0 85 81 80 84 73.00 64.00 72.00 23.002 8 1 92 92 92 80 52.00 54.00 56.00 44.002 9 0 75 79 76 79 81.00 75.00 69.00 50.002 10 1 42 43 37 33 59.00 51.00 54.00 75.002 11 0 40 46 39 40 78.00 75.00 73.00 75.002 12 0 76 80 75 71 48.00 44.00 51.00 56.002 13 0 40 42 45 40 50.00 54.00 52.00 65.002 14 1 76 75 77 78 50.00 60.00 60.50 48.002 15 0 83 80 79 79 82.00 70.00 64.00 60.002 16 0 34 28 35 20 56.00 70.00 74.00 78.002 17 0 80 86 84 88 41.00 43.00 39.00 46.002 18 1 21 27 31 33 44.00 49.00 43.00 79.002 19 0 70 68 74 68 69.00 58.00 43.00 52.002 20 1 68 60 60 68 60.00 61.00 45.00 60.002 21 0 62 49 53 66 76.00 86.00 55.33 46.002 22 0 31 29 37 33 80.00 70.00 55.33 68.002 23 0 77 68 71 72 52.00 51.00 46.00 68.002 24 1 58 42 60 61 86.00 77.00 76.00 67.002 25 0 91 84 85 87 45.00 69.86 62.00 50.002 26 0 80 88 80 94 64.00 86.00 60.00 45.003 27 0 31 45 55 47 64.00 62.00 50.00 50.003 28 1 93 100 100 100 55.00 54.00 51.00 30.003 29 0 69 71 66 68 58.00 60.00 58.00 48.003 30 1 64 68 69 71 80.00 64.00 64.00 47.003 31 0 50 45 50 50 68.00 65.00 64.00 58.003 32 0 51 62 61 48 79.00 83.00 99.00 64.003 33 1 33 27 27 26 70.00 66.00 72.00 76.003 34 0 54 56 54 63 44.00 42.00 64.00 66.003 35 1 63 70 66 80 58.00 62.00 54.00 54.003 36 0 28 21 20 20 52.00 20.00 21.00 89.003 37 1 77 77 78 76 48.00 50.00 59.20 60.003 38 0 57 56 47 50 84.00 100.00 88.00 58.003 39 0 94 89 88 88 84.00 65.00 76.00 46.003 40 1 56 54 56 52 56.00 47.00 49.00 58.003 42 0 46 60 71 72 44.00 68.00 62.00 51.00

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Appendix I

Histograms of Test Scores

Conceptual and Procedural Understanding

605040302010

score on Test 1

12

10

8

6

4

2

0

Freq

uenc

y

Mean = 41.78Std. Dev. = 9.213N = 41

706050403020

score on Test 2

20

15

10

5

0

Freq

uenc

y

Mean = 52.57Std. Dev. = 7.516N = 41

Page 174

706050403020

score on Test 3

12

10

8

6

4

2

0

Freq

uenc

y

Mean = 53.8Std. Dev. = 7.936N = 41

706050403020

score on Test 4

15

12

9

6

3

0

Freq

uenc

y

Mean = 52.71Std. Dev. = 7.26N = 41

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Aiken Mathematics Attitude Scores

10080604020

Raw aiken score 1

10

8

6

4

2

0

Freq

uenc

y

Mean = 61.63Std. Dev. = 20.119N = 41

10080604020

Raw aiken score 2

10

8

6

4

2

0

Freq

uenc

y

Mean = 61.43Std. Dev. = 20.313N = 41

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10080604020

Raw aiken score 3

10

8

6

4

2

0

Freq

uenc

y

Mean = 62.5Std. Dev. = 19.17N = 41

10080604020

Raw aiken score 4

10

8

6

4

2

0

Freq

uenc

y

Mean = 63.24Std. Dev. = 21.001N = 41

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Semantic Differential Scores

4.003.00

2.001.00

time

100.0080.0060.0040.0020.00

semdiff

121086420

Freq

uenc

y

121086420

Freq

uenc

y

121086420

Freq

uenc

y

121086420

Freq

uenc

y

Page 178

Appendix J

Box Plots

Conceptual and Procedural Knowledge Scores

score on Test 4score on Test 3score on Test 2score on Test 1

80

60

40

20

0

161616

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Aiken Scores

Raw aiken score 4Raw aiken score 3Raw aiken score 2Raw aiken score 1

100

80

60

40

20

Documents

Version Acstl-csm.semo.edu/cjmcallister/mainpage/Appendixestotal.doc · Web viewVersion A. Version B. Fraction Assessment – MA318. The purpose of this test is to determine what