Version A
Appendix A
Version A
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Version B
Fraction 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 [pic] shaded in. You may need to draw in lines to determine if [pic] are shaded.
[pic]
(2) What fraction of the circle is part c? __________________
[pic]
(3) If shape A is the unit, what fraction can you give to 2 triangles? __________________
[pic]
(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:
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| | | |
| | | |
_____ = ______
(7) Using pictures of circles for chips or tiles, draw a picture to show the fraction [pic] using 15 circles.
(8) The whole circle is the unit. Write 2 fraction names for the picture for the amount shaded.
[pic] ________ _______
(9) Imagine that you have a bag of chips (tiles) to work with. You are to show the fraction [pic] with the chips in two different ways. Draw pictures of your two models in the boxes below.
[pic]
(10) Write the number that should go in the box on this number line: _________________
[pic](((((((((((((((((((((((((((((((((((((((((((
0 1 ( 2
(11) Write the number that should go in the box on this number line: _______________
(((((((((((((((((((((((((((((((((
0 [pic] ( [pic]
(12) This rectangle below is [pic] 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:
[pic]
15) Show where the number [pic] should be on this number line by placing an “x” at the appropriate spot.
(((((((((((((((((((((((((((((((((
0. 1
(16) Name three fractions equal to [pic].
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) [pic] [pic] (18) [pic] [pic]
(19) [pic] [pic] (20) [pic] [pic]
(21) [pic] [pic] (22) [pic] [pic]
(23) [pic] [pic] (24) [pic] [pic]
For each of the next three problems, use a model of the situation to illustrate the operation indicated and solve each problem.
(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 [pic] of his allowance on video games. Jim spent [pic] 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 [pic] meter in length. She then cut a piece [pic] 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?
(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) [pic] Word problem:
(32) [pic] Word problem:
(33) [pic] Word problem:
(34) [pic] Word problem:
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 |
|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 |SD D U A SA |
|mathematics. | |
|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 |
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
Appendix C
Traditional Lecture/Discussion Format Lessons
Introduction to Fractions -Lesson 1
Note: 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: [pic] means 3 parts out of 4 equal parts of a unit whole
2. Operator: [pic] of something – multiply by 3 and divide by 4 OR divide by 4, then multiply by 3.
3. Ratio and Rates: [pic]means 3 parts compared to 4 parts
4. Quotient: [pic] means 3 divided by 4. [pic] is the amount each person receives when 4 people share a 3-unit of something (concrete model)
5. Measure: [pic] 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 [pic] or a/b means ‘a’ amount of ‘b’ equivalent parts
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 [pic] 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.
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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. [pic] (Discuss cross products and cross multiplication)
1. Example: Determine which of [pic], [pic], or [pic] 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
D. Faster way to find a fraction in-between 2 given fractions – add numerators and add denominators.
1. 8/15b 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: [pic], [pic]
D. Represent the following improper fractions as mixed numerals: [pic],[pic]
VII. If time - p. 230 problems for writing and discussion Assign #1,2,3 to different tables – talk about responses.
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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.
I. Adding Fractions
A. With a common denominator
1. Let a/b and c/b be fractions then: [pic]
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 : [pic]
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. [pic]
IV. Subtraction of fractions
A. With common denominators
1. Let a/b and c/b be fractions then: [pic]
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 : [pic]
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 [pic].
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 [pic] and [pic]
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.
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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 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.
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 [pic]
1. It is helpful to teach students to simplify first – why?
2. Example: [pic]
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 [pic].
This is a very useful property in algebra: [pic]
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 [pic]?
B. Common denominator approach
1. Allows division of fractions to be viewed as an extension of whole number division
2. [pic] 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 [pic]
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
[pic]
Note: d/c is the reciprocal of c/d (this is where invert and multiply comes from.)
3. Example: [pic]
D. Maintaining the quotient (a little know property)
a. If [pic], then [pic]
b. So [pic]
c. Example: [pic]
E. Missing factor approach to fraction division
1. If [pic] with b, d, and f not equal to zero, then it must be true that
[pic]. 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 [pic]
2. Examples using missing factor approach
a. [pic]
b. [pic]
E. Rectangular shaded regions to model fraction division
a. [pic]
b. [pic]
F. Divide numerators, divide denominators approach
1. [pic], becomes a complex fraction
a. complex fractions [pic]
2. [pic]
3. [pic]
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), [pic]
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.
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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 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
[pic]
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 word problem that goes with this math fact.
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 word problem that goes with this math fact.
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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.
Appendix D
Hands-on Laboratory/Discussion Format Lessons
Introduction to Fractions - Objective 1
Note: 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: [pic] means 3 parts out of 4 equal parts of a unit whole
7. Operator: [pic] of something – multiply by 3 and divide by 4 OR divide by 4, then multiply by 3.
8. Ratio and Rates: [pic]means 3 parts compared to 4 parts
9. Quotient: [pic] means 3 divided by 4. [pic] is the amount each person receives when 4 people share a 3-unit of something (concrete model)
10. Measure: [pic] 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 [pic] or a/b means ‘a’ amount of ‘b’ equivalent parts
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.)
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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.
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1. How does the figure above represent [pic]? What is the whole? What do the smaller subdivisions of the rectangle represent? In the symbolic representation [pic], what does the 5 mean? What does the 3 mean? Could this figure represent [pic]?
2. Can you now look at the same figure and see it representing [pic] of something? What is the whole in this case? What do the smaller subdivisions of the rectangle represent? Interpret the symbolic representation [pic]in this situation.
3. Can you see [pic] of [pic] in this situation? Explain.
4. Can you see [pic] of [pic] in this situation? Explain.
5. Can you see [pic] 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.
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 [pic] 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.
[pic]
9. Benton used [pic] 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 [pic] cup of butter. How could Benton cut the dough? Explain.
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10. Mira has a recipe that calls for 5 cups of flour. Mira wants to make [pic] of the recipe. Instead of figuring out what number [pic] of 5 is, Mira measures [pic] of a cup of flour 5 times, and uses this amount of flour for [pic] 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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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 [pic] of a pie is the same amount of pie as [pic] of pie, which is [pic] of the pie. Draw pictures to support your explanation. Discuss how your pictures show the process of multiplying both the numerator and denominator of [pic] by 4.
2. Discuss how to modify your explanation for Problem 1 to show that [pic] of a pie is the same amount of pie as [pic] of the pie.
3. Suppose A and B are whole numbers and B is not 0. Also suppose that N is any natural number. Explain why [pic] of some object is the same amount of that object as [pic] of the object.
4. Anna says [pic] because starting with [pic], you get [pic] 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 [pic] 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. [pic] (Discuss cross products and cross multiplication)
1. Example: Determine which of [pic], [pic], or [pic] 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 Teachers. Boston, MA: Peasron/Addison Wesley.)
Simplifying Fractions
1. Use the next diagram to help you explain why the following equations that put [pic] in simplest form make sense: [pic].
( ( ( ( (
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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 [pic].”
Chrissy says, “The part left over is 1/3 cup of rice, but the answer should be [pic]. Did we do something wrong?” Help Tonya and Chrissy.
3. a. Write a “how many groups?” story problem for [pic]. Choose your story problem so that a mixed-number answer makes sense.
b. Write a “how many groups?” story problem for [pic]. (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 [pic] in a simple and concrete way, for example, by drawing pictures. Explain your solution.
d. Use the “invert and multiply” procedure to calculate [pic]. Verify that the answer agrees with your answer in part c.
4. Do steps b, c, and d above for the problem [pic].
5. Write a “how many in one group?” (sharing model) story problem for [pic]. Using this as a model, write a “how many in one group?” story problem for [pic] 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 [pic]. 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 [pic] 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
2. [pic] 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 [pic]
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
[pic]
Note: d/c is the reciprocal of c/d (this is where invert and multiply comes from.)
3. Example: [pic]
E. Maintaining the quotient (a little known property)
a. If [pic], then [pic]
b. So [pic]
c. Example: [pic]
E. Missing factor approach to fraction division
1. If [pic] with b, d, and f not equal to zero, then it must be true that
[pic]. 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 [pic]
2. Examples using missing factor approach
a. [pic]
b. [pic]
F. Divide numerators, divide denominators approach
1. [pic], becomes a complex fraction
a. complex fractions [pic]
2. [pic]
3. [pic]
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), [pic]
B. Mental arithmetic (p. 251)
C. Estimation (p. 251)
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.
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 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.
Appendix F
MA318 – Outside reading assignment
You 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 Selections
1. 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.
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 Format
Summaries 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)
Appendix G
After School Mathematics Program Curriculum
|Session |Learning Objective |Skill Builder Activity |Challenge Problem |Practice Activity |Extra |
| | | | | |Activity |
|1 |Development of part/whole |Lessons 3-5 from Cramer, et |Benny’s Bakery |Fraction Capture Game from | |
| |definition of fractions |al., Level 1* | |Everyday Mathematics | |
|2 |Equality of fractions |1) Lesson 8 from Cramer, et |Fair share candy bar|1) Fraction Fill game | |
| | |al., Level 1* |problem |2) Fraction Concentration | |
| | |2) Equivalent fraction chart | |game | |
|3 |Ordering fractions |Lessons 6 and 11 from Cramer, |Exploring shaded |Fraction War game |Fraction Circle |
| | |et al., Level 1* |regions models of | |Activity |
| | | |fractions | | |
|4 |Mixed numerals and |Lesson 17 from Cramer, et al., |That’s the way the |Fraction Concentration game |Fractions with |
| |improper fractions |Level 1* |cookie crumbles |with improper and mixed |Tangrams Activity|
| | | | |fractions | |
|5 |Review of equivalent |Lesson 14 from Cramer, et al., |Break the code on |Students may choose any of | |
| |fractions |Level 1* |equivalent fractions|the games introduced so far | |
|6 |Addition and subtraction |Lesson 19 from Cramer, et al., |Problem solving with|Lesson 19 Student Page A, | |
| |of fractions |Level 1* |Dr. Loyd’s Fraction |Cramer, et al., Level 1* | |
| | | |Kit | | |
|7 |Addition and subtraction |Lesson 20 from Cramer, et al., |Pattern Block |Fraction Addition and |Fraction Squares |
| |of fractions |Level 1* |Problem #1 |Subtraction Worksheet |A and B |
|8 |Multiplication of |Lesson 23 from Cramer, et al., |Pattern Block |Fraction Multiplication |Fraction Squares |
| |fractions |Level 2* |Problem #2 |Worksheet |C and D |
|9 |Locating fractions on a |Students will use sidewalk | | | |
| |number line |chalk, string, and paper clips | | | |
| | |to make a fraction number line | | | |
*Full citations in reference list at the end of this Appendix.
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 session
2. Fraction circle kits
3. Pencils
4. Paper
5. 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-defined
2. each part must be of equal value
3. the denominator of the fractional representation indicates the number of equal sized pieces the whole has been ‘cut’ into
4. 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 |
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 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 used
2. Fraction circle kits
3. 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
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? ________________________
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. [pic]
Central Middle School
After School Mathematics Program
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 chart
1. 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 first
Rules for Fraction Fill Up
1. 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 pencil
3. 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.
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’.
[pic]
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
Team Challenge Problem for Session 2
List Team Leader: ________________________________________
List Team Members: __________________________________
__________________________________
__________________________________
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.
Central Middle School
After School Mathematics Program
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
Reference to District Mathematics Curriculum: 1, 3, 11
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.
[pic] [pic] [pic] [pic] [pic] [pic]
From the problems above, what can you say about comparing fractions that have the same denominator?
[pic] [pic] [pic] [pic] [pic] [pic]
From the problems above, what can you say about comparing fractions that have the same numerator?
[pic] [pic] [pic] [pic] [pic] [pic]
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.
[pic] [pic] [pic] [pic] [pic] [pic]
[pic] [pic] [pic] [pic] [pic] [pic]
Team Challenge Problem for Session 3
List Team Leader: ________________________________________
List Team Members: __________________________________
__________________________________
__________________________________
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.
[pic]
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
Central Middle School
After School Mathematics Program
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
Reference to District Mathematics Curriculum: 1, 3, 11
Team Challenge Problem for Session 4
List Team Leader: ________________________________________
List Team Members: __________________________________
__________________________________
__________________________________
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?
Central Middle School
After School Mathematics Program
Skill 5
Supplies needed: One set of fraction circles.
One set of Dr. Loyd’s fraction kit.
Fraction numeral cards
Worksheets A1, A2, B, C for each student
Scratch 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
Reference to District Mathematics Curriculum: 1, 3, 11
Central Middle School
After School Mathematics Program
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 answer
Is the sum less than ½ or bigger than ½?
Is the sum less than 1 or bigger than 1?
b. model with manipulatives
c. 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?
a. estimate the answer
Is the sum less than ½ or bigger than ½?
Is the sum less than 1 or bigger than 1?
b. model with manipulatives
c. find the correct answer in simplest form
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?
a. estimate the answer
Is the sum less than ½ or bigger than ½?
Is the sum less than 1 or bigger than 1?
b. model with manipulatives
c. find the correct answer in simplest form
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.
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)
Reference to District Mathematics Curriculum: 1, 3, 11
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 Jack
1. 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 points
If 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.)
Team Challenge Problem for Session 6
List Team Leader: ________________________________________
List Team Members: __________________________________
__________________________________
__________________________________
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?
Central Middle School
After School Mathematics Program
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 answer
Is the difference less than ½ or bigger than ½?
b. model with manipulatives
c. find the correct answer in simplest form
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.)
a. estimate the answer
Is the difference less than ½ or bigger than ½?
b. model with manipulatives
c. find the correct answer in simplest form
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)
a. estimate the answer
Is the difference less than ½ or bigger than ½?
b. model with manipulatives
c. find the correct answer in simplest form
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.)
a. estimate the answer
Is the difference less than ½ or bigger than ½?
b. model with manipulatives
c. find the correct answer in simplest form
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.
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
Reference to District Mathematics Curriculum: 1, 3, 11
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?
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. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
Team Challenge Problem for Session 7
List Team Leader: ________________________________________
List Team Members: __________________________________
__________________________________
__________________________________
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.
Fraction square
[pic]
Central Middle School
After 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 answer
Is the product more than 1 or less than 1? less than ½ or bigger than ½?
e. model with manipulatives or a drawing
f. 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 answer
Is the product more than 1 or less than 1? less than ½ or bigger than ½?
b. model with manipulatives or a drawing
c. 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 answer
Is the product more than 1 or less than 1? less than ½ or bigger than ½?
b. model with manipulatives or a drawing
c. find the correct answer in simplest form
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.
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
Reference to District Mathematics Curriculum: 1, 3, 11
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?
Team Challenge Problem for Session 8
List Team Leader: ________________________________________
List Team Members: __________________________________
__________________________________
__________________________________
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.
Fraction Square
[pic]
Central Middle School
After School Mathematics Program
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 [pic]
(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:
[pic]
4. Extra Challenge:
A. Locate the following on your number as accurately as possible: [pic]
B. Which is bigger, 4/5 or 4/6? Which is bigger, 5/8 or 4/5?
Reference to District Mathematics Curriculum: 1, 3, 11
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
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
The Math Forum (1994-2005). Fraction Squares. Retrieved 3/10/2005 from
Simon, M. (2003). Frack Jack: A magnet game to teach fractions. Retrieved March 14, 2005
from .
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
Appendix H
Raw Data
Section |ID no. |Tutoring |Score 1 |Score 2 |Score 3 |Score 4 |cum hour |ACT sub |math 118 |math 318 | |2 |1 |1 |51 |60 |58 |58 |25 |27 |3 |4 | |2 |2 |0 |51 |57 |57 |56 |70 |20 |4 |3 | |2 |3 |0 |51 |60 |62 |59 |61 |19 |4 |3 | |2 |4 |1 |52 |57 |61 |60 |31 |29 |4 |4 | |2 |5 |0 |30 |58 |46 |46 |70 |20 |3 |3 | |2 |6 |0 |50 |58 |55 |51 |15 |22 |4 |3 | |2 |7 |0 |50 |58 |59 |51 |54 |21 |3 |4 | |2 |8 |1 |48 |61 |63 |63 |31 |24 |4 |4 | |2 |9 |0 |45 |55 |47 |54 |33 |24 |4 |4 | |2 |10 |1 |25 |43 |43 |50 |98 |18 |3 |3 | |2 |11 |0 |33 |45 |46 |41 |42 |18 |3 |2 | |2 |12 |0 |40 |47 |52 |49 |40 |18 |3 |3 | |2 |13 |0 |35 |50 |54 |47 |80 |19 |3 |3 | |2 |14 |1 |49 |57 |57 |57 |65 |20 |3 |4 | |2 |15 |0 |53 |56 |61 |62 |60 |27 |4 |4 | |2 |16 |0 |20 |26 |29 |29 |62 |18 |2 |2 | |2 |17 |0 |48 |49 |56 |50 |45 |17 |3 |3 | |2 |18 |1 |42 |51 |52 |53 |39 |17 |3 |3 | |2 |19 |0 |50 |57 |62 |61 |21 |25 |3 |4 | |2 |20 |1 |34 |55 |48 |41 |33 |18 |4 |3 | |2 |21 |0 |33 |49 |56 |48 |73 | |3 |4 | |2 |22 |0 |37 |38 |45 |44 |110 |16 |2 |2 | |2 |23 |0 |39 |55 |61 |60 |41 |26 |4 |4 | |2 |24 |1 |46 |58 |61 |57 |15 |23 |4 |4 | |2 |25 |0 |52 |63 |62 |56 |24 |29 |4 |4 | |2 |26 |0 |43 |54 |61 |56 |45 |25 |4 |4 | |3 |27 |0 |47 |55 |59 |57 |66 |21 |2 |3 | |3 |28 |1 |52 |55 |60 |56 |24 |24 |3 |4 | |3 |29 |0 |46 |60 |62 |62 |34 |25 |4 |4 | |3 |30 |1 |34 |43 |50 |48 |81 | |4 |3 | |3 |31 |0 |46 |62 |52 |59 |91 |21 |2 |3 | |3 |32 |0 |39 |56 |53 |43 |15 |21 |3 |1 | |3 |33 |1 |27 |51 |46 |50 |76 |17 |3 |2 | |3 |34 |0 |39 |49 |48 |51 |73 |20 |2 |1 | |3 |35 |1 |35 |47 |48 |57 |64 |17 |4 |3 | |3 |36 |0 |35 |48 |52 |55 |48 |19 |2 |3 | |3 |37 |1 |50 |58 |59 |58 |50 |23 |3 |4 | |3 |38 |0 |18 |37 |31 |39 |28 | |3 |3 | |3 |39 |0 |45 |54 |57 |54 |15 |27 |4 |4 | |3 |40 |1 |49 |53 |55 |58 |23 |23 |4 |3 | |3 |42 |0 |44 |50 |60 |55 |43 |23 |3 |3 | |
Section |ID no. |Tutoring |Aiken 1 |Aiken 2 |Aiken 3 |Aiken 4 |Semantic Diff 1 |Semantic Diff 2 |Semantic Diff 3 |Semantic Diff 4 | |2 |1 |1 |68 |67 |67 |67 |70.00 |84.00 |85.00 |49.00 | |2 |2 |0 |45 |40 |50 |41 |56.00 |48.00 |52.00 |90.00 | |2 |3 |0 |74 |81 |76 |83 |75.00 |70.00 |78.00 |54.00 | |2 |4 |1 |86 |72 |85 |85 |90.00 |78.00 |54.00 |50.00 | |2 |5 |0 |47 |52 |46 |63 |23.00 |37.00 |60.50 |38.00 | |2 |6 |0 |60 |59 |61 |69 |52.00 |54.00 |52.00 |52.00 | |2 |7 |0 |85 |81 |80 |84 |73.00 |64.00 |72.00 |23.00 | |2 |8 |1 |92 |92 |92 |80 |52.00 |54.00 |56.00 |44.00 | |2 |9 |0 |75 |79 |76 |79 |81.00 |75.00 |69.00 |50.00 | |2 |10 |1 |42 |43 |37 |33 |59.00 |51.00 |54.00 |75.00 | |2 |11 |0 |40 |46 |39 |40 |78.00 |75.00 |73.00 |75.00 | |2 |12 |0 |76 |80 |75 |71 |48.00 |44.00 |51.00 |56.00 | |2 |13 |0 |40 |42 |45 |40 |50.00 |54.00 |52.00 |65.00 | |2 |14 |1 |76 |75 |77 |78 |50.00 |60.00 |60.50 |48.00 | |2 |15 |0 |83 |80 |79 |79 |82.00 |70.00 |64.00 |60.00 | |2 |16 |0 |34 |28 |35 |20 |56.00 |70.00 |74.00 |78.00 | |2 |17 |0 |80 |86 |84 |88 |41.00 |43.00 |39.00 |46.00 | |2 |18 |1 |21 |27 |31 |33 |44.00 |49.00 |43.00 |79.00 | |2 |19 |0 |70 |68 |74 |68 |69.00 |58.00 |43.00 |52.00 | |2 |20 |1 |68 |60 |60 |68 |60.00 |61.00 |45.00 |60.00 | |2 |21 |0 |62 |49 |53 |66 |76.00 |86.00 |55.33 |46.00 | |2 |22 |0 |31 |29 |37 |33 |80.00 |70.00 |55.33 |68.00 | |2 |23 |0 |77 |68 |71 |72 |52.00 |51.00 |46.00 |68.00 | |2 |24 |1 |58 |42 |60 |61 |86.00 |77.00 |76.00 |67.00 | |2 |25 |0 |91 |84 |85 |87 |45.00 |69.86 |62.00 |50.00 | |2 |26 |0 |80 |88 |80 |94 |64.00 |86.00 |60.00 |45.00 | |3 |27 |0 |31 |45 |55 |47 |64.00 |62.00 |50.00 |50.00 | |3 |28 |1 |93 |100 |100 |100 |55.00 |54.00 |51.00 |30.00 | |3 |29 |0 |69 |71 |66 |68 |58.00 |60.00 |58.00 |48.00 | |3 |30 |1 |64 |68 |69 |71 |80.00 |64.00 |64.00 |47.00 | |3 |31 |0 |50 |45 |50 |50 |68.00 |65.00 |64.00 |58.00 | |3 |32 |0 |51 |62 |61 |48 |79.00 |83.00 |99.00 |64.00 | |3 |33 |1 |33 |27 |27 |26 |70.00 |66.00 |72.00 |76.00 | |3 |34 |0 |54 |56 |54 |63 |44.00 |42.00 |64.00 |66.00 | |3 |35 |1 |63 |70 |66 |80 |58.00 |62.00 |54.00 |54.00 | |3 |36 |0 |28 |21 |20 |20 |52.00 |20.00 |21.00 |89.00 | |3 |37 |1 |77 |77 |78 |76 |48.00 |50.00 |59.20 |60.00 | |3 |38 |0 |57 |56 |47 |50 |84.00 |100.00 |88.00 |58.00 | |3 |39 |0 |94 |89 |88 |88 |84.00 |65.00 |76.00 |46.00 | |3 |40 |1 |56 |54 |56 |52 |56.00 |47.00 |49.00 |58.00 | |3 |42 |0 |46 |60 |71 |72 |44.00 |68.00 |62.00 |51.00 | |
Appendix I
Histograms of Test Scores
Conceptual and Procedural Understanding
[pic]
[pic]
[pic]
[pic]
Aiken Mathematics Attitude Scores
[pic]
[pic]
[pic]
[pic]
Semantic Differential Scores
[pic]
Appendix J
Box Plots
Conceptual and Procedural Knowledge Scores
[pic]
Aiken Scores
[pic]
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Shape A
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