MATH 330- Final Exam

Describe a situation where 1/4 is greater than 1/2.

(when the whole of 1/4 is bigger. FRACTION SIZE IS RELATIVE.) Monica has 1/4 of a large pizza left, and Tay has 1/2 of a personal pizza left.

Why are fractions so difficult?

-Many meanings make it difficult to compare size and apply rules -Written in a unique way -Students overgeneralize their whole number knowledge

How could you add without using the traditional algorithm (finding a common denominator? Explain in detail. Use diagrams throughout.

-using fraction circles -using cuisenaire rods -drawing pictures

problem-based number sense approach: (4 steps)

1 begin with simple contextual tasks (story problems) 2 models 3 estimation and invented method 4 address common misconceptions

What are 4 common misconceptions of fractions? What are solutions to those misconceptions?

1. Students have a hard time seeing the numerator and denominator as one value. Solution: Find fractions on a number line. 2. Student's do not understand that 2/3 means two equal-sized parts, although not necessarily equal-shaped objects. Solution: Create representations of fractions on paper and with manipulatives. 3. Students think that a fraction like 1/5 is smaller than a fraction like 1/10 because 5 is less than 10. Solution: Teach the rule and explain why it works, or students may think 1/5 is bigger than 2/7. Use many visuals and contexts that show parts of the whole. (such as time and food) 4. Students use operation rules for whole numbers to compare with fractions--1/2 + 1/2 = 2/4. Solution: Use many visuals and contexts. Focus on whether answers are reasonable or not.

What is a fraction?

A fraction is an indicated division : a/b = a divided by b

What does CCSS mean by extend understanding to show equivalence and ordering?

Comparing fractions is a crucial step in extending understanding and showing equivalence and ordering. Being able to know that the bigger the number on the bottom, the smaller the pieces are, is important to understand when comparing one fraction to another fraction.

What does iteration mean?

"counting fractional parts"

A student adds and gets . How will you help the student understand that this is incorrect? How will you redirect him or her to do it correctly?

I would begin by explaining that when we add fractions, we are only adding the pieces (the top number), not the whole (the bottom number). If the bottom numbers are not the same, it would not be correct for us to add them because we would not have the same sized pieces in the whole. It sounds like in this case, the student needs a visual in order to better understand the problem. That being said, I would ask the student to show me how he or she would add those fractions using the hands-on materials (Cuisenaire Rods, fraction strips, etc.) Using the Cuisenaire Rods, the student would find a whole. Next, the student would use the manipulatives to represent 4/5. Then, the student would use the manipulatives to represent 2/3. Finally, the students are able to find the answer by figuring out the denominator using manipulatives. Once the student solves the problem, I would record what he or she is doing step by step using fractions to help him or her better understand each step and why it works.

Why is it important to teach fraction operations with contextual problems and informal strategies?

It is important to teach fraction operations with contextual problems and informal strategies to students because the task focuses attention on the mathematics of the problem, is accessible to students, and requires justification and explanation for answers or methods.

change unknown 1/6, 2/4

Jim ate 1/6 of a sandwich. His sister ate some more. If together they ate a total of 2/4 of the sub, how much did his sister eat?

Create a contextual story problem for 1 1/2 - 5/8

John has 1 1/2 of a cheese pizza left over from his birthday. he eats 5/8 of the leftover pizza. how much pizza does John have left?

start unknown 1/4, 1 1/2

Katie has some cookie batter. It is too runny, so she adds 1/4 cup of flour to it. If she uses a total of 1 1/2 cups of flour, how much flour is in the batter at first?

result unknown 3/4, 1/2

Logan has 3/4 cup of flour. His mom gives him 1/2 cup more. How much flour does he have now?

create a "take away" subtraction story problem for 3 1/4 - 1 1/2

Mark has 3 1/4 slices of his birthday cake leftover. His friend, Ashlee, eats of the total leftover birthday cake. How much birthday cake is left?

create a "comparison" subtraction story problem for 3 1/4 - 1 1/2

Monica's pet snake is 3 1/4 inches long. Taylor's pet snake is 1 1/2 inches. long. How much longer is Monica's pet snaked than Taylor's?

What does CCSS mean by explore and develop an understanding of fractions?

Understanding what the numbers mean and represent in a fraction is a big step in developing an understanding of fractions. The top number represents how many parts we have, while the bottom number represents how many parts the whole number is divided into.

What does CCSS mean by unit fractions?

Unit fractions are important because students need them to know the size of a single part of the whole. These are very helpful because they can be used to compare sizes of other fractions.

name technique and explain reasoning: 2/4 vs 3/8

benchmark numbers: 2/4 is half, while 3/8 would need 1/8 to be half. therefore, 2/4 is greater than 3/8

define the following meaning of fraction: ratio

can be part-whole or part-part part-whole: those wearing jackets to those in the class part-part: those wearing jackets to those not 1/12 the ratio of men to women in this class

comparing fractions: residuals

compare what is missing (bigger denominator has smaller parts)

comparing fractions: benchmarks

comparing given fractions to these 0, 1/2, 1, etc

The ________________ tells us the number of pieces in the whole

denominator

when the _______________ is the same, we compare the number of pieces.

denominator

What does partitioning mean?

equal size fractional pieces. the pieces have to be the same size but not the same shape

define the following meaning of fraction: measurement

identifying a length and using that length as a measurement piece to determine the length of an object. focuses on "how much" rather than "how many" given 5/8 use 1/8 (unit fraction) to determine length

iteration: the ________________ counts and the ________________ tells what is being counted

numerator, denominator

_______________ are the numbers of parts, ________________ are the size of parts

numerators, denominators

What are three similarities between fraction computation (addition and subtraction) and whole number computation? Explain in each case.

o When the denominator is the same, the top number is added or subtracted as if they are whole numbers. Instead of adding or subtracting whole numbers, however, it's just adding or subtracting the parts of a whole. o When the denominators are not the same, the students can use different invented strategies to solve the problem. These examples may include: benchmarks, estimation, illustrations (such as fraction circles), and many others to develop a "why it works" understanding of fractions. These strategies may also be utilized in whole number computation. o When adding and subtracting fractions, there are different types of problems like result unknown, start unknown, and change unknown. Similarly, those different types of problems also appear in whole number computation.

define the following meaning of fraction: part-whole

part of a shaded region or group ex: 3/4 of the class went on a field trip

What are the meanings of fractions? (5)

part-whole measurement division operator ratio

what are the four techniques for comparing fractions without common denominators?

residuals benchmark same numerator same denominator

name technique and explain reasoning: 5/6 vs 7/8

residuals: both still need one more part to reach a whole. however, 5/6 needs 1/6 to reach a whole, while 7/8 needs 1/8. that being said, since 1/6 is greater than 1/8, 5/6 is greater than 7/8.

name technique and explain reasoning: 2/4 vs 3/4

same denominator: same size of parts, different number of parts. because the denominator is the same, we compare numerators. 2 < 3 so 2/4 < 3/4

comparing fractions: same numerators

same number of parts vs size of parts

name technique and explain reasoning: 2/4 vs 2/5

same numerators: same number of parts, diff size. since the numerator is the same, we compare the size of parts. when the denominator is larger, it has smaller parts. so 2/4 is greater than 2/5

comparing fractions: same denominators

same size of parts vs number of parts

define the following meaning of fraction: division

sharing $10 with 4 people. they each get 1/4 (or 2.50) not necessarily part-whole situation, but the pieces are still divided equally among 4 people.

what are contextual tasks also known as?

story problems

define the following meaning of fraction: operator

used to indicate an operation 4/5 of 20 sq feet

Describe two ways to compare and (not using common denominators or cross products).

• One way to compare 5/12 and 5/8 is to recognize that because the numerators are the same, the student can easily compare the two numbers in their head. The problem is like saying that Ashlee is hungry and has a size large pizza cut into 12 slices. She is offered five of them. However, she has another size large pizza that is the exact same size as the other pizza that is cut into 8 slices. She is offered five of them. Because it is the same size pizza (whole) the fraction with the smaller numerator would have larger pieces. Therefore, Ashlee would want to choose 5/8 because it would have 5 larger pieces of pizza. • Another way to compare 5/12 and 5/8 is recognize that 5/8 is over a half, and 5/12 is under a half. That being said, 5/8 is closer to the whole; therefore, it is the larger fraction.