Transcript Fractions- Grade 4 - Nevada Mathematics Project

th
4
Grade Number &
Operations-Fractions
Alicia Shaw, Rachel Stewart,
Extend Understanding of Fraction
Equivalence and Ordering

4.NF.A.1: Explain why a fraction a/b is equivalent to a fraction (n × a)/(n ×
b) by using visual fraction models, with attention to how the number and
size of the parts differ even though the two fractions themselves are the
same size. Use this principle to recognize and generate equivalent
fractions.
Foundational Standard
3.NF.A.1: Understand a fraction 1/b as the quantity
formed by 1 part when a whole is partitioned into b
equal parts; understand a fraction a/b as the
quantity formed by a parts of size 1/b.
•
From 3rd grade students already understand what
a fraction is and how to represent it on a number
line as well as what an equivalent fraction is.
•
4th grade extends this knowledge to understand
the concept ½ = 4/8 or 3/6, etc…

By subdividing the fraction students
learn this creates smaller parts.

Students then learn that by
subdividing they are multiplying
both the numerator and
denominator by n
Extend Understanding of Fraction
Equivalence and Ordering

Conversely, students then learn they can divide the
numerator and denominator equally to generate
equivalent fractions.

They use a model, to help them understand this concept
by equally combining smaller parts to create larger parts.
Misconceptions

If you use the term reducing, it can seem as though one fraction is less than
the other. For example 2/8 reduced to ¼, students might think ¼ is less than
2/8. Use the term “simpler form”

Simpler form doesn’t always equate to easier to understand. Example: 6/10
or 3/5
4.NF.A.2 Comparing Fractions

Compare two fractions with different numerators and different
denominators, e.g., by creating common denominators or numerators, or
by comparing to a benchmark fraction such as 1/2. Recognize that
comparisons are valid only when the two fractions refer to the same
whole. Record the results of comparisons with symbols >, =, or <, and
justify the conclusions, e.g., by using a visual fraction model.
Comparing Fractions

Students will use what they have already learned about fractions to
compare fractions with like and unlike numerators and denominators.

Students may use equivalent fractions to be more efficient at
comparing fractions.

When comparing 1/3 and ½ it may be easier to think of
1/3 as 2/6 and ½ as 3/6
Comparing Fractions

Fraction towers

Talk at your table about how
you could rename 4/8 to
compare it to ¾
What are some student
misconceptions about comparing
fractions?
Comparing Fractions

Number line

Relate the numerator to the denominators

Use benchmarks
Misconceptions
•
Using improper fractions
•
Using student made visual models
•
Comparing fractions using the numerator
when the denominator is different
References
Barlow, A. T., & Harmon, S. (2012). CCSSM: Teaching in Grades 3 and 4. Teaching
Children's Mathematics, 18(8), 498-507.
Ding, M., Li, X., Capraro, M. M., & Kulm, G. (2012). A Case Study of Teacher Responses to a Doubling Error and Difficulty in Learning
Equivalent Fractions. Investigations in Mathematics Learning, 4(2), 42-73.
Hecht, S. A., & Vagi, K. J. (2012). Patterns of strengths and weaknesses in children’s knowledge about fractions. Journal of experimental child
psychology, 111(2), 212-229.
Jigyel, K., & Afamasaga-Fuata'i, K. (2007). Students' conceptions of models of fractions and equivalence. Australian Mathematics Teacher, 63(4), 1725.
Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., ... & Wray, J. (2010). Developing Effective Fractions Instruction for
Kindergarten through 8th Grade. IES Practice Guide. NCEE 2010-4039.What Works Clearinghouse.
Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive
psychology,62(4), 273-296.
Generalizing operational
sense with whole
numbers to rational
numbers
Sarah Roggensack
Brie Gant
“Success with fractions, in particular
computations in all four operations, is
directly related to success in algebra. The
inverse is also true, that student with a poor
understanding of operations with fractions
are at risk of difficultly in algebra and that
weakness will have long-term impact
affecting college majors and career
opportunities.”
-Van de Walle, et al. (2014) Teaching Student-Centered Mathematics. p. 232
4.NF.B. Build fractions from unit fractions by
applying and extending previous understandings
of operations on whole numbers.

Properties of Addition:
 Commutative: 3 4 +
¼ = ¼+ ¾ =
4
4
=1
 Distributive:
¾+¼=
 Associative:
(¾ + ¼) + ¼ = ¾ + (¼ + ¼ )
2
4
+( ¼ + ¼ ) = 2
4
+2 4= 4
4
=1
Standard & Progression
4.NF.3 Understand a fraction a/b with a > 1 as a sum of
fractions 1/b.
b. Decompose a fraction into a sum of fractions with the same denominator in more
than one way, recording each decomposition by an equation. Justify decompositions,
e.g., by using a visual fraction model.
Underlying Ideas:
3
8

Thinking of mixed numbers as sums: 23

Renaming mixed numbers as improper fractions: 3
8=2+
3
5
=
5
5
5
3
+ + +
5
5
5
5
Underlying Ideas:

Renaming improper fractions as mixed numbers:
26
9
=
9
9
9
9
+ +
8
9
Standards & Progression
4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts referring to the same
whole.
c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an
equivalent fraction, and/or by using properties of operations and the relationship between addition and
subtraction.
d. Solve word problems involving addition and subtraction of fractions referring to the same whole and
having like denominators, e.g., by using visual fraction models and equations to represent the problem.
Underlying ideas:

Fraction Number Sense: Fractions can be either a sum or a difference

Thinking of the unit when adding and subtracting (same with mixed numbers)
4.NF.B. Build fractions from unit fractions by
applying and extending previous understandings
of operations on whole numbers.

Problem Types for Addition & Subtraction:
Standards & Progression
4.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole
number.
a. Understand a fraction a/b as a multiple of 1/b.
b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction
by a whole number.
c. Solve word problems involving multiplication of a fraction by a whole number by using visual
fraction models and equations to represent the problem.
4.NF.B. Build fractions from unit fractions by
applying and extending previous understandings
of operations on whole numbers.

Properties of Multiplication:
 Commutative: 3 4
 Associative:

x ¼ = ¼ x ¾ =3
16
(¾ x ¼) x ¼ = ¾ x (¼ x ¼ ) = 3
64
Distributive: 15 x ½ = (10 x ½) + (5 x ½ ) = 7 ½
Underlying idea

Fraction number sense: Thinking of a single fraction as a product
4
5
1
5
1
5
1
5
1
5
= + + + =4𝑥
1
5
4.NF.B. Build fractions from unit fractions by
applying and extending previous understandings
of operations on whole numbers.

Problem Types for Multiplication:
Fraction Misconceptions

Students may only be used to dividing traditional shapes – circles, squares, and rectangles
– they begin to think that all shapes are divided similarly.

Children often do not recognize groups of objects as a whole unit.


Students may still be struggling with is that the denominator stays the same when
repeatedly adding a unit fraction such


For example, there may be 2 cars and 4 trucks in a set of 6 vehicles. The student may
mistakenly confuse the set of cars as 4 instead of 6
For example, 2/5 +1/5= 3/10 instead of 2/5 +1/5= 3/5
Some students may still be discovering the relationship between the numerator and
denominator in relation to 1. When the numerator is greater than the denominator we
have passed the benchmark of 1 whole.

For example, 6/4 is greater than 4/4, which is 1 whole, meaning that 6/4 is 1 whole and 2/4
remaining, equaling 1 2/4 or 1 1/2
Activity:
Fraction Cookies Bakery
~from Georgia Math
There are 7 halves that
make a total of 3 and ½
cookies ordered.
4 cookies would have to
be made to provide 3 ½
cookies.
Activity
Discussion Opportunities
- Improper fractions and mixed number
connection
- Accounting for the remainder of the
cookie without toppings and how that
affects the total amount
- Misconceptions
- Connect back to the standard
(adding/subtracting mixed numbers with
like denominators, using fraction visual
models, decomposing fractions
Interventions and Extensions


Interventions

Allow students to use more concrete pre-made circle fraction pieces to
create the cookies

Students may need more examples modeled before starting task
independently
Extensions

Challenge students with different versions of the order form


Incorporating trading for equivalents
Ask students to create orders of their own, then switch with a partner
References
•
Bean salad using algebraic reasoning to determine quantities of ingredients. National council of
teachers of mathematics.
•
Dixon, J. K., & Tobias, J. M. (2013). The whole story: understanding fraction
computation. Mathematics teaching in the middle school,19(3), 156-163.
•
Fennel, F. (2007). Fractions are foundational. National council of teachers of mathematics.
•
Georgia standards of excellence framework. Unit 4: Fractions and operations.
https://www.georgiastandards.org/Georgia-Standards/Frameworks/4th-Math-Grade-Level-Overview.pdf
•
Glidden, P. L. (2002). Build your own. Mathematics teaching in the middle school, 204-208.
•
Mack, N. K. (2004). Connecting to develop computational fluency with fractions. Teaching children
mathematics, 226-232.
•
Newton, K. J., Willard, C., & Teufel, C. (2014). An examination of the ways that students with learning
disabilities solve fraction computation problems. The elementary school journal, 115(1), 1-21.
•
Olive, J. (2002). Bridging the gap using interactive computer tools to build fraction schemes. Teaching
children mathematics, 356-361.
•
Small, M. Uncomplicating fractions to meet common core standards in math, K-7.
•
Van de Wall, J. K.-W. (2014). Building strategies for fraction computation. In J. K.-W. Van de Wall,
Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (2
ed., pp. 231-253). Upper Sadle River, NJ: Pearson.
th
4
grade decimal fractions
Ashleigh Phariss AND Amy Williams
standards

Understand decimal notations for fractions, and compare decimal fractions.

CCSS.Math.Content.4.NF.C.5
Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and
use this technique to add two fractions with respective denominators 10 and 100.2 For
example, students must realize that 3/10 is equivalent to 30/100, and then add 3/10 +
4/100 = 34/100.

CCSS.Math.Content.4.NF.C.6
Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62
as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

CCSS.Math.Content.4.NF.C.7
Compare two decimals to hundredths by reasoning about their size. Recognize that
comparisons are valid only when the two decimals refer to the same whole. Record the
results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using
a visual model.
Big Ideas

Kids have had an exposure to decimals because of money (cents)

Decimals seem friendlier to kids over fractions (to enter in a calculator, to compare)

Can represent visually easily (up to hundredths)
(on hundreds unit block, pennies and dimes, number line, and place value chart)

Relate to place value chart and the base ten system- 10 times as many as.

Use metric measurement to see the relationship to different whole units. For example, 1
centimeter as 1 hundredth of a meter.

Show with expanded form (both as fraction and decimal form) (helps with comparing and exposes to
adding/subtracting with unlike denominators)

Want to teach decimals and fractions together because it gives them the opportunity to see the
connection and helps with their understanding of rational numbers. (rational numbers are more
complex than whole numbers because there are several ways to represent it)
suggestions

Use their equivalent fraction knowledge to know that 7/10 =70/100
So then they are able to solve 3/10+ 27/100=30/100+27/100=57/100 (they can think
of 3 dimes with 27 cents) (3 dimes=30 pennies)
Models start 10x10 grid (relate to money) then move into number line etc.
Indicators if students are understanding
decimals

1. Use correct vocabulary/precise language

2. Accurately use models

3. Decompose and compose decimals (mentally) to help compare and order.

4. Understand the size

5. Decompose and compose using a model to demonstrate adding/subtracting.
activities

Art- to help students understand the connection between fraction and decimal (can be modified for
older grades with percents as well).

10x10 grid
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Design a pattern

Record the color, fraction, and decimal

Easily modified for older grades


Percents

Other fractions

Reducing the grid size
Benefits

Connections

Engaging

patterns
Other activities

Decimal top it (comparing)

Race to a $1.00

Decimal of the day- multiple representations

Collect receipts, sports statistics, weather reports

Use a deck of playing cards- create greatest number
possible with decimals, compare

Practice on slates, reading and writing decimal
notation
Resources

Scaptura, C., Suh, J., & Mahaffey, G. (2007). Masterpieces to Mathematics:
Using Art to Teach Fraction, Decimal, and Percent Equivalents.
Mathematics
Teaching in the Middle School, 13. Retrieved from
http://mason.gmu.edu/~jsuh4/math%20masterpiece.pdf

Cramer, K., Monson D., Ahrendt, S., Colum, K., Wiley, B., & Wyberg, T. (2015).
5 Indicators of Decimal Understandings. Teaching Children
Mathematics(22).

We., H. H. (2014). Teaching Fractions According to the Common Core
Standards. Retrieved from https://math.berkeley.edu/~wu/CCSSFractions_1.pdf

Barnett, C., Far West Lab. for Educational Research and Development, S. C., &
And, O. (1994). Fractions, Decimals, Ratios, & Percents: Hard To Teach and Hard
To Learn? Mathematics Teaching Cases.

D'Ambrosio, B. S., & Kastberg, S. E. (2012). Building Understanding of Decimal
Fractions. Teaching Children Mathematics, 18(9), 558-564.

Pramudiani, P., Zulkardi, Hartono, Y., & van Amerom, B. (2011). A Concrete
Situation for Learning Decimals. Indonesian Mathematical Society Journal On
Mathematics Education, 2(2), 215-230.