Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, 100. 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. 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. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. 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. 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. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4). 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. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
Anchor Standard/Mathematical Practice(s)
MP.1. Make sense of problems and persevere in solving them. MP.2. Reason abstractly and quantitatively. MP.4. Model with mathematics. MP.5. Use appropriate tools strategically. MP.6. Attend to precision. MP.7. Look for and make use of structure. MP.8. Look for and express regularity in repeated reasoning.
Information Technology Standard
Use technology tools and skills to reinforce classroom concepts and activities.
Revised Bloom's Level of thinking
Remembering Understanding
Learning Target/Task Analysis
4.NF.3 A fraction with a numerator of one is called a unit fraction. When students investigate fractions other than unit fractions, such as 2/3, they should be able to decompose the non-unit fraction into a combination of several unit fractions. Example: 2/3 = 1/3 + 1/3 Being able to visualize this decomposition into unit fractions helps students when adding or subtracting fractions. Students need multiple opportunities to work with mixed numbers and be able to decompose them in more than one way. Students may use visual models to help develop this understanding. Example: 1 ¼ - ¾ = 4/4 + ¼ = 5/4 5/4 – ¾ = 2/4 or ½ Example of word problem:
Mary and Lacey decide to share a pizza. Mary ate 3/6 and Lacey ate 2/6 of the pizza. How much of
the pizza did the girls eat together? Solution: The amount of pizza Mary ate can be thought of a 3/6 or 1/6 and 1/6 and 1/6. The amount of pizza Lacey ate can be thought of a 1/6 and 1/6. The total amount of pizza they ate is 1/6 + 1/6 + 1/6 + 1/6 + 1/6 or 5/6 of the whole pizza. A separate algorithm for mixed numbers in addition and subtraction is not necessary. Students will tend to add or subtract the whole numbers first and then work with the fractions using the same strategies they have applied to problems that contained only fractions. Example:
Susan and Avery need 8 3/8 feet of ribbon to package gift baskets. Susan has 3 1/8 feet of ribbon
and Avery has 5 3/8 feet of ribbon. How much ribbon do they have altogether? Will it be enough to complete the project? Explain why or why not. The student thinks: I can add the ribbon Susan has to the ribbon Avery has to find out how much ribbon they have altogether. Susan has 3 1/8 feet of ribbon and Avery has 5 3/8 feet of ribbon. I can write this as 3 1/8 + 5 3/8. I know they have 8 feet of ribbon by adding the 3 and 5. They also have 1/8 and 3/8 which makes a total of 4/8 more. Altogether they have 8 4/8 feet of ribbon. 8 4/8 is larger than 8 3/8 so they will have enough ribbon to complete the project. They will even have a little extra ribbon left, 1/8 foot. Example:
Timothy has 4 1/8 pizzas left over from his soccer party. After giving some pizza to his friend, he
has 2 4/8 of a pizza left. How much pizza did Timothy give to his friend? Solution: Timothy had 4 1/8 pizzas to start. This is 33/8 of a pizza. The x’s show the pizza he has left which is 2 4/8 pizzas or 20/8 pizzas. The shaded rectangles without the x’s are the pizza he gave to his friend which is 13/8 or 1 5/8 pizzas. Mixed numbers are introduced for the first time in Fourth Grade. Students should have ample experiences of adding and subtracting mixed numbers where they work with mixed numbers or convert mixed numbers into improper fractions. Keep in mind Concrete-Representation-Abstract (CRA) approach to teaching fractions. Students need to be able to ―show‖ their thinking using concrete and/or representations BEFORE they move to abstract thinking. 4.NF.4 Students need many opportunities to work with problems in context to understand the connections between models and corresponding equations. Contexts involving a whole number times a fraction lend themselves to modeling and examining patterns. This standard builds on students’ work of adding fractions and extending that work into multiplication. (4.NF.4a) This standard extends the idea of multiplication as repeated addition (4.NF.4b) For example, 3 x (2/5) = 2/5 + 2/5 + 2/5= 6/5 = 6 X (1/5). Students are expected to use and create visual fraction models to multiply a whole number by a fraction. This standard calls for students to use visual fraction models (Area, Linear and Set Models) to solve word problems related to multiplying a whole number by a fraction. (4.NF.4c)
I can... I can join fractions with a numerator of 1 and the same denominator by adding them. I can separate fractions with the same denominator by subtracting them. I can dissect fractions using fraction models. I can write a decomposed fraction (broken apart) using an equation. I can add and subtract mixed numbers with like denominators. I can determine which operation (addition or subtraction) to use when solving word problems involving fractions with like denominators. I can multiply a fraction with a numerator of 1 by a whole number. I can multiply a fraction with a numerator greater than one by a whole number. I can solve word problems that involve multiplying a fraction by a whole number.
Illustrate adding and subtracting of fractions and mixed numbers using number lines, fraction strips, area models, set models, rulers, etc. Illustrate decomposing of fractions and mixed numbers with number lines, fraction strips, area models, set models, rulers, etc.
Solve. Simplify your answer.
5 x 1/4 =
a×16=76
6×23=
Problem Task:
Draw a picture to explain why 85=8×15.
Kathy is having a party. She wants 23cups of trail mix per guest. She expects 6 guests. How much trail mix should Kathy prepare? Write an equation and justify your solution with a visual model. • Draw a picture to show why these equations are true, and explain your reasoning: 7/8 = 4/8 + 3/8 214 = 34 + 64 • Sally said that 1/10 + 7/10 + 4/10 is the same as 1210. Is she correct? Explain and use a model to illustrate your explanation. (This is for Standard 4.c.) • Draw a model of the garden plot according to the data table below. The plot is divided into 15 sections. What fraction of the plot will be potatoes? Crop Number of sections Corn 4 Peas 2 Strawberries 2 Tomatoes 3 Potatoes The rest
Letter R--Have students work in groups to make fraction/decimal dominoes. One end has a picture; the other has the fraction and/or decimal notation. On the other side they could write the number/fraction out in word or expanded form and they could also write a story problem.
Have students write in their journal how they could share lunch with two or friends (Or) write how knowing fractions helps you play a musical instrument.
Near Zero, About ½ and Close to 1. Pass out index cards with decimal numbers. Students would work with partners to convert the decimal to a fraction and place your card in the correct bucket.
Have students write in their journal how they could share lunch with two or friends (Or) write how knowing fractions helps you play a musical instrument.
Common Core Standards
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, 100.4.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.
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. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
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.
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. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
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. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
Anchor Standard/Mathematical Practice(s)
MP.1. Make sense of problems and persevere in solving them.MP.2. Reason abstractly and quantitatively.
MP.4. Model with mathematics.
MP.5. Use appropriate tools strategically.
MP.6. Attend to precision.
MP.7. Look for and make use of structure.
MP.8. Look for and express regularity in repeated reasoning.
Information Technology Standard
Use technology tools and skills to reinforce classroom concepts and activities.Revised Bloom's Level of thinking
RememberingUnderstanding
Learning Target/Task Analysis
4.NF.3A fraction with a numerator of one is called a unit fraction. When students investigate fractions other than
unit fractions, such as 2/3, they should be able to decompose the non-unit fraction into a combination of
several unit fractions.
Example: 2/3 = 1/3 + 1/3
Being able to visualize this decomposition into unit fractions helps students when adding or subtracting
fractions. Students need multiple opportunities to work with mixed numbers and be able to decompose
them in more than one way. Students may use visual models to help develop this understanding.
Example:
1 ¼ - ¾ =
4/4 + ¼ = 5/4
5/4 – ¾ = 2/4 or ½
Example of word problem:
- Mary and Lacey decide to share a pizza. Mary ate 3/6 and Lacey ate 2/6 of the pizza. How much of
the pizza did the girls eat together?Solution: The amount of pizza Mary ate can be thought of a 3/6 or 1/6 and 1/6 and 1/6. The amount
of pizza Lacey ate can be thought of a 1/6 and 1/6. The total amount of pizza they ate is 1/6 + 1/6
+ 1/6 + 1/6 + 1/6 or 5/6 of the whole pizza.
A separate algorithm for mixed numbers in addition and subtraction is not necessary. Students
will tend to add or subtract the whole numbers first and then work with the fractions using the
same strategies they have applied to problems that contained only fractions.
Example:
- Susan and Avery need 8 3/8 feet of ribbon to package gift baskets. Susan has 3 1/8 feet of ribbon
and Avery has 5 3/8 feet of ribbon. How much ribbon do they have altogether? Will it be enough tocomplete the project? Explain why or why not.
The student thinks: I can add the ribbon Susan has to the ribbon Avery has to find out how much ribbon
they have altogether. Susan has 3 1/8 feet of ribbon and Avery has 5 3/8 feet of ribbon. I can write this as
3 1/8 + 5 3/8. I know they have 8 feet of ribbon by adding the 3 and 5. They also have 1/8 and 3/8 which
makes a total of 4/8 more. Altogether they have 8 4/8 feet of ribbon. 8 4/8 is larger than 8 3/8 so they will
have enough ribbon to complete the project. They will even have a little extra ribbon left, 1/8 foot.
Example:
- Timothy has 4 1/8 pizzas left over from his soccer party. After giving some pizza to his friend, he
has 2 4/8 of a pizza left. How much pizza did Timothy give to his friend?Solution: Timothy had 4 1/8 pizzas to start. This is 33/8 of a pizza. The x’s show the pizza he has left which
is 2 4/8 pizzas or 20/8 pizzas. The shaded rectangles without the x’s are the pizza he gave to his friend
which is 13/8 or 1 5/8 pizzas.
Mixed numbers are introduced for the first time in Fourth Grade. Students should have ample experiences
of adding and subtracting mixed numbers where they work with mixed numbers or convert mixed numbers
into improper fractions. Keep in mind Concrete-Representation-Abstract (CRA) approach to teaching
fractions. Students need to be able to ―show‖ their thinking using concrete and/or representations BEFORE
they move to abstract thinking.
4.NF.4
Students need many opportunities to work with problems in context to understand the connections between
models and corresponding equations. Contexts involving a whole number times a fraction lend themselves
to modeling and examining patterns. This standard builds on students’ work of adding fractions and
extending that work into multiplication. (4.NF.4a)
This standard extends the idea of multiplication as repeated addition (4.NF.4b) For example,
3 x (2/5) = 2/5 + 2/5 + 2/5= 6/5 = 6 X (1/5). Students are expected to use and create visual fraction
models to multiply a whole number by a fraction.
This standard calls for students to use visual fraction models (Area, Linear and Set Models) to solve word
problems related to multiplying a whole number by a fraction. (4.NF.4c)
I can...
I can join fractions with a numerator of 1 and the same denominator by adding them.
I can separate fractions with the same denominator by subtracting them.
I can dissect fractions using fraction models.
I can write a decomposed fraction (broken apart) using an equation.
I can add and subtract mixed numbers with like denominators.
I can determine which operation (addition or subtraction) to use when solving word problems involving fractions with like denominators.
I can multiply a fraction with a numerator of 1 by a whole number.
I can multiply a fraction with a numerator greater than one by a whole number.
I can solve word problems that involve multiplying a fraction by a whole number.
Essential Vocabulary
fraction, addition, subtraction, joining and separating parts, decompose, decomposition, mixed numbers, numerator, denominator
Sample Assessments
Illustrate adding and subtracting of fractions and mixed numbers using number lines, fraction strips, area models, set models, rulers, etc. Illustrate decomposing of fractions and mixed numbers with number lines, fraction strips, area models, set models, rulers, etc.
Solve. Simplify your answer.
5 x 1/4 =
a×16=76
6×23=
Problem Task:
Draw a picture to explain why 85=8×15.
Kathy is having a party. She wants 23cups of trail mix per guest. She expects 6 guests. How much trail mix should Kathy prepare? Write an equation and justify your solution with a visual model.• Draw a picture to show why these equations are true, and explain your reasoning:
7/8 = 4/8 + 3/8 214 = 34 + 64
• Sally said that 1/10 + 7/10 + 4/10 is the same as 1210. Is she correct? Explain and use a model to illustrate your explanation. (This is for Standard 4.c.)
• Draw a model of the garden plot according to the data table below. The plot is divided into 15 sections. What fraction of the plot will be potatoes?
Crop Number of sections
Corn 4
Peas 2
Strawberries 2
Tomatoes 3
Potatoes The rest
Adding and Subtracting FractionsDifferentiation
Intervention:
DPI Classroom Strategies p. 22
Letter R--Have students work in groups to make fraction/decimal dominoes. One end has a picture; the other has the fraction and/or decimal notation. On the other side they could write the number/fraction out in word or expanded form and they could also write a story problem.
http://daretodifferentiate.wikispaces.com/Choice+BoardsEnrichment:
Have students write in their journal how they could share lunch with two or friends (Or) write how knowing fractions helps you play a musical instrument.Near Zero, About ½ and Close to 1. Pass out index cards with decimal numbers. Students would work with partners to convert the decimal to a fraction and place your card in the correct bucket.
Have students write in their journal how they could share lunch with two or friends (Or) write how knowing fractions helps you play a musical instrument.
Instructional Resources
http://studyjams.scholastic.com/studyjams/jams/math/fractions/add-sub-common-denom.htmhttp://www.ncpublicschools.org/docs/acre/standards/support-tools/unpacking/math/4th.pdf
Notes and Additional Information
http://maccss.ncdpi.wikispaces.net/file/view/4thGradeUnit.pdf