2 weeks
In this unit students will:
- Identify visual and written representations of fractions
- Understand representations of simple equivalent fractions
- Understand the concept of mixed numbers with common denominators to 12
- Add and subtract fractions with common denominators
- Add and subtract mixed numbers with common denominators
- Convert mixed numbers to improper fractions and improper fractions to mixed fractions
- Understand a fraction as a multiple of . (for example: model the product of as 3 x ).
- Understand a multiple of as a multiple of , and use this understanding to multiply a fraction by a whole number.
- 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.
- Multiply a whole number by a fraction
- Make a line plot to display a data set of measurements in fractions of a unit ( , , )
- Solve problems involving addition and subtraction of fractions by using information presented in line plots
- Solve multi-step problems using the four operations
Unit 4 Overview video Parent Letter Number Talks Resources Vocabulary Cards
Prerequisite Skills Assessment Sample Post Assessment
Topic 1: Operations with Fractions
Big Ideas/Enduring Understandings:
- Fractions can be represented in multiple ways including visual and written form.
- Fractions can be decomposed in multiple ways into a sum of fractions with the same denominator.
- Fractional amounts can be added and/or subtracted.
- Mixed numbers can be added and/or subtracted.
- Mixed numbers and improper fractions can be used interchangeably because they are equivalent.
- Mixed numbers can be ordered by considering the whole number and the fraction.
- Proper fractions, improper fractions and mixed numbers can be added and/or subtracted.
- Fractions, like whole numbers can be unit intervals on a number line.
- Fractional amounts can be added and/or multiplied.
- If given a whole set, we can determine fractional amounts. If given a fractional amount we can determine the whole set.
- When multiplying fractions by a whole number, it is helpful to relate it to the repeated addition model of multiplying whole numbers.
- A visual model can help solve problems that involve multiplying a fraction by a whole number.
- Equations can be written to represent problems involving the multiplication of a fraction by a whole number.
- Multiplying a fraction by a whole number can also be thought of as a fractional proportion of a whole number. For example, x 8 can be interpreted as finding one-fourth of eight.
- Data can be measured and represented on line plots in units of whole numbers or fractions.
- Data can be collected and used to solve problems involving addition or subtraction of fractions.
- How are fractions used in problem-solving situations?
- How can equivalent fractions be identified?
- How can a fraction represent parts of a set?
- How can I add and subtract fractions of a given set?
- How can I find equivalent fractions?
- How can I represent fractions in different ways?
- How are improper fractions and mixed numbers alike and different?
- How can you use fractions to solve addition and subtraction problems?
- How do we add fractions with like denominators?
- How do we apply our understanding of fractions in everyday life?
- What do the parts of a fraction tell about its numerator and denominator?
- What happens when I add fractions with like denominators?
- What is a mixed number and how can it be represented?
- What is an improper fraction and how can it be represented?
- What is the relationship between a mixed number and an improper fraction?
- Why does the denominator remain the same when I add fractions with like denominators?
- How can I model the multiplication of a whole number by a fraction?
- How can I multiply a set by a fraction?
- How can I multiply a whole number by a fraction?
- How can I represent a fraction of a set?
- How can I represent multiplication of a whole number?
- How can we model answers to fraction problems?
- How can we write equations to represent our answers when solving word problems?
- How do we determine a fractional value when given the whole number?
- How do we determine the whole amount when given a fractional value of the whole?
- How is multiplication of fractions similar to repeated addition of fraction?
- What does it mean to take a fractional portion of a whole number?
- What strategies can be used for finding products when multiplying a whole number by a fraction?
- How do we make a line plot to display a data set?
Content Standards
Content standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
MGSE4.NF.3 Understand a fraction with a numerator >1 as a sum of unit fractions 1.
- Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
- 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.
- 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.
- 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.
- 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).
- 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.)
- 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?
MGSE4.MD.4 Make a line plot to display a data set of measurements in fractions of a unit (,,). Solve problems involving addition and subtraction of fractions with common denominators by using information presented in line plots. For example, from a line plot, find and interpret the difference in length between the longest and shortest specimens in an insect collection.
Use the four operations with whole numbers to solve problems.
MGSE4.OA.3 Solve multistep word problems with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a symbol or letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
Vertical Articulation
Third GradeFraction, Measurement, & Problem-SolvingStandards
Develop understanding of fractions as numbers
- MGSE3.NF.1 Understand a fraction as the quantity formed by 1 part when a whole is partitioned into b equal parts (unit fraction); understand a fraction as the quantity formed by a parts of size. For example, means there are three parts, so = + + .
- MGSE3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that
each part has size. Recognize that a unit fraction is located whole unit from
0 on the number line.
b. Represent a non-unit fraction ab on a number line diagram by marking off a
lengths of (unit fractions) from 0. Recognize that the
resulting interval has size ab and that its endpoint locates the non-unit
fraction on the number line.
- MGSE3.NF.3 Explain equivalence of fractions through reasoning with visual fraction models. Compare fractions by reasoning about their size.
the same point on a number line.
b. Recognize and generate simple equivalent fractions with denominators of 2,
3, 4, 6, and 8, e.g., = , = . Explain why the fractions are equivalent, e.g.,
by using a visual fraction model.
c. Express whole numbers as fractions, and recognize fractions that are
equivalent to whole numbers. Examples: Express 3 in the form 3 = (3 wholes
is equal to six halves); recognize that = 3; locate and 1 at the same point of
a number line diagram.
d. Compare two fractions with the same numerator or the same denominator by
reasoning about their size. Recognize that comparisons are valid only when
the two fractions refer to the same whole. Record the results of comparisons
with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual
fraction model.
Represent and interpret data.
- MGSE3.MD.4 Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.
- MGSE3.OA.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
Use equivalent fractions as a strategy to add and subtract fractions
- MGSE5.NF.1 Add and subtract fractions and mixed numbers with unlike denominators by finding a common denominator and equivalent fractions to produce like denominators.
- MGSE5.NF.2Solve word problems involving addition and subtraction of fractions, including cases of unlike denominators (e.g., by using visual fraction models or equations to represent the problem). Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + ½ = 3/7, by observing that 3/7 < ½.
- MGSE5.NF.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. Example: can be interpreted as “3 divided by 5 and as 3 shared by 5”.
- MGSE5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
- Apply and use understanding of multiplication to multiply a fraction or whole number by a fraction. Examples: ×?as ×and ×=
- Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths.
- MGSE5.NF.5 Interpret multiplication as scaling (resizing), by:
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
- MGSE5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
- MGSE5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
Represent and interpret data.
- MGSE5.MD.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.
Instructional Strategies
BUILD FRACTIONS FROM UNIT FRACTIONS BY APPLYING AND EXTENDING PREVIOUS UNDERSTANDINGS OF OPERATIONS ON WHOLE NUMBERS
Students extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number. Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: operations, addition/joining, subtraction/separating, fraction, unit fraction, equivalent, multiple, reason, denominator, numerator, decomposing, mixed number, rules about how numbers work (properties), multiply, multiple.
Standard NF.3
In Grade 3, students added unit fractions with the same denominator. Now, they begin to represent a fraction by decomposing the fraction as the sum of unit fraction and justify with a fraction model. For example, =++
Students also represented whole numbers as fractions. They use this knowledge to add and subtract mixed numbers with like denominators using properties of number and appropriate fraction models. It is important to stress that whichever model is used, it should be the same for the same whole. For example, a circular model and a rectangular model should not be used in the same problem.
Understanding of multiplication of whole numbers is extended to multiplying a fraction by a whole number. Allow students to use fraction models and drawings to show their understanding.
NF.3a – Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
A fraction with a numerator of one is called a unit fraction. When students investigate fractions other than unit fractions, such as , they should be able to join (compose) or separate (decompose) the fractions of the same whole.
Example: = + - Being able to visualize this decomposition into unit fractions helps students when adding or subtracting fractions.
Students need multiple opportunities to work with mixednumbers and be able to decompose them in more than one way. Students may use visual models to help develop this understanding:
Example: 1 – =? → + = → − = or
Example of word problem:
Mary and Lacey decide to share a pizza. Mary ate and Lacey ate of the pizza. How much of the pizza did the girls eat together?
Possible solution: The amount of pizza Mary ate can be thought of as or + + . The amount of pizza Lacey ate can be thought of as + . The total amount of pizza they ate is + + + + or of the pizza.
NF.3b - 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.
Students should justify their breaking apart (decomposing) of fractions using visual fraction models. The concept of turning mixed numbers into improper fractions needs to be emphasized using visual fraction models and decomposing.
Example:
NF.3c - 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.
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 8feet of ribbon to package gift baskets. Susan has 3feet of ribbon and Avery has 5feet 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 3feet of ribbon and Avery has 5feet of ribbon. I can write this as 3+5. I know they have 8 feet of ribbon by adding the 3 and 5. They also have and which makes a total of more. Altogether they have 8feet of ribbon. 8is larger than 8so they will have enough ribbon to complete the project. They will even have a little extra ribbon left, foot.