Students learn that a fraction is a product of a whole number and a unit fraction , or , and that . Students learn to add and subtract fractions with like denominators, and to add and subtract tenths and hundredths.
In this unit, students deepen their understanding of how fractions can be composed and decomposed, and they learn about operations on fractions.
In grade 3, students partitioned a whole into equal parts and identified one of the parts as a unit fraction. They learned that non-unit fractions and whole numbers are composed of unit fractions. They used visual fraction models, including tape diagrams and number lines, to represent and compare fractions. In a previous unit, students extended that work and reasoned about fraction equivalence.
Here students multiply fractions by whole numbers, add and subtract fractions with the same denominator, and add tenths and hundredths. They rely on familiar concepts and representations to do so. For instance, students had represented multiplication on a tape diagram, with equal-size groups and a whole number in each group. Here they use a tape diagram that shows a fraction in each group.
In earlier grades, students used number lines to represent addition and subtraction of whole numbers. Here, they use number lines to represent the decomposition of fractions into sums, and to reason about addition and subtraction of fractions with the same denominator, including mixed numbers.
Students then apply these skills in the context of measurement and data. They analyze line plots showing fractional lengths and find sums and differences to answer questions about the data.
Lastly students use fraction equivalence to find sums of tenths and hundredths. For instance, to find , they reason that is equivalent to , so the sum is , which is .
Create and analyze line plots that display measurement data in fractions of a unit ().
Represent and solve problems that involve the addition and subtraction of fractions and mixed numbers, including measurements presented in line plots.
Use various strategies to add and subtract fractions and mixed numbers with like denominators.
Section Narrative
In this section, students learn to add and subtract fractions by decomposing them into sums of smaller fractions, writing equivalent fractions, and using number lines to support their reasoning.
Students begin by thinking about a fraction as a sum of unit fractions with the same denominator, and then as a sum of other smaller fractions. They represent different ways to decompose a fraction by drawing “jumps” on number lines and writing different equations.
Number line. 21 evenly spaced tick marks. First tick mark, 0. Eleventh, 1. Twenty first, 2. Arrow, labeled ten tenths, from first tick mark to eleventh tick mark. Arrow, labeled three tenths, from eleventh tick mark to point at fourteenth tick mark, labeled thirteen tenths.
Number line. 21 evenly spaced tick marks. First tick mark, 0. Eleventh, 1. Twenty first, 2. Arrow, labeled five tenths, from first tick mark to sixth tick mark. Arrow, labeled eight tenths, from sixth tick mark to point at fourteenth tick mark, labeled thirteen tenths.
Working with number lines helps students see that a fraction greater than 1 can be decomposed into a whole number and a fraction, and then be expressed as a mixed number. This in turn can help us add and subtract fractions with the same denominator. For example, to find the value of , it helps to first decompose the 3 into , and then subtract from the .
Later in the section, students organize fractional length measurements ( inch, inch, and inch) on line plots. They apply their ability to interpret line plots and to add and subtract fractions to solve problems about measurement data.
What is the difference between the longest and the shortest shoe lengths?
Explain or show your reasoning.
Reason about equivalence to add tenths and hundredths.
Reason about equivalence to solve problems involving addition and subtraction of fractions and mixed numbers.
Section Narrative
In this section, students apply their understanding of fraction equivalence to add tenths and hundredths.
In the previous unit, students learned that . They use this reasoning to add tenths and hundredths by generating equivalent fractions. They also apply what they learned in the previous section to strategically use decomposition and the associative and commutative properties to add three or more tenths and hundredths, including mixed numbers.
This section ends with an optional lesson that allows students to apply what they have learned about multiplication, addition, and subtraction of fractions and mixed numbers to solve a design problem.
Represent and explain that a fraction is a multiple of , namely .
Represent and solve problems involving multiplication of a fraction by a whole number.
Section Narrative
In this section, students extend their earlier understanding of multiplication as equal groups of whole numbers of objects to now include equal groups of fractional pieces.
How many do you see? How do you see them?
Students begin by reasoning about groups containing unit fractions. For instance, they interpret the 5 plates with half an orange each as , which is . Later, they also reason about groups of non-unit fractions and write expressions to represent the quantities. For instance, 5 groups of can be expressed as or .
Later, students reason with diagrams and equations. Through repeated reasoning, they see regularity in the product of a whole number and a fraction (MP8). The numerator in the resulting fraction is the product of the whole number and the numerator of the fractional factor, and the denominator is the same as in the fractional factor.
These diagrams also help students see that some fractions can be represented by more than one multiplication expression. Students can reason that is , which also is equivalent to and , and is therefore equivalent to and , respectively.
By circling the diagram in various ways, students can visualize the different combinations of groups, understand their equivalence, and observe the associative property of multiplication. In doing this work, students practice looking for and making use of structure (MP7).
Students then solve problems that involve fraction multiplication, using diagrams and equations to show their reasoning. These diagrams also will be useful in later grades, when students make sense of fractions as quotients.
