Lesson 2

Sums and Differences of Fractions

Warm-up: Number Talk: Wholes and Units (10 minutes)

Narrative

This Number Talk encourages students to think about sums that make 10, 100, and one whole and look for ways to use these sums to mentally find the value of different expressions with whole numbers and fractions. The understandings elicited here will be helpful throughout this unit as students add and subtract whole numbers fluently and add and subtract fractions with the same denominator.

When students identify ways to make 1, 10, or 100, they look for and make use of the properties of operations and the structure of whole numbers and fractions (MP7).

Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”

Activity

• 1 minute: quiet think time
• Keep expressions and work displayed.
• Repeat with each expression.

Student Facing

Find the value of each expression mentally.

• $$38 + 62$$
• $$38\frac{2}{6} + 62\frac{3}{6}$$
• $$38\frac{2}{6} + 62\frac{3}{6} + 17\frac{1}{6}$$
• $$138\frac{2}{6} + 162\frac{3}{6} + 17\frac{2}{6}$$

Activity Synthesis

• “How were these expressions the same? How were they different?” (They each have $$38 + 62$$, so you can look for ways to make 100 in each. The last three expressions all have a mixed number with sixths. Some expressions did not have fractions. One expression had fractions that had a sum that was equivalent to 1 whole.)
• As needed, “How could we look for ways to make 100 in each expression? How could we look for ways to make 1?”

Activity 1: Straws for A Roller Coaster (15 minutes)

Narrative

The purpose of this activity is to represent and solve a measurement problem with fractions. Students may approach this activity in multiple ways and are invited to apply what they know about operations with fractions, comparing fractions, and fraction equivalence to make sense of and solve the problems (MP2). Throughout the activity, listen for the ways students use what they know about comparing fractions and fraction equivalence as they make sense of the problem. Although students may consider ways to subtract fractions with unlike denominators, this is not a requirement for grade 4. Focus the conversation during the activity and the synthesis on how students can solve the problem by reasoning about equivalent fractions and the representation they use to make sense of the problem.

MLR8 Discussion Supports. Synthesis: Create a visual display of the diagrams. As students share their strategies, annotate the display to illustrate connections. For example, next to each representation, write how the diagram relates to the situation.

• Groups of 2

Activity

• 5 minutes: independent work time
• 5 minutes: partner work
• Monitor for different representations of the situation (for example, equations, drawings, and number lines).

Student Facing

In science class, Noah, Tyler, and Jada are building a model of a roller coaster out of 1-foot-long paper straws.

• Noah needs a piece that is $$\frac{7}{12}$$ foot long.
• Tyler needs one that is $$\frac{1}{4}$$ foot long.
• Jada needs one that is shorter than the other two.

Jada says, “We can just use one straw for all these pieces.”

1. Draw a diagram to represent this situation and explain to your partner how it matches the situation. Then, find the length of the piece of straw that could be Jada’s piece.

2. Did Noah use more than $$\frac{1}{2}$$ foot or less than $$\frac{1}{2}$$ foot of straw? Explain or show your reasoning.
3. Tyler says, “If Jada uses a piece that is $$\frac{1}{6}$$ foot long, there would be a piece of straw that is $$\frac{1}{12}$$ foot left.”

Do you agree or disagree with Tyler? Explain your reasoning.

Student Response

If students create representations that do not match the quantities in the problem, consider asking:

• “How does your representation show the length of the straw? How does it show Noah and Tyler’s pieces?”
• “How does your representation show the action in the situation?”

Activity Synthesis

• Invite previously selected students to share their arguments and allow peers to extend and add ideas to the conversation.
• “How does each representation match this situation about the straw?” (They each show the total length of the straw and ways it could be broken into smaller parts. They each show the length of Noah and Tyler’s pieces. They show how long Jada’s piece could be.)

Activity 2: Tall Enough for a Ride? (20 minutes)

Narrative

In this activity, students practice solving word problems that involve adding and subtracting mixed numbers. Students interpret fractions in the context of comparing heights and use what they know about decomposing whole numbers and equivalent fractions to make sense of and solve each problem (MP2). Look for the different ways students use what they know about the structure of whole numbers and fractions as they reason about how to solve each problem and share their thinking with others.

Action and Expression: Develop Expression and Communication. Provide access to a variety of tools, such as fraction strips and meter sticks (or if possible, tape measures).
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing, Attention

Launch

• Groups of 2
• “Have you ever ridden a ride at an amusement park or a fair? Have you ever not been able to ride a ride because you were too young or not tall enough?”
• As needed, explain that some rides require the riders to be a certain height for their safety.
• “Let’s solve some problems about height and amusement park rides.”

Activity

• 10 minutes: independent work time
• 5 minutes: partner discussion
• “Compare your strategy with your partner’s strategy.”
• Monitor for students who use the relationship between addition and subtraction to represent the situations.

Student Facing

Lin’s class is on a trip to the amusement park. Visitors must be at least a certain height to get on rides. Use the table to answer questions about four students’ height.

ride height requirement
tilt and spin 52 inches
roller coaster 54 inches
bumper cars 44 inches
1. Andre is $$3\frac{3}{8}$$ inches shorter than the height requirement for the roller coaster. How tall is Andre?
2. Lin is $$\frac{18}{8}$$ inches taller than Andre. How tall is Lin?
3. Elena was $$1\frac{3}{4}$$ inches too short to ride the bumper cars last year. Since then she grew $$4\frac{1}{2}$$ inches. How tall was Elena last year? How tall is she now?

4. Mai is tall enough to ride all the rides this year. Mai was $$51\frac{7}{8}$$ inches tall last year. At least how many inches did Mai grow?

Student Response

If students add or subtract quantities in each problem in ways that do not match the situation, consider asking:

• “Who should be taller _____ or _____? How do you know?”
• “Does your answer match the heights in the situation? How do you know?”
• “How could you use a diagram to help you represent the different heights in this problem?”

Activity Synthesis

• Invite students to share their equations and explain their solution for Lin’s height.
• Display: $$50\frac{5}{8} + \frac{18}{8}$$
• “How might you reason about how to find the value of Lin’s height?” (Think about adding $$\frac{3}{8}$$ to $$\frac{5}{8}$$ to make 51, then it'd be $$51\frac{15}{8}$$. We are adding more than 1 because $$\frac{18}{8}$$ is more than $$\frac{8}{8}$$.$$\frac{18}{8} = \frac{8}{8} + \frac{8}{8} + \frac{2}{8}$$ or $$2\frac{2}{8}$$, and $$50\frac{5}{8} + 2\frac{2}{8} = 52\frac{7}{8}$$)

($$50\frac{5}{8} + \frac{18}{8} = 50\frac{5}{8} + 2\frac{2}{8} = 52\frac{7}{8}$$)

• Select students to share both addition and subtraction equations for Mai’s current height and discuss how both equations ($$54 - 51\frac{7}{8} = 2\frac{1}{8}$$ and $$54 = 51\frac{7}{8} + 2\frac{1}{8}$$) could be used to represent this situation.

Lesson Synthesis

Lesson Synthesis

“Problem solving is about reasoning. Today we solved problems involving addition and subtraction of fractions and mixed numbers.”

“What did you use to make sense of the problems? What helped you make sense of the strategies that others shared?” (The diagrams and representations helped me visualize the situation and make sense of the math in the problems. It helped to see others' equations and compare them to what I wrote.)

“How did understanding fraction equivalence help you solve the problems?” (It helped me write equations that made it easier to add or subtract. It helped me compare fractions and make sense of what the problems were asking.)