The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer, based on experience and known information. In this Estimation Exploration, students apply what they know about fractions to estimate the length of an insect that is less than 1 inch.
Launch
Groups of 2
Display the image.
“What is an estimate that’s too high? Too low? About right?”
1 minute: quiet think time
Activity
“Discuss your thinking with your partner.”
1 minute: partner discussion
Record responses.
What is the length of this ladybug?
Record an estimate that is:
too low
about right
too high
Student Response
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Advancing Student Thinking
Activity Synthesis
Consider asking:
“Is anyone’s estimate less than inch? Is anyone’s estimate greater than inch?”
“Based on this discussion, does anyone want to revise their estimate?”
Activity 1
Standards Alignment
Building On
Addressing
3.NF.A.3.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.
The purpose of this activity is for students to compare two numbers in context, to explain or show their reasoning, and to record the results of the comparison with the symbol >, <, or = (MP2). The numbers may be fractions with the same numerator or the same denominator, or a fraction and a whole number.
Students are likely to generate different comparison statements for the same situation. For example, they may write or to represent as the greater fraction. During the Activity Synthesis, discuss how both statements capture the comparison and are valid.
MLR8 Discussion Supports. Synthesis: As students share the similarities and differences between the representations and the comparison statements, use gestures to emphasize what is being described. For example, show with your fingers the partitions, such as fourths or eighths, that are the same in the representations being compared. Advances: Listening, Representing
Engagement: Provide Access by Recruiting Interest. Synthesis: Optimize meaning and value. Invite students to share a connection between the activity content and their own lives. Ask “How can I use this in my own life?” Supports accessibility for: Conceptual Processing
Launch
Groups of 2
“Let’s use what we learned about comparing fractions and recognizing equivalent fractions to solve problems about lengths.”
Activity
“Work independently to solve the problems. For each one, be sure to show your thinking and to write a comparison statement.”
6–8 minutes: independent work time
“Share your responses and your reasoning with your partner.”
2–3 minutes: partner discussion
Monitor for:
Different representations or reasoning strategies used for the same problem, such as diagrams, fraction strips, number lines, or explanations in words.
Different statements written for the same problem, such as and , or and .
Activity Synthesis
Select 2 or 3 students to share their reasoning strategies or representations for at least one of the situations.
“How did you use what you’ve learned in earlier lessons to compare fractions?” (If the fractions had the same numerator, I thought about the size of the denominators. If they had the same denominator, I compared the numerators.)
Select students, who wrote different but equally valid comparison statements (for instance, and ), to share.
Discuss how to read each statement, and ask students whether both accurately represent the comparison.
Emphasize that we can write comparison statements in more than one way, but we need to check that the statements make sense given the numbers we write and the symbols we use.
Activity 2
Standards Alignment
Building On
Addressing
3.NF.A.3.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.
The purpose of this activity is for students to generalize what they have learned about comparing fractions to complete comparison statements and to generate new ones, using the symbols >, <, and =. Students first consider all numbers that could make an incomplete comparison statement true. Then they find fractions greater than, less than, and equivalent to a given fraction, and write statements to record the comparisons. As in the previous activity, students see that there are different ways to record the same comparison of two numbers.
Launch
Groups of 2
“Now that we have practiced comparing fractions, let's come up with fractions that are greater than, less than, or equivalent to a given fraction.”
Activity
“Noah was working with fractions when some juice spilled. Now he can’t tell what some of the numbers are. Help him figure out what was written before the juice was spilled.”
5 minutes: partner work time
Pause for a discussion and invite students to share the numbers that they think make sense in the first statement ().
Display or write the comparison statements, using students’ numbers.
“Do all of these statements make sense? How do you know?”
“Are there any more statements that we could write?”
If time permits, repeat with the next two parts.
“Now work independently on the last set of problems.”
5 minutes: independent work time
Oh, no! Some juice spilled on Noah’s fractions. Help him figure out what was written before the juice was spilled.
Find as many numbers as you can to make each statement true. Explain or show your reasoning.
Find a fraction that is greater than, a fraction that is less than, and a fraction that is equivalent to each fraction. Then write a statement that uses the symbol >, <, or = to record each comparison.
Greater than : __________
Statement:
Less than : __________
Statement:
Equivalent to : __________
Statement:
Greater than : __________
Statement:
Less than : __________
Statement:
Equivalent to : __________
Statement:
Student Response
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Advancing Student Thinking
If students don’t find more than one number that would make the statements in the first problem true, consider asking:
“You found one number that made the statement true. How did you find that number and know that it made the statement true?”
“How could you use a similar strategy to find another number that would make the statement true?”
Activity Synthesis
Invite students to share their responses to the last set of problems.
“How did you find a fraction that was less than (or greater than or equivalent to) the given fraction?”
Activity 3
Optional
Standards Alignment
Building On
Addressing
3.NF.A.2
Understand a fraction as a number on the number line; represent fractions on a number line diagram.
The purpose of this activity is for students to use their knowledge of fractions to locate fractions with different denominators on the number line. Students may use a variety of reasoning to locate the fractions, including their knowledge of equivalence, strategies about the same numerator or the same denominator, or benchmark numbers with which they are familiar. The Activity Synthesis focuses on the variety of strategies that make sense, and students should be encouraged to use different strategies for different fractions as needed.
Although students have represented fractions on number lines (including those with two different denominators, when reasoning about equivalence), this activity is optional because representing multiple fractions of different denominators on the same number line involves a deeper understanding than is required by the standards.
Launch
Groups of 2
Activity
“Work independently to start placing these fractions on the number line.”
3–5 minutes: independent work time
“Share your strategies with your partner, and place any fractions you have left together.”
5–7 minutes: partner work time
Monitor for students who:
Place each fraction separately by partitioning for that single fraction.
Compare the fraction they are placing to others they’ve already placed.
Use equivalent fractions.
Use strategies about the same numerator or the same denominator.
Use benchmarks, such as whole numbers or halves.
Locate and label each fraction on the number line. Be prepared to share your reasoning.
Student Response
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Advancing Student Thinking
Activity Synthesis
Invite students to share a variety of strategies for placing fractions on the number line.
Consider asking:
“Which fractions were easier to place on the number line?”
“Which fractions were more difficult?”
“Did anyone have the same strategy but would explain it differently?”
Lesson Synthesis
“We have compared a lot of different fractions. Fractions with the same denominator, fractions with the same numerator, and in this lesson, we again saw fractions that were equivalent.”
“What do you think would be some of the most important things to tell a friend who wanted to learn about comparing two fractions?” (I would tell my friend to think about whether they can draw a representation, like a number line or a diagram, to see which fraction is greater. I think they need to know whether the fractions have the same numerator or the same denominator. They can check to see if the fractions are the same size or are at the same location because that means they are equivalent.)
Consider asking: “Does your strategy for comparing fractions change, depending on the fractions?
Student Section Summary
We compared fractions with the same numerator or the same denominator, and used the symbol , , or to record the comparison. We used diagrams and number lines to represent our thinking.
Standards Alignment
Building On
3.NF.A.2
Understand a fraction as a number on the number line; represent fractions on a number line diagram.
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.
