The purpose of this Warm-up is to elicit what students know about the numbers 15 and 30, preparing them to work with fractions whose denominators are factors of 15 and 30 later in the lesson. While students may bring up many things about these numbers, relating the two numbers by their factors and multiples is the important discussion point.
Launch
Display the numbers.
“What do you know about 15 and 30?”
1 minute: quiet think time
Activity
Record responses.
What do you know about 15 and 30?
Student Response
Loading...
Advancing Student Thinking
Activity Synthesis
If no students mentioned factors of 15 and 30, ask them about it.
“What are the factors of 15?” (1, 3, 5, 15)
“What are the factors of 30?” (1, 2, 3, 5, 6, 10, 15, 30)
“What factors do they have in common?” (1, 3, 5, 15)
“Do 15 and 30 have any common multiples? What are some of them?” (30, 60, 90)
Activity 1
Standards Alignment
Building On
Addressing
4.NF.A.1
Explain why a fraction is equivalent to a fraction by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
In earlier lessons, students compared fractions by rewriting one fraction as an equivalent fraction with the same denominator as the second fraction. In this activity, students see that—although it’s still possible to compare the fractions—this particular strategy doesn’t work if neither of the denominators of the two fractions is a factor or multiple of each other. Students learn that in such a case, both fractions can be expressed as equivalent fractions with a common denominator that is different from either fraction’s original denominator and is a multiple of both.
MLR8 Discussion Supports. Synthesis: Display sentence frames to support partner discussions: “First, I _____ because . . . .”, “I noticed _____ so I . . . .” Advances: Speaking, Conversing
Representation: Develop Language and Symbols. Activate background knowledge. Provide students with access to a visible display that shows definitions or reminders of the terms “factor” and “multiple.” Supports accessibility for: Memory, Language
Launch
Groups of 2
Activity
“Take a few quiet minutes to work on the first two questions.”
6–7 minutes: independent work time
“Share your responses to both questions with your partner. Be sure to explain how you compared the fractions in the first question.”
3–4 minutes: partner discussion
Pause for a brief whole-class discussion. Invite students to share their responses for the first two questions.
If no students suggest that the second pair of fractions are harder to compare because their denominators have no factors in common (or one does not multiply or divide to make the other), ask them about it.
“Now work with your partner on the last question.”
3–4 minutes: group work time
In each pair of fractions, which fraction is greater? Explain or show your reasoning.
or
or
Han says he can compare and by writing an equivalent fraction for . He says he can’t use that strategy to compare and . Do you agree? Explain your reasoning.
Priya and Lin show different ways to compare and . Make sense of what they did. How are their strategies alike? How are they different?
Priya
is greater than ,
so is greater than .
Lin
is greater than ,
so is greater than .
Student Response
Loading...
Advancing Student Thinking
Activity Synthesis
Display Priya and Lin’s reasoning for all to see. Select students to share their observations on how the two are alike and how they are different.
Highlight the fact that both students rewrote the two fractions so that they have a common denominator.
“Why might it be helpful to write equivalent fractions with the same denominator?” (It is easier to compare the fractions when the fractional part is the same size.)
“Can we choose any number to be the common denominator?” (No, it must be a multiple of both of the original denominators.)
“Does it matter if we choose a smaller or a larger common multiple?” (No, but it could work better to multiply by a smaller number.)
Activity 2
Standards Alignment
Building On
Addressing
4.NF.A.1
Explain why a fraction is equivalent to a fraction by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
This activity serves two main goals: to prompt students to rewrite pairs of fractions as equivalent fractions with a common denominator, and to consider this newly developed skill as a possible way to compare fractions.
To write equivalent fractions, many students are likely to reason numerically (by multiplying or dividing the numerator and denominator by a common number). Some may, however, find equivalent fractions effectively by continuing to reason about how many of this fractional part is in that fractional part.
To compare the fractions in the second question, students may choose to write equivalent fractions with a common denominator because they were just learning to do so. The fractions, however, were chosen so that students have opportunities to choose an approach strategically, rather than writing equivalent fractions each time. For instance, students may notice that:
In part a, one fraction is away from , and the other is from .
In part b, one fraction is greater than 2 and the other is less than 2.
In part c, writing an equivalent fraction for only one given fraction (rather than for both) is sufficient for comparing.
In part d, one fraction is less than , and the other is greater than .
Launch
Groups of 2
Activity
“Work with your partner to write equivalent fractions for the question.”
7–8 minutes: group work time on the first set of fractions
Pause for a brief whole-class discussion.
Poll the class on the common denominator they chose for each pair of fractions. Record their responses. (Likely denominators for each part:
24 or 12
24
60 or 30
40 or 20)
Some students are likely to suggest multiplying one denominator by the other. Discuss whether there are other ways to find a common denominator.
“Work independently to compare the fractions in the second set of questions. Be prepared to explain how you know which fraction is greater. ”
7–8 minutes: independent work time on the second set of questions
Monitor for students who are strategic in how they compare the pairs of fractions in the second question (not exclusively writing equivalent fractions).
For each pair of fractions, write a pair of equivalent fractions with a common denominator.
and
and
and
and
For each pair of fractions, decide which fraction is greater. Be prepared to explain your reasoning.
or
or
or
or
Activity Synthesis
See Lesson Synthesis.
Lesson Synthesis
Invite students to share their responses to the last set of questions of Activity 2 and how they went about making comparisons. Record their responses.
Select students who made strategic choices when making comparisons to share their thinking.
Emphasize that, while it is possible to compare every pair of fractions by rewriting them so that they have a common denominator, all the fractions could be compared by reasoning in other ways.
Standards Alignment
Building On
4.OA.B.4
Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols , =, or , and justify the conclusions, e.g., by using a visual fraction model.
If students multiply the denominators to find equivalent fractions for all the problems in the second problem set, consider asking:
“How did you decide to compare the fractions?”
“How could you use benchmark numbers, such as one-half or one whole to compare the fractions?”
4.NF.A.2
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols , =, or , and justify the conclusions, e.g., by using a visual fraction model.
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols , =, or , and justify the conclusions, e.g., by using a visual fraction model.
