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Activity 1

Standards Alignment

Building On

Addressing

Instructional Routines

None

Materials

To Gather

Straightedges

Activity Narrative

The purpose of this activity is to remind students of a key insight from grade 3—that the same point on the number line can be named with fractions that don’t look alike. Students see that those fractions are equivalent, even though their numerators and denominators may be different.

Students have multiple opportunities to look for regularity in repeated reasoning (MP8). For instance, they are likely to notice that:

Fractions that have the same number for the numerator and denominator all represent 1.
In fractions that describe the halfway point between 0 and 1, the numerator is always half the denominator, or the denominator twice the numerator.
In fractions that describe , the denominator is 4 times the numerator.

These observations will help students to identify and generate equivalent fractions later in the unit.

Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were most important to solve the problem. Display the sentence frame, “The next time points are in the same place on different number lines, I will . . . .“
Supports accessibility for: Language, Attention

Launch

Groups of 2
Give students access to straightedges. Display the first set of number lines.
“What do you notice? What do you wonder?” (I notice each number line has different fractions represented. The first number line has a point that is half-way between 0 and 1 labeled , but if you label all the tick marks you won’t have 2 in the denominator for all of them. I wonder what fraction goes on each mark? Can you have a number line with both halves and fourths? How many fourths are at the line?)
1 minute: quiet think time
“Share what you noticed and wondered with your partner.”
1 minute: partner discussion

Activity

“Take a moment to work independently on the task. Then discuss your work with your partner.”
“The labels that you write for the points on different number lines should be different.”
7–8 minutes: independent work time
Monitor for students who:
- Partition each number line into as many parts as the denominator before naming a fraction for the point on the number line.
- Use multiplicative relationships between denominators to name a fraction. (For instance, , so the line showing twelfths has 3 times as many parts as the one showing fourths.)

Activity Synthesis

Select students to share their responses and reasoning for the first set of questions. Highlight explanations that convey that:
- Any fraction with the same number for the numerator and denominator has a value of 1.
- Equivalent fractions share the same location or are the same distance from 0 on the number line.
Invite previously selected students to share their responses for the second set of questions.
Consider asking:
- “Who can restate _______ 's reasoning in a different way?”
- “How are these ways for labeling the points the same? How are they different?“

Activity 2

Standards Alignment

Building On

Addressing

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students reason about whether two fractions are equivalent in the context of distance. To support their reasoning, students use number lines and their understanding of fractions with related denominators (where one number is a multiple of the other). The given number lines each have only one tick mark between 0 and 1, so students need to partition each line strategically to represent two fractions with different denominators on the diagram.

To help students intuit the distance of 1 mile, consider preparing a neighborhood map that shows the school and some points that are a mile away. Display the map during the Launch.

Launch

Groups of 2
“Who has walked a mile? Who has run a mile?”
“How far is 1 mile? How would you describe it?”
Consider showing a map of the school and some landmarks or points on the map that are 1 mile away.

Activity

6–8 minutes: independent work time
Monitor for the different ways students reason about the equivalence of and . For instance, they may:
- Know that 1 fourth is equivalent to 3 twelfths and reason that 3 fourths must be 9 twelfths.
- Note that and are both halfway between and 1 on the number line.
- Locate and on the same number line (or separate ones) and show that they are in the same location.
2–3 minutes: partner discussion

Han and Kiran plan to go for a run after school.
- Han says, “Let’s run mile. That’s how far I run to my soccer practice.”
- Kiran says, “I can only run mile.”
Which distance should they run? Explain your reasoning. Use one or more number lines to show your reasoning.
Mai wants to join Han and Kiran on their run. She says, “How about we run mile?”

Is the distance Mai suggests the same as what her friends wanted to run? Explain or show your reasoning.

Student Response

to access Student Response.

Advancing Student Thinking

Activity Synthesis

Invite previously selected students to share their responses and how they knew that is equivalent to but is not.
To facilitate their explanations, ask students to display their work, or display blank number lines for them to annotate.
Consider asking:
- “Who reasoned the same way but would explain it differently?”
- “Who thought about it differently but arrived at the same conclusion?”