Lesson 5

The purpose of this warm-up is to remind students of doubling as a strategy for multiplication in which a factor in one product is twice a factor in another product. The reasoning that students do here with the factors 2, 4, 8, and 16 will support them as they reason about equivalent fractions and find multiples of numerators and denominators.

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

• Record answers and strategy.
• Keep problems and work displayed.
• Repeat with each expression.

• “How did the first three expressions help you find the value of the last expression?”

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 \(\frac{1}{4}\), the denominator is 4 times the numerator.

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

• 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 \(\frac{1}{2}\), 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 \(\frac{1}{2}\) line?)
• 1 minute: quiet think time
• “Share what you noticed and wondered with your partner.”
• 1 minute: partner discussion

• “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, \(4 \times 3 = 12\), so the line showing twelfths has 3 times as many parts as the one showing fourths)

• 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.
• Select students to share their responses for the second set of questions. 

MLR3 Clarify, Critique, Correct

• If students show the following partially correct idea, display this explanation:
  “To know what fraction a point represents, I counted the tick marks from 0. Then, I used the denominator of the fraction for 1. For example, for question 2 part b, the point is on the first tick mark from 0 and the label for 1 says \(\frac{10}{10}\), so I’d label the point \(\frac{1}{10}\).”
• Read the explanation aloud.
• “What do you think the student understands well? What do you think they might have misunderstood?”
• 1 minute: quiet think time
• 2 minutes: partner discussion
• “With your partner, work together to write a revised explanation.”
• Display and review the following criteria:
  • Explain: How would one know what numerator and denominator the fraction can have?
  • Write in complete sentences.
  • Use words such as: “first,” “next,” or “then.”
  • Include the number line diagram.
• 3–5 minutes: partner work time
• Select 1–2 groups to share their revised explanation with the class. To facilitate their explanation, display blank number lines for students to annotate. Record responses.
• “What is the same and what is different about the explanations?”

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.

• 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 a mile away.

• 6–8 minutes: independent work time
• Monitor for the different ways students reason about the equivalence of \(\frac{9}{12}\) and \(\frac{3}{4}\). For instance, they may:
  • know that 1 fourth is equivalent to 3 twelfths and reason that 3 fourths must be 9 twelfths
  • note that \(\frac{3}{4}\) and \(\frac{9}{12}\) are both halfway between \(\frac{1}{2}\) and 1 on the number line
  • locate \(\frac{3}{4}\) and \(\frac{9}{12}\) on the same number line (or separate ones) and show that they are in the same location
• 2–3 minutes: partner discussion

• Select students to share their responses and how they knew that \(\frac{9}{12}\) is equivalent to \(\frac{3}{4}\) but \(\frac{7}{8}\) is not.
• To facilitate their explanation, ask them 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?”