Lesson 3

Same Denominator or Numerator

Warm-up: Number Talk: Hundreds More (10 minutes)

Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for adding and subtracting multi-digit numbers. These understandings help students develop fluency and will be helpful in later units as students add and subtract multi-digit numbers fluently using the standard algorithm.

When students decompose addends to support mental addition they are looking for and making use of the base-ten structure of numbers (MP7).

Launch

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

Activity

• Keep expressions and work displayed.
• Repeat with each expression.

Student Facing

Find the value of each expression mentally.

• $$136 + 100$$
• $$136 + 300$$
• $$136 + 370$$
• $$136 + 378$$

Activity Synthesis

• “Who can restate _____’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone approach the expression in a different way?”
• “Does anyone want to add on to _____’s strategy?”

Activity 1: Fractions with the Same Denominator (15 minutes)

Narrative

The purpose of this activity is to prompt students to reason about the relative sizes of two fractions with the same numerator and articulate how they know which one is greater. Students have done similar reasoning work (and used similar tools to support their reasoning) in grade 3, but here the fractions include those with denominators 5 and 10. When students observe that 5 equal parts are greater than 3 of the same equal part, regardless of the size of those parts, they see regularity in repeated reasoning (MP8).

To add movement to this activity and if time permits, assign each group a pair of fractions in the second question and ask them to create a visual display showing their reasoning. Then, allow a few minutes for a gallery walk. Ask students to identify any patterns they notice on the displays.

Engagement: Provide Access by Recruiting Interest. Provide choice and autonomy. Provide access to colored pencils students can use to label each rectangle.
Supports accessibility for: Attention, Organization

• Groups of 2

Activity

• “Take a few quiet minutes to answer the first three questions. Discuss your work with your partner before moving on to the last question.”
• 5–7 minutes: independent work time
• 3–4 minutes: partner discussion
• 2–3 minutes: independent work time on the last question

Student Facing

1. This diagram shows a set of fraction strips. Label each rectangle with the fraction it represents.

2. Circle the greater fraction in each of the following pairs. If helpful, use the diagram of fraction strips.

1. $$\frac{3}{4}$$  or  $$\frac{5}{4}$$
2. $$\frac{3}{5}$$  or  $$\frac{5}{5}$$
3. $$\frac{3}{6}$$  or  $$\frac{5}{6}$$
4. $$\frac{3}{8}$$  or  $$\frac{5}{8}$$
5. $$\frac{3}{10}$$  or  $$\frac{5}{10}$$
3. What pattern do you notice about the circled fractions? How can you explain the pattern?
4. Which one is greater: $$\frac{7}{3}$$ or $$\frac{10}{3}$$? Explain your reasoning.

Activity Synthesis

• “What do you notice about each pair of fractions in the second question?” (They all have 3 and 5 for the numerators, and they have the same denominator.)
• “What does it mean when two fractions, say $$\frac{3}{8}$$ and $$\frac{5}{8}$$, have the same denominator?” (They are made up of the same fractional part—eighths in this case.)
• “How can we tell which fraction is greater?” (Because the fractional parts are the same size, we can compare the numerators. The fraction with the greater numerator is greater.)

Activity 2: Fractions with the Same Numerator (20 minutes)

Narrative

The purpose of this activity is for students to reason about the relative sizes of two fractions with the same numerator. As before, a diagram of fraction strips can be used to help students visualize the sizes of various fractional parts. When students discuss and improve their explanation for why $$\frac{70}{20}$$ is greater than $$\frac{70}{100}$$ they develop their mathematical communication skills (MP3).

This activity uses MLR1 Stronger and Clearer Each Time. Advances: Reading, Writing

Launch

• Groups of 2
• “What do you notice about the fractions in the first question?” (Each pair has the same numerator and has 3 and 5 for the denominators.)

Activity

• “Think quietly for a moment about how you can find out which fraction in each pair is greater. Then, share your thinking with your partner.”
• 1 minute: quiet think time
• 2 minutes: partner discussion
• Monitor for students who use the size of a unit fraction or one fractional part to help them make comparisons.
• “Take a few quiet minutes to work on the questions.”
• 7–8 minutes: independent work time

Student Facing

1. Circle the greater fraction in each of the following pairs. If helpful, use the diagram of fraction strips.

1. $$\frac{1}{3}$$  or  $$\frac{1}{5}$$
2. $$\frac{2}{3}$$  or  $$\frac{2}{5}$$
3. $$\frac{3}{3}$$  or  $$\frac{3}{5}$$
4. $$\frac{4}{3}$$  or  $$\frac{4}{5}$$
5. $$\frac{9}{3}$$  or  $$\frac{9}{5}$$
2. What pattern do you notice about the circled fractions? How can you explain the pattern?
3. Which one is greater: $$\frac{70}{100}$$ or $$\frac{70}{20}$$? Explain your reasoning.
4. Tyler is comparing $$\frac{4}{10}$$ and $$\frac{4}{6}$$. He says, “Ten is greater than 6, so $$\frac{4}{10}$$ is greater than $$\frac{4}{6}$$.” Explain or show why Tyler’s conclusion is incorrect.

Activity Synthesis

• Select students to share their responses to the first two questions.
• “In the set of fractions you saw, why are the fractions with 3 for the denominator always greater than fractions with 5 for the denominator?” (A third is always greater than a fifth, so some number of thirds will always be greater than the same number of fifths.)

MLR1 Stronger and Clearer Each Time

• “Share your response to the third question with your partner. Take turns being the speaker and the listener. If you are the speaker, share your explanation. If you are the listener, ask questions and give feedback to help your partner improve their explanation.”
• 3–5 minutes: structured partner discussion
• Repeat with 2–3 different partners.
• “Revise your initial explanation based on the feedback from your partners.”
• 2–3 minutes: independent work time

Lesson Synthesis

Lesson Synthesis

“Today we looked at fractions with the same denominator and those with the same numerator.”

Select students to share their explanations on the last question in the second activity. “What might have Tyler misunderstood? What would you say to help clear it up for him?”

“Based on your work today, how would you complete these sentence starters?”

• “If two fractions have the same denominator, I can tell which one is greater by . . . .”

(looking at which one has a greater numerator, because it would mean more of the same fractional parts)

• “If two fractions have the same numerator, I can tell which one is greater by . . . .”

(looking at which denominator is smaller, because the smaller denominator would mean a larger fractional part)