Lesson 7

The purpose of this Estimation Exploration is for students to practice estimating a given length on a number line. Students are given the length of a longer segment as a point of reference and apply their understanding of equal parts to the number line to estimate a shorter length.

• Groups of 2
• Display the image.
• “What number could go in the box? What is an estimate that’s too high? Too low? About right?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

If students do not explain why the number in the box is going to be less than half of 5, ask them to identify the approximate location of the numbers 1, 2, 3, and 4 on the number line.

• “Is the number that goes in the box more or less than half of 5? How do you know?” (Less, since it is less than half the way to 5.)
• “How does this help you estimate the value?” (I know half of 5 is \(2 \frac{1}{2}\),  so it is less than that.)
• Optional: Reveal the actual value and add it to the display.

The purpose of this activity is for students to use the structure they noticed in the previous lesson to solve real world problems in which a whole number is multiplied by a unit fraction. Students may use a variety of strategies to solve these problems. They may relate the situations to multiplication of a whole number by a fraction or division of two whole numbers. During the synthesis, connect the different interpretations of the situations.

Students share their different representations and expressions and explain to each other how they relate to the running situation (MP3).

This activity uses MLR7 Compare and Connect. Advances: Representing, Conversing.

• Groups of 2

• 1–2 minutes: independent think time
• 5 minutes: partner work time
• Monitor for:
  • students who draw diagrams including continuous number line representations and discrete rectangular representations like the ones used in earlier lessons
  • students who use different multiplication or division expressions
  • students who write their final answer as a fraction or as a mixed number

MLR7 Compare and Connect

• “Create a visual display that shows your thinking about the second problem. You may want to include details such as notes, diagrams, drawings, etc., to help others understand your thinking.”
• 5–7 minutes: gallery walk
• “What is the same and what is different between the different approaches to solving the problem?”
• 30 seconds quiet think time
• 1 minute: partner discussion
• Additional questions could include:
  • “Does anyone have a question they would like to ask about a strategy or solution?”
• Consider asking students if they would like to revise their work before the synthesis.

• Ask previously selected students to share their solutions.
• “What are some expressions that represent the distance Han ran?” (\(7 \div 4\), \(1 \frac{3}{4}\), \(\frac{7}{4}\), \(\frac{1}{4}\times7\))
• “How does each of these expressions represent the distance Han ran?” (He ran \(\frac{1}{4}\) of 7 miles and we can write that as \(\frac{1}{4} \times 7\). We can figure out how many miles that is if we divide 7 into 4 equal parts. That’s \(7\div4\) or \(\frac{7}{4}\) or \(1\frac{3}{4}\).)

The purpose of this activity is to match different expressions and diagrams with one situation. Some students may match the expressions, diagrams, and situation by finding the solution to the problem and the value of each expression. Some may match the representations without finding the value of the expressions. During the activity synthesis, highlight how the different expressions relate both to the situation and to the diagrams, and connect the relationships to the meaning of the expressions and diagrams.

Students reason abstractly and quantitatively (MP2) when they relate the story to the diagrams and expressions. All of the diagrams and expressions involve the same set of numbers so students need to carefully analyze the numbers in the story, the diagrams, and the expressions in order to choose the correct matches.

• Create a set of cards from the blackline master for each group of 2.

• Groups of 2

• 8 minutes: partner work time
• Monitor for students who match the expressions and diagrams by thinking about the meaning in each case.

If students do not match the diagrams and expressions correctly, ask students to explain how each diagram and expression represents each part of the situation.

• Display cards J and K.
• “How are these diagrams the same? How are they different?” (They both show \(\frac{3}{4}\) or \( 3 \div 4\). The first one shows \(\frac{1}{4}\) of each whole. In the second one, the shaded parts are all together, so they show \(\frac{3}{4}\) of a single whole.)
• Display the expression: \(\frac{1}{4} \times 3\).
• “How does this expression relate to the situation?” (The race is 3 miles long and each person will run \(\frac{1}{4}\) of the race.)
• “How do the diagrams represent the expression?” (In the first diagram \(\frac{1}{4}\) of each whole is shaded. It is harder to see in the second diagram because I can't tell that the shaded parts are \(\frac{1}{4}\) of the 3 rectangles.)
• “How can we adapt card K to show there are 4 equal sections of \(\frac{3}{4}\)?” (We could show the sections.)
• Mark card K to show the 4 equal sections of \(\frac{3}{4}\). For example, use different colors to show the other 3 sections of \(\frac{3}{4}\).