Lesson 9

In this number talk, students compare quantities involving division with whole numbers, decimals, and fractions. In each case, a strategy is available that does not require calculating the quantities. When the quantities are complex, there is motivation for using the structure of the expressions to compare (MP7) since the actual calculations would be more time-consuming. 

Display one problem at a time. Give students 1 minute of quiet think time per problem and ask them to give a signal when they have an answer and a strategy. Follow with a whole-class discussion. 

Students may misinterpret the last question as \(15 \boldcdot \frac{1}{3}\) or \(15 \boldcdot \frac{1}{4}\). Point out that one way to interpret the first expression is “How many one-thirds are there in 15?”

Make sure to bring out different approaches for comparing the quantities, avoiding direct calculation where possible:

• \(\frac{7}{13}\) and \(\frac{11}{23}\) can both be compared with \(\frac{1}{2}\), or students can find a common numerator or denominator, but this requires more calculations.
• \(0.63\) is greater than \(0.55\), and 2 is less than 3, or students might notice that \(0.63 \div 2\) is greater than 0.3 while \(0.55 \div 3\) is less than 0.2. 
• Dividing by \(\frac{1}{4}\) is equivalent to multiplying by 4 while dividing by \(\frac{1}{3}\) is equivalent to multiplying by 3, so \(15 \div \frac{1}{4}\) is greater than \(15\div \frac{1}{3} \).

In previous lessons in this unit, students have investigated the meaning of scale drawings and have used them to solve problems. The purpose of this activity is to prepare students for creating their own scale drawing. The discussion highlights that a scale can be expressed in different ways, or that different pairs of numbers may be used to show the same relationship. For example, a scale of 4 cm to 1 m is equivalent to a scale of 1 cm to 0.25 m. 

As students work, notice those who use the language of scale appropriately, e.g., by saying that “every 4 cm on the drawing represents 1 m,” or “every 0.25 m shows up as 1 cm on the drawing.” Also notice those who do and do not attend to the relationship between actual and scaled lengths in finding missing measurements.

Students may see that one value is 4 times the other and write the scale backwards, as “1 cm to 4 m.” Prompt students to pay attention to the units and the meaning of each number.

Focus the class discussion on two things:

1. Different ways to express the same scale
2. The relationship between scaled and actual lengths

Invite a couple of students to share how they determined the scale of the drawing. Students are likely to come up with several variations, e.g., 4 cm to 1 m, 1 cm to 0.25 m, 16 cm to 4 m, etc. Discuss how all of these express the same relationship and are therefore equivalent, especially how 4 cm to 1 m is equivalent to 1 cm to 0.25 m (or 1 cm to \(\frac14\) m).

Explain that although we can express a scale in multiple but equivalent ways, 1) scales are often simplified to show the actual distance for 1 scaled unit, and 2) it is common to express at least one distance (usually the scaled distance) as a whole number or a benchmark fraction (e.g., \(\frac 14\), \(\frac12\)) or a benchmark decimal (e.g., 0.25, 0.5, 0.75).

Given their work on scaled copies, students may be inclined to say that the scaled and actual lengths are related by a scale factor of 4. Ask: “Are the actual lengths four times the lengths on the drawing? Why or why not?” Point out that because the units for the two quantities are different, multiplying a scaled length in centimeters (e.g., 2.5 cm) by 4 will yield another length in centimeters (10 cm), which is not the actual length. It is not essential for students to know that the scale factor here is 250. That work will be explored in an upcoming lesson.

In the previous activity, students calculated the scaled distances they would need to create a scale drawing, but did not actually create the scale drawing. In this activity, they create two different scale drawings of the state of Utah and notice how the scale impacts the drawing. One of the reasons choice of a scale is important is that we want to see the appropriate level of detail within a fixed space.

Display the outline of Utah and ask students “What do you notice? What do you wonder?”

Ask students to describe the shape: a rectangle with a smaller rectangle removed in the upper right corner.

Give students 5–6 minutes of quiet work time followed by whole-class discussion.

Some students may get a shape that is not closed or does not have right angles if they did not measure carefully enough. Prompt them to double-check their measurement for a particular side of the state if you can easily tell which side is drawn incorrectly.

Students may think that a scale of 1 centimeter to 50 miles will produce a smaller scale drawing than a scale of 1 centimeter to 75 miles (because 50 is less than 75). Ask them how many centimeters it takes to represent 75 miles if 1 centimeter represents 50 miles (1.5) and how many centimeters it takes to represent 75 miles if 1 centimeter represents 75 miles (1).

Ask students what the two scale drawings share in common. Answers include: they both represent Utah, they both have the same shape, and they both can be used to measure distances in the actual state of Utah.

Ask students how the two scale drawings differ. The one at a scale of 1 centimeter to 50 miles is larger than the one at a scale of 1 centimeter to 75 miles.

Some students may notice that the scale drawing at a scale of 1 centimeter to 75 miles is actually a scaled copy of the other drawing, with a scale factor of 1.5. If so, ask them to share their observation linking scale drawings with scaled copies.