Lesson 2

The purpose of this activity is to prime students for locating negative fractions on a number line. Students discern the value of a number by analyzing its position relative to landmarks on the number line. In this case, students estimate that the point is halfway between 2 and 3 and use their understanding about fractions and decimals to identify numbers equal or close to 2.5. In later activities, students do the same process when describing negative rational numbers, except with those numbers increasing in magnitude going from right to left.

Notice students who argue that 2.49 is correct or incorrect.

Arrange students in groups of 2. Give students 2 minutes of quiet think time and then 2 minutes for partner discussion.

The goal of this discussion is for students to understand that they can use landmarks on the number line (in this case, 2 and 3) and their knowledge of fractions to identify equivalent expressions of a number on the number line. Ask students:

• “Were there any responses you could tell right away were not correct? How?” (Sample response: \(\frac25\) is less than 1, but \(B\) is between 2 and 3.)
• “Were there any responses you had to think harder about? How did you decide those ones?” (Sample response: \(\frac{25}{10}\) seemed too large at first because the numbers are bigger, but after thinking, I saw it is equivalent to \(\frac52\), which I already knew to be correct.)

If time allows, select students to share their thinking about whether 2.49 could represent \(B\).

The purpose of this task is to use the previously introduced context of temperature to build understanding of the negative side of the number line, both by reading values and assigning values to equally spaced divisions. Non-integer negative numbers are also used. Students reason abstractly and quantitatively as they interpret positive and negative numbers in context (MP2).

Notice the arguments students make to decide whether Elena or Jada are correct in question 2. Some students may defend Elena because they see the liquid is above -2 and conclude that the temperature is -2.5 degrees. Other students will defend Jada by noting the temperature is halfway between -1 and -2 degrees, concluding that it must be -1.5 degrees.

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

Some students may have difficulty identifying the non-integer temperatures on the thermometers. This difficulty arises when students are unable to identify the scale on a number line. This may be significantly more challenging on the negative side of the number line as students are accustomed to the numbers increasing in magnitude on the positive side as you go up. This issue is addressed in task item number 2. It may be helpful to draw attention to the tick mark between 1 and 2 and its label. This previews the idea of opposites addressed in the next activity.

The purpose of the discussion is to use temperature to explore the concept of negative numbers and introduce the vocabulary of rational numbers. Select students to share their reasoning as to whether they agreed with Jada or Elena in question 2. If not mentioned by students, connect this question to the warm-up by pointing out that the temperature is halfway between -1 and -2 on the number line, and so it must be -1.5 degrees.

Tell students that rational numbers are like fractions except they can also be negative. So rational numbers are all fractions and their opposites. The term “RATIOnal number” comes from the fact that ratios and fractions are closely related ideas. Display some examples of rational numbers like 4, -3.8, \(\text-\frac{4}{3}\), and \(\frac12\) for all to see. Ask students whether they agree 4 and 3.8 are fractions. Tell them these might not look like fractions, but they actually are fractions because they can be written as \(\frac{16}{4}\) and \(\frac{38}{10}\). All rational numbers can be plotted as points on the number line and can be positive, zero, or negative just like temperature.

The purpose of this task is both to build an understanding of the symmetry across zero on the number line and to start introducing the notion that we can compare the distance from zero, or the absolute value, of numbers (MP7). Though this activity does not explicitly introduce the vocabulary of absolute value, it seeds the idea that a positive and a negative number can each have the same absolute value.

This is also the first time students work with negative numbers on a horizontal number line. If students have difficulty, remind them of the previous activities where they worked on a vertical number line. It might be helpful to have a vertical number line to display in order to compare and connect.

Provide access to tracing paper and rulers marked by centimeters. If the tracing paper is less than 20 cm wide, instruct students to make their number lines from -7 to 7 instead of -10 to 10 or instruct them to make their number line on the diagonal of the tracing paper. Allow 10 minutes for students to construct their folded number line and answer question 2. Check student work on question 2 and allow 5 more minutes for all to complete question 3, followed by whole-class discussion.

Some students may have difficulty creating a number line. This includes creating equal interval tick marks. Also, students may label the space between the tick marks rather than the tick marks. Have students compare their number line to a peer’s or the previous activities in which a number line was used.

With the new language of opposites, return to the definition of a rational number. We can now think of a rational number as a fraction or the opposite of a fraction. So 6, -6, \(\frac27\), \(\text-\frac27\), 5.8, and -5.8 are all examples of rational numbers. In future grades, students will encounter numbers that are not rational.

The main goal of the discussion is to check that students understand what it means for numbers to be opposites, and to take that a step further in thinking about opposites of opposites. During discussion, it may be useful to provide these sentence frames:

• “The opposite of __ is __.”
• “The opposite of the opposite of __ is __.”

Ask students to identify and name a point on their folded number line and find the opposite of that number. Challenge students to find fractions like \(\frac{5}{2}\) and their opposites on the number line. Then ask them to find the opposite of the opposite. Do this for positive and negative numbers, including numbers written as fractions and decimals. Connect those sentence frames to equations. For example, the opposite of -4 is 4, so \(\text-(\text-4) = 4\). Point out that the opposite of the opposite of a number is always the number itself. We can write, for example, \(\text-(\text-\frac23)=\frac23\) to express that the opposite of the opposite of \(\frac23\) is itself and verify this fact using the number line.