Lesson 3

The purpose of this warm-up is for students to review strategies for comparing whole numbers, decimal numbers, and fractions as well as the use of inequality symbols. The numbers in each pair have been purposefully chosen based on misunderstandings students typically have when comparing. Since there are many pairs of numbers to compare, it may not be possible to share all of the students’ strategies for each pair. Consider sharing only one strategy for each pair if all of the students agree and more than one if there is a disagreement among the students. 

Give students 3 minutes of quiet work time followed by a whole-class discussion. 

Some students may not remember the inequality symbols that represent the phrases: greater than, less than, and equal to. Show these students each of the inequality symbols in an example that they can refer back to as they work.

For each pair of numbers, ask one or two students to share their reasoning. Record and display their reasoning for all to see. If the whole class agrees, move on to the next question, but if there is a disagreement, ask students to explain their thinking until an agreement is reached. If possible, spend more time on the questions with numbers expressed with decimals and fractions. 

In this activity, students order rational numbers from least to greatest in 2 steps. They first order positive rational numbers, and then negative ones. The numbers are written as fractions, decimals, and integers. By manipulating physical cards, students get a tangible sense of how rational numbers relate to each other.

Notice conversations students have when deciding how to place fractions and decimals, especially on the negative side of the number line. Pay attention to proper use and understanding of “less” or “greater,” and improper use of “bigger” or “smaller.” One strategy to look for is fitting new numbers between known numbers (e.g., \(\text-\frac98\) is between -1 and -2).

Arrange students in groups of 2. Distribute the first set of cards to each group. Give students 5 minutes to order the first set of cards, taking turns to place each number. When a group finishes ordering, check their ordering before giving them the second set of cards. To speed up the checking process, consider referring groups finishing the first set to compare their ordering to a group you have already checked. Give students 5 minutes to order the second set of cards followed by whole-class discussion. When collecting the cards, ask groups to separate the negative set of numbers and randomize each set for the next class.

Some students may place negative numbers in order of increasing absolute value on the left side of 0. Ask these students to draw a number line that goes 5 units right and 5 units left of 0. Point out that negative numbers progress as -1, -2, -3, -4, -5 as they move outward from 0.

The purpose of the discussion is to solidify students’ understanding of the order of rational numbers. Select previously identified students to share how they decided how to place numbers like \(\text-\frac98\), \(\frac98\), \(\frac83\), and \(\text-22\frac12\). Here are some questions to consider:

• Which numbers were hardest to place and which were the least difficult?
• How does placing negative numbers compare to placing positive numbers?
• How did you use numbers you had already placed to reason about where to place new numbers?

Students practice using relational language “greater than” and “less than” to describe order and position on number line.

Arrange students in groups of 2. Give students 7 minutes of quiet work time for both problems before 3–5 minutes for partner discussion, followed by whole-class discussion.

The purpose for discussion is to give students the opportunity to use precise language as they compare the relative positions of rational numbers. Give students 3–5 minutes to discuss their responses with a partner before whole-class discussion. Ask students to share their partner’s reasoning, especially if it was different than their own. Here are some questions to consider for whole-class discussion:

• “Did you ever have a different answer than your partner? If so, were you both correct? If not, how did you work to reach agreement?”
• “How did your partner decide the value of each unit on the number line in problem 2? Did you think of it a different way?”
• “How can we tell if two numbers are opposites?” (They are the same distance from 0.)
• “How can we tell if one number is greater or less than another number?” (Numbers toward the right are considered greater, and numbers toward the left are considered less.)

Students interpret positive and negative numbers in the context of a changing inventory (MP2).

Keep students in the same groups of 2. Give students 8 minutes of quiet work time and 3 minutes of partner discussion. Follow with whole-class discussion.

The goal of the discussion is to allow students to share their thoughts on the meaning of positive and negative numbers in context. Ask students to share their responses with their partner and work to reach agreement. Monitor groups’ discussions and select students to share their partner’s reasoning about what the numbers in the table mean. Ask students to summarize the story the numbers in the table tell. Tell students that tables like this (but perhaps more complicated) are used all the time to tell stories about what is happening in the world.