Lesson 7

In this warm-up, students partition a number line and locate fraction and decimal equivalents in preparation for working with double number lines in this unit. Students are purposely not asked to locate 1 on the number line to see how they reason about locating the \(\frac12\) and \(\frac14\). It is important for students to be able to identify the fractions or decimals and label tick marks correctly, interpreting the distance between tick marks, rather than the number of tick marks, as the fractional size. As students discuss with their partner, select students to share their answers to the first question during the whole-class discussion.

Arrange students in groups of 2. Display the number line for all to see. Give students 2 minutes of quiet think time and ask them to give a signal when they have an answer and a strategy. Ask students to compare their number line with a partner and share the fractions or decimals they chose to place on the number line for the second question.

Students may place \(\frac12\) in the center of the number line, reasoning that it is half of the number line. Explain to the students they are placing the number \(\frac12\), which has a specific value and location on the number line.

Select students to explain how they reasoned about the location of each number on the number line. After each number, ask the class whether they agree or disagree, and if anyone else had a different way of thinking about that number. If time permits, ask students to share the fractions or decimals they located for the second question. Discuss why they chose those numbers and how they decided on their location.

Students continue to use double number lines to reason about equivalent ratios. Here students’ attention is directed to the \(1:3\) blue-to-yellow ratio in the green water recipe, which can then be used to determine any equivalent ratio. The task is also the beginning of students’ exploration of finding and using ratios containing a 1.

One key idea to convey here is that finding a ratio associated with 1 unit of a quantity can be very helpful. Another is that the intervals on double number lines can be subdivided to help us find such ratios.

As students work, identify those who use division to determine the \(1:3\) ratio, and then use multiplication to determine the ratios for 8 ml and 13 ml of blue water. This is a key insight for a type of reasoning that is broadly useful and will be developed further.

Ask students to recall what double number lines are and how they can be used to represent problems involving equivalent ratios. Explain that they are going to investigate the structure of double number lines in more detail. Give students 5 minutes of quiet think time, and then time to discuss their responses with a partner.

Students may have trouble figuring out that the length of a segment between consecutive tick marks is \(\frac15\) of the interval from 0 to 5, especially since there are four tick marks (not five). When focusing on blue, students’ first guess about the tick marks is generally correct. For yellow, remind them that the numbers on the tick marks are made by skip counting; they are then likely to try 3’s and 5’s since both can make 15. Students who label the spaces between tick marks rather than the tick marks themselves may need additional work with important measurement conventions.

Debrief as a class after students have a chance to share their work with a partner. Focus discussions on how students determine the amount of yellow water for 1 ml of blue, and how they determine the amounts of yellow for 8 ml and 11 ml of blue. Select students who used division (to find the former) and multiplication (to find the latter) to share.

If students do not do so, frame the relationship of blue to yellow using phrases such as “for every 1 ml of . . . ” or “per milliliter of . . .”

• “There are 3 milliliters of yellow water for every 1 milliliter of blue water.”
• “There are 3 milliliters of yellow water per milliliter of blue water.”

The word per means “for every.” Ask students to think of any other situation in which they may use the word “per” as it is used here (e.g., price per bottle of water, cost per ticket, etc.) and discuss why knowing the value of one item would be helpful.

In the previous lesson, students were given blank double number line diagrams and were only responsible for labeling them to match the situation. In this activity, students draw their own double number line diagram from scratch and identify which elements are important to create a useful double number line diagram.

“You just used a double number line to solve some problems. Now, you’ll create a double number line from scratch. Once you know how to make double number lines, you can use them for any situation with equivalent ratios.”

Arrange students in groups of 2. Ensure each student has access to a ruler. Have students check with a partner and come to an agreement about how to draw the diagrams before moving on to question 3.

Students may not label tick marks with equal increments or may not align the tick marks.

Select students to explain how they used their double number line diagram to answer the last question. Ask students how they can indicate the number of cups of flour per pint of water on the double number line.

If desired, capitalize on ways students might have incorrectly constructed their double number line. For example:

• Do all increments on each line need to be equal? Why or why not?
• Do the tick marks on the top line need to match those on the bottom line? Why/why not?
• Does it matter what number we use to start each line? Why or why not?

In this activity, students revisit familiar contexts they represented with discrete diagrams in previous lessons. Here, they see how double number line diagrams are helpful for answering more questions about these situations.

Monitor for students who use a discrete diagram and for students who use a double number line diagram.

Arrange students in groups of 2. 

Select students to present their solutions. Sequence discrete diagrams first and double number line diagrams second. Help students see connections between the two representations.