Illustrative Mathematics Accelerated Grade 6, Unit 5.5 - Teachers (2024)

Do you see a pattern? What predictions can you make about future rectangles in the set if your pattern continues?

Students have already looked at measurement conversions that can be represented by proportional relationships, both in grade 6 and in previous lessons in this unit. This task introduces a measurement conversion that is not associated with a proportional relationship. The discussion should start to move students from determining whether a relationship is proportional by examining a table, to making the determination from the equation.

Note that some students may think of the two scales on a thermometer like a double number line diagram, leading them to believe that the relationship between degrees Celsius and degrees Fahrenheit is proportional. Ifnot mentioned by students, the teacher should point out that when a double number line is used to represent a set of equivalent ratios, the tick marks for 0 on each line need to be aligned.

Arrange students in groups of 2. Provide access to calculators, if desired. Give students 5 minutes of quiet work time followed by students discussing responses with a partner, followed by whole-class discussion.

Representation: Internalize Comprehension. Activate or supply background knowledge. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing

The other day you worked with converting meters, centimeters, and millimeters. Here are some more unit conversions.

1. Use the equation \(F =\frac95 C + 32\), where \(F\) represents degrees Fahrenheit and \(C\) represents degrees Celsius, to complete the table.

   temperature \((^\circ\text{C})\)	temperature \((^\circ\text{F})\)
   20
   4
   175

2. Use the equation \(c = 2.54n\), where \(c\) represents the length in centimeters and \(n\) represents the length in inches, to complete the table.

   length (in)	length (cm)
   10
   8
   3\(\frac12\)

3. Are these proportional relationships? Explain why or why not.

After discussing the work students did, ask, “What do you notice about the forms of the equations for each relationship?” After soliciting students’ observations, point out that the proportional relationship is of the form \(y = kx\), while the nonproportional relationship is not.

Speaking: MLR8 Discussion Supports. To provide a support students in producing statements when they compare and contrast equations with proportional and nonproportional relationships. Provide sentence frames such as: “I noticed that _____,” “What makes _____ different from the others is _____,” or “The patterns show _____.” Invite students to use these sentence frames when students comprehend orally the patterns they notice with proportional equations.
Design Principle(s): Support sense-making

This activity builds on the perimeter and area activity from the warm-up. Its goal is to use a context familiar from grade 6 to compare proportional and nonproportional relationships. The units for the quantities are purposely not given in the task statement to avoid giving away which relationships are not proportional. However, discussion should raise the possible units of measurement for edge length, surface area, and volume.

The focus of this task is whether or not relationships between the quantities are proportional, but there is also an opportunity for students to reinforce their understanding of geometry. Students should not spend too much time figuring out the surface area and volume. Watch carefully as students work and be ready to provide guidance or equations as needed, so students can get to the central purpose of the task, which is noticing the correspondences between the nature of relationships and the form of their equations. Strategic pairing of students and having snap cubes on hand can help struggling students complete the tables and determine equations.

Arrange students in groups of 2. Display a cube (e.g. cardboard box) for all to see and ask:

• “How many edges are there?”
• “How long is one edge?”
• “How many faces are there?”
• “How large is one face?”

Give students 5 minutes of quiet work time followed by students discussing responses with a partner, followed by whole-class discussion.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention

Here are some cubes with different side lengths. Complete each table. Be prepared to explain your reasoning.

1. How long is the total edge length of each cube?

   side length	total edge length
   3
   5
   \(9\frac12\)
   \(s\)

2. What is the surface area of each cube?

   side length	surface area
   3
   5
   \(9\frac12\)
   \(s\)

3. What is the volume of each cube?

   side length	volume
   3
   5
   \(9\frac12\)
   \(s\)

4. Which of these relationships is proportional? Explain how you know.

5. Write equations for the total edge length \(E\), total surface area \(A\), and volume \(V\) of a cube with side length \(s\).

1. A rectangular solid has a square base with side length \(\ell\), height 8, and volume \(V\). Is the relationship between \(\ell\) and \(V\) a proportional relationship?
2. A different rectangular solid has length \(\ell\), width 10, height 5, and volume \(V\). Is the relationship between \(\ell\) and \(V\) a proportional relationship?
3. Why is the relationship between the side length and the volume proportional in one situation and not the other?

Some students may struggle to complete the tables. Teachers can use nets of cubes (flat or assembled) partitioned into square units to reinforce the process for finding total edge length, surface area, and volume of the cubes in the task. Snap cubes would also be appropriate supports.

If difficulties with the fractional side length \(9\frac12\) keep students from being able to find the surface area and volume or write the equations, the teacher can tell those students to replace \(9\frac12\) with 10 and retry their calculations. Their answers for surface area and volume will be different for that row in the table, but their equations and proportionality decisions will be the same. That way they can still learn the connection between the form of the equations and the nature of the relationships.

Ask, “What do you notice about the equation for the relationship that is proportional?” Make sure students see that it is of the form \(y = kx\), and the others are not. This realization is the central purpose of this task.

Ask students:

• "What could be possible units for the side lengths?" (linear measurements: centimeters, inches)
• "Then what would be the units for the surface area?" (square units: square centimeters, square inches)
• "What would be the units for the volume?" (cubic units: cubiccentimeters, cubic inches)

Connect the units of measurements with the structure of the equation for each quantity: Theside length and the units are raised to the same power.

Arrange students in groups of 2. Provide access to calculators. Give students 5 minutes of quiet work time followed by students discussing responses with a partner, followed by whole-class discussion.

Engagement: Internalize Self Regulation. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to choose and complete tables for four of the equations. Encourage students to complete tables for the equations \(y=\frac{x}{4}\) and \(y=4x\).
Supports accessibility for: Organization; Attention

Here are six different equations.

1. Predict which of these equations represent a proportional relationship.
2. Complete each table using the equation that represents the relationship.
3. Do these results change your answer to the first question? Explain your reasoning.
4. What do the equations of the proportional relationships have in common?

Invite students to share what the equations forproportional relationships have in common, and by contrast, what is different about the other equations. At first glance, the equation \(y=\frac x4\) does not look like our standard equation for a proportional relationship, \(y = kx\). Suggest to students that they rewrite the equation using the constant of proportionality they found after completing the table: \(y = 0.25x\) which can also be expressed \(y=\frac14 x\). If students do not express this idea themselves, remind them that they can think of dividing by 4 as multiplying by \(\frac14\).

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share what they identified as similarities between the proportional and nonproportional relationships, present an incorrect explanation. For example, “There are no commonalities among the equations because they’re all different. ” Ask students to identify the error, critique the reasoning, and write a correct description. As students discuss with a partner, listen for students who clarify that equations\(y = 4x\)and\(y=\frac14 x\)are of the form\(y = kx\)and both represent proportional relationships. Invite students to share their critiques and corrected responses with the class. Listen for and amplify the language students use to describe the similarities and differences between equations that do and do not represent proportional relationships. This helps students evaluate, and improve upon, the written mathematical arguments of others, as they clarify how their understanding of proportional relationships and how they can be represented with equations.
Design Principle(s): Maximize meta-awareness

Review the findings from the activities and make explicit the fact that proportional relationships are characterized by equations of the form \(y = kx\). Be sure to point out that this includes equations with other variables. For example:

\(y = 5.2x\) \(d = 58t\) \(a = 0.12B\) \(W = 205n\)

This form characterizes proportional relationships due to a property we examined in previous lessons: if a table represents a proportional relationship between \(x\) and \(y\), then the unit rates \(\frac yx\) are always the same.

If \(\frac yx = k\), then \(y = kx\) (as long as \(x \ne 0\), but you don’t need to mention this now unless a student brings it up.)

If two quantities are in a proportional relationship, then their quotient is always the same. This table represents different values of \(a\) and \(b\), two quantities that are in a proportional relationship.

Notice that the quotient of \(b\) and \(a\) is always 5. To write this as an equation, we could say \(\frac{b}{a}=5\). If this is true, then \(b=5a\). (This doesn’t work if \(a=0\), but it works otherwise.)

If quantity \(y\) is proportional to quantity \(x\), we will always see this pattern:\(\frac{y}{x}\) will always have the same value. This valueis the constant of proportionality, which weoften refer to as \(k\). We can represent this relationship with the equation \(\frac{y}{x} = k\) (as long as \(x\) is not 0) or \(y=kx\).

Note that if an equation cannot be written in this form, then it does not represent a proportional relationship.