Two Related Quantities, Part 1
10 min
Teacher Prep
Setup
Narrative
This Warm-up invites students to describe an additive relationship between two quantities, first using specific numbers, and then more generally, using descriptions. During the Activity Synthesis, students learn that they can write equations with two variables to represent the relationship between the quantities.
The familiar context of age difference allows students to see that two equations are possible because we can describe the age of a younger person based on the age of the older person (such as, “A is some years younger than B”) and the other way around (such as, “B is some years older than A”). Students have an opportunity to attend to precision as they use words and equations to represent relationships (MP6).
Launch
Arrange students in groups of 2. Give students 2–3 min of partner work time. Follow with a whole-class discussion.
Student Task
The table shows the relationship between Han’s age and the age of a neighbor, a high school student.
| Han's age (years) | neighbor's age (years) |
|---|---|
| 6 | |
| 12 | 17 |
| 18 | |
| 25 | |
| h | |
| n |
- Complete the table to show their ages.
- Describe the relationship between the ages of the two students in two ways.
Sample Response
-
Han's age (years) neighbor's age (years) 6 11 12 17 18 23 20 25 h h+5 n−5 n - Sample response:
- Han’s neighbor is 5 years older than Han.
- Han is 5 years younger than his neighbor.
Activity Synthesis (Teacher Notes)
Invite students to share their descriptions of the relationship between Han’s and his neighbor’s ages, and to explain how those relationships can be seen in the table.
To highlight that two equations can be written to represent the same relationship, ask students:
- “Let’s say h represents Han’s age in years and n represents his neighbor’s age. What equations can we write to represent the relationship between Han’s and his neighbor’s ages?” (n=h+5 and h=n−5.)
- “If we know Han’s age, which equation would help us find his neighbor’s age?” (n=h+5)
- “If we know Han’s neighbor’s age, which equation would help us find Han’s age?” (h=n−5)
Next, display the following two graphs for all to see. Give students a minute to observe the graphs.
Ask students:
- “Which graph represents the relationship between Han’s age and his neighbor’s age? How do you know?” (Both graphs. The coordinates of the points match the values in the table but the coordinate values are switched.)
- “Which graph corresponds to which equation? How can we tell?” (Graph A corresponds to n=h+5. Adding 5 to a horizontal coordinate value gives the vertical value. Graph B corresponds to h=n−5 because subtracting 5 from the first coordinate value gives the second value.)
Tell students that they will explore other relationships in which one quantity affects the other and describe them using words, tables, equations, and graphs.
Standards
- 6.EE.9·Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. <em>For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.</em>
- 6.EE.C.9·Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. <span>For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation <span class="math">\(d = 65t\)</span> to represent the relationship between distance and time.</span>
25 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Two Related Quantities, Part 1
10 min
Teacher Prep
Setup
Narrative
This Warm-up invites students to describe an additive relationship between two quantities, first using specific numbers, and then more generally, using descriptions. During the Activity Synthesis, students learn that they can write equations with two variables to represent the relationship between the quantities.
The familiar context of age difference allows students to see that two equations are possible because we can describe the age of a younger person based on the age of the older person (such as, “A is some years younger than B”) and the other way around (such as, “B is some years older than A”). Students have an opportunity to attend to precision as they use words and equations to represent relationships (MP6).
Launch
Arrange students in groups of 2. Give students 2–3 min of partner work time. Follow with a whole-class discussion.
Student Task
The table shows the relationship between Han’s age and the age of a neighbor, a high school student.
| Han's age (years) | neighbor's age (years) |
|---|---|
| 6 | |
| 12 | 17 |
| 18 | |
| 25 | |
| h | |
| n |
- Complete the table to show their ages.
- Describe the relationship between the ages of the two students in two ways.
Sample Response
-
Han's age (years) neighbor's age (years) 6 11 12 17 18 23 20 25 h h+5 n−5 n - Sample response:
- Han’s neighbor is 5 years older than Han.
- Han is 5 years younger than his neighbor.
Activity Synthesis (Teacher Notes)
Invite students to share their descriptions of the relationship between Han’s and his neighbor’s ages, and to explain how those relationships can be seen in the table.
To highlight that two equations can be written to represent the same relationship, ask students:
- “Let’s say h represents Han’s age in years and n represents his neighbor’s age. What equations can we write to represent the relationship between Han’s and his neighbor’s ages?” (n=h+5 and h=n−5.)
- “If we know Han’s age, which equation would help us find his neighbor’s age?” (n=h+5)
- “If we know Han’s neighbor’s age, which equation would help us find Han’s age?” (h=n−5)
Next, display the following two graphs for all to see. Give students a minute to observe the graphs.
Ask students:
- “Which graph represents the relationship between Han’s age and his neighbor’s age? How do you know?” (Both graphs. The coordinates of the points match the values in the table but the coordinate values are switched.)
- “Which graph corresponds to which equation? How can we tell?” (Graph A corresponds to n=h+5. Adding 5 to a horizontal coordinate value gives the vertical value. Graph B corresponds to h=n−5 because subtracting 5 from the first coordinate value gives the second value.)
Tell students that they will explore other relationships in which one quantity affects the other and describe them using words, tables, equations, and graphs.
Standards
- 6.EE.9·Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. <em>For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.</em>
- 6.EE.C.9·Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. <span>For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation <span class="math">\(d = 65t\)</span> to represent the relationship between distance and time.</span>