Staying in Balance
10 min
Teacher Prep
Setup
Narrative
Students encounter and reason about a concrete situation, two clothes hangers with equal and unequal weights on each side. Students then see diagrams representing balanced and unbalanced hangers and think about what must be true and false about the situations. In subsequent activities, students will use the hanger diagrams to develop general strategies for solving equations.
If possible and if time allows, demonstrate the balancing concept with a real clothes hanger, clothespins, socks, and different weights, as shown in this Warm-up image. Or provide those materials for groups of students to experiment as they work through the activities in this lesson.
Launch
Tell students to close their books or devices (or to keep them closed). Display the photo of the socks for all to see. Give students 1 minute of quiet think time and ask them to be prepared to share at least one thing they notice and one thing they wonder. Record and display responses without editing or commentary. If possible, record the relevant reasoning on or near the photo.
Things students may notice:
- There are two pink socks and two blue socks.
- The socks are clipped to either ends of two clothes hangers. The hangers are hanging from a rod.
- The hanger holding the pink socks is level, but the hanger holding the blue socks is not level.
Things students may wonder:
- Why is the hanger holding the blue socks not level?
- Is something inside one of the blue socks to make it heavier than the other sock?
- What does this picture have to do with math?
Use the word “balanced” to describe the hanger on the left and “unbalanced” to describe the hanger on the right. Tell students that the hanger on the left is balanced because the two pink socks have an equal weight, and the hanger on the right is unbalanced because one blue sock has something in it that makes it heavier than the other blue sock.
Tell students to open their books or devices and look at the two diagrams with shapes. Point out that the diagrams are like the clothes hangers in the photo except that they have shapes instead of socks. Explain that students will now reason about the weights of the shapes just like they reasoned about the weights of the socks.
Give students 3 minutes of quiet work time followed by whole-class discussion.
Student Task
For Diagram A, make:
- One statement that must be true
- One statement that could be true or false
- One statement that cannot possibly be true
For Diagram B, find:
- One statement that must be true
- One statement that could be true or false
- One statement that cannot possibly be true
Sample Response
Sample responses:
Diagram A:
- The triangle is heavier than the square.
- The triangle could weigh 10 ounces and the square could weigh 6 ounces.
- The square and the triangle weigh the same.
Diagram B:
- One triangle weighs the same as three squares.
- The triangle weighs three pounds and each square weighs one pound.
- One square is heavier than one triangle.
Activity Synthesis (Teacher Notes)
Ask students to share some things that must be true, could be true, and cannot possibly be true about each diagram. Ask students to explain their reasoning.
The purpose of this discussion is to help students understand how the hanger diagrams work.
- When the diagram is balanced, or level, there is equal weight on each side. For example, since Diagram B is balanced, we know that one triangle weighs the same as three squares.
- When the diagram is unbalanced, one side is heavier than the other, making that side lower. For example, since Diagram A is unbalanced, and the side with a triangle is lower, we know that one triangle is heavier than one square.
Standards
- 6.EE.7·Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
- 6.EE.B.7·Solve real-world and mathematical problems by writing and solving equations of the form <span class="math">\(x + p = q\)</span> and <span class="math">\(px = q\)</span> for cases in which <span class="math">\(p\)</span>, <span class="math">\(q\)</span> and <span class="math">\(x\)</span> are all nonnegative rational numbers.
10 min
15 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Staying in Balance
10 min
Teacher Prep
Setup
Narrative
Students encounter and reason about a concrete situation, two clothes hangers with equal and unequal weights on each side. Students then see diagrams representing balanced and unbalanced hangers and think about what must be true and false about the situations. In subsequent activities, students will use the hanger diagrams to develop general strategies for solving equations.
If possible and if time allows, demonstrate the balancing concept with a real clothes hanger, clothespins, socks, and different weights, as shown in this Warm-up image. Or provide those materials for groups of students to experiment as they work through the activities in this lesson.
Launch
Tell students to close their books or devices (or to keep them closed). Display the photo of the socks for all to see. Give students 1 minute of quiet think time and ask them to be prepared to share at least one thing they notice and one thing they wonder. Record and display responses without editing or commentary. If possible, record the relevant reasoning on or near the photo.
Things students may notice:
- There are two pink socks and two blue socks.
- The socks are clipped to either ends of two clothes hangers. The hangers are hanging from a rod.
- The hanger holding the pink socks is level, but the hanger holding the blue socks is not level.
Things students may wonder:
- Why is the hanger holding the blue socks not level?
- Is something inside one of the blue socks to make it heavier than the other sock?
- What does this picture have to do with math?
Use the word “balanced” to describe the hanger on the left and “unbalanced” to describe the hanger on the right. Tell students that the hanger on the left is balanced because the two pink socks have an equal weight, and the hanger on the right is unbalanced because one blue sock has something in it that makes it heavier than the other blue sock.
Tell students to open their books or devices and look at the two diagrams with shapes. Point out that the diagrams are like the clothes hangers in the photo except that they have shapes instead of socks. Explain that students will now reason about the weights of the shapes just like they reasoned about the weights of the socks.
Give students 3 minutes of quiet work time followed by whole-class discussion.
Student Task
For Diagram A, make:
- One statement that must be true
- One statement that could be true or false
- One statement that cannot possibly be true
For Diagram B, find:
- One statement that must be true
- One statement that could be true or false
- One statement that cannot possibly be true
Sample Response
Sample responses:
Diagram A:
- The triangle is heavier than the square.
- The triangle could weigh 10 ounces and the square could weigh 6 ounces.
- The square and the triangle weigh the same.
Diagram B:
- One triangle weighs the same as three squares.
- The triangle weighs three pounds and each square weighs one pound.
- One square is heavier than one triangle.
Activity Synthesis (Teacher Notes)
Ask students to share some things that must be true, could be true, and cannot possibly be true about each diagram. Ask students to explain their reasoning.
The purpose of this discussion is to help students understand how the hanger diagrams work.
- When the diagram is balanced, or level, there is equal weight on each side. For example, since Diagram B is balanced, we know that one triangle weighs the same as three squares.
- When the diagram is unbalanced, one side is heavier than the other, making that side lower. For example, since Diagram A is unbalanced, and the side with a triangle is lower, we know that one triangle is heavier than one square.
Standards
- 6.EE.7·Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
- 6.EE.B.7·Solve real-world and mathematical problems by writing and solving equations of the form <span class="math">\(x + p = q\)</span> and <span class="math">\(px = q\)</span> for cases in which <span class="math">\(p\)</span>, <span class="math">\(q\)</span> and <span class="math">\(x\)</span> are all nonnegative rational numbers.