Symbolic Representation

Symbolic Representation

History of Scales

Opening

History of Scales

The history of balance scales begins in ancient Egypt, possibly as far back as 5000 BCE. The Egyptians used balance scales to compare the weights of goods. They placed goods on each side of the scale; when the scale became balanced, they knew that the weights of the goods were equal.

  • What type of scales have you used, or seen someone else use, to measure things?
  • How are the scales you've used similar to, or different from, the balance scale?

Hint:

  • Think about places you visit, like the grocery store or the doctor's office. Have you seen scales there?
  • Scales are used to measure weight. What are some of the things we weigh?

Balancing the Scale

Opening

Balance Scales

An equation consists of two equivalent expressions that are linked by an equal sign (=).

An inequality consists of two nonequivalent expressions that are linked by a less than sign (<) or a greater than sign (>).

Using the Balance Scale interactive, place the numbers 2 and 3 on the left side of the scale.

  • What number(s) do you need to place on the right side of the scale in order to balance the scale?
  • Write an equation to represent the balanced scale.
  • Now, “unbalance” the scale so that what is on the left side is greater than what is on the right side.
  • Write an inequality to represent the unbalanced scale.

INTERACTIVE: Balancing the Scale

Hint:

  • Think about the definitions of equation andinequality :
    • An equation consists of two equivalent expressions that are linked by an equal sign (=).
    • An inequality consists of two nonequivalent expressions that are linked by a less than sign (<) or a greater than sign (>).

Math Mission

Opening

Set up balanced and unbalanced scales, and represent the scales using equations and inequalities.

Equations and Inequalities

Work Time

Equations and Inequalities

Set up numbers on both sides of the scale to match the descriptions below.
After you set up each scale, write the equation or inequality that represents the scale.

  1. Use Balance Scale A to set up a balanced scale (both sides equal) that has two numbers on both sides, using four different numbers.
  2. Modify your scale from the previous step to make the right side less than the left side.
  3. Use Balance Scale B to set up an unbalanced scale with one number on one side and two numbers on the other side.
  4. Modify your scale from the previous step to make it balanced.
  5. Use Balance Scale C to set up an unbalanced scale that has three numbers on one side and two numbers on the other side, such that the side with two numbers is greater than the side with the three numbers.
  6. Modify your scale from the previous step to make it balanced.
  7. Modify your scale again by placing the same number on both sides. What happens?

INTERACTIVE: Balance Scale A

INTERACTIVE: Balance Scale B

INTERACTIVE: Balance Scale C

Hint:

  • Think about the definitions of equation andinequality :
    • An equation consists of two equivalent expressions that are linked by an equal sign (=).
    • An inequality consists of two nonequivalent expressions that are linked by a less than sign (<) or a greater than sign (>).
  • If the scale is balanced, write an equation.
  • If the scale is not balanced, write an inequality.

Prepare a Presentation

Work Time

Prepare a Presentation

  • Choose one equation or inequality that you created in the Balance Scale interactives that you think is particularly interesting.
  • Use this equation/inequality to explain your understanding of equations and/or inequalities. In your explanation, refer to the corresponding scale you created using the interactive.

Challenge Problem

NOTE: Ignore the instructions in the interactive and follow the steps below.

  1. Place a number on the left side of the scale.
  2. Place a different number on the right side.
  3. Is there any one number you can now add on both sides that will balance the scale? Explain.

INTERACTIVE: Balance Scale B

Hint:

  • Think about the definitions of equation andinequality :
    • An equation consists of two equivalent expressions that are linked by an equal sign.
    • An inequality consists of two nonequivalent expressions that are linked by a less than sign (<) or a greater than sign (>).

Make Connections

Performance Task

Ways of Thinking: Make Connections

Take notes as your classmates explain their understanding of equations and inequalities.

Hint:

As your classmates present, ask questions such as:

  • Should the pointed side of the inequality sign point to the greater or lesser number?
  • How does the equation or inequality you wrote represent the balance scale you made?
  • What can you do to an unbalanced scale to balance it? How can you represent the balanced scale using an equation or an inequality?
  • What can you do to a balanced scale to make the right side greater than the left side? How can you represent this scale using an equation or an inequality?

Equations and Inequalities

Formative Assessment

Summary of the Math: Equations and Inequalities

Write a summary about equations and inequalities based on what you learned today.

Hint:

Check your summary:

  • Do you explain the difference between an equation and an inequality?
  • Do you explain the meaning of the signs <, >, and =?

Reflect On Your Work

Work Time

Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

A balance scale helps me understand equations and inequalities because …