Lesson 3: Balanced Moves
Let’s figure out unknown weights on balanced hangers.
Illustrative Math Unit 8.4, Lesson 3 (printable worksheets)
Lesson 3 Summary
An equation tells us that two expressions have equal value. For example, if 4x + 9 and -2x - 3 have equal value, we can write the equation
4x + 9 = -2x - 3
Earlier, we used hangers to understand that if we add the same positive number to each side of the equation, the sides will still have equal value. It also works if we add negative numbers! For example, we can add -9 to each side of the equation.
4x + 9 + -9 = -2x - 3 + -9 (add -9 to each side)
4x = -2x - 12 (combine like terms)
Since expressions represent numbers, we can also add expressions to each side of an equation. For example, we can add 2x to each side and still maintain equality.
4x + 2x = -2x - 12 + 2x (add 2x to each side)
6x = -12
If we multiply or divide the expressions on each side of an equation by the same number, we will also maintain the equality (so long as we do not divide by zero).
6x • 1/6 = 12 • 1/6 (multiply each side by 6)
Now we can see that x = -2 is the solution to our equation.
We will use these moves in systematic ways to solve equations in future lessons.
Lesson 3.1 Matching Hangers
Figures A, B, C, and D show the result of simplifying the hanger in Figure A by removing equal weights from each side.
- Write the equation that goes with each figure:
- Each variable (x, y, and z) represents the weight of one shape. Which goes with which?
- Explain what was done to each equation to create the next equation. If you get stuck, think about how the hangers changed.
Lesson 3.2 Matching Equation Moves
Your teacher will give you some cards. Each of the cards 1 through 6 show two equations. Each of the cards A through E describe a move that turns one equation into another.
- Match each number card with a letter card.
- One of the letter cards will not have a match. For this card, write two equations showing the described move.
Lesson 3.3 Keeping Equality
- Noah and Lin both solved the equation 14a = 2(a - 3).
Do you agree with either of them? Why?
- Elena is asked to solve 15 - 10x = 5(x + 9). What do you recommend she does to each side first?
- Diego is asked to solve 3x - 8 = 4(x + 5). What do you recommend he does to each side first?
Are you ready for more?
In a cryptarithmetic puzzle, the digits 0–9 are represented with letters of the alphabet. Use your understanding of addition to find which digits go with the letters A, B, E, G, H, L, N, and R.
HANGER + HANGER + HANGER = ALGEBRA
Lesson 3 Practice Problems
- In this hanger, the weight of the triangle is x and the weight of the square is y.
- Match each set of equations with the move that turned the first equation into the second.
- Andre and Diego were each trying to solve 2x + 6 = 3x - 8. Describe the first step they each make to the equation.
a. The result of Andre’s first step was -x + 6 = -8.
b. The result of Diego’s first step was 6 = x = 8.
- a. Complete the table with values for x or y that make this equation true: 3x + y = 15.
b. Create a graph, plot these points, and find the slope of the line that goes through them.
- Select all the situations for which only zero or positive solutions make sense.
A. Measuring temperature in degrees Celsius at an Arctic outpost each day in January.
B. The height of a candle as it burns over an hour.
C. The elevation above sea level of a hiker descending into a canyon.
D. The number of students remaining in school after 6:00 p.m.
E. A bank account balance over a year.
F. The temperature in degrees Fahrenheit of an oven used on a hot summer day.
The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.
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