Unit 3, Lesson 6
Reasoning about Solving Equations (Part 1)
Accelerated 7
6.1
Warm-up
In the two diagrams, all the triangles weigh the same and all the squares weigh the same.
For each diagram, come up with . . .
- One thing that must be true
- One thing that could be true
- One thing that cannot possibly be true
Two hanger diagrams. First hanger, unbalanced with left side lower, left side, green triangle, right side, blue square. Second hanger, balanced, left side, green triangle, right side, three blue squares, one red circle.
6.2
Activity
On each balanced hanger, shapes with the same variable have the same weight.
- Match each hanger to an equation. Complete the equation by writing , , , or in the empty box.
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- Find the solution to each equation. Use the hanger to explain what the solution means.
A Diagram A, left side, 7 squares, 1, right side, three circles, w, 1 square, 1.B Diagram B, left side, 2 triangles, z, 3 squares, 1, right side, 5 squares, 1.C Diagram C, left side, 3 pentagons, x, 2 squares, 1, right side, 3 squares, 1.D Diagram D, left side, 6 squares, right side, 1, 2 crowns, y, 3 squares, 1.
6.3
Activity
Here are some balanced hangers diagrams where each piece is labeled with its weight in the same units. For each diagram:
- Write an equation.
- Explain how to figure out the weight of a piece labeled with a variable by reasoning about the diagram.
- Explain how to figure out the weight of a piece labeled with a variable by reasoning about the equation.
A Diagram A, left side, rectangle, 7, right side, 3 circles, x, square, 1.B Diagram B, left side, 2 diamonds, y, square, 10, right side, rectangle, 31.C Diagram C, left side, rectangle 6.8, right side 2 pentagons, z, square 2.2.D Diagram D, left side, 4 triangles, w, square, 3 over 2, right side, rectangle, 17 over 2.
Student Lesson Summary
In this lesson, we worked with two ways to show that two amounts are equal: a balanced hanger and an equation. We can use think about the weights on a balanced hanger to understand steps we can use to find an unknown amount in a matching equation.
This hanger diagram shows a total weight of 7 units on one side that is balanced with 3 equal, unknown weights and a 1-unit weight on the other. An equation that represents the relationship is .
We can remove a weight of 1 unit from each side and the hanger will stay balanced. This is the same as subtracting 1 from each side of the equation.
Balanced hanger, left side, 6 blue squares and one red squared being removed. Right side, 3 green squares, and one red square being removed. To the side, an equation 7 minus 1 = 3 x + 1 minus 1, with each minus 1 written in red.
An equation for the new balanced hanger is .
We can make 3 equal groups on each side and the hanger will stay balanced. This is the same as dividing each side of the equation by 3 (or multiplying each side by ). In other words, the hanger will balance with of the weight on each side.
Balanced hanger, left side 6 blue squares, right side, three green circles. A dotted line is drawn around each of 3 groups, each groups consists of two blue square s from the left side and one green circle from the right side. To the side, an equation says 1/3 * 6 = 1/3 * 3 x.
The two sides of the hanger balance with two 1-unit weights on one side and 1 weight of unknown size on the other side. So, the unknown weight is 2 units.
Here is a concise way to write the steps above:
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