options: Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Perform arithmetic operations on polynomials.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Understand and apply properties of operations and the relationship between addition and subtraction.

Represent and analyze quantitative relationships between dependent and independent variables.

Multiply and divide within 100.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Work with equal groups of objects to gain foundations for multiplication.

Interpret the structure of expressions.

Create equations that describe numbers or relationships

Gain familiarity with factors and multiples.

Use properties of operations to generate equivalent expressions.

Analyze patterns and relationships.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Represent and solve problems involving addition and subtraction.

Solve systems of equations.

Understand the connections between proportional relationships, lines, and linear equations.

Add and subtract within 20.

Work with addition and subtraction equations.

Solve equations and inequalities in one variable.

Write and interpret numerical expressions.

Rewrite rational expressions.

Represent and solve problems involving multiplication and division.

Understand properties of multiplication and the relationship between multiplication and division.

Represent and solve equations and inequalities graphically.

Use polynomial identities to solve problems.

Use the four operations with whole numbers to solve problems.

Understand solving equations as a process of reasoning and explain the reasoning.

Generate and analyze patterns.

Reason about and solve one-variable equations and inequalities.

Work with radicals and integer exponents.

Understand the relationship between zeros and factors of polynomials.

Write expressions in equivalent forms to solve problems.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
problem_activity: Activity
The purpose of this lesson is to increase fluency in solving equations. Students will solve equations individually and then compare differing, though accurate, solution paths in order to compare their work with others. This will help students recognize that while the final solution will be the same, there is more than one path to the correct answer that uses principles of balancing equations learned in previous lessons.
Launch
Arrange students in groups of 3–4. Give students quiet think time to complete the activity and then tell groups to share how they solved the equations for
\(x\)
and discuss the similarities and differences in their solution paths.
Representation: Internalize Comprehension.
Activate or supply background knowledge about solving equations. Encourage students to use previously solved equations as guidelines to determine appropriate steps.
Supports accessibility for: Memory; Conceptual processing
Representing, Conversing: MLR2 Collect and Display.
As groups discuss their work, circulate and listen for the language students describe the similarities and differences in their solution paths. Write down different solution paths that led to the same result in a visual display. Consider grouping words and phrases used for each step in different areas of the display (e.g., “multiply first”, “divide first”, “subtract
\(x\)
from the right”, “add
\(x\)
from the left”). Continue to update the display as students move through the activity, and remind them to borrow from the display while discussing with their group. This will help students develop their mathematical language around explaining different solution paths to solving equations.
Design Principle(s); Maximize meta-awareness
Student Facing
Solve these equations for
\(x\)
.
1.
\(\frac{12+6x}{3}=\frac{5-9}{2}\)
2.
\(x-4=\frac13(6x-54)\)
3.
\(\text-(3x-12)=9x-4\)
Student Response
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Student Facing
Are you ready for more?
I have 24 pencils and 3 cups. The second cup holds one more pencil than the first. The third holds one more than the second. How many pencils does each cup contain?
Student Response
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Activity Synthesis
Students should take away from this activity the importance of using valid steps to solve an equation over following a specific solution path. Invite students to share what they discussed in their groups. Consider using some of the following prompts:
“How many different ways did your group members solve each problem?”
“When you compared solution paths, did you still come up with the same solution?” (Yes, even though we took different paths, we ended up with the same solutions.)
“How can you make sure that the path you choose to solve an equation is a valid path?” (I can use the steps we discovered earlier when we were balancing: adding the same value to each side, multiplying (or dividing) by the same value to each side, distributing and collecting like terms whenever it is needed.)
“What are some examples of steps that will not result in a valid solution?” (Performing an action to only one side of an equation and distributing incorrectly will give an incorrect solution.)


correct_option_index: 13