options: Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
problem_activity: Activity
The purpose of this activity is for students to practice generating equivalent equations. The structure of taking turns and justifying your thinking to your partner supports students to check their work as they go, and the class has a chance to check for equivalence when the total score is recorded.
Students also have to justify, whether formally or informally, why each move keeps the equation equivalent. This will be helpful for students in an associated Algebra 1 lesson when they must determine what moves are acceptable to make to an equation without changing its solution.
Monitor for students who solve the equation first and then work to create new equations with the same answer, and for students who simply create new equivalent equations from the given equation. Monitor for students who add or multiply by the same number on each side of an equation and students who use properties of operations (distributive, commutative, associative).
Launch
Arrange students in groups of 2. Explain to students that they are going to play a cooperative game in which the class tries to come up with as many different equations with the same solutions as they can. You give them an equation, and their job is to come up with other equations with the same solution as the original equation. The partner’s job is to check that the new equation is equivalent to the original by listening to their partner’s reasoning and making sure they agree. Each partner should create their own equations before moving to the next question.
At the end of each round, you will ask them to share the equations they came up with, and keep track of how many different equations the class came up with. Set a goal for the second round of coming up with 4 more equations than the first round, and continue to set meaningful challenge goals with each round.
Calling on previously identified students to share strategies may help the class be more successful.
Display these equations, one at a time:
\(x = 5\)
\(x + 1 = 0\)
\(2(x + 1) = 10\)
\(\frac12x + \frac32 = \frac72\)
Student Facing
Your teacher will display an equation. Take turns with your partner to generate an equivalent equation—an equation with the same solution. Generate as many different equations with the same solution as you can. Keep track of each one you find.
For each change that you make, explain to your partner how you know your new equation is equivalent. Ask your partner if they agree with your thinking.
For each change that your partner makes, listen carefully to their explanation about why their new equation is equivalent. If you disagree, discuss your thinking and work to reach an agreement.
Student Response
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Activity Synthesis
Remind students that today they did a lot of thinking about equivalent equations and moves they could do to equations that wouldn’t change the solution.
Make a semi-permanent display of “moves that won’t change the solutions to equations.” Sort the list by moves that are done to each side, and moves that are done to one side. Use students’ language, adding formal language if that is an emphasis in your school.
Possible list:
Add the same value to both sides.
Subtract the same value from both sides.
Multiply both sides by the same value (but not zero!). Divide both sides by the same value (but not zero!).
Change the order of terms (on one side) being added or multiplied (commutative property).
Change the grouping of terms (on one side) being added or multiplied (associative property).
Distributive property:
\(a(b + c) = ab + ac\)
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correct_option_index: 7