options: 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.

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

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.

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.

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.

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

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

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 identify why equations are not equivalent. In an associated Algebra 1 lesson, students rewrite equations to express one variable in terms of the other. The work in this activity prepares students to be successful in the associated Algebra 1 lesson by allowing them to gain a deeper understanding of what makes two equations equivalent and to practice making appropriate moves to maintain the balance in an equation.
Student Facing
[IMG]
Tyler is practicing finding different equivalent equations that match the story. For each of the problems below, he gets one equation right but the other equation wrong. For each one, explain the error, give the correct equivalent equation, and explain your reasoning.
Situation: The yogurt at Sweet Delights costs \$0.65 per pound and \$0.10 per topping. The total cost of a purchase was \$1.70. Let
\(p\)
be the weight of the yogurt in pounds and
\(t\)
be the number of toppings bought.
Tyler’s first and correct equation:
\(0.65p+0.10t=1.70\)
Tyler’s second and
incorrect
equation:
\(t= (1.70 - .65p) \boldcdot 0.10\)
What is the error?
What is a correct second equation Tyler could have written?
What might Tyler have been thinking that led to his mistake?
Situation: The perimeter of a rectangle (twice the sum of the length and width) is 13.5 inches. Let
\(l\)
be the length of the rectangle and
\(w\)
be the width of the rectangle.
Tyler’s first and correct equation:
\(2(l + w) = 13.5​​​​​​​​​​​​​​\)
Tyler’s second and
incorrect
equation:
\(w = 13.5 - 2l\)
What is the error?
What is a correct second equation Tyler could have written?
What might Tyler have been thinking that led to his mistake?
Situation: For a fundraiser, a school is selling flavored waters for \$2.00 each and pretzels for \$1.50 each. The school has a fundraising goal of \$200. Let
\(w\)
be the number of waters sold and
\(p\)
be the number of pretzels sold.
Tyler’s first and correct equation:
\(2w + 1.5p = 200\)
Tyler’s second and
incorrect
equation:
\(1.5p = 198w\)
What is the error?
What is a correct second equation Tyler could have written?
What might Tyler have been thinking that led to his mistake?
Student Response
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Activity Synthesis
The purpose of this discussion is to reinforce the concept that different equations can represent the same relationship.
Display the school fundraiser situation from the practice activity for all to see.
Here are discussion questions:
What are some of the true equations that describe the school fundraiser situation? (
\(2w + 1.5p = 200, w = 100 - 0.75p, p = \frac{200 - 2w}{1.5}\)
)
What can be seen in each equation students share, and why might these equations be useful? (
\(2w + 1.5p = 200\)
shows how the water and pretzel revenue should equal
$
200. The other two equations help if you want to know how many waters you need to sell or how many pretzels you need to sell, if you know the other amount.)
Explain that because all these equations describe the same relationship, just in different ways, they are
equivalent
. Discuss how equivalent equations can be found by thinking about the situation in different ways, as well as by applying acceptable moves for solving equations, such as adding the same amount to each side or multiplying each side by the same amount.


correct_option_index: 3