options: Represent and solve equations and inequalities graphically.

Work with addition and subtraction equations.

Rewrite rational expressions.

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

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

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

Multiply and divide within 100.

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

Use properties of operations to generate equivalent expressions.

Work with radicals and integer exponents.

Analyze patterns and relationships.

Write expressions in equivalent forms to solve problems.

Use polynomial identities to solve problems.

Represent and solve problems involving addition and subtraction.

Write and interpret numerical expressions.

Use the four operations with whole numbers to solve problems.

Add and subtract within 20.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Understand the relationship between zeros and factors of polynomials.

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

Reason about and solve one-variable equations and inequalities.

Perform arithmetic operations on polynomials.

Represent and analyze quantitative relationships between dependent and independent variables.

Create equations that describe numbers or relationships

Apply and extend previous understandings of arithmetic to algebraic expressions.

Interpret the structure of expressions.

Gain familiarity with factors and multiples.

Generate and analyze patterns.

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

Solve systems of equations.

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

Solve equations and inequalities in one variable.

Represent and solve problems involving multiplication and division.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.
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: 25