options: Work with addition and subtraction equations.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Interpret the structure of expressions.

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

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

Perform arithmetic operations on polynomials.

Solve equations and inequalities in one variable.

Use polynomial identities to solve problems.

Use the four operations with whole numbers to solve problems.

Create equations that describe numbers or relationships

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

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

Add and subtract within 20.

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

Represent and solve problems involving addition and subtraction.

Rewrite rational expressions.

Represent and solve problems involving multiplication and division.

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

Analyze patterns and relationships.

Understand the relationship between zeros and factors of polynomials.

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

Generate and analyze patterns.

Reason about and solve one-variable equations and inequalities.

Write and interpret numerical expressions.

Use properties of operations to generate equivalent expressions.

Write expressions in equivalent forms to solve problems.

Gain familiarity with factors and multiples.

Represent and solve equations and inequalities graphically.

Work with radicals and integer exponents.

Multiply and divide within 100.

Solve systems of equations.

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

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

Represent and analyze quantitative relationships between dependent and independent variables.
problem_activity: Activity
The goal of this activity, and the two that follow, is for students to solve an equation in a real-world context while previewing some future work solving systems of equations. Here, students first make sense of the situation using a table of values describing the water heights of two tanks and then use the table to estimate when the water heights are equal. A key point in this activity is the next step: taking two expressions representing the water heights in two different tanks for a given time and recognizing that the equation created by setting the two expressions equal to one another has a solution that is the value for time,
\(t\)
, when the water heights are equal.
Launch
Give students 2–3 minutes to read the context and answer the first problem. Select students to share their answer with the class, choosing students with different representations of the situation if possible. Give 3–4 minutes for the remaining problems followed by a whole-class discussion.
Representation: Internalize Comprehension.
Represent the same information through different modalities by using diagrams. If students are unsure where to begin, suggest that they draw a diagram to help illustrate the information provided.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Representing, Conversing: MLR2 Collect and Display.
To begin the whole-class discussion, give students the opportunity to discuss their solutions to the first question in groups of 3–4. Circulate through the groups and record language students use to describe what is happening in each tank. Listen for language related to rate of change, differences between rates, initial amounts of water, etc. If groups are stuck, consider asking, “What happens to the amount of water in Tank 1 (or 2) as time goes on?” and “Can you draw a picture of Tank 1 (or 2) at 5 minutes and then another picture at 10 minutes? What do you notice?” Post the collected language in the front of the room so that students can refer to it throughout the rest of the activity and lesson. This will help students talk about the relationship between the two tanks prior to being asked to find the time when they are equal.
Design Principle(s); Maximize meta-awareness
Student Facing
The amount of water in two tanks every 5 minutes is shown in the table.
| time (minutes) | tank 1 (liters) | tank 2 (liters) |
|----------------|-----------------|-----------------|
|       0        |       25        |      1000       |
|       5        |       175       |       900       |
|       10       |       325       |       800       |
|       15       |       475       |       700       |
|       20       |       625       |       600       |
|       25       |       775       |       500       |
|       30       |       925       |       400       |
|       35       |      1075       |       300       |
|       40       |      1225       |       200       |
|       45       |      1375       |       100       |
|       50       |      1525       |        0        |
Describe what is happening in each tank. Either draw a picture, say it verbally, or write a few sentences.
Use the table to estimate when the tanks will have the same amount of water.
The amount of water (in liters) in tank 1 after
\(t\)
minutes is
\(30t + 25\)
. The amount of water (in liters) in tank 2 after
\(t\)
minutes is
\( \text-20t + 1000\)
. Find the time when the amount of water will be equal.
Student Response
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Activity Synthesis
The purpose of this discussion is to elicit student thinking about why setting the two expressions in the task statement equal to one another is both possible and a way to solve the final problem.
Consider asking the following questions:
“What does
\(t\)
represent in the first expression? The second?” (In each expression,
\(t\)
is the time in minutes since the tank’s water level started being recorded.)
“After we substitute a time in for
\(t\)
and simplify one of the expressions to be a single number, what does that number represent? What units does it have?” (The number represents the amount of liters in the water tank.)
“How accurate was your estimate about the water heights using the table?” (My estimate was within a few minutes of the actual answer.)
“If you didn’t know which expression in the last problem belonged to which tank, how could you figure it out?” (One of the expressions is increasing as
\(t\)
increased, which means it must be Tank 1. The other is decreasing as
\(t\)
increases, so it must be Tank 2.)
“How did you find the time the two water heights were equal using the expressions?” (Since each expression gives the height for a specific time,
\(t\)
, and we want to know when the heights are equal, I set the two expressions equal to each other and then solved for the
\(t\)
-value that made the new equation true.)


correct_option_index: 31