Unit 4, Lesson 4
Equations and Their Solutions (Optional)
Integrated Math 1
4.1
Warm-up
Standards Alignment
Building On
Addressing
HSA-REI.A
Understand solving equations as a process of reasoning and explain the reasoning.
HSA-REI.B.3
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Building Toward
Instructional Routines
NoneMaterials
NoneActivity Narrative
This Warm-up prompts students to recall what they know about the solution to an equation in one variable. Students interpret a given equation in the context of a situation, explain why certain values are not solutions to the equation, and then find the value that is the solution.
Launch
Give students continued access to calculators.
Activity
NoneCement is made by mixing water and concrete mix. The amount of water, , added to the concrete mix can affect the strength of the material. One liter of water weighs 2.2 pounds.
The equation represents the relationship between these quantities.
- What could the 80 represent in this situation?
- Priya said that neither 2 nor 3 could be the solution to the equation. Explain why she is correct.
- Find the solution to the equation.
Student Response
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Building on Student Thinking
Activity Synthesis
Focus the discussion on how students knew that 2 and 3 are not solutions to the equation and how they found the solution. Highlight strategies that are based on reasoning about what values make the equation true.
Ask students: "In general, what does a solution to an equation mean?" Make sure students recall that the solution to an equation in one variable is a value for the variable that makes the equation a true statement.
4.2
Activity
Instructional Routines
MLR5: Co-Craft QuestionsMaterials
NoneActivity Narrative
In this activity, students write equations in one variable to represent the constraints in a situation. They then reason about the solutions and interpret the solutions in context.
To solve the equation, some students may try different values of until they find one that gives a true equation. Others may perform the same operations to each side of the equation to isolate . Identify students who use different strategies and ask them to share later.
4.3
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
In the previous activity, students recalled what it means for a number to be a solution to an equation in one variable. In this activity, they review the meaning of a solution to an equation in two variables.
Launch
Give students continued access to calculators.
Activity
NoneOne gallon of gasoline produces about 20 pounds of carbon dioxide. One gallon of pure ethanol produces about 13 pounds of carbon. A car engine that can run on gasoline or ethanol produces 100 pounds of carbon dioxide from gallons of gasoline and gallons of ethanol.
The equation represents the relationship between these quantities.
- Determine if each pair of values could be the number of gallons of gasoline and ethanol burned in the car engine. Be prepared to explain your reasoning.
- 5 gallons of gasoline and 7.7 gallons of ethanol
- 3.05 gallons of gasoline and 3 gallons of ethanol
- 2 gallons of gasoline and 4.5 gallons of ethanol
- If the car engine burned 4 gallons of gasoline, how many gallons of ethanol were burned? Show your reasoning.
- In this situation, what does a solution to the equation tell us? Give an example of a solution.
Lesson Synthesis
To summarize the lesson, refer back to the activity about carbon dioxide. Remind students that a gallon of gasoline produces 20 pounds of carbon dioxide and a gallon of ethanol produces 13 pounds. Discuss questions such as:
- “What does the equation tell us about the carbon dioxide created?” (100 pounds are produced, some from burning gasoline and some from burning ethanol.)
- “In this situation, what does it mean to solve the equation?” (To find the combination of gasoline and ethanol that produce 100 pounds of carbon dioxide.)
- “Is the combination of 3 gallons of gasoline and 5 gallons of ethanol a solution to the equation? Why or why not?” (No, they don't add up to 100 pounds. Substituting 3 for and 5 for into the equation doesn't lead to a true equation.)
- “Consider the equation . What does it tell us about the fuel?” (The car engine burned 4.5 gallons of gasoline and produced a total of 100 pounds of carbon dioxide.)
- “What does it mean to solve this equation?” (To find the gallons of ethanol that, when combined with 4.5 gallons of gasoline, burns to produce 100 pounds of carbon dioxide. To find the value of that would make the equation true.)
Student Lesson Summary
An equation that contains only one unknown quantity or one quantity that can vary is called an equation in one variable.
For example, the equation represents the relationship between the length, , and the width, , of a rectangle that has a perimeter of 72 units. If we know that the length is 15 units, we can rewrite the equation as:
.
This is an equation in one variable, because is the only quantity that we don't know. To solve this equation means to find a value of that makes the equation true.
In this case, 21 is the solution because substituting 21 for in the equation results in a true statement.
An equation that contains two unknown quantities or two quantities that vary is called an equation in two variables. A solution to such an equation is a pair of numbers that makes the equation true.
Suppose Tyler spends \$45 on T-shirts and socks. A T-shirt costs \$10 and a pair of socks costs \$2.50. If represents the number of T-shirts and represents the number of pairs of socks that Tyler buys, we can can represent this situation with the equation:
This is an equation in two variables. More than one pair of values for and make the equation true.
and
and
and
In this situation, one constraint is that the combined cost of shirts and socks must equal \$45. Solutions to the equation are pairs of and values that satisfy this constraint.
Combinations such as and or and are not solutions because they don’t meet the constraint. When these pairs of values are substituted into the equation, they result in statements that are false.