The purpose of this Warm-up is to elicit observations about possible unknown quantities in a situation and to make explicit that a variable can be used to represent such a quantity. The reasoning here builds on students’ previous work in using a “?” to represent an unknown value or quantity.
This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.
Student Lesson in Spanish
Launch
Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2. Introduce the context of daily reading routine or reading a certain number of pages regularly to finish a book. Use Co-Craft Questions to orient students to the context and elicit possible mathematical questions.
Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation before comparing questions with a partner.
Activity
None
Jada reads 25 pages of a book every day. After several days, she has read the entire book.
Student Response
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Building on Student Thinking
Activity Synthesis
Invite several partners to share one question with the class and record responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “how many days,” “how many pages,” and “how long.”
To connect the idea of an unknown quantity in a situation to a variable, display the equation for all to see and ask students:
“Suppose Jada finishes a 200-page book by reading 25 pages a day. If this equation represents the situation, what might the variable represent?” (The number of days Jada reads the book.)
“What value of makes the equation true?” (8)
“What does that number mean in this situation?” (Jada reads the book in 8 days.)
If time permits, display another equation, , and ask students:
“Suppose Jada spends 50 minutes reading 25 pages each day. If this equation represents the situation, what might the variable represent?” (The number of minutes Jada spends reading one page)
“What value of makes the equation true?” (2)
“What does it mean in this situation?” (Jada reads at a rate of 2 minutes a page.)
Highlight that a variable can also be used to represent an unknown quantity in a situation, such as the number of days or the number of minutes in Jada’s reading example.
5.2
Activity
Standards Alignment
Building On
Addressing
6.EE.B.6
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
In this activity, students select an equation with a variable that can represent a situation with an unknown quantity. Students are presented with three stories. Each story involves the same three quantities: 5, 20, and an unknown quantity . Students think about the actions (cycling a number of miles, splitting up cups of cat food) and relationships (five times as many clubs) in the situations and consider the operations needed to describe them. For each situation, monitor for one student who chooses a correct equation and has a way to explain their reasoning, either verbally or by using a diagram.
As students make sense of what the unknown quantity represents in each story and think about how to show its relationship to the other two quantities, they practice reasoning abstractly and quantitatively (MP2).
Launch
Tell students that they will see three situations and six equations. For each situation, the task is to find one equation that represents it. Some of the equations will go unused.
Give students 5 minutes of quiet work time, followed by a whole-class discussion.
Representation: Develop Language and Symbols. Invite students to orally explain their thinking about each situation, using tape diagrams before selecting the equations that match. Supports accessibility for: Language, Conceptual processing
Activity
None
Here are three situations and six equations. Which equation best represents each situation?
If you get stuck, consider drawing a diagram.
After Elena rode her bike for 5 miles on Friday, she had biked a total of 20 miles for the week. She biked miles before Friday.
Andre’s school has 20 clubs, which is five times as many clubs as his cousin’s school has. His cousin’s school has clubs.
Jada volunteers at the animal shelter. She divides 5 cups of cat food equally to feed 20 cats. Each cat receives cups of food.
Activity Synthesis
For each situation, ask a student to share which equation they selected and the reason they chose it. If any students drew a diagram to help them reason about the situation, ask them to present their diagram and draw connections between the situation, equation, and diagram. Consider these questions for discussion:
“All of the equations and situations had the same numbers. How did you decide which equations represented which situations?”
“Did any of the words in the stories confuse or mislead you? How did you move past the confusion?”
“What does the variable in each equation represent?”
5.3
Activity
Standards Alignment
Building On
Addressing
6.EE.B.5
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
Solve real-world and mathematical problems by writing and solving equations of the form and for cases in which , and are all nonnegative rational numbers.
The purpose of this activity is for students to practice matching equations to situations and then solving those equations using their choice of strategy. Monitor for students who reason about the solution to each equation by:
Drawing diagrams (tape, hanger, or their own creations) that describe the relationships.
Thinking about values of the variable that would make the equation true and would make sense in the situation.
Doing the same operation to each side of an equation.
In the Activity Synthesis, students analyze similarities in the structure of the situations and the equations that represent them (MP7).
Launch
Explain to students that they will look at descriptions of four situations. For each situation, they are to identify equations that match the relationship described.
One of the situations is about using electricity from solar panels. Invite students to share any knowledge they have about solar panels. Explain that solar panels are considered a “clean” source of energy because they create energy from sunlight rather than from burning fossil fuels like coal or oil, and that kilowatt-hour (kWh) is a unit of energy. (With 1 kilowatt-hour we can, for instance, toast 160 slices of bread or keep cool for 30 hours with a ceiling fan.)
Allow students 8–10 minutes of quiet work time, followed by a whole-class discussion. Remind students that they can draw a diagram to help them visualize the relationships between the quantities in the situation.
Activity
None
Circle all of the equations that describe each situation. Then find the solution for those equations and tell what it means in the situation.
Kiran scores 223 fewer points in a computer game than Mai. Mai scores 409 points. How many points does Kiran score?
Clare learned that on a sunny day, one solar panel produces 1.5 kilowatt hours (kWh) of electricity. That’s enough electricity to charge a smartphone 100 times! How many kWh of electricity is used to charge a smartphone one time?
Han is charging a school tablet with a new charging cable, which is feet longer than the old cable. The new cable is 6 feet long. How long is the old cable?
Noah is cutting a 12-inch strip of tape into pieces that are each inches long. How many pieces can he make?
Student Response
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Building on Student Thinking
If students continue to focus on key words instead of the relationships described in each situation, consider asking:
“How did you choose an equation for this problem?”
“How can you describe the relationships between the quantities in your own words?
“How can you draw a diagram to represent those relationships?
Activity Synthesis
Invite students to share their strategies for matching equations to the stories and for solving the problems. Include students who used different ways to reason, as noted in the Activity Narrative.
For students who used equations to solve the problems, ask which of the chosen equations they decided to solve and why. For each representation or strategy, ask where students see information from the situation. Regardless of the solving strategy used, students should express each solution in the context of the problem, including the appropriate units.
If no students bring it up, ask if any of the situations have a similar structure. If needed, point out:
The cable and game situations share a similar structure, where both the larger quantity and the difference between the smaller and larger quantities are known, while the smaller quantity is unknown. This relationship is expressed with “longer” in the cable situation and with “fewer” in the game situation.
The electricity and tape situations are both about equal-size groups making a total, but differ in the quantities that are known and unknown. In the electricity situation, the total and size of each group is known, and the number of groups is unknown. In the tape situation, the size of each group and the total are known, and the number of groups is unknown.
Focusing on structure in this way helps students reason about the relationships between quantities in a situation. This will help students represent and solve the problems.
Lesson Synthesis
Students used equations with variables to represent and solve real-world problems. To encourage students to reflect on the reasoning involved in this process, consider asking some of the following questions:
“Why do you think people write equations to represent real-world problems?”
“Why is it important to clearly state what a variable represents when you use an equation to represent a situation?”
“How do you know when a situation can be represented by an addition (or a multiplication) equation?”
“When you solve an equation to solve a real-world problem, how can you check your answer?”
Student Lesson Summary
Writing and solving equations can help us answer questions about situations.
A scientist has 13.68 liters of oil and needs 16.05 liters for an experiment. How many more liters of oil does she need for the experiment?
We can represent this situation with the equation:
We can solve the equation by subtracting 13.68 from each side. This gives us some new equations that also represent the situation:
The solution means the scientist needs 2.37 more liters of oil.
Volunteers at a food pantry divide a 54-pound bag into portions that each weigh pound. How many portions can they make?
We can represent this situation with the equation:
We can find the value of by dividing each side by . This gives us some new equations that represent the same situation:
The solution means the volunteers can make 72 portions.
Standards Alignment
Building On
Addressing
6.EE.B.6
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
Students who focus on key words might be misled in each situation. For the first situation, students might see the word “total” and decide they need to add 5 and 20. In the second situation, the words “five times as many” might prompt students to multiply 5 by 20. The third story poses some additional challenges: students see the word “divided” but there is no equation with division. Additionally, students might think that division always means divide the larger number by the smaller. To help students make sense of the situations, encourages students to:
Represent each situation by acting it out, drawing a picture, or creating a tape diagram.
Focus on what quantity in the story each number or variable represents, and on the relationships among them.
Think about where a situation describes a total and where it describes parts of the total.
