4.1: Notice and Wonder: Equations (5 minutes)
The purpose of this warm-up is to elicit the idea that equations are used to represent situations and relationships, which will be useful when students create equations from a situation in a later activity. While students may notice and wonder many things about the equation, how it describes a relationship between two quantities and the meaning of a solution are the important discussion points.
Through articulating things they notice and things they wonder about the equation, students have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly. Listen to the language students use to describe the equation. Do they refer to 12 as “the answer”? Do they see the operations of multiplication and addition in the expression \(2x + 3y\)? Do they see a relationship between \(x\) and \(y\)? Do students think about possible values for \(x\) and \(y\)?
Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the equation and ordered pairs for all to see. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.
What do you notice? What do you wonder?
- \(2x + 3y = 12\)
- \((0,4)\) and \((6,0)\)
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the equation. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If the ideas of what the equation represents, the meaning of its solutions, and possible values for \(x\) and \(y\) do not come up during the conversation, ask students to discuss these ideas.
If students’ answers included phrases like “\(2x\)” or “\(2x\) plus \(3y\),” tell students that you are going to remind them of another way to describe \(2x\) and \(2x + 3y\). Say, “\(2x\) is the product of 2 and \(x\).” Remind students that they noticed that 2 was multiplied by \(x\), and connect the word product to their noticing of the term \(2x\) and the idea that 2 and \(x\) are multiplied.
Say, “\(2x\) plus \(3y\) is the sum of \(2x\) and \(3y\).” Remind students that they noticed the addition, and connect the word sum to their noticing of the + sign and addition.
4.2: Raffles and Snacks (15 minutes)
The mathematical purpose of this lesson is to prepare students to make sense of equations in the form \(ax + by = c\). The work of this lesson connects to previous work done in grade 8, when students explored linear relationships expressed as \(ax + by = c\). This lesson prepares students to begin the work in their Algebra 1 lesson by supporting them to make sense of contexts that can be represented with equations in the form \(ax + by = c\), and to build up a sense of the relationship between \(x\) and \(y\) in these contexts as well as the operations represented by \(ax + by\).
Students express regularity in repeated reasoning (MP8) as they find the total of many different combinations of raffle tickets and healthy snacks, and then generalize when they represent the relationship between tickets purchased, snacks purchased, and total cost with an equation.
Encourage students to talk with their classmates about what they notice and wonder about the story before they jump into calculating.
- For a fundraiser, a school club is selling raffle tickets for $2 each and healthy snacks for $1.50 each. What is the cost of:
- 3 tickets?
- 5 tickets?
- \(x\) tickets?
- 2 snacks?
- 6 snacks?
- \(y\) snacks?
- 10 tickets and 8 snacks?
- 7 tickets and 5 snacks?
- \(x\) tickets and \(y\) snacks?
- Lin bought some tickets and some snacks, and paid $22.
- Write an equation representing this situation.
- What are some combinations of tickets and snacks that Lin might have purchased?
Ask students to list all the quantities in the problem—anything they could count or measure. If students tell you numbers like $1.50, encourage them to also say what that number means in the story, e.g., the cost of one snack. Make a list of all the quantities in the problem. Call on a student to tell you the equation they wrote for question 2. Ask students about each coefficient, variable, and constant in the problem, and label them with the names of the quantities in the problem, e.g.,:
- $2: cost of one raffle ticket
- \(x\): number of raffle tickets bought
- $1.50: cost of one snack
- \(y\): number of snacks bought
- $22: total spent
Then ask students about different parts of the equation, e.g.,:
- \(\)\(2x\): total spent on raffle tickets
- \(1.5y\): total spent on snacks
- \(2x + 1.5y\): total spent on raffle tickets and snacks
As you ask students about \(2x, 1.5y\), and \(2x + 1.5y\), use phrases like, “What does the product of 2 and \(\)\(x\) represent in the story?,” or “What does the sum of \(2x\) and \(1.5y\) represent in the story?,” so students continue to hear, and get a chance to practice using sum and product in an algebraic context. This can help students make meaning of the operations and how they connect to the story. Also, use sum and product with the quantities from the story: “The product of the cost of one ticket and the number of tickets bought is the total spent on raffle tickets.” “The sum of the amount spent on raffle tickets and the amount spent on snacks is the total amount spent.”
Prompt students to use the language of sum and product themselves. For example, when you ask students which part of the equation represents the total spent on snacks, celebrate the answer \(1.5y\), but also prompt students to use “the product of 1.5 and \(y\).”
4.3: Row Game: Solving Equations (20 minutes)
In this activity, students get a chance to practice applying their skills in representing situations symbolically, and finding and interpreting solutions.
The structure of a row game supports students to check their thinking because their partners should get the same answer to different questions. When students’ answers disagree, they must make sense of each other’s questions and thinking to figure out whose answer is correct.
The practice prepares students to solve equations and solve for one variable in terms of another in their Algebra 1 lessons.
Monitor for students:
- calculating some specific values before generating an equation
- analyzing the quantities and relationships in the text
- looking for and using mathematical structures, such as \(ax + by\) or \(mx + b\)
Students will work independently to complete their designated column, and will work with their partners in the event that an answer is different from what their partners got.
Partner A completes only column A, and partner B completes only column B. Your answers for each problem should match. Work on one problem at a time, and check whether your answer matches your partner’s before moving on. If you don’t get the same answer, work together to find your mistake.
- Lin’s teacher has a daughter that is \(\frac13\) of his age. Write an expression to represent the daughter’s age. Let \(z\) represent the teacher’s age, in years.
- Han wants to save $40. He hasn’t met his goal yet. Write an expression to represent how far Han is from his goal, in dollars. Let \(q\) represent the amount of money, in dollars, he’s saved so far.
- Priya has some money to spend at a fair. It costs $6 to get in and $0.50 per ride ticket. Write an expression to represent how much Priya spends at the fair, in dollars. Let \(x\) represent the number of ride tickets Priya buys.
- Diego is inviting some friends over to watch movies. He is buying popcorn and peanuts. Popcorn costs 6 cents per ounce and peanuts cost 17 cents per ounce. Write an expression to represent the total cost of peanuts and popcorn, in cents. Let \(j\) represent how many ounces of popcorn Diego buys and \(k\) represent how many ounces of peanuts he buys.
- Jada leaves the beach with some seashells. One out of every three of the shells turns out to contain a hermit crab. Write an expression to represent the number of hermit crabs Jada found. Let \(z\) represent the total number of seashells she collected.
- Tyler started the school year with 40 pencils, but he’s lost some. Write an expression to represent how many pencils Tyler has left. Let \(q\) represent the number of pencils he’s lost so far.
- When Clare bought her plant, it was 6 inches tall. Each week, it’s been growing \(\frac12\) of an inch. Write an expression to represent how tall Clare’s plant is, in inches. Let \(x\) represent the number of weeks since Clare bought her plant.
- Mai is packing care packages. She is putting in boxes of granola bars that weigh 6 ounces each and paperback books that weigh 17 ounces each. Write an expression to represent the total weight of a care package, in ounces. Let \(j\) represent the number of boxes of granola bars and \(k\) represent the number of books.
The goal of this discussion is for students to name and clarify strategies they have for writing algebraic expressions to describe given situations.
Strategies students might have include:
calculating some specific values before generating an expression
analyzing the quantities and relationships in the text
looking for and using mathematical structures, such as \(ax + by\) or \(mx + b\)
Call on previously identified students to share their strategies for writing expressions, and support students to share insights or tips that will help other students. Possible questions for discussion include:
Why did you do calculations with some specific numbers first? How did that help you?
How did you record your calculations? What patterns did you notice?
When you read the situation, how many times did you read it?
What did you notice the first time you read the situation?
What did you notice when you looked at it again?
How did you decide what operations to use in your expression?
What made you think that your final expression would look like \(mx + b\) (or whatever structure is appropriate)?