Unit 2, Lesson 1
Expressing Mathematics
Extra Support Materials for Algebra 1
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The purpose of this Warm-up is to elicit the idea that there can be several quantities in a given situation, which will be useful when students practice representing quantities and the relationships that exist between them in a later activity. While students may notice and wonder many things about the situation, known and unknown quantities are the important discussion points.
This Warm-up prompts students to make sense of a problem before solving it, by familiarizing themselves with a context and the mathematics that might be involved (MP1).
Arrange students in groups of 2. Display the situation for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.
Kiran is helping his aunt and uncle plan a party. His family has a lot of experience planning parties.
Kiran’s uncle is in charge of the furniture. He tells Kiran that he plans to have 1 table for every 4 people and 1 chair per person.
Kiran’s aunt is getting plates and paper towels. She plans on buying one plate per person, plus 10 extra plates just in case, and she’s going to buy one roll of paper towels for every 10 people.
What do you notice? What do you wonder?
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the situation. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If representing the quantities with numbers and variables does not come up during the conversation, ask students to discuss this idea.
The goal of this activity is for students to continue to notice quantities and relationships in written situations and to connect those quantities and relationships to operations on variables. Students can use the notice and wonder strategy to help them understand the context. They can write in words how they calculate the number of mice for each snake. This might also help students see patterns in calculations and generalize them to an expression. As students use patterns to generalize expressions, they are looking for and expressing regularity in repeated reasoning (MP8).
Monitor for students who develop incorrect expressions, especially expressions using the wrong operation, for example for the number of mice needed. Ask students to put their information in a table, but to write out each calculation like this:
| number of snakes | number of mice |
|---|---|
| 10 | 10 + 2 |
| 6 | 6 + 2 |
Continue to display the information about Kiran’s party from the earlier activity. Ask students, “If I told you that Kiran’s aunt and uncle were expecting 100 people to attend the party, what else could you tell me?” (They need 25 tables. They need 100 chairs. They need 110 paper plates. They need 10 rolls of paper towels.)
Follow up by asking what calculations they did to answer the question. Record their calculations and results, for example or . Ask students, “How did you decide it made sense to use that operation? What does the 100 represent here? What does the 1 (or ) represent here?”
Ask students, “Did anyone use different operations but get the same answer?” (I divided the number of people by 4 instead of multiplying by .)
Ask students, “If I told you that Kiran’s aunt and uncle were expecting 40 people, what else could you tell me?” (They need 40 chairs. They need 10 tables. They need 50 plates. They need 4 rolls of paper towels.)
Record students’ operations for 40 guests below what you recorded for 100 guests. The operations should look exactly the same, just with a 40 in place of the 100 and the other numbers adjusted accordingly.
Ask students, “What is different between the calculations that involved 40 guests and the ones that involved 100? What is the same?” (Only the number of guests changes. It’s always 10 more than the number of guests for the plates needed, for example.)
Ask: “What if we wanted a way to compute the number of tables for any number of guests? Like, for guests? What could we write?”
Record students’ operations for guests below what you recorded for 100 guests and 40 guests. The operations should look exactly the same, just with an in place of the 100 or 40.
Tell students that they are going to study another situation and write expressions based on that situation.
As students work, direct them to focus on how they know what operation to use.
The goal of this discussion is for students to share the key words or phrases that indicate what operation to use.
Direct students’ attention to the reference created using Collect and Display. Ask students to share the expressions they came up with. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. (For example, the display may have “the expression shows what happened” already on it and can be updated with the more precise phrase “the expression models the situation.”)
For each expression, ask students:
Display this expression for all to see: . Tell students someone in another class came up with this expression for the amount of mice needed in the first question. Ask students to turn to a partner to discuss why this expression doesn’t make sense. Instead of , consider using (anonymous) incorrect answers used by multiple students.
Here are more questions for discussion:
Tell students that the strategy of checking a number in the story to see if it works in your expression is one that mathematicians use all the time, and one that they can practice today.
The purpose of this activity is for students to practice identifying the different quantities involved in a situation, both known and unknown. In the associated Algebra 1 lesson, students will create expressions to represent situations. This lesson prepares students for the associated Algebra 1 lesson by asking them to determine what information is important and what quantities would be helpful (MP4) when given information about a situation in written form.
Allow students to complete the Task Statement individually.
Monitor for students who create expressions or equations to represent the situations. Students will create equations in the associated Algebra 1 lesson, so if time permits, allow students to explain their reasoning for the expressions or equations they create.
To understand the situation, what is some information you would like to know? What information is already given?
The goal is to practice thinking through a situation and identifying quantities that are involved. Allow students to share the quantities they described for each situation. Invite students to share their responses. Here are some questions for discussion:
"How did you identify the important quantities in each situation?" (I thought about the type of problem I can solve with the information I was given. Then I thought about what missing information I still need to solve the problem.)
"Did anyone identify a quantity for [a particular situation] that has not been shared yet?"
"Why is [a particular quantity] important, or what types of problems can we solve with this quantity?"
If time permits, allow students to explain their reasoning for any expressions or equations they created.
A zookeeper is preparing to care for snakes in an exhibit. For each question, write an expression representing the supplies needed.
She needs 4.5 ounces of crickets for each snake. How many ounces of crickets are needed if the number of snakes is:
She needs one mouse for each snake, plus two extra mice. How many mice are needed if the number of the snakes is:
For every 2 snakes, she needs 1 bowl of water. How many bowls of water are needed if the number of snakes is:
There is one male snake, and the rest are female. She needs one vitamin pill for every female snake. How many vitamin pills does she need if the number of snakes is: