The warm-up for this problem is pulled from evidence of recent student work. I noticed many students are making similar errors so this warm-up was created to address that issue.
Lift Tickets for Skiing
"My Grammy always buys her grandchildren a gift card for a day at the mountain. She has 10 grand-children who like to ski. The table shows the different options for purchasing tickets. What combination of tickets should she purchase to get the best deal? How much will she save?"
Before we begin, I ask a student to read the problem out loud. The class then discusses the importance of interpreting the problem. I facilitate by asking key guiding questions.
"What information do I know now that I read this question?"
"What am I going to need to figure out?" Then I let the students work.
This is a great warm-up problem because it has multiple entry points. When I was circulate the classroom, I notice most of the students using the same entry point. They want to determine how much 10 tickets will cost, using the different packet combinations. Some students start with the 5 pack first, others start with the 1 pack.
I need to push students to think about this problem another way. As students wrap up their discussions, agreeing that Grammy should purchase 2 packs of 5 to save $90, I ask them to use another approach to prove their answer is accurate. Some groups are able to get right to determining the price per ticket. Others needed prompting.
In the end, I call students to the board to explain their thinking in answering the question. I ask students to write a number sentence to match their thinking.
As you can see in the picture of our work on the whiteboard, students often make mistakes when they are dividing with decimals. This is a good thing, in my class we make sure that we embrace mistakes. Mistakes help us learn. The student in this picture has provided the class an opportunity to review reasonableness, checking work and estimation.
Vo-Back-Ulary is a game that students really enjoy. I play it at the beginning of a new topic to help students make connections to the new words they will use. In this game, each student gets exposure to a list of new vocabulary words and builds a connection to one of the words. I emphasize that the words on the list fit together like a puzzle. The word that becomes “your word” is an important piece of the puzzle. It is your responsibility to figure out how/why your piece belongs on this list.
For this particular list, I ask the students to make sure their questions did not have to do with the meaning of the words. This lesson is designed to introduce the terms to the students. Many of the words are new, therefore, if a student is asked to answer a yes or no question about the meaning of a word, he or she is likely to have to answer "I don't know" or make a guess that could result in the student who is questioning to get thrown off. Before we begin the game, I make this point clear to the students so they can strategize appropriately.
When three students are left playing, I ask them to come to the front of the room. They take turns asking the whole class yes or no questions about the word on their back. When they discover their words too, we have a discussion about strategy.
Next, I ask a few students to share. Here is an example of a strategic student's share.
First I asked, is it in the first column? When my friend said no, I knew that my word was on this half of the paper. Then I asked if my word was more than one word, my friend said yes. I knew I only had two words to choose from. I was done quickly."
To build on the experience of the game, I spend time working with students to activate the prior knowledge they have with variables. Understanding variables and expressions is the foundation for the upcoming topic, so it is important to build a strong common understanding.
What words on this list do you already know? As students share words they are comfortable with, I ask them to explain why. I accept feedback from the rest of the group before highlighting words with the yellow highlighter. If some students are so/so on feeling comfortable with a word, I use green. When there are few hands up, showing that only a few are participating, I switch to the blue highlighter and ask the kids to point out words that make them say "whhhhaaatt".
A lot of hands shoot up right away to share these new "scary" words. Those are highlighted in blue. All words that fit into both categories came out green. (Attached is a screen capture of the vo-back-ulary list).
I let the students talk to each other about these questions before we talk as a group. This helps make everyone feel comfortable and confident with the discussion topic.
I used this visual to demonstrate that when we learn something new, it builds on what we already know. This topic looks like it will be a balance between what we know already and what we get to learn together.
Create a number sentence using a variable. Show a friend. Share.
As students share, write each equation on the board. As I expected, most pairs share equations with the variable at the end. I draw attention to the few examples that do not place the variable at the end. I like having the students make up their own examples, because many times I am thinking about basic examples when I introduce something new. One student used a variable to represent an exponent. This provides a nice window for a rich conversation that I didn't plan on.
Next, I ask the students to rewrite their expression to change the way it looks on the paper, without changing the value of the variable. I walk around the room to each desk, asking, "Can you make another expression?". I encourage talking at this time. Students help each other realize that they can use their knowledge of fact families and operations to write multiple expressions.
I share two examples of these on the board and ask the students to determine the correct position of a variable in an expression. Using the evidence from these examples, students are able to agree that a variable can be found in any location in a number sentence. They are reminded that any letter can serve as a variable, and that variables are appropriate in an expression with any operation.
One student asks if a variable can take the place of an operation, "To make you guess what operation you should use". Great question. It provides another opportunity for clarification.
From the list of expressions on the board, I choose 8 x n = 24 to discuss further. "What part of this expression is the variable?" Most students say "n" right away. I respond with, "I am confused." You told me a variable is a letter that is in an expression to take the place of a number. But I am looking at this, and I feel confused. What can be confusing me?" Students recognize that the multiplication sign looks like a variable.
"This is a problem that mathematicians ran into long ago. They had to come up with a solution. Talk to each other and see if you can think of a solution too." The students make suggestions, we talk about the advantages and disadvantages of each, and I let them know when they are on the right track by saying "You're on to something" and writing their thoughts on the board. Eventually, I come back to the students' suggestion to make the multiplication symbol "x", smaller. I make the x on the board smaller and smaller until it can't be seen any more. This introduces the notation 8n.
I like to show the students a preview of end goal at the beginning of a topic. I think it helps them put each of the pieces together and understand the significance of their learning journey. Today, I complete the exit ticket, the students sit back and enjoy!
3 x [(18x 5.5) / c] when c= 3