When students enter the room, I give them three sample problems on a small piece of paper that I often refer to as the entrance ticket. I like to hand entrance tickets as students enter. This is usually faster than letting them find their seats and then either passing out the entrance tickets or having them read a prompt that directs them to grab a sheet. The time difference is about 2-3 minutes, but I try and utilize each minute of every lesson.
The problems are written to include student names and involve situations that relate to their interests. I suggest rewriting each problem by using the names of your students and use products that those students are likely to actually purchase. These details remind your class that each learning experience is connected to the community you have built in your classroom.
Here are the questions I used:
Apples etc, A linear Systems practice sheet
Shaun bought 4 pounds of apples and 1 pound of bananas at the local supermarket for $9. He went back later that day and bough another 6 pounds of apples and 1 pound of bananas for $13. If the prices didn’t change, what was the price per pound of the apples and bananas? Write equations to model this situation.
Ruby bought 10 crates of neon socks and 2 crates of neon shirts for $320. Stella bought 5 crates of neon socks and 2 crates of neon shirts for $220. If both Stella and Ruby paid the same price, how much did they pay per crate? Write equations to model this situation.
Kyle bought 2 pairs of shoes and 4 basketballs for $360. Chris bought 3 pairs of shoes and 2 basketballs for $480. If Chris and Kyle both paid the same price, what was the cost of each basketball and pair of shoes? Write equations to model this situation.
As students are getting ready I always circulate and make sure each student has what they need to get started, this might be a pen, pencil, calculator or even something like a tissue. The idea is to help them succeed and let them know you are there to support them in any way you can. As always, these little teaching moves establish your presence in the room and help unify your class around the content of the day. I give them two minutes to work and struggle with the problems on the page. I follow by giving a brief introduction into the algebraic nature of each problem, being careful to extract the meaning of whatever variable we choose to loose. In these problems, it is important that students know that the variables represent the cost of a unit and not just the unit itself. For example, in 4x + 1y = 9, the 4x represents 4 apples times x, which is the cost of a single pound of apples. Students often think that the x in 4x represents apples an the y in 1y represents bananas. Help them realize this by playing with the value of x. Guide them with questions like, “if a pound of apples was 5 dollars, how much would 4 pounds cost? What if a pound was 10 dollars? 100 dollars? What are we doing each time? How could we write this multiplicative relationship as an expression or equation?” As you play with the poundage of apples, set up a table to support the reasoning.
Work with the class through the next two problems and involve them in each step. Many students are quick to set up each equation but struggle with the necessary steps to solve the system. Help them see the basis of every algorithm and not just the steps in each example. My students succeed when they remember that the goal of elimination (often the preferred method when systems are in standard form) is to eliminate one of the variables. The way many students accomplish this is by creating an additive inverse or opposite term and then subtracting. In the first example, we have 4x + 1y = 9 and 6x + 1y = 13 and we can simply subtract 1y from 1y to isolate x and solve. The other examples build on this. The second requires multiplying the second equation and then subtracting and the last example requires two multiplication steps (certainly the toughest in the set).
Also prepare for the wonderful intuition that students will offer in their solutions. In the first example, students look at 4x + 1y = 9 and 6x + 1y = 13 and realize that x must equal 2 by comparing the quantities of apples and bananas purchased in each situation. Their algorithm usually requires the observation that both equations have the same number of bananas but a different number of apples. Students then realize that since the customer is buying 2 more pounds of bananas and paying 4 more dollars, those 2 pounds must cost 2 dollars each. Help students realize that this solution is equivalent to the formal process of elimination through the subtraction of equations.
When you are finished with review, introduce the concept of the lesson. Ask students to create a problem of their own. I like to spend a minute clarifying the type of problem that they should create. Students an help you and point out that each problem will require a common unit like money or weight and two variables like apples and bananas and at least two scenarios to compare. Push them a bit to think about if two scenarios will always be enough to solve a system (yes, as long as the equations are not collinear). After I define the problem, I set up the requirements of the activity and request that students create a problem that will be challenging and humorous. I point out their entrance ticket as an example and mention that they can certainly create funnier problems. The idea is to have a cover sheet with the problem and then the solution on the back or on a separate page so that we can swap problems with another partnership. I tell students that they will have about 20 minutes to complete the activity. I ask them to start by setting up the problem first and then following with the story (which they get really into. They know we will share these and want to see their audience laugh).
As students begin, I circulate and distribute paper, markers, crayons or color pencils. Students can also send a partner to get supplies, but I like to help with supplies as it gives my plenty of chances to engage students and get them on task. If a partnership is struggling with the math, I might ask a specific question or have them switch and talk to me about to the storyline. Usually students have a general idea of what their story might be and this is a fun way to access the math. The biggest two issues with the math that students have is around the concept of eliminating opposite terms and how to deal with a variable that equals a negative value. When a variable results in a negative value, you can have fun and ask them how they could change their scenario so that the variable would be positive and ask them what a negative value might mean in the context of their problem. Students usually also reveal misconception through the process of creating a story. A struggling student will create a story with too many variables or with no connection to the type of linear system needed to solve these problems. Read their stories to help get at the math content. If a student is done early, I like to review their work and give quick feedback about the accuracy and complexity of their work. Simple comments help them improve their work, like “did you need to use whole numbers here?” You could ask them to work a bit more, “Could you create a similar problem that doesn’t involve money?” The idea is for students to challenge their understanding of the problem by creating a problem. If the problem is set up quickly and correctly, you could also ask them to annotate their solution and explain why and how they know they are write. This will help readers understand their work.
This lesson extends over several days, but you can usually finish class by sharing a few observations that you made during the lesson. I find that divergent and creative activities will help many mathematical misconceptions emerge. Use this opportunity to share those misconceptions. Share only the equations with and mathematical observations (you will share the stories in the following lesson). This summary will look a lot like the introduction. Present a system and describe the issue you noticed (like x < 0) and solve the system with the class. Discuss why the example is interesting and then present a solution. I like to follow with a similar example (just change the coefficient values) and use their work as an exit ticket. This will give you a snapshot of how well they understood the content and will help you direct instruction in the following lesson.