The goal here is to begin review content that spans the entire year. To help students, we start class by collecting old textbooks and materials. We ask them to clear out their binders and make room to focus on their upcoming task. We discuss the types of materials they would keep in their binders, like study sheets and summaries of their notes, and we list out materials to submit for collection, like old projects. We also set the atmosphere of transition of shifting from a typical lesson model to a review model by handing them a workbook full of sample problems. We order these from a variety of companies and make sure each student has one. If your school is short on funds you can print out review pages that are inserted into the binder to create their own review workbook. While students are clearing out their binders and looking and working on their review workbooks, we hand back exam booklets from our diagnostic. We end the starting session by reviewing the goals for the day. I often have a student read the start up within the first few minutes of class and then have another student later read the prompt that outlines the goals for class.
Sample start up: Good Morning Wonderful students! Today we begin to review for the year! Lets start off by clearing the clutter out of our binders and backpacks. Get rid of unnecessary scraps and forms. Donate any math work that you aren’t keeping. Save review notes and study sheets that you created throughout the year. Also return your old texts and get a new one. Write your name on the cover and start to work through the problems.
Sample prompt: Today you and your partner are going to teach each other about your ideas and algorithms from the diagnostic quiz. Be sure to discuss what you got for an answer, how you solved the problem and why you believe your work is correct. Avoid the temptation to simply compare answers. Your goal is to compare algorithms!
It is also important to mention that if they finish review algorithms early, they can work on their workbook that reviews problems from the year.
The goal is to circulate and see how students are doing with the problems. I often join student conversations and highlight different algorithms. Many students are hesitant to explain and truly listen to another algorithm. If a partnership is struggling, I jump in.
“I noticed that you both got the same answer.”
“Did you notice that you both have two great and completely different strategies?”
“I didn’t notice that.”
“Could I listen to you explain your work?”
Then I jump in and often ask the other student to rephrase or summarize parts of the algorithm that they just listened to. If they struggle, I will ask specific questions, like “how did she get the number 5 in her first equation?” My goal is to gather stories of learning that can be shared later in the summary. Even if I am unable to reach all students, I will have some powerful stories and algorithms that can be shared with the group. These stories resonate with others because students know they are authentic and realize that they too have similar algorithms.
I start by asking if there are any pressing questions that I missed. After listening to the question, I decide to take it on with the whole class (if it is relevant for all students and easy to share with whatever resource I am doing) or I deflect and explain that the goal is to reach that question later in class or in a future lesson (with the option of meeting at lunch of afterschool or class for immediate feedback). I like to pick the first three or so questions in the diagnostic and then proceed with other questions throughout the review lesson series. The goal is to constantly stress powerful algorithms and share the value of the discussions between partners. I often start a question by having a student read the question as I or another student project or redraw the problem for all to see. Here is one example:
“Question 1: A coffee shop buys from a distributor at a wholesale price and then sells the coffee for twice the wholesale price. This graph shows the relationship:
The coffee shop determines that they must increase the selling price to three times the wholesale price. Which graph represents this new price?”
You want to have some students help explain the meaning of wholesale and selling price with some type of example. I gave this question as multiple choice, but it might be even more powerful to omit the multiple choice. If you decide to use the multiple choice format, here are some basic guidelines that will help your class discussion:
1. Include a choice in which the slope is less then the given graph, preferably something like y = x. This allows you to ask a question like “is the graph of this business model profitable?” Students should recognize that the answer is no since they are breaking even and selling an item for the price at which they purchased the item (when y = x)
2. Include a choice with a slope and a higher y intercept and another with the same slope but a lower y-intercept. This will allow you to question students on the meaning of these graphs. Specifically, if you use y = 2x + 1, you could ask, “I noticed that the selling price is $3 when they wholesale price is $1. Does this graph represent the answer?” Students will realize that the selling price is only triple the wholesale price at this point and that has a different relationship elsewhere. You then have options to question about proportional vs. linear relationships, about the equation of the line in standard form or y = mx + b format and opportunities to question the meaning of the y-intercept at +1 and at -1 (if another choice is y = 2x – 1). Students are quick to recognize the meaning of +1 as “starting with 1 dollar even when nothing is sold.” The meaning of -1 is tougher, but it will help students to discuss the meaning of debt and profit. You could also mention the x-intercept here (and in any example) as the point at which the company has zero profit.
When you present the answer, encourage students to explain how they could solve this visually by looking for a steeper slope and check their answer by finding the equation as y = 3x or by testing certain input/output pairs. This will give the students multiple entry points and allow you to spiral back and discuss linear functions and functions in general.
I like to ask students if they found the value of w in a way that felt manageable. Students are often excited to share that they could solve this in two steps by adding 2 and then multiplying both sides by 5/3. I would ask students if they needed to add 2 first and show that you could multiply by the reciprocal and then add 10/3 to both sides. You could ask students other pointed questions about other algorithms, like “what would happen if I multiplied both sides by 20?” Students are quick to notice that this “gets rid of the fractions.” Ask them to explain why 20 works and follow up with “is there a better choice?” The goal is for students to realize that you could multiply by 10, the LCM of 5 and 2. You might even mention this as a general strategy (if not named already).
“Question 3: Ted is thinking of some number. He knows that if he multiplies his number by 3/8 and subtracts 2, he will get the same answer he would get if he multiplied the number by 5/8 and added 6. Is he correct?”
This is a great question. At first glass it might seem to be routine, but it highlights the power of intuitive algorithms. Here are three different and useful strategies.
Algorithm 1: Plug in multiple choice responses.
Many students plug in choices to see which number “works.” I have students present how they do this and use this as a step into the algebraic thinking. Students will literally plug the number into the operations listed in the problem. When they write these out have them or some student write the algebraic equation.
Algorithm 2: Balancing the equation.
Students set up the problem in terms of x and demonstrate how to find the answer (x = -32)
Algorithm 3: Seeing the answer.
On our diagnostic test, the first choice was the answer. But more importantly, some students recognized that this equation has to have a negative solution and therefore knew that the first choice (the only negative choice) was correct. Encourage students to explain why they know this solution is negative and have as many students rephrase the logic. Offer this solution as an elegant challenge. Here are some general ways in which students can explain why x must be negative. Use these as prompts if your class is stuck.
Student explanations: They will explain that the expression on the left takes a smaller fraction of a number and then losses value. They will then explain that the expression on the right takes a larger fraction and then adds value. The only way this can happen is if you are dealing with negative numbers. You can support this with a general number line, showing that a smaller fraction is worth more value with negative numbers. Therefore it makes sense that a smaller negative minus 2 is equal to a more negative negative plus 6. Students also can explain the two expressions as linear functions that clearly meet as some negative value.