The goal is to circulate and see how students are doing with the problems from the diagnostic exam. It is important to set specific goals here. I tell students to have a certain chunk of problems done by the end of class. For example, if most students have had a few days to work on their workbook, I ask them to have the first 40 problems done by the following day. They know that if they are unable to finish the class work, then they have homework. I circulate and have students read answers to the questions they have done so far. If I find an error, I have students at the table explain how to fix the work. I give them markers and a clean sheet of paper to teach from. I watch the mini lesson and then ask students to consider presenting on that question.
Partners continue to check the what, how and why of their algorithms. 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 the summary by reminding them of the goal of these review sessions. Students need to know that this work should be very easy and that it is important to be honest if something is not clicking. We hand out post-it notes and ask students to write their name and any question numbers that still aren’t clicking for them. Students hand these to me after the summary discussion as a sort of exit question. I like to post these questions on our question board and respond through video or by meeting with the student before or after school. I often don’t answer these questions in class as we move forward and try more examples. I am also able to frequently have students answer questions from the question board. For example, if a student is done early they know to walk to the question board, grab a question and write a response. Here are some typical questions that come up when we are this far into review.
Which of the following expressions does not equal 1/16?
4^-9 x 4^7
4^11 x 4^-13
4^0 x 4^0 x 4^0 x 4^2
4^-2 x 4^-2
I often highlight laws of exponent problems as they allow students to proudly demonstrate how simple this type of problem really is. I also like to have them follow up by explaining I often highlight laws of exponent problems as they allow students to proudly demonstrate how simple this type of problem really is. I also like to have them follow up by explaining why the laws of exponents apply in the way they do. With a problem like this students will describe why it was helpful to first identify that 4^-2 is 16 and then test each case until they find an expression that is equivalent to 4^-2
If a student states, “I added the exponents,” I ask them why they are able to add.
If a student identifies 4^-2 as 1/16, I ask them why they know this. If they state that it is the reciprocal of 4^2, I ask them to use the concepts of repeated multiplication and repeating division to explain why this happens.
If 3x + 12 and 6x + 24 have infinitely many solutions and a = 3x + 12 and b = 6x + 24, what is the relationship between a and b?
I like to bring up poorly worded and mathematically deficient questions in our discussions. This question had an answer of a = 2b, but is clearly written incorrectly. Help students describe why this question is incorrect and discuss how to fix the wording. This question should read, “If a = 3x + 12 and b = 6x + 24, what values of a and b make the expressions 3x + 12 and 6x + 24 have infinitely many solutions?” Students enjoy naming why a question has no correct answer and are quick to correct the mistakes of test makers. This question will give you a chance to mention the transitive property and the critical concepts and meanings of linear systems having one, none or infinite solutions in both algebra and on a graph. Its important to name zero solutions as have functions that never meet, one solution as one crossing point and infinite as being collinear and show this on a graph. Students also know that if the two expressions start with infinitely many solutions that they are equal and thus a should also equal b. Of course the problem here is that these two expressions are not equal.