Fixing Errors, Translations and Dilations
Lesson 3 of 8
Objective: Students will be able to recall the basics of translations and dilations as you continue to review content from the year
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.
Now that we are on the third day of review, we need to take a moment to reflect. 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.
Question 1: Find the slope and y-intercept of the linear function below:
Even though we start class by discussing translations and dilations, we let students work through a large library of problems and always spiral back to other topics. Questions invariably come up in regard to linear functions and this one is pretty typical. Sometimes I set the wording of the question to be a bit more ambiguous. Instead of asking for the slope and y-intercept of the linear function, I might ask “what is the slope and y-intercept of this function.” Or even less clear, “what is the slope and y-intercept of this table?” I do this for classes that like to find errors in the wording of a question. State test questions often omit the key phrase “linear” and sometimes even omit the word “function.” This is often a good opportunity to discuss the critical importance of the words “linear” and “function.” Students need to know if a function isn’t linear, then the slope can change at any time and might not be consistent over any interval. It is fun to have them draw non-linear functions with the three points given in the table. They can be very creative if you give them this opportunity. Playing with the semantics of a problem can help lead to some rich conversations around mathematics. I also like to play with the word function and discuss how the answer would change based on the type of function we have and the important fact if we have a function or not. I show them that if x mapped to 2 points and we didn’t know more about the relationship, we could never answer this question. -1 could map to 3 and also 4,5, 6 and any other number. In that case how would we define any particular slope or change in y when there are many different slope in a given interval.
If you don’t have time to discuss the intricacies of slope and function, jump right to the solution of this problem. I often make sure to use multiple choice here to give all students a way to access the mathematics. You could even make the choices all have the same slope or y-intercept. This would help struggling students locate the answer that must be correct and you could still discuss how they would find the slope or y-intercept if it wasn’t listed as a multiple choice option. Students usually plug points from the table into y=mx + b format and see which point balances the equation. This problem also presents a great chance to discuss the meaning of slope. An advanced and intuitive approach is to use the idea of a slope of 3 as meaning we go up 3 values for y for every 1 value increase in x. Students can then go up from -1 on x and go up 3 on y from 4 to 7 or go down from 1 on x and go down by 3 on y from 10 to 7. As always, we try to discuss the location of all y-intercepts as (0,b) and all x-intercepts as (x,0).
How many times larger is the measurement 1.5 x 10^-10 than 2.5 x 10^-11?
I help students approach this problem by giving a simpler example, like “how many times larger is 8 than 2?” Then follow up with the question “how did you know that?” or “what operation helps you find how many times larger one number is than another?” Its important that students recognize that division will work for all types of problems, for any type of number. Typically students set up the division and use the laws of exponents to solve, but sometimes students rewrite 2.5 x 10^-11 as .25 x 10^-10 and then divide by canceling out the exponent terms altogether. I encourage teachers to show this technique even if a student doesn’t bring it up. It requires a deep intuitive understanding of numbers and scientific notation.