Expressions Evaluated

As students enter the classroom today, I project the Launch questions Each of the questions in the launch stress the need to agree upon the rules for how to evaluate expressions. I have chosen them based on some patterns that I see in mistakes that my students make. 

Question 1: I expect that many of my students will get -97 for Question 1. When we discuss this problem, I will help the class to perform the operations in the correct order and get 99. I will then ask students for some suggestions about how to make sure that another student will perform the operations in the correct order. We'll eventually come around to using parentheses to make sure that we perform the subtraction  in the correct order.

Question 2: This task demonstrates how written expressions can be ambiguous. There is no way to know for sure whether you should divide or multiply first. So I tell students that it's quite normal to feel confused here. Sixty divided by two times three, written in one sentence would be 60 ÷ 2 x 3 and the answer should be 90, because we proceed from left to right. But, who is to really know what was meant?  

Question 3: I anticipate that some students may write 3 as an answer. I remind students that 8x means 8 times x, unless otherwise indicated. 

When we begin working on new content today, I will pair up students. I'll give each pair a copy of New Info Sheet Evaluating Expressions. Evaluating expressions should be review for students. (In my school there will be some exceptions.) I find that it tends to be quite easy for students approaching this for the first time.

I will introduce the three questions without telling students the mathematical property involved. I want them to figure it out arithmetically, then add their own examples showing that these really work with numbers. As students are working, I walk around listening to their conversations guiding their thinking with questions without giving answers away.

After students are done with Part 1, I will ask students to write the name of each property next to its general (variable) representation. I never need to actually tell the class which name goes where. I can count on my students to call these out. 

Before moving on to the Application section of the lesson, I will work with my students to name the property shown in Options iii and iv of Part 2. I will begin by asking volunteers to go up to the board and write a numerical expression demonstrating each of these properties.  

As I distribute the Application worksheet for today's lesson, I will ask my students to try to work as independently as possible.

One of the things I  try to establish early in the year is a culture of "try your best" and don't give up easily. I want students to gradually see that perseverance is essential to improvement (MP1). Effective discussion and learning through collaboration results from students sharing their own ideas. Obtaining answers from other students without reflecting on the differences between their classmates' thinking and their own is unproductive. These ideas are not always easy for students to learn, so it's really my job to continually enforce them. It is very difficult to get students off the "grade" mindset and into the "learning and improving" (growth) mindset. But, I find that it can be done through formative assessment and feedback. By helping students know where they need to improve and then helping them in these areas. 

It is also helpful to be aware of potential trouble spots in advance. Here are a few that I expect to arise during today's lesson: 

• Question 3: I make sure that students see the fraction bar as representing two sets of implicit parentheses (in the numerator and in the denominator). Students are then better able to follow the order of operations. 
• Questions 6: One of my goals of this question is to help students see that properties of real numbers (Associative Properties for Addition and for Multiplication in this case) work in algebra as well. Students may not be willing to take this problem on. I am ready to encourage them to take a risk and use what they know. 
• Question 7: I find it important to tell students that the Transitive Property is both important and common. After answering the first part of the Launch section, I expect students to be successful with both Part a and Part b, but I am looking out for unusual responses. 
• Question 8: I include Question 8 because it targets a theme that I push in my class: putting parenthesis in place correctly when using calculators. I observe carefully to make sure that my students recognize that parenthesis must be used to group 3p - q, before dividing by 2p.