This is a culminating activity in which "math family" groups compile the work they have done on the 4 fours problem. It is one of their early opportunities to engage in some simple argumentation and collaborative work. Students have been working with order of operations for a few days at this point and are beginning to critique each other's work.
The warm up in this lesson purposely surfaces some common mistakes as a way of initiating disagreement and argumentation. Drawing attention to mistakes is really important for a couple of reasons. I want mistakes to be seen as normal and as an important part of learning. I would rather surface those mistakes than try to avoid them.
Students begin work on the following problems that are projected on the screen as they enter.
1. Does 4 + 4 x 4 + 4 equal 36?
2. Does 4 x 4 - 4 / 4 = 3?
3. Does 4 ^ (4 + 4) / 4 = 16?
The first two expressions were student created from the assignment (4 fours from the "Sharing multiple methods" lesson in the First Days unit) they have been working on for a few days that they are finishing up today. About midway through the warm up I bring their attention to number 3, which I know has been the source of some confusion and, hopefully curiosity. Many of them have been trying to use exponents in their 4 fours assignment, but not an exponent in the form of an expression, so this is the first they've seen. I ask if they notice that the (4+4)/4 is written smaller than the first 4 in the front and a little bit higher up on the page. This generates a bunch of ah-has "ooh, ooh, that's an exponent!" and they dive back in to work out the problem. Some of them are so excited to notice this they are acting the expert and sharing their ideas with the group and getting them involved in the problem. I love the team building!
When we go over the warm up I first poll the class to see if they think each problem is true or false, they walk us through the solution. After each of the first two, which are both incorrect, I ask them to imagine what someone might have done in order to arrive at the given wrong answer. I am hoping that someone who DID arrive at that answer will speak up as the expert who knows what "someone" might have done.
In the back of the room I have a large chart paper that is similar to the homework chart they have been working on. Students all have their four fours homework out in which they were asked if they could make all the numbers 1-20 using exactly 4 fours.(from "Sharing multiple methods" lesson in First Days unit) They have had several days to work on it at home as well as a little class-time and were encouraged to get ideas from family members of friends. Some of them worked on it over a weekend. They were told ahead of time that they were not expected to be able to complete all of the numbers as there were a few numbers I was unable to come up with myself. Happily, some students take this as a challenge!
Their task today is to check the expressions that they and their "math family" group came up with. When they have agreed that an expression is correct they fill out the team chart, which I make on a different color paper - one for each math family team, and send someone in their group to write it on the class chart in the back of the room while other members continue checking another expression.
I really want groups to keep trying to write expressions for the numbers that their team did not come up with yet. To encourage this I go to the large chart on the back wall that students have been filling in and point out to the class that some of the numbers are still blank meaning we haven't come up with an expression to make that number yet. I try to challenge them by saying things like: "Wow, I wonder who will find one!"
Each team has a different color marker and periodically I will go check for mistakes on the large chart and go get that team to figure it out. Students from other teams also notice mistakes and bring them to my attention.
Students take out their white boards to do the following problems:
I do the last two because students often put in unnecessary parentheses.
When I do white boards I have them all raise their boards at the same time. This way I don't miss someone and it takes less time. What that means though is that some students finish earlier than others. Later in the year I will put up "stretch" problems for them, but this early in the year I want them to get into the practice of helping each other. While I go around I look for boards that have been turned over and I tell students they don't have to hide what they did from their group. They instead should be looking to see if they all agree or if they or someone needs help.
Students are allowed to begin their homework in class as soon as they get it. I ask the class why they think I did not tell them in the directions to use the correct order of operations. If they don't offer "because we are supposed to use the correct order all the time" I ask when they are expected to use the correct order. Some will be confused by the word evaluate and need to be reminded that it means to find the value or the answer.