I ask students to take about 2 minutes to do a Quiet Write about what they may have struggled with in their first math homework assignment last night:
Can you make all the numbers 1-20 using exactly 4 fours?
I ask them to write about any surprises that they didn't expect.
I expect many of my students will be concerned that they didn't complete the task, even though I made clear that it was not expected. Some may have struggled because they did not entirely make sense of the task before jumping into it. Another source of frustration results from trying to work in order, since some are more difficult than others, this is not always the best strategy. One of the surprises that usually comes up is the satisfaction of finding some of the numbers by accident, just by playing around with the numbers. Of course, this is exactly what I wanted them to do.
After they write, I have them discuss their responses with their Math Family Group and share some of their solutions. At this point I expect some will realize they did it wrong and used numbers other than 4 or used more or less than four of them. Before moving on to the next exploration I tell them it may be easier to work on this activity after we have spent some time reviewing order of operations. I point out that some of the methods of Making 8 (sharing multiple methods in the "First days" unit) that we found earlier in the week, violate the correct order reminding them that these are like the rules of the road in that I am always expected to follow traffic rules every time I drive.
In this lesson I don't want to just remind my students of what they learned about order of operations last year. I don't want to just review the rules to follow. I want students to think about the order.
In my classes, many of my students begin the year simplifying expressions by beginning at the left side of an expression and continuing to the right until they run out of terms and operators. I want them to develop the habit of having a "look around". I want them to suspend the tendency to go left to right, as if reading rotely.
I will do half of this lesson today and other half tomorrow. I start by showing the class an equation on the screen and telling them that the calculation is performed correctly:
2 + 3 x 2 = 8
I ask them first to identify the operations they see in the equation and then ask them to figure out in what order we have to do the math to make the expression on the left side equal 8. It will not be long until they tell me, "We have to multiply first, then add." I will follow their suggested order to confirm their idea. Then, I will try it the other way, adding first and multiplying next:
2 + 3 x 2 = 10
We'll then discuss why this is incorrect, beyond simply stating that it does not follow the order of operations.
Following this, I ask, "What can we say about the correct order of operations from this equation?" When students say, "We multiply before we add" I will write it down. Oftentimes, a student may try to state all the rules for order of operations. Here, I remind them that we can't observe all of those from this equation. I want to model making a good mathematical argument (MP3).
I'll now put up another equation and tell the class we may or may not have to modify our list of rules to include more information.
5 x 2+3 = 13
I will continue the string of equations until we get to a partial rule of "do operations inside parentheses before multiplication & division and multiplication & division before addition & subtraction.
15 - 5 x 2 = 5
3 - 2/2 = 2
6 + 9/3 = 9
2(2+2) = 8
At each new equation students discuss with each other how the math must be done to get the correct answer (MP3). As I circulate around the room, I encourage them to test by trying the math in a left-to-right order. This usually takes a while because students are not used to doing this type of activity, or, in some cases thinking that it is important to try different ways.
After milking the string sufficiently, I pull out some giant flash cards made of recycled manila folders on which I have written some expressions. I display one for the class, do a 3-finger silent count down, then have them call out in unison which operation must be done first. If they are in unison I can hear it if some call the wrong one and go back to it. We can usually get through about 10 examples really quickly using this strategy.
Following our discussion, the remainder of class is spent working on the Four 4s assignment that was sent home last night. I give them a chart with a box for each number 1-20 and tell them to fill in a box with whatever solutions they may have found already.
I remind them about "playing around" with the fours to see what they come up with (MP1). During this time I am visiting each of the groups and checking to see that students understand the task. I expect at least one person from each group will ask if one of their solutions is correct (if not I will ask to take a look at what they have found so far). Before I respond, I will draw the attention of all the group members to take a look (MP3).
I first ask where we need to start in their expression and ask if they are using the correct order of operations and if they are using exactly four 4s. Once I have visited each group I ask if anyone has been able to make one of the solutions I haven't been able to solve yet. Many of them probably forgot about that and they get excited again. If they haven't done so already they start paying more attention to what their group members are doing and double checking. I love it when students shout out very excited that one of their partners got it!
I give students a choice as to whether they want to leave their work with me until the next day or if they want to take it home and work on it with their families. Most take it.