Traditionally, order of operations is taught by telling students the rules and having them practice following them. This is not enjoyable for me or for them. It has been hard to find some way of involving students in figuring something out when order of operations is really more of a conventional notation system than a concept. I really want to engage students in figuring something out as much as possible.
Starting with an equation, in which the correct answer is given, and asking students to figure out what the rules must be makes them more actively involved, allowing them to discover the rules for themselves. While there may not be a lot to understand here, I think that just shifting the approach helps students remember the convention. Also, I think that taking the pressure off of getting the right answer and giving them a mystery to solve puts students in a better mindset for learning.
The structure of this lesson is really accessible to ELL students, because the written rules are created by them and so, are not subject to misinterpretation.
We pick up where we left off in the last lesson (Writing the Rules Part 1) in which we had been writing the rules for the order of operation. We begin with the rule as we left it, which is partially complete. So far we have come up with:
Do operations inside parentheses first, then do multiplication and division, then do addition and subtraction.
Today, we need to explore where exponents fit in to the rules as well as the relationship between addition/subtraction and multiplication/division. I remind students they will use the following pairs of equations to modify their rule as needed, just like they did in the last lesson.
Today I will give two equations at a time. So, as in the last lesson, students have to figure out what order the math has to be done to get the correct answer. I put the following pairs of equations up, one pair at a time.
18/3^2=2 with 5x2^3=40; 15-10+5=10 with 7+3-5=5; 15/5x3=9 with 5x3/15=1
The first set I expect students to add "do exponents before multiplication and division." Since the last set of rules was learned mostly in order, students may need prompting by asking them where does the new information fit in to our rule. The last two sets of equations should lead them to remember the left to right rule.
Distribute a set of expression cards to each math family (groups of 3 or 4). Have them separate the expressions into categories by which operation is done first. I give give them an example by holding up two expression cards and tell them if both expressions require addition to be done first then they belong in the same category, but if one expression card requires addition first and the other requires division first then they don't belong in the same category. I leave the rule they came up with from the warm up displayed on the screen for them to refer to.
There are really two reasons I have students take the time to do this before practicing solving the expression, maybe even three. One reason is to get students used to the expectation and process of arguing about the math and making decisions cooperatively (MP3). Another is that the main mistake my students seem to make with order of operations is just jumping straight in before taking the time to decide where they are supposed to jump in first. One last reason is also that I want to separate their calculation errors from the mistakes in operation order.
Students will often have questions about the various symbols in the expressions including variables. The different ways of indicating multiplication as well as expressions with only variables has typically been a source of confusion for my students in the past. Common questions that come up every year are: "what does it mean when a letter is next to a letter" or "a number is next to a letter" or "when a number is next to a parentheses". There may also be questions about whether the dot is a decimal point or means multiplication. I want as many of these questions to be answered within their groups.
One way to scaffold this activity is to modify the sorting cards so they have solutions. This way students can "rediscover" the rules as they work.
Students each work on personal white boards during this segment in the class. As they begin, the expressions are still sorted into categories in each group. We will explore the categories one at a time. I will give the class a category and ask each group to agree on one expression to solve on white boards.
Within each group, students are allowed to check and correct each other before they show their boards. I have students raise their boards at the same time, so it is easy for me to give quick feedback. Since groups may be solving different expressions, I have done all the problems for each section I call out ahead of time, so I can easily assess and give feedback. I also figure out what the answers would be if they make some common errors, so that I can give the most specific and corrective feedback possible. I can't really avoid giving the "nice job" feedback from time to time, but I like to give more specific feedback whether it is corrective or not. I may say that I like that they showed me some of their thinking on their boards or I might need to remind them of a rule or what exponents represent, etc.
If I call "subtraction first" I know that they are choosing between a multistep problem [4^2+8(6-3)], a left to right rule problem [160-20+8], or a problem that involves an exponent with a large base[(30-1)^2]. I know they should have the answer of 40, 148, or 841. If they don't then I would expect to see an answer that results from a common mistake. I know if they got 32 that they did [4^2+8(6-3)], but multiplied the base with the exponent, if they got 72 I know that they did the power correctly, but they added before multiplying, If they get 48 they made both mistakes and I can give them specific corrective feedback so they can fix it. If they get 132 I know they did [160-20+8], but they added before subtracting and forgot the left to right rule. If they got 58 they did [(30-1)^2], but multiplied by the exponent and if they got 741 or 941 they made a carrying mistake and if they got 261 they only multiplied by the ones digit in 29. I work with them in later lessons on mental math to develop flexibility and numeracy and to avoid mistakes when operations are digitized (beginning in "Let's talk addition" lesson later in this unit. I always keep my notes with me so that I can have my feedback ready. If students have an incorrect answer that I didn't predict I will often take their board and hold it up after giving feedback to everyone else. I don't want the student to feel ridiculed for a wrong answer, so I won't do it this way if no one else made any mistakes. When I hold it up I will tell the class that he/she stumped the teacher and I need help figuring this one out. I tell the student who made the mistake not to tell anyone what he/she did and that sort of makes them feel better that he/she knows something the rest of us don't. It also inspires some of the others to try to figure it out. It usually ends up being a calculation error, so we can determine that the student did the order or some of the order correctly.
Once students have done 3-4 practice problems on white boards I have them continue working on the 4 fours problem (which they started last night) together and take it home for one more night.