Tuesday, December 13, 2016

Order of Ops WODB, 5 Practices with 3 Act

I had a substitute give all my students the following Which One Doesn't Belong prompt on a quarter sheet of paper to all my students as a warm-up. I saw some of my 8th graders got some wrong, so I assumed my 6th graders struggled as well. So, I passed out some brand new copies of it. They spent a lot of time on it.

As you can see below, most students' first instinct is to simplify it all from left to right. In the bottom right they really want to subtract 6 to 20 rather than subtract 6 from 38.
We discussed some of the mistakes. But before doing so, for one of them there were almost 5 different answers. I made the analogy that if people in New York get one answer and people here in California get a different answer, we have to agree on a method so we are not getting different answers for the same problem. This is why mathematicians came up with the order of operations. We also confronted the myth that you add before subtracting because "A" comes before "S" in PEMDAS.

We then dove into Graham Fletcher's 3-act task "Gassed." To be honest I did not prepare for alternative incorrect solution methods I did reflect and know that if a student got an answer by adding or multiplying incorrectly I could ask for an estimate or how much 2 gallons of gas would cost, or 3? When I asked them if it seemed reasonable, they thought so. Therefore, I have to have a backup question to that initial one.

After watching Act 1 of the video, students wondered how much money was given to the gas station. I told them that I never go inside, I just swipe my card because I want to fill it up all the way.

Part of a 3 act is then asking students what they want to know to answer the initial Act 1 question. They came up with asking how many gallons could the car hold and how much does it cost per gallon. I showed them these clues (9.52 gallons and $2.09 respectively).

Many students just added the numbers together. Others unsuccessfully got past multiplying one factor by the hundredths place (not understanding the placement of the factor multiplying by the tenths place).

This student for some reason thought that the $0.23 shown in Act 1 had something to do with the answer.

This is the student that was able to estimate the answer was near $20. As you can see above, they did not place their 3rd row of multiplication correctly lined up below the ones place.

Once again, struggling with 3 digit by 3 digit multiplication.
This student ended up lining it up correctly, as you can see with the zeros with the line through them representing the placeholder zeroes. They made one small calculation error for 9*5+1, coming up with 43 there. This affected the rounding.
This person did not make any calculation error here.

The best part of the conversation was when they lined up the decimal in their product with the decimals in the factors. Then they said, 1,989.58?? That is WAYY too much money. So, instead of telling them a rule, I asked them where it made SENSE to place the decimal. Then they quickly said after the 19. Unfortunately, we didn't discuss rounding to the nearest penny because we were looking at a students work that miscalculated it slightly. They did come up that their answer was off by a penny.

Students also loved watching the money display increase quickly. This relatively simple problem provided a rich discussion in our 6th grade math support class. Thanks Graham!


  1. Hey Martin!

    I really like this write-up! I'll keep it in my back pocket and may try to use it in an upcoming lesson inquiry. Thanks for sharing.

    Some things I'm wondering...

    Did you see any students try repeat addition? Or use visual models to try and calculate the sum? Seems like they all reached for the algorithm, and I wonder what that means about their reasoning.

    Tracy Zager gave a great talk at CMC North about intuition. She noticed that 3-Acts were good at leveraging student intuition for learning in Act 1 and a little in Act 3, but that we (teachers) weren't using it enough in Act 2. But it seems like this lesson is primed for it as your students were able to identify errors and/or unreasonable answers using intuition ("that's way too much!"). She may dig reading this write-up too and have some insights.

    Thanks again for sharing!

  2. I totally agree. I hope to see students see ways of making link from repeated addition to multiplication or some sort of groupings. The level of engagement makes these type of deep conversations possible because students are motivated to know what works. Also, I didn't really mention it in the post but part of what incorporated was the sequencing of student attempts and solution methods from incorrect to correct to most efficient, etc. I definitely highlighted the estimation with rounding before the algorithm used for the exact answer.

    This lesson would have been even better had I anticipated ALL the types of solution methods and incorrect methods that I could prepare pocket questions for to move students in a new direction rather than stay stuck. Thanks for the comment and compliment Chase.