Day 48: Dan Meyer's Taco Cart

Regular Math Objective: Apply the Pythagorean theorem in a real-world context

Regular Math Standards: 8.G. 6a. Understand the relationship among the sides of a right triangle. b. Analyze and justify the Pythagorean Theorem and its converse using pictures, diagrams, narratives, or models.

8.EE.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Regular Math Lesson Sequence: Today's (Wednesday) lesson came off the heals of a professional development we had in the district all day Tuesday so the kids had the day off. We had to pick up from where we left off on Monday, which unfortunately was an awkward place. These five things had happened Monday.

1. I had shown the classes Act I of Dan Meyer's Taco Cart. 
2. In partners, students provided the question that we needed to consider. Who will arrive to the taco cart first - Ben or Dan? 
3. They made a prediction to who was going to be first to the Taco Cart.
4. I asked them and their partner to determine what information would be needed and they gave me the bits that made solving the problem a possibility. 
5. I revealed the information about the speed on sand versus the speed on land as well as the distance of the legs of the triangle courtesy of the link above. 

Almost 48 hours later I knew that to ask students to recall one of those five things let alone all of them was a tall order. So we did a quick recall of all five things which helped get the absent kids up to speed and also set the stage for the rest of the class. From that point, I spent the entire lesson getting the students to answer four questions. 

What is the distance Dan will walk? 

I thought for sure this was a gimme, but it wasn't. What was very revealing was that students who drew a diagram did much better than those who did not. Some students subtracted for some reason that I do not know. The most common error though was that they wrote down the Pythagorean Formula. That's what their instinct is. And to be honest I cannot blame them or even scold them. If they get just one picture of a triangle on a standardized test this year, I would put my mortgage on Señor Pythagoras being called upon. This is exactly why I made this a question within the context of this lesson though. I wanted to see students think about this. Per usual I made every effort not to tell students what to do. Unlike other days though where I hurriedly send them in the right direction, today I was determined to take my time with just these four questions so the students were able to uncover their own misconception. Onto Question 2:

How much time will Dan take to get to the taco cart? You can use a calculator if you want but show what numbers are being calculated. 

At the professional development yesterday we spent about an hour discussing writing in math, which morphed into students explaining their work in some way. Everyone in the room was familiar with the same reactions.

T: What do you think you'll do? 

S: Multiply. Teacher gives confused look. Divide. Add? 

And with this "guess to get by" line of answering in mind, I was asking students that answered this question about Dan's time incorrectly and correctly the same questions. Why did you do that? About literally every number and operation in the problem. Some students were dividing the total distance that Dan walked by five. They were ignoring that he is slower on the sand. Some students had the correct steps on their paper, but they got that information from their partner and had not bothered to inquire why the numbers were the way that they were. Some students had actually done out the correct steps on their own, but had a much harder time articulating it. Other students were just plain old stuck. For these students I reminded them of George Polya's problem solving strategies. We broke down what was already known and what we were trying to find. Then we discussed strategies. We ha already drawn a picture. Had we tried a chart? Had we tried a similar number? And next thing you know we were trying both as students were determining how much time moving two feet, four feet, and six feet would take in the sand. From there we could attack the more intimidating numbers. 

My favorite part though was the unintended consequence of the math in this problem that seemed to happen once per class today. Students were successful in everything they did until it came to converting seconds to minutes.

Again, we made a chart. We looked at how 240 seconds was 4 minutes and 300 seconds was 5 minutes. How was 275 seconds so close to 5 minutes? We also discussed what 4.5 minutes was a fraction. And finally I had to show them a proportion for a cherry on top of what the error was. Deep Breath. Question 3:

What was the distance that Ben traveled?

They did really well with this. Again they were dying to use a-squared plus b-squared equals c-squared from the moment they saw the right triangle. It's like running into Johnny Drama from Entrourage in real-life and restraining yourself from yelling 'Victory!' True story that really happened to me, but I didn't restrain myself. 

The beautiful unintended consequence was that they had no clue where to find the square root button on some of the calculators. Standard for Math Practice 5 - learning to use appropriate tools strategically. Check. On to Question 7:

How much time will it take Ben to arrive at the taco cart? 

By this time, they were ready for this question. It was easier than the question they had done with Dan's time because Ben was on the sand the whole time. The issue for many students though was that the difficulty of Dan's time had caused them to avoid this. At this point in the lesson going from each partnership and really digging deep on the questions I asked was taking it's toll on classroom management. I kind of let things go today. I did not want to stop the class for risk of revealing information that they discovered. The lesson itself was extremely engaging and some kids took off and ran. The ones that did not would have struggled regardless of what I did, so for today I just kind of let them drift a little until I had the chance to intervene. 

Overall this lesson was well worth the time - even though I had already quizzed the students. Many of them were excited to be able to solve a real-world problem with something as complicated as what we had done with a fancy formula. 

Honors Math Objective: Identify if ordered pairs, input out tables, and graphs are functions; use function notation to solve problems in a mathematical and real-world context; give the domain and range of discrete and continuous functions

Honors Math Standards: A1.F-IF

1. Understand that a function from one set (called the domain) to another set (called the range) assigns to
   each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output (range) of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
   
2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 

Honors Math Lesson Sequence: The entire lesson was the study guide. Students had the biggest issue with defining the range for a problem in which an equation and the domain were given.

Quote of the Day: “Which relationship is most strained in your life right now? What would it look like if you began focusing on that person’s best moments and sought to affirm them?” – Dale Carnegie