Thursday, April 21, 2016

Day 143: Dan Meyer's Taco Cart & Imaginary Numbers

I saw this awesome succulent in the sunset. Anyone know what it is called? It was rather large.
Students estimated how many shopping carts were in a long line. Students used the pillars as reference points and sectioned it off.

 This was after I went over exterior angle theorem and why an acute triangle's small and medium side lengths squared add up to greater than the longest side squared. I had students draw an equilateral or equiangular triangle. They reasoned 180/3 is 60 degrees. I then supposed the sides lengths were 3, and squaring them all results in 9, 9, and 9. 9+9=18 and 18>9 so this generalization must be true for all acute triangles. Therefore, obtuse triangles the inequality faces the opposite direction.

Then I extended an exterior angle. Students said it was obtuse, which was true, but I made sure they realized that it could be an acute angle as well. They saw the 2 angles 60 degrees and the exterior angle as a linear pair or supplementary angles. 180-60 resulted in 120. When you add the 2 remote interior angles, 60 and 60, you get 120, the measure of the exterior angle.

Then students wrangled with Dan Meyer's Taco Cart. For each class I recorded their Act 1 questions, notice, and wonderings. Below that I had them focus on who would arrive to the taco cart first. I modified the question in later periods to who would arrive first and by how much time? Here they are:
2nd period
Act 1:

They both want to go to the same place (taco cart), but want to take separate ways. - Lucas

What are we trying to figure out?

The routes they went formed a triangle.

Which route is faster to get to the taco cart?

There is concrete and sand pathways. Sand is hard to walk in. So it’s slower.

I think the other guy’s route sand and then the concrete is longer.

I think it will even out. Sand will go slower. Other guy is on sand, then going faster on the concrete.

Act 2:

How far is it from the taco cart to where they are (in the sand)?

How far is where they are from the concrete?

How far is the taco cart from the concrete?

How fast do they go in the sand? How fast on the concrete?

3rd period

Act 1:

Are they going at the same pace? - Heidi (no)
I notice they made a triangle. - Amir
What is the distance in the sand from where they are to the street and the sidewalk to the taco cart?

What is distance from them to the taco cart? - Mikaela

The person in the sand was going slower. Concrete was normal. - Makenna
When they showed the dotted lines it formed a slope. - Kyle
Did they get there at the same time? (NO) {I don't know why I answered that question}
DId they start at the same time? (yes)

Who got to the taco cart first?

Act 2: What do you want to know?
How long did it take them?

What is the speed on sand?
What is their speed on concrete?
The distance each person went in the sand, and in the sand and concrete?

4th period:
Act 1:

Is it a right triangle? (Yes) - Roxanna
How slowly do you walk in the sand compared to the street? - Louis
I noticed there were 2 plus signs in the sand. - Bing
How far is the taco cart from the start position of Ben and Dan? - Andrew
Are they walking at the same pace? - Natalie (no)
Will weather slow them down? - Jonathan (no)
Can ben and dan run? No, they are walking. (Sami)
What is the pace?
Do they both have the same leg strides ? -Jonas (let’s assume yes)
Which one is the faster pace?
Dan is going to the sidewalk. Then on the concrete forming a right angle.
Ben is going diagonally.
Who gets to the taco cart first?

Act 2:

How fast are each of them going?
What is the distance between the starting point and the taco cart?
What is the distance between the start point and the concrete and the concrete sidewalk and the taco cart?
How much faster do you walk on concrete than on sand?

Period 5:
Act 1:
What is the speed when you walk on the sand and the speed when you walk on the sidewalk? -Eyad
What is the distance from where they are standing to the taco cart? - Ethan
Who got there first? - Joey
How far is it from the sand to the sidewalk? Gracyn
What’s so good about the tacos?
Is it a race of who gets there first?
Are they walking at the same speed?
Is there anything delaying them?

Act 2:
What is the distance from where they are standing and the taco cart?
What is the distance from the sand to the sidewalk?
How fast are they walking?
What is the distance from the taco cart to where he gets off the sand and walks on the concrete?

What is the speed when you walk on the sand and the speed when you walk on the sidewalk?

Some students looked up the pythagorean theorem to use. Some thought to add up the sides and divide by 2. Most solved Dan's route first. The key there was finding the time using the distance and rate using separate rates. More students last year thought the diagonal was the 2 sides added together. This year very few thought that. I did suggest some to take out their data sheets where they investigated acute, obtuse, and right triangles.

Some realized the side lengths needed to be squared to find the area of the squares that form the side lengths. Once they squared both distances and added them together, they reasoned they had to square root that sum to get the 3rd side, or hypotenuse.

Like the 5 Practices book suggest, I tried to get a student that hadn't figured out the diagonal to present how they found Dan's route using their knowledge of rates. Then I had another student present about how they found the diagonal's length and divide it by the rate they walk in the sand.

I think that Estimation 180 helped students be more successful with interpreting the seconds, but students still sometimes interpret decimals as portions of minutes. When I encouraged long division to double check they realized the remainder was the number of seconds.

In this class I had time to show how the diagram related to the pythagorean theorem on the right.
Here you can see the quotient 4.58 minutes. Some students mistook this as 4 minutes 58 seconds. Also, I remember telling students before this, if I go for a walk do I say it took me 275 seconds?
The larger numbers made some of the students' eyes get a little wider.
In accelerated I reviewed the part c of the problem. Students were having trouble simplifying the square root of -81. I modeled that =81=81 *-1 so I can substitute that. Then I can square root 81, followed by square rooting -1, and replacing that with i. Therefore, the answer is plus or minus 9i.

I broke it down into building blocks with the square root of 4 times itself. I simplified each factor to 2, multiplied 2 by 2 to get 4. I reasoned that I also could have multiplied the radicands to get a product of 16, and the square root of 16, is also 4. I don't recall the textbook or myself discussing radicals and simplifying them. (note to self: include that in my paternity sub plans).

Unfortunately I forgot to take a picture of 2 different students who solved quadratics and used "i" in their answers.

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