Monday, May 2, 2016

Day 150: Intro to Cube Root & Abs Value Inequalities

Today the estimation was the flight from Dallas, Texas to Vancouver, Canada. Most students estimated it to be more than the distance from the west coast to the east coast. In reality, it was about 400 miles less than that.

Students started on 10.1.1, about cube roots. I did a participation quiz to reinforce study team norms. It's important to remind students of this at the beginning of the week and towards the end of the year.

They started with coming up with problem situations where you'd need the surface area of a cube of side length 3 feet. Students came up with paint needed for painting a cube, wrapping paper for a present, square shaped stickers on a cube, and more.

For volume they thought of water in a fish tank, space in a suitcase, mini rubiks cube's in a giant rubik's cube, and more.

Then they solved for volume and surface area showing work. They were prompted to show how they found the volume using an exponent. When they found the surface area, they reasoned that one face (some students called it a side) was 9 because 3 times 3 is 9. There are 6 faces, so 6 times 9 is 54.

I pointed out that surface area was 3 feet by 3 feet for one face, which is 9 square feet. That multiplied by 6 is 54 square feet. Volume was different. It was 3 feet by 3 feet by 3 feet which is 27 feet cubed, or cubic feet.

I asked students to tell me the highest number you could roll on a number cube. They said 6. I said that's not a coincidence, because the smallest you can roll is 1, so the numbers count how many faces there are.

Then they were given volume and had to find the side length. They used mental math until they got to 40 cubic inches. Students reasoned it must be between 3 and 4 because 3 cubed was 27 and 4 cubed was 64. I hinted they could try pressing the MATH button in the TI calculator, which brings up the cube root option. I also showed how to do it using the iPhone built in calculator.
Here is an image I want to show at Open House this Thursday night to parents.
Participation quiz, weak expo marker.
One group thought of ice cubes that could fit in a drink. Good idea, not fleshed out fully.
In 5th period I used a lot of wait time because the same 3 people were raising their hands. It worked because it got awkward and more students raised their hands to end the silence.
In accelerated students solved an inequality and reasoned if the boundary point was part of the solution.

Then came a tougher problem that I consulted other teachers on twitter about. It was an absolute value inequality, |x-2| > 3. The first instinct was for students to write x-2 > 3 and x - 2 > -3 just like with equations. That resulted in a strange solution set, shown in the bottom right and graphed, that is incorrect.

Teachers around the country suggested I conceptually teach it to students using distance on a number line and pointed out that the Common Core standards were de-emphasizing this skill.

I started by asking what does the absolute value of -6 mean? They said it means how far -6 is from 0 on a number line, so the answer is 6. Good. I then wrote |x-4| = 6 and asked them to solve it. They told me to write the 2 equations, and told me how to solve it. I told them that that equation literally meant that "the distance between a number, x, and 4, is 6." So when you graphed x=10 and x=-2, I showed with curly brackets how -2 was 6 units from 4, and 10 was also 6 units from 4.

Then I revisited our absolute value inequality. I asked students for a sentence that represented this problem. Arthur said it was "the distance between a number x and 2 is greater than 3". So, when we looked at 2 on a number line, 5 was indeed 3 away from 2. Then you can see that that the other boundary point was -1, because -1 was 3 away from 2 also. Students then reasoned that when you do plus or minus on the right side, the positive keeps the inequality direction the same. When you do the negative expression on the right, the inequality sign direction changes.

Reasoning about absolute value inequalities.
Then students reasoned about a quadratic equation of a football's flight path. They had to figure how long the ball was above the San Francisco fog if it was at 140 feet. They reasoned the quadratic was greater than or equal to 140. Students divided all the terms by a common factor to make it simpler. Some factored, which was a bit tedious. Some used the quadratic equation. In the photo below Jaden completed the square and graphed the quadratic. Students found that the ball was above 140 between 2.5 and 3.5 seconds.

No comments:

Post a Comment