Thursday, May 21, 2015

Rugs Re-Engagement: Perimeter, triangles, and circles

I gave my 8th grade students this task to work on for the last 15 minutes of a class period. I told them that we would revisit it. And they'd get a chance to revise it and finish it after a lesson.

All of the resources are available at http://www.insidemathematics.org/assets/common-core-math-tasks/rugs.pdf for free. It definitely hits on lots of topics.

I feel that the most common mistake was English standard system measurements being converted to decimals incorrectly by my students. This had come up in the Dan Meyer Taco Cart 3 Act lesson that we had previously done.

GOALS:
1. Address a common mistake as a class and analyze it.
2. Give reason for your own mistake after analyzing it and write it down.
3. Cross out your incorrect answer, and correct it with work shown.
4. Turn in for a final grade.

So, let's analyze the first most common mistaken answer for #1: students getting an answer of 13.2.

  • What mistake are they making? 
  • How can you show your work using a formula?


In the next problem, for the perimeter of the triangle, the most common mistake was getting an answer of 7. There were A LOT of people that had no response also.

  • What are students incorrectly assuming?
  • How does the fact of knowing what an isosceles triangle help you in this problem?
  • What does perpendicular mean? What does perpendicular height mean?

In the third problem, a lot of students made the mistake of not following the directions of giving their answer in whole feet. The main wrong answer we will analyze is the answer of 31.4.
  • What are the students incorrectly assuming?
  • What do they need to know about parts of circles to not make this mistake?

In the final problem, most students either did not give a response. The most common mistake was 7.85.
  • What are students not accounting for if they gave that answer?

Monday, May 11, 2015

Pythagorean theorem in 2D then 3D

On Thursday instead of completing practice problems out of CPM course 3 we worked on Dan Meyer's taco cart lesson. It's a great lesson that covers rates and in addition to the theorem. I was impressed with how my students demonstrated their thinking:

Andre demonstrates his understanding of right triangles. As you can see he refuses to write the square root sign. My goal was to continually compliment yet demand more clarity and evidence from every student to show where they got numbers from and how labeling could help.

CPM introduced acute, obtuse, right, and non-triangles through the use of squares and their side lengths you can put on card stock. You then compare the small and medium squares areas to the large squares areas to determine what type of triangle it is. Here's a pic of students working on the activity:



Many rich discussions arose: why does adding the legs not equal the hypotenuse? When I got answers in seconds I asked if I'm going to see you in 5 minutes do I say I'll see you in 300 seconds? Also, some students after using their calculators said 4.58 minutes was 4 minutes 58 seconds.... Great conversations. They groaned at the slowness of video then were pleasantly surprised by the fast forward. Haha. 

That was on Thursday. Now on Monday we started with looking at the Pythagorean Theorem in 3 dimensions. My 1st period students seemed super sleepy so I decided to ask if anyone had a fish tank at home. I then proceeded to reveal that the current Autozone store in the neighborhood used to be a pet store (1 student knew that) and I had worked there as a high schooler (when I was a few years older than them). I told them how I would buy feeder goldfish from work and feed them to my Oscars. They were enthralled, and some grossed out.

I also modified this task, Fish Tale (get it?? told students it's a pun),  by yummy math by taping paper strips over the dimensions so students specifically asked for what they needed to solve. I also required students to draw the diagram with me step by step. I first asked them what shape it was called. I asked them why it wasn't a cube. How did they KNOW it was a rectangular prism like they said?

The key point to helping them draw it (some had difficulty)was making a rectangle (the front face) with another congruent rectangle spaced up and to the left from it. I had them color code the different pathways. I also demonstrated how erasing the lines and making them dashed showed it was behind your view. By not giving students the dimensions, they had to come up with asking for the length, height, and width, which helped them differentiate between them.

Kehl used the same method as I did. It takes more steps but was how we visualized it.

But Jerod's method caused my brain to explode with clarity once he showed me the more efficient way that I sheepishly hasn't worked out before but was delighted that worked. 

Galvin neatly organized his work to show how he got his solutions. Haha, he even labeled the problem title with (pun!). This was a great lesson too because I didn't use any of my precious copies on the copy machine.

Saturday, May 9, 2015

7th Grade Ducklings Re-Engagement Task

Following the guidelines of SVMI after having 7th graders working on Ducklings MARS task:




http://www.insidemathematics.org/assets/common-core-math-tasks/ducklings.pdf (credit to Insidemathematics.org)

This gives students a chance to find a median from a data table, calculate the mean, and interpret all of this in context when a duck family is added later on that does not change the mean.

Analyze errors in filling in question 1's frequency chart:

1. Filled out 7, 8, 9, 10, 11, 12
2. Filled out 7.5, 9, 10.5, 12, 13.5, 15

How does filling out the chart help you find the mean number of ducklings in a family?



Focus questions for Question 2 (finding the median):

  1. How do we ensure we have the correct amount of data? How can you verify it in more than 1 way? 
  2. Is there an even or odd amount of data?

Many students said median was 5.5 because 5+6/2 = 5.5.

Also, a few students said the median was 2, because 2 is the median of 1,2,2,2,2,4,6.

I know the median is ____ because there are _____ total ducklings which is an (even/odd) number. Since it's ______, there (is/isn't) a true median.

For question 3, show student methods for finding average without making mistakes.

Sentence Frame for question 4:

If there are _____ total ducklings and _____ families, one more family of ____ ducklings totals ______ ducklings in ____ families which has an average of _____ ducklings because _____/_______ = _____.

For students completely successful, how could you use a stem and leaf plot in this problem? Also a good follow up task is from the 2000 MARS tasks called "Supermarket" comparing estimations of median and mean.

Notes to self:

  • Highlight Chase's method of finding median and mean at the same time in an organized fashion.
  • Mention Rion's explanation for number 4 and also Joey's understanding of how mean works.

Wheel or Spiral of Theodorus

Resources courtesy of yummymath.com

http://www.yummymath.com/wp-content/uploads/Irrational-numbers.pdf

Extra credit for turning in a colored and labeled Wheel.

Monday, May 4, 2015

Can these sides form a triangle? If so, acute, obtuse, or right?

These notes were taken after collecting data using sides of squares to form triangles in lesson 9.2.1. After discovering some of the patterns we synthesized our learning with these notes.

The illuminati sign at the top of the triangle grabbed my students attention. They are all obsessed with it for some reason. 

I asked my students if they could think of a way to remember how to differentiate the inequalities for acute and obtuse. I just thought of a triangle everyone can identify. What's true about all of them? They are similar, three 60 degree angles, and three congruent side lengths. 3 sides of 3 would be 3 squared plus 3 squares which is 9+9 which is 18 and greater than the third side squared: 3 squared which would be 9. 


Sunday, May 3, 2015

Common Core Math 8 Activity - Perfect Squares and Irrational Numbers on Geoboards

After reviewing perfect squares and their areas and perimeters, we will be using geoboards and rubber bands to find other squares and their irrational side lengths.

We will also learn a technique for estimating irrational numbers as mixed numbers and decimals without using calculators. Calculators may be used to verify estimations and approximations.


If you were absent pay close attention to the bottom right where we approximate the square root of 14 as a decimal rounded to the nearest tenth.

A wealth of information can be found at https://www.mathsisfun.com/irrational-numbers.html

This lesson idea was adopted from NCTM's web site http://illuminations.nctm.org/Lesson.aspx?id=3096.

This video explains how to approximate irrational numbers on a number line, we will go into a little more detail. https://www.youtube.com/watch?v=6htWeCE2wro