The Box Investigation

Shown below is the picture of a word problem that is often found in homework problems when discussing relative maximums and minimums. This year, I decided to give it as an investigation to my Algebra 2 Honors class with very little help. I broke them up into groups and then gave them the document below. What followed was a 40 minute class of strong discussion, aha! moments, and deep learning of how what we do in class can be applied to the real world. Students recently learned how to find the maximum and zeros on their calculator, but were not adept at setting their window before this activity. We had 40-minute classes due to a special assembly, but 50 minutes would have been perfect for students to have created the boxes in class. it is interesting how, even after solving everything, some students were unsure how to make the boxes. Seeing the whole problem through was important. I assigned the creating of the boxes for homework.


(also found here.)

Graphing calculator pictures:

A picture of the maximum volume

A picture of one non-extraneous value of x where the volume = 400.

I think this problem tied together many things:

• Why domain of a function is important in the real world
• How to adjust windows on the graphing calculator
• How to find the relative maximum on a graphing calculator
• Why the relative minimum of the graph didn't matter (out of the domain.)
• What an extraneous root looks like (the graphing calculator gave a third answer outside of the domain for a volume of 400 cubic cm.)
• How to work in groups as a team (though each student did have to create their own boxes)
• How letter e) could have been solved two ways: by using the so far unknown intersect feature of the graphing calculator or by setting the equation equal to 400, subtracting 400 and finding zeros...this was cool for kids to see!
• The eventual tie-in to polynomial graphs
• The actual making of the boxes

Perhaps my favorite comment (several mentioned that they loved doing this) was when a student said that when he heard we were doing an investigation today, he was nervous, but then during and afterward, he was excited and really enjoyed himself.

I was observed during this class, and in the write-up, my department chair noted that every student was focused, on-task, and working together for the entire period. He wrote that students exclaimed: WE GOT IT!!!! and that "the teacher smiles and laughs with glee." Ha. Yep. Every damn day.

Quizlet and Quizizz for Practice or Formative Assessments

As I sat down to write my lessons for Monday, I started looking for quick formative assessments online. In a matter of minutes, I was able to tailor some uniquely to my classes using the sites http://quizlet.com and https://quizizz.com.

QUIZLET

For one class, I wanted students to review parent functions.
1.  I went to http://quizlet.com.
2. I searched "parent functions".
3. I found several quizzes and one was very close to what I wanted. I clicked on it.
4. I clicked on "copy".
5. I added/deleted things I wanted. When I needed a picture or equation of a cubed root function quizlet automatically suggested graphs and equations it had in its memory!
6. I clicked on "create."
7. I clicked on "add to class."
8. I created the name of my class.
9. I clicked on "add members" and then "automatic join link" which I will post to my class.

Students can use flashcards, take a "test," match, and play quizlet live (my favorite.)

Here is the link to my quizlet: https://quizlet.com/_2na0r8

(Side note to quizlet: I reallllllyyy miss space race (replaced by Gravity.))

QUIZIZZ

1. I went to https://quizizz.com.

2. I searched for vertex form quadratic functions.

3. I found a quizizz I liked by looking at the questions on the right and scrolling. I clicked on this quiz and then clicked "duplicate."

4. I clicked on "edit" deleted questions I didn't like and added by using the search button. This was BEAUTIFUL because it gave me questions from other quizizzes (try and spell that on the computer!) questions already premade! I added them.

5. I clicked "finish."

6. I saved the link and shared it with my colleagues.

Here is the link to my quizizz: https://quizizz.com/admin/quiz/5802351dab98727c4ff85d48

I like that you can also assign it for homework!

What are your favorite websites for formative assessments? I have blogged about others here.

MATHO! as a review game

This blog post was crossposted at mathequalslove. Thanks, Sarah!

I learned how to play this tried-and-true game with my classes in my first year of teaching. It was when I wrote answers on an overhead transparency using those felt tip transparency markers that smelled sour when they got too old. I remember cleaning the transparencies with Windex or even sometimes in the sink, math dreams swirling away.

Now, I write the answers on a piece of paper, snap a picture with my iPad, and project it on the wall with my Apple TV. A lot has changed in 25+ years technology-wise, but not the love of this game. I don't know a kid who does not enjoy playing it. It's low key and yet there is a bit of friendly competition with who gets MATHO! first.

Here's what you do to prepare for the game.

• Write out about 30 review problems. I wrote them out using Notability on my iPad ahead of time. MATHO! is probably better when you don't have a ton of long problems. Here is an example of the first few for review of functions...the rest are linked here as a pdf.

• Write out the answers to these questions in a different order on a separate piece of paper. I literally write the answers all over the place and boxed them. Here's an example of what it looks like (with a link here.)

• Have MATHO! sheets (linked here) ready to go. That's it for preparation.

Here's what you do right before playing the game. 

• Give students blank MATHO! sheets. Tell them to fill in the 24 spaces with 24 of the 30 review problems. They should scatter the answers in different boxes to ensure that everyone has a different MATHO! card. This takes a bit of time, but if you play a song and tell them that after the song ends, they should be done, they are usually on task and copy the answers quickly. There are more answers than there are spaces, and this spices it up a bit because some of the answers will not be on everyone's cards. 

• Have students check that they did not copy any answers twice. I ask them to switch with a partner who can look and double check for them.

• Ask students to take out paper for doing work. If you have students that are not self-motivated, you may want to collect this paper for a formative grade after the game. Students must show work to get credit.

• Tell students that even if they know what the answers is before they see the question, they have to do the work and not call out the answer. (Sometimes, if an answer is obvious, i.e., there is only one graph, I will put a WRONG but similar graph as an "answer" so that they will not just pick the graph without thinking.) This also spices things up, as not every answer will be used. 

• Project question 1 on the board. Have students write the work down and put an X in the box if they have it. Walk around to help struggling students (I put mine in groups to help each other.) Go over on board if necessary. Continue with the next question and repeat until someone gets MATHO!

• I have students continue this game after the first MATHO! so that many students are able to get MATHO! I only allow a student to win twice if they get blackout (all squares have an X).

• I give my students a choice of candy. I get fruit roll ups at the dollar section from Target or lollipops or any kind of candy. 

That's it! I think 50 minutes is a good amount of time for the game. There are probably websites that will scramble the cards for you, but I think the kids enjoy filling in the spots. They get excited when the one they wrote is on the card and chosen and mad when they didn't write one down that gets picked. It's just good wholesome fun :)

How to Solve the Rubik's Cube 2.0

When Art Benjamin visited our school, I bought his DVD called "The Mathematics of Puzzles: From Cards to Sudoku." "Mastering the Rubik's Cube" was one of the 12 lectures on the DVD. I watched it this summer and was surprised how easy his method is. I broke it down for my students in my Problem-Solving class, and here are the 8 steps. I do think it's easier (and clearer instruction-wise) than my former blog post on solving the Rubik's Cube. Also, here is my post on why I teach it in my class and a link to the mosaic contest in which we won 3rd place.

8 Steps

First Layer

1. The Daisy

·      https://youtu.be/QzP1cI-1Oks

·      The yellow face is on top

·      White petals will surround the daisy.

2. The Easy (bottom edges) 

·      https://youtu.be/Cp8G1W6LNmw

·      The yellow face is on top

·      Twist cube so the white petal finds its center

·      Rotate face 180˚

·      Repeat for the other three white petals

·      Flip Rubik’s cube upside down and look at white cross 

3. The 123 (bottom layer) 

·      https://youtu.be/icnjPpdIdZA

·      With white daisy on the bottom, find a white corner on the top rim

·      Twist it until its other color finds it’s center (it matches in color with the center color)

·      Hold the cube so that this white cube is on a side.

·      ONE: Point with your index finger to the white. If you’re pointing with your right hand, you will twist the right face away from you. If you’re pointing with your left hand, you will twist the left face away from you. (UP)

·      TWO: Right hand, twist the top CLOCKWISE. Left hand, twist the top COUNTERCLOCKWISE. Either way, whites will match up.

·      THREE: Right hand, bring right face back down toward you. Left hand, bring left face back down toward you. Either way, now white corner is in proper place on the “daisy.”

Note: a 1-2-3 always starts with an “up”

Complications:

What if there are no white corners on the top rim, but there is one in top face?

            1. Rotate white so it’s directly above a non-white (unsolved) on bottom face.

            2. Perform 1-2-2-3 move

            3. Now the white is on the rim, so you can repair with the 1-2-3 move.

What if you don’t have any whites on a tip rim or face? Do a 1-2-3 move, and it will move a white piece into the top rim or face, and then you can repeat. 

Second Layer

4. The Middle Layer 

 https://youtu.be/xP8ywEBpxB0

·      With white daisy on the bottom (or yellow face on top—same thing), practice this move:

UP	TWIST	DOWN	TWIST	UP	TWIST	TWIST	DOWN
1	2	3	2	1	2	2	3
R	U	R’	U	R	U	U	R’

(NOTE: the above does not have anything to do with the middle layer. It is just for practice for later)

·      Only have to change 4 edge pieces.

·      Start by looking at the top layer and see if there are any edge pieces on the rim that do not use the color yellow.

·      Find one that does not use the color yellow and twist it so that it matches its center color.

·      The color of the top edge will either match the left or right face. Whichever side it matches, give that side a mental “slap.”

·      Twist the top in the direction of the slap.

·      With same slapping hand, perform the 1-2-3 move.

·      Lost a white, fix it by moving the color on the white cubie to match its center.

·      Perform the 1-2-3 with the side that contained that white face (white should be on the side, NOT facing you.)

If all four-edge pieces have yellow, we need to move a piece out of its wrong place. 

·      Find the piece that needs to be moved and perform a 1-2-3 move (no slapping required.)

·      Fix the white piece by moving it to match its center and follow with a 1-2-3 (note: the white will be on the side)

Third Layer

5. FUR U’R’F’ move (top cross) 

·      https://youtu.be/zyP4sP9U7Uk

·      F = front, B = back, R = right, L = left, U = up, and D = down

·      When you see these letters, always move the face clockwise

·      If you see these letters with a ‘ by it, always move the face counterclockwise, for example R’ is moving the right face counterclockwise.

·      Up face is always yellow, down face is always white

·      L moves toward you, R moves away from you

·      Perform at “9:00”

·      If you don’t have “9:00,” do FUR U’R’F’. You may have to do this twice, but then you want “9:00

·     

·      Do FUR U’R’F’ one more time to get the yellow cross.

Just a reminder: R is up, R’ is down, L is down, L’ is up

6. The Fun move (top face yellow) 

·      https://youtu.be/xEsrHF9YJF0

·      Count the number of yellow corners on the top. Best case is one. You can only have zero, one, or two.

·      With 0 or 2 yellow corners, rotate the cube so that a yellow is in the upper left corner of the front face.

UP	TWIST	DOWN	TWIST	UP	TWIST	TWIST	DOWN
1	2	3	2	1	2	2	3
R	U	R’	U	R	U	U	R’

·      Repeat until you have only one yellow corner

·      Orient the cube so that the one yellow cube is in the bottom left corner

·      Repeat the “fun” move and the one yellow cube in the bottom right corner until you have the top face yellow.

   

7. The R’F move (top corners) 

·      https://youtu.be/5j-3uDsoTZs

·      Look around the top rim and see if any of the top sides have matching corner colors

·      If not, then do: R’ F R’ B²      R F’ R’ B²       R²

·      When you have matching corner pieces, move them to the back face

·      Repeat R’ F R’ B²      R F’ R’ B²       R²

·      Orient so that all corners should now be matched

(remember, R’ is “down”)

 (or, know that Up or Down refers only to the right side, and the mantra is:

D F D B²      U F’ D B²       U²

8. The “FFURL” move (top edges) 

·      https://youtu.be/edToGuRKlqY

·      All should be in proper orientation except for three or four edges.

·      If you only have three edges out of position, then you have one perfect side, which you want to move to the back

·      FFU   R’L   FF   RL’  UFF (or, FFU, both sides down, FF, both sides up, UFF)

  

All on one page:

1. The daisy

2. Bottom edges

3. The 123 move (bottom layer) (123 always starts with an up)

4. Middle layer (slap 123, adjust white 123)

5. FURU’R’F’ (look for 9:00)

6. The fun move (start with right up)

7. R’ F R’ B²  R F’ R’ B²       R² (where R’ is down) or D F D B²      U F’ D B²       U²

8. FFU, both sides down, FF, both sides up, UFF

I heard from a former student who is a freshman in college and who told me today, "It's so much fun to know...I always use it as a fun fact during ice breakers!" Love. I hope you'll have as much fun, too! My fastest time is 2:45, but normally it's around 3 minutes. There are definitely faster ways, but this method is, to me, easy to teach...as long as you have willing students. 

My plan is to have my class teach some of the 5th graders at our school how to solve the first and possibly the second layers, so that maybe they will want to learn the rest on their own. I've joined a national Rubik's Cube club, and we may host a competition...we will see! Just like the Rubik's Cube, one step at a time :)