Day 29: Day 2 of Cubed Roots

Quote of the Day: “Part of our problem is that we think about ourselves way too much. The more we obsess about our problems, our weaknesses, and our deficiencies, the more we perpetuate them. It’s ironic but true.” - Judah Smith

Question of the Day: "Why did you get 21 for the square root of 1/441?" In reference to my own mistake

Regular Math Objective: Find square roots and cube roots of rational numbers

Regular Math Standards: 8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that is irrational.

Regular Math Lesson Sequence:

1. Visual Pattern #23
2. Review HW 
3. Day two of Engage New York lessons (what was originally the Engage New York Day 3 Lesson HW) 
4. Exit Ticket
5. Assign new homework

Honors Math Objective: Solve multiple step absolute value and compound inequalities

Honors Math Standards: A1-A-REI B3 Solve linear equations and inequalities in one variable; including equations with coefficients represented by letters

Honors Math Lesson Sequence: I wasn't in class this class because I had a TenMarks professional development. Thus I left the students 7 problems that could be done in their notebooks and checked on the Chromebook plus homework. It would have been great to be there too because students continue to struggle with absolute value, but we're making serious progress.

Day 28: Absolute Value Inequalities Are Stifling

Quote of the Day: “Csikszentmihalyi reports that people are happier when they are at work than they’re caring for a child, and researchers observe that happiness increases when children leave home. Yet many people also say that bringing up their children was the best experience of their life. Happiness is not simply the aggregate of happy moments.” – John Kay

Regular Math Objective: Find the cubed root and square root of low numbers

Regular Math Standard: 8.EE.3

Regular Math Lesson Sequence:

1. Jumpstart via Engage New York. This included two charts. One chart was x to the second power and the other was x to the third (or m and p respectively as the chart chose those variables instead of the world famous x). I really thought that this was engaging and a goldilocks task as students struggled, but did not quit in doing this. They were especially perplexed about how to express the relationship when it was a variable and not a constant that they needed to find a rule fo
2. QSSQ
3. Partner Work via Engage New York. I had to reteach how to solve basic equations and explain the merits behind that in terms of multiple step equations. Students were very ignorant of doing something to both sides of the equations. 
4. Exit Ticket. This was really difficult to fit in because students were unable to complete more than five of the nine questions in most classes which left them stranded on negative exponents which was a component of this exit ticket. Instead I verbally went over negative exponents in the last two minutes of class, which is not typically what I like to do. 

Honors Math Objective: Solve absolute value inequalities

Honors Math Standard: A1-A-REI B3 Solve linear equations and inequalities in one variable; including equations with coefficients represented by letters 

Honors Math Lesson Sequence: 

1. QSSQ 
2. Basic absolute value inequalities |x| > 2, |x| < 3 and two that were similar. Students were able to  recognize the pattern with absolute value that started out as greater as being "or" compound inequality and less than to mean that they were and compound inequalities. 
3. Advance toward higher problems that involve moving constants around such as 5 |x - 4| - 3 > 7 

Students were able to conquer the very basic inequalities such as the ones listed in problem two. As soon as we start to introduce the questions with more meat on them, students have a phobia. We could not get to a third problem in this day and completely go over it. Obviously this is where we will enter tomorrow. It's amazing to me how much they understood absolute value and yet could not muster a solid answer on a multiple step absolute value inequality. 

Day 27: Password Hacking

Quote of the Day: “Try to encourage a kid to learn math by paying her for each workbook page she completes - and she’ll almost certainly become more diligent in the short term and lose interest in math in the long term.” - Daniel Pink

Question of the Day: "Why is it that 0.7 is 7/10 but 0.7 repeating is not 7 repeating over 10?"

Regular Math Objective: Give an example of where exponents are used in everyday life.

Regular Math Standards: 8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. 

Regular Math Lesson Sequence:

1. Students completed a self-assessment. I asked them two things. "Do you believe at some date in the future you can answer all ten questions?" I wanted to see who among them was brave enough to say no to this question. These are the students that are truly crushed when it comes to confidence in math. From there I know that I need to make it my own personal mission to prove them wrong and get these kids to see that it is possible to change their mindset in math. The other question was "Can you think of some real-world examples of exponents?" This has kind of been lost on me in teaching this entire unit. Many students struggled with this question. The thing I heard all day was space exploration or some version of that, but we missed other obvious answers. 
2. Review quiz errors in pairs. I really liked this idea. Students sat in partners and had to put the correct answer to a problem that they had answered incorrectly on a marker board. They exchanged ideas to one another. I heard great discussion as it was taking place. It was a great way to ditch stand and deliver in going over the quiz and a much more measurable way of knowing that students are recognizing their misconceptions than a fist of five or asking if they understand now. They always say yes when you ask that question.
3. Password hacking. 

I knew that students were lacking a basis for why we are teaching exponents and saw that on their self-assessments. Robert Kaplinsky's lesson really hits home especially on a level that students can relate to. What I did to introduce this was have students leave their seats and tell each other the five vowels (we did not include y). It might seem silly to have students rehearse what the vowels are, but it got them out of their seat and for students that are WIDA level 1 or 2, knowing the vowels is not a given.

Second, I had students write a password on a marker board so that their partner could not see it. The password had to be only one letter that was a vowel. Students then took turns guessing one another's password. It's an amazingly simple password, but I wanted them to guess it quickly so that we could move to a more advanced password and see the value in that. I asked what the maximum guesses are for a password are under these constraints. Students recognized that five guesses would be the maximum assuming of course that the same letter was not guessed twice.

Third, I told students to pick a new password that was two vowels long and then exchange guesses with their partner to see if they could hack the password. Some students could do it while others could not. As the students were guessing, I was putting all possible combinations on the board. After about a minute or two, I had students raise their hands if they got hacked. Only a handful had been hacked. We discussed why. Students were quick to point out by looking at my chart that there were now 25 password combinations instead of only 5 with two letters.

Fourth, I had a chart which compared characters to password combinations. I asked students based on the pattern, how many password combinations would there be if we added a third character. Students discussed with their partners and then were able to tell me that 125 password combinations were possible. For the skeptics in the room, I showed all 25 passwords that would have started with the letter A.

After all of this, we were able to get a deeper and more invested conversation of the idea that there are actually 70 potential characters when we consider uppercase, lowercase, numeric, and special characters. The lesson got somewhat frustrating from here as I had to rehearse for students that a computer could check for 350 billion passwords in one second. To students (and teacher) that have no background knowledge of this it's hard to say if this is really all that impressive. We live in an era where technology has seemingly no limit so some students even guessed that a computer could guess an infinite number of passwords in one second. We eventually closed the lesson with the Desmos link and formula. I manipulate the numbers of characters that were allowed so that it would be an exceedingly long password and explained to students that it would take centuries to crack this password. Of course, computers are probably capable of testing even more passwords now than they were two years ago.

Honors Math Objective: Graph absolute value inequalities of the form |x| is greater than or less than an integer c

Honors Math Standards: A1-A-REI B3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. a. Solve linear equations and inequalities in one variable involving absolute value.

Honors Math Lesson Sequence:

1. QSSQ
2. We looked at yesterday's exit ticket which was a compound inequality. I then had students do a similar problem and when it seemed that the room felt comfortable (I did a fist of five and circumvented to certain students that I knew had struggled yesterday) we moved to absolute value. 
3. Students were put in groups of three to four and given a two foot by two foot marker board. 
4. I gave them the instructions to pass the marker when I told them to and that whoever the writer was could not talk and could only write what the others in the group asked. I had done something similar two days prior with great success. Today I wish I had not done this because students really struggled with the concept. As a result, I had to go around and do cueing questions and failed to pass the marker as much as I would have liked. I think students would have benefitted from having everyone in the group be able to communicate at all times as a result of the level of difficulty absolute value inequalities gave us. 
5. The first two problems were extremely basic in terms of absolute value, but they were deep enough to last the entire class. They were in the form that the objective states. Students had problems with graphing and setting the inequality to the negative with a flipped sign. The mistakes graphing was perplexing. The inequality |x| < 3 was graphed correctly on the positive side, but then students were putting an open dot at -2. 

Day 26 Compound Inequalities

Quote of the Day: “My advice for you is figure out what you enjoy doing most in life, and then try to do it full-time. Life is short. Follow your passion.” - Will Shortz

Question of the Day: "Is there such a thing as a fake number?"

Regular Math Objective: Simplify exponential expressions; rewrite exponential expressions

Regular Math Standards: 8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. 

Regular Math Lesson Sequence:

1. QSSQ
2. Quiz 
3. TenMarks Weekly Quiz 
4. Retakes
5. Get to ten or 2048 

The quiz was about what I would expect. Some silly mistakes but overall I was pleased with the growth that students have made on this standard.

I saw some of the mistake here where students were multiplying before they were simplifying. This will undoubtedly be something we have to reexamine each month this year for students to master what they have not yet and retain it even if they did well.

Honors Math Objective: Graph and write compound inequalities

Honors Math Standards: A1-A-REI B3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. a. Solve linear equations and inequalities in one variable involving absolute value.

Honors Math Lesson Sequence:

1. Review the two-step inequality homework
2. QSSQ 
3. Compound inequality practice in the Big Ideas workbooks
4. Trying two compound inequalities on their own

The nice thing with the compound inequalities was that the workbooks served as a nice little appetizer for the main course. If students were to just try a problem that said 4 < -2x + 3 < 8 or something of that format they would have been overly challenged. As it were, a little over half the class could do the exit ticket and the rest could not. 

Day 25: Exponents Skit

Regular Math Objective: Divide exponential expressions; multiply exponential expressions; rewrite exponential expressions that involve a power to a power

Regular Math Standards: 8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. 

Regular Math Lesson Sequence:

1. Skit
2. Study Guide
3. Review the Study Guide 

The skit took only about ten minutes, but it was a great way to break up the monotony of the class. I made name tags for the characters that are pictured below. I would certainly use it again even if the math portion of it is missed by the students because they enjoyed it. This is the script from the skit:

Narrator: There once was a happy little exponential expression. The birth name was 5 to the third power, but friends just called it Five Cubed for short.

Five Cubed: Hi I’m 5 to the third power, but my friends call me Five Cubed for short.

Narrator: Five Cubed was always told that if he ate his vegetables he could visit the squaring fairy and be magically grow into a much bigger body. After many broccoli popsicles, carrot strings, and spinach gum he got a visit from the squaring fairly.

Squaring Fairy: Five Cubed you’ve been a splendid little exponent. It’s time to shed your five times five times five home and move into a much more spacious world. I’m going to reward you with another five for every five you already have.

Narrrator: The squaring fairy then performed magic. Suddenly five cubed had morphed into a giant 5 to the sixth power. Five to the sixth power felt a surge of power.

Five Cubed: I must be the most powerful exponent in all Rational Numberhood!

Narrator: Unfortunately the new found power resulted in a curse. Five to the sixth rejected old friends because they were not “big enough” anymore. The Number Psycho decided to flip five to the sixth into a negative exponential universe.

Number Psycho: I bet you wish you could just go back to being five to the third power now. Muahahaha. You are the smallest little fraction I’ve ever seen.

Five Cubed: Oh no. I’m smaller than the size of an ant now!

Narrator: This was the worst possible set of circumstances that the base of 5 could have ever imagined. Luckily, this exponent did not have a fixed mindset. Each day the exponent did what could be done to win the favor of the all powerful number wizard called the Numba Wizard. He ate proteins, fruits, veggies, and plenty of vitamins. Within a month the wizard had granted five to the fourth power growth.

Numba Wizard: I grant you five times five times five times five the size that you currently are. Unfortunately at this time, you have not done enough to come out of the land of negative exponents.

Five Cubed: Ok so I’m a little bigger than an ant. I’m still just barely as big as a mouse in negative land. I need to keep growing.

                            

Narrator: The night after the Wizard visited him, Five to the Negative Second thought about what could still be done to grow just a little more. All five to the negative second wanted was to be one. With that thought in mind, Five to the Negative Second threw an irrational number (the square root of 12) into the wishing well. After waking up that night, five to the zero power had realized a dream.

Five Cubed: I’ve never been so happy that five to the zero power is one. 

Honors Math Objective:

Honors Math Standards: A1-A-REI B3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. a. Solve linear equations and inequalities in one variable involving absolute value.

Honors Math Lesson Sequence:

1. QSSQ
2. Multiple step inequalities done on over-sized marker boards.

My colleague gave the suggestion that he stole from someone else to have a group of four try eight problems. The rules were that the person writing could not talk and must do exactly what the group told him or her. It worked really well. I started out by putting two problems on the board and circumvented the class confirming if problems were right or wrong. By the end of class some groups had correctly done all eight while others were close. I was able to pass out the homework for the groups that finished early. I really enjoyed this lesson.

Day 23: Flipping the Sign Continued

Quote of the Day: "Discipline is doing what you really don't want to do so that you can do what you really want to do." - John Maxwell

Question of the Day: How do you solve a problem like (2^3) to the second power?

Regular Math Objective: Solve or simplify exponential expressions

Regular Math Standards: 8.SP.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions.

Regular Math Lesson Sequence: It was a little mixed throughout the day, but overall the most effective way that the material was presented was in this order.

1. Students get mini whiteboards and work on the problem that was the Question of the Day
2. We continue to do problems such as 8m to the fourth power over 2m to the third power and 6a x 4a
3. Exit Ticket
4. Pass out the homework

I gave this problem throughout the day and I never once had a student write 12 times a squared. It was mostly the answer that is pictured up above.

The good news was that many students were able to divide exponents by rewriting it in expanded form. I'm not even going to tell them that they can subtract the exponents. If they discover it on their own that's fine. I think by doing it this way, they are getting the why behind the math.

Honors Math Objective: Analyze when and why an inequality sign gets flipped

Honors Math Standards: A1.A-CED A1 Create equations and inequalities in one variable and use them to solve problems. (Include equations arising from linear, quadratic, and exponential functions with integer exponents.

Honors Math Lesson Sequence:

1. Give students practice to see how signs change in different situations (adding to both sides, subtracting to both sides, multiplying, dividing, multiplying by a negative, and dividing by a negative).
2. QSSQ
3. Explore with Big Ideas workbooks
4. Exit ticket asking students when and why the inequality symbols change places
5. Pass out the homework 

Unlike the exponent rule, I will by reminding students to flip the sign in the coming days without saying why. The exit ticket to me is a clear indication that students understand why, so we can simply put that in our memory. 

Day 22: TenMarks Procedures Rehearsed

Regular Math Lesson Objective: Rewrite and solve exponent expressions that raise a power to a power

Regular Math Standard: 8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. 

Regular Math Lesson Sequence:

1. Which One Doesn't Belong (1, 9, 36, and 45)
2. Logged into Chromebook and did a TenMarks rounding decimals lesson 
3. Had four students do problems from the homework on the board 
4. Did a quick exponent power of a power exploration (which I skipped in one class due to time constraints)
5. Exit Ticket. This consisted of three things. First, rewrite (5^2)^4 in expanded form. Second take that same expression and put it into exponential form. Third, rewrite the expression 7^6/7^3 and then write it in standard form. 

Only one class needed to be prompted to bring up the perfect squares and rational numbers with Which One Doesn't Belong. It was refreshing to hear students also talk about composite numbers and divisibility. 

Some students really struggled with rounding on the computer. I was not surprised given how difficult it can be to do problems on computers and given what I know from teaching sixth grade surrounding place value and rounding. My colleague and I discussed how there is just too much on the plate for these students in math now. I can only remember working on the operations, fractions, and place value when I was in elementary school. Today there is so much more that they have to know and as a result their mastery of what I would perceive to be the basics is no longer a guarantee. 

The power of a power exercise was good once my co-teacher told me to write (3^2)^4 as 3^2 x 3^2 x 3^2 s 3^2. I was initially writing out three eight times to explain this to kids. The tickets to leave were pretty on target. As the pictures below illustrate some students made errors in putting the exponent (one student used 6 and another used 9) on problem two, but overall they are seeing the big picture. 

Honors Math Objective: Solve addition and subtraction inequalities

Honors Math Standard: A1-REI-D12 Graph the solutions of a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set of a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Honors Math Lesson Sequence:

1. Which One Doesn't Belong (x squared, 4 - 2y, x - 1, and 3 (x + 1)
2. Login to TenMarks and complete the work on rounding decimals
3. Review the homework
4. Exit Ticket on inequalities

There was a great question about an absolute value equation in the form of c |x - 4| + d = e where a student asked if we could divide by c before subtracting by d on the homework. We went through what could be done in this scenario to make sure that c/d was a fraction if you did divide by c. I honestly had never really considered doing the division as I had been trained away from this at a young age and never thought to question it. It was enlightening for me to learn too. 

The which one doesn't belong is a great conversation about what it means to simplify because students could argue that 4 - 2y can be simplified and so can 3 (x + 1). 

The exit ticket went pretty well for the most part. From what I can tell students know what they are doing with the inequalities. It is subtracting by a negative or subtracting from a negative that threw them off if anything threw them off. Only one student left the dot open when graphing.