The newest foreign language

This clip is from the movie The Core. While the movie is of questionable quality, this clip encapsulates the meaning behind a conversation I had with one of my math teachers.

The math teacher came to me asking my experience with computer programming. Now, while it was a while ago, I did have some experience with computer programming. In elementary school, I had some experiences with BASIC and creating simple programs. When I was in junior high school, I took a summer class in LOGO and our computer class added to work with BASIC. After junior high, it was mandated, as a part of my math education, I had to take a little more advanced class in BASIC. My class was the last class to take BASIC before the evolution into True Basic and the loss of line numbers.

Sometime between my freshman year of high school and beginning teaching in high schools, the requirement of a computer programming course seemed to have disappeared. While most people might have not had many chances to apply their computer programming skills, I actually used my understanding of programming to make my wonderful TI-81 a more useful tool and created programs of algorithms so I did not have to remember them. (This was before programs could be shared and beamed to one another with TI calculators.) So when I started teaching freshman biology and asked students to use their calculators in this method, they looked at me like I was crazy and I had to change tactics quickly.

Flash forward to a few months ago and the conversation with this math teacher. She was telling me how, as a math major, she was require to take computer science courses and even pointed out the programming courses still listed on our course taxonomy. The conversation quickly morphed into more current applications of programming (Scratch, app development, JAVA, Flash, etc.) and how might she implement these in her classroom.

I then posed the question, why is programming part of the math department and not part of the world language department. While Spanish and Chinese are languages spoken by large portions of the world population, it is arduous to find anyone on the planet not affected by some type of computer programming.

Of course, after I had this conversation, neural priming went into effect and I saw multiple articles in my Twitter feed about this conversation, which I have linked below:

Is coding the new second language? -- From Smithsonian
Should coding be the new "foreign language" requirement? -- From Edutopia
Programming as a second language -- From Learning & Leading with Technology

So I ask you, will colleges accept 4 years of computer programming to meet the world language requirement? Is being able to create new programs more important that being able to ask "Donde esta el bano"? (Which I can tell you it is much more important to understand the directions than ask the question...) Do computer programming languages meet the 21st century need more than Spanish or French?

What do I wish that I had known?

The counseling office is one of the departments I have the honor of overseeing. I am willing to admin that, when I was in the classroom as a teach, I did not really understand what the counselors did. I did not utilize my guidance counselor much when I was in high school, so I did not realize the resource that was available to me. Counselors are educators who go through extensive training to do exactly what their name says: counsel. Whether it is about the courses to take for the next school year, where to apply to college, dealing with difficult emotional issues, bullying, troubles at home, teen pregnancy, eating disorders...pretty much anything there has been an after school special about, counselors are on the front line. They are a wonderful resource and connecting point to a multitude of additional resources. That is one thing I wish I had know when I was in high school, or as a teacher.

We have had college nights in the past, but typically it was in January and for seniors in high school. By that point, college applications and financial aid forms should have already been completed and it makes the usefulness of the evening moot for most parents and students. The counselors came to me with an idea...have a college night much earlier in the year...and invite parents of juniors to attend as well. I told them to run with it. They came up with the sessions, the speakers, the plan, and the logistics for the entire evening. In fact, I am writing this post as the parents and students are in one of the sessions. To improve things, we have an evaluation form for the parents and students to complete so we can make sure that future nights will be designed to better fit the needs and desires of the our families.

I thought it was fantastic that the counselors wanted to get families involved earlier in the process and not wait until the student was in 12th grade to begin thinking about college. I want to take it a step further.When is the time that students should being thinking about their post-high school plans? As a parent, I think that it should begin before the child is conceived, but if we start them thinking about specifics when they are freshman, that can be very useful. There is a slogan being kicked around the have a College 101 night. I say make it College and Career 101and let's go with it.

College and Career 101 can be for the 9th grade students and parents. Further events in the year can be named College and Career 102, 103, etc. For our 10th grade students and parents, we advance to College and Career 201; 11th graders will get College and Career 301; 12th graders get College and Career 401. Which brings me to the title of the post...What do I wish that I had known?

This is the question I want to pose to our parents and our students who have gone through this series of events this year. Their wishes for information will help us design the 9-12 College and Career preparation curriculum. I want to include the student voices to help guide our underclassmen through this high school process to help them be the best prepared for whatever their post-secondary plans may hold.

What do you wish you had known?

Coaching by listening

One of the best parts of my position is discussing what teachers are doing with students in their classrooms. I enjoy the conversations had between colleagues about the choices, practices, and results in their classes. It is a great chance to reflect and I encourage the teachers with whom I work to see the observation/evaluation cycle as the most honest professional development in which they will participate.

As I reflect on my own practices, I stumbled across the Partner Discussion Protocol from a conference I attended. When there are partners working together, here is how it generally works:

·        A shares, B is an active listener (then switch)

·        A shares need, goal, or ideas for evidence

·        B’s role is to probe and push with ?’s

o   What’s going on now?

o   Why do you want this to change?

o   What exactly is your goal (desired change)? 
o   How will you know is changed?

 In the administrator/teacher dynamic, there may not necessarily be the reversal of roles, but idea of identifying the wanted change, defining the reason for the change, and being able to assess if the change has actually occurred is definitely valuable.

As the concept of peer evaluators gains popularity, this model can be applied after a pair of observations has been completed. The role of the active listener is key.

As it has been said: "We have two ears and one mouth so we can listen twice as much as we speak".

I have a dream...50 years later

On a hot August 28th, 1963, a quarter of a million people gathered on and around the steps of the Lincoln Memorial for the March on Washington for Jobs and Freedom. The reverend Martin Luther King, Jr. took the podium to address the crowd. His speech from 50 years ago is still regarded as one the best pieces of spoken rhetoric in history. While I work in a district that has a majority of African-American families and will take the opportunity to share this message of needed work and hope, this message can be shared with all students.

Here are some resources for you:

Full text of the speech 

Some facts about the speech

Please mark this 50th anniversary by providing students that the chance to read or listen (or both) to the speech and think about what they can do to make that dream a reality.

You have 1 week to plan! Share your experiences or ideas in the comments!

http://www.tubechop.com/watch/1431982

Don't live in the 10 seconds

Welcome back to another school year. I am sure that there have been changes all over the place. Change can be a wonderful thing, but very difficult to accept and deal with in the moment. We become very comfortable with the status quo, sometimes even when we know there are things that can be improved.

For students, change occurs all of the time causing them to have to continually adapt to new situations. Over the summer, when discussing how to help students express themselves better, about their academic, emotion, physical, psychological, etc. needs, a teacher told me that kids today tend to "live in the 10 seconds". They react before they think about the larger picture, understand the root of the issue, what is the real cause, and what some of the effective solutions could be. When this happens, sometimes students will yell and scream, hit, withdraw from communicating, and other non-effective actions to help remedy the issue. If we, as educators, accept that we must educate the whole child, then we must educate them on social emotional needs, conflict resolution, goal setting and achieving, in addition to their academic needs. But we know this...

As we open the school year, we, the adults, need to keep the big picture in perspective. Teachers might not like having to switch classrooms, when their planning period is, classes are full, teaching assignments change, etc. With the excitement and hiccups that always accompany the opening of school, we need to NOT live in the 10 seconds. We need to look at the larger picture to solve the immediate issues and once things have settled down and reached equilibrium, then we need to reflect on the system and look for improvement. Changes that happen will disrupt equilibrium and that is uncomfortable. Adults need to discern the difference between the discomfort of change and an actual problem. If it is discomfort, give it a chance to work and know that reflection and evaluation will occur to make improvements. If it is an actual problem, we need to develop a solution and then monitor if that is the best scenario for the big picture.

As leaders, we need to exercise our listening abilities to help teachers discern those differences and identify them ourselves. We need to effectively communicate with teachers about the big picture and prioritizing how issues will be handled.

Change is never easy. But when the need for change is communicated, team members listen to one another, and the change is understood to help the system improve, it can be an easier pill to swallow and might allow people to see the 11th second and beyond.

Have a great start to the year!

Happy birthday, America

237 years ago, on this date, 56 men committed a heinous act of treason that we call our Declaration of Independence. So how has it been going?

Please take this short survey. Results will be shared.

This is my boss

In the never ending series of changes to my role and responsibilities (which I can appreciate because if we cannot change and adapt, we go extinct), I am currently in charge of the summer bridge program. This is a program where students who had some difficulty getting through 8th grade are provided support in reading or math in order to support them in their transition to high school. The students are reluctant at first to accept the ideals of the program and every once in a while, a student will be sent to my office to discuss choices that they have made that might have led them to this program and the choice they most recently made. The discussions with the students usually go pretty well and we are able to get them to reflect on their choices and be cognizant of the triggers that lead to them.

Every once in a while, one of the teachers will tell a student "This is my boss". While I would like to think that, for the teacher, the meaning is colleague, supporter, and fellow learner, I get the feeling that the teacher is indicating to the student that this is the person in charge and if you won't behave for me, you had better do it for the boss. This has left me perplexed.

On one hand, some of these exchanges will progress out of the developing teacher-student relationship. Essentially, the teacher is saying "Look, I know you are having a difficult time right now, but please, for MY sake, don't make me look bad in front of my boss". If that initiates the desired behavioral change, then that indicates to me that the teacher has made a connection with the student, and vice versa, and for each others sake, a change in choice will be made. This is a good thing that relationships are being developed, but is it truthful? Why is the external motivator of "the boss" needed?

On the other hand, the "this is my boss" comment seems to give up control, power, and authority (for lack of better terms) that the teacher of a class should have. It reminds me of when a teacher writes a discipline referral for a minor classroom issue. The referral sends the unintended message to the student that the teacher no longer cares to deal with this issue and will pass it off to someone else. Is that what the "this is my boss" comment is going for? Does the teacher lack some confidence or not feel the immediate support to enact a decision that they feel is best for the student and the class?

This situation reminded me of the "Front Porch Leader" by Jimmy Casas (@casas_jimmy). "It is in these moments that leaders must decide to leave through the back door or take a seat on the front porch". Am I being overly critical of a teacher who uses the boss as a deflection? Are they leaving through the back door? How often do I use my boss as a deflection of why we have to do something? This is something I will have to contemplate in this new school year.

Why is it different in school?

I have been in the workforce since the age of 15 and a half. When I first began, it was part time work as a cashier and stock boy in a local pharmacy. If there was a large project that needed to be completed, especially in a short amount of time, a few of us working that day would work together, divide the tasks, and, at the end of the day, report out what was completed and what still needed to be done. The people working the next day would pick up where we left off until the project was complete.

This process continues in my work as an administrator. When a large project comes in, we will sit down, discuss the goals and expected outcomes, divide up the tasks, assign realistic deadlines, and provide progress reports. If one of us runs into an issue or has a question, we ask our colleagues for their expertise, guidance, and ideas. When complete, it is OUR project. While individuals might be recognized for a specific contribution, it is the work of the whole that gets evaluated. We stand or fall together.

As a child, I participated in multiple team sports and learned the skills of cooperation, teamwork, problem solving, settling disputes, providing guidance, accepting assistance. These skills were mirrored at home because I have a sister and we would have to get along. When disputes arose, we had to figure out how to co-exist peacefully.

But, as I grew up, I also received conflicting training and information. For 8 hours a day, 5 days a week, 36 weeks a year, I was told that I needed to work only by myself. I could not confer with others, share ideas, or seek out advice from someone with more experience. In the rare instance that I was able to work with someone else, when it came time for the task completion, I had to complete my own and the credit or fault was all mine. The same was true for anyone with whom I may have worked. Work was duplicated and it was evaluated on an individual basis.

Why the two different systems? Why is it different in school than out in the "real world"?

I understand that in the early grades it is a matter of ensuring that each student has assimilated essential skills in order to effectively participate in a team structure and share in the problem solving. It is even essential to do so in the upper grades as new information is introduced in newer, more complex content areas. But if we focus on skill development, once the basics have been introduced, should we, in the educational world, model the skills and project completion models that the business world is asking for?

Having been a part of our career and college readiness, I have heard from many local business leaders that they would prefer a generalist, who has good communication, teamwork, and problem solving skills, and the business can provide them more job-specific training. That models what Thomas Friedman discusses in The World is Flat. Schools will need to meet the flexibility that the working world has discovered the need for.

Why do schools force isolation, in both practice and assessment? Why does the federal government mandate it...without funding.

Let's try something different! Flip the class, PBL, cooperative assessments...something has to give. Leaders need to be supportive as they push instruction out of the standard comfort zone of the adults to move the learning into the comfort zone of the students!

Make the gloom go away

I believe it was Gandhi who said "Be the change you wish to see in the world". It is in that light that I would like to share some low-cost and/or free things that can be done to make the gloom of rainy mornings go away to help improve the school climate within your buildings.

Last night, the #iledchat (9pm CST every Monday) discussed school climate. We had a great turn-out and the moderating/development team of Kathy Melton, Kevin Rubenstein, Jill Maraldo, and I are ecstatic about the increasing involvement of people across the world coming to participate in the weekly chat. If you would like to see the archive of the chat, please check out the storify link.

The discussion began with what can you do as a teacher or leader to create a positive school climate and went from there. It was refreshing to see that so many educators, both classroom teachers and building leaders, saw it was their responsibility to contribute to a positive school climate. They really lived the Gandhi quote. As Mr Z (Josh) stated: "[I] tell myself every day I decided how my day starts".

During the chat we asked the question about what programs/activities did you do that have added to a positive learning environment. There were some great ideas. Below, some of those ideas are included with some of my ideas that were low cost or free that can help improve school climate. While my high school economics teacher would remind me, "Nothing in life is free. There is cost associated with everything". I am focusing on ideas that have low monetary costs.

• 15 minute meetings - Teacher share successes and challenges in a 1:1 meeting.
• Surveys of staff mood with a report out and solution generating sessions
• Surprising a group with treats (homemade or store bought)
• Administrators sub for teachers to let them go home early
• Administrators go out an start cars/scrape ice on snowy, icy days
• Meet teachers in the parking lot with an umbrella on a rainy day
• Hold a staff talent-show for the kids. The experience will bring many people together.
• Know when to listen and when to try and solve a problem, but provide TIME and OPPORTUNITY for this
• Send a quick birthday email
• 'Atta Boy notes when someone does something of note...you decide what is something of note
• Staff friendly competitions (athletic, artistic, trivia)
• Community service projects

These are some ideas. Obviously, there are MANY MORE. Start here to improve your school climate. A principal of mine worked to build a climate around the Fish Philosophy: Be there, Make their day, Play, and Choose your attitude. http://www.charthouse.com/content.aspx?nodeid=22610 How often do you feel you are playing at work? You made the CHOICE to be there? A local (self) change in attitude can permeate throughout the building.

Overall, find a way to make sure EVERYONE get recognized. Everyone is special in some manner.

Where is the line?

I am a big proponent of educators utilizing a variety of social media outlets to participate in professional development, engage students in the learning, and maintain open communication lines with stakeholders concerning important events. Having said that, I have had an occurrence that has got me asking the question of where is the line?

My 2 boys (4 1/2 and 2 1/2) both go to school/daycare. It is a WONDERFUL facility with an amazing staff who genuinely care about the social, emotional, and cognitive growth of all of the children who attend. My youngest son is in the 2-3 year old room and, with any child who is exploring the world but can not fully express himself, there are instances where two children will get into a scuffle. Sometimes it is a little hitting or pushing, sometimes biting, but these are isolated incidents and parents are informed of when they occur. All of the rooms are under video surveillance and in order to protect all parties involved, a parent is notified that their child was involved in an incident but is not informed of who the other child is.

I pick my boys up from school in the afternoon and notice that my youngest son has a scrape on his nose and by his eye. I ask the teacher in the room what happened and she did not know. This is not necessarily uncommon because of the shifts that the teachers have during the work day. She went to go check in the office to see if there was a notification and came back to tell me that there was not one. At this point, I go to the office and speak with the directors asking them to check the video to see what had happened because there is one child in the class who has been having some issues in respecting the personal space and belongings of other children. (The only reason why I know this child is because my son tells me who did it when something happens.) The directors apologized that there was no incident report and said that they would look at the video and get back to me.

Here is where my dilemma started...through Facebook, I am connected to many of the teachers' personal pages that my kids had at the school. As I said, they are wonderful people there and like keeping up with the goings on of the school's families, even after they have left a particular class. My quandary was do I contact the teacher directly through Facebook to see what happened.

I remember when I was a second year teacher and a parent called me a home to yell at me about their child's progress (or lack thereof) and to challenge what I was teaching in class. I remained calm, answered all of her concerns, and then politely told her that if she has further questions or concerns that she should contact me at school via phone or email and do not call me at home again. When I received that phone class at my home, I felt attacked and felt that this parent had broken a line of decency, for lack of a better term, because she had made no attempt to contact me at school.

As I was trying to decide if I should send her a private message about the incident, I received one from the director of the daycare indicating that he saw my son fall down (with no one around him) and that seemed to be the only event that could have caused this incident. He further explained in the message that he would speak with the teachers in the room and remind them of reporting policies and procedures.

Since I had a resolution to this incident and it was cause by my son's inherited grace and balance, I did not contact the teacher via Facebook. When I dropped the kids off the next day, the teacher came directly to me and told me what she knew about the incident and showed me the report that was completed, but had not been filed yet.

Should I have contacted the teacher via her personal page? If she had a work email or classroom page, I would have no issue in initiating the contact. When I thought of my own experience, I felt that contacting her via her personal page would be akin to the phone call that I received at home. But what of the director contacting me?

I viewed this as a contact from the school to a parent in which, as a teacher, I would call the home or business number or email an available address to discuss any issue. I did question why he did it via his personal Facebook account, but did not push the matter.

It just raised some questions. Where is the line of appropriate contact? As an educator, I would not want people posting items to my personal page nor calling me at home uninvited. How much training do we provide for our staffs about issues like this? Connecting with parents and students through personal pages? Are mandates and policies needed? Guidelines?

Regardless, education of all is needed in appropriate ways to establish lines of communication in the hyper connected world.

Revisiting the falling coconuts

In late November, there was a chance for people to win $550 million in the PowerBall Lottery. I wrote a post with some interesting statistics about the chances of winning. You can read that post here: http://misterabrams.blogspot.com/2012/11/watch-out-for-coconuts.html

With all that being said, March Madness is upon us and I have filled out my bracket. I make not pretenses, when I complete this every year, I always end up in the middle of the pack of my friends in "scoring" the bracket. It got me thinking, could I ever pick a bracket that is 100% correct?

If we take a look at the design of the bracket (ignoring the play-in games of yesterday and today), there are 63 games that will happen. (If you need a bracket, you can get one here)

64 teams start the playing tomorrow in 32 games
32 winners will then play each other in 16 games
16 winners will then play each other in   8 games (Sweet 16)
8 winners will play each other in 4 games (Elite 8)
4 winners will play each other in 2 games (Final 4)
The 2 winning teams will then play 1 games for the National Championship

32+16+8+4+2+1=63 games

If we take power rankings and knowledge of teams out of if, the chance of any team winning is 1/2 (although we all know that no #16 seed has ever gotten out of the first round...)

If we take the probability of winning a game and then expand that through the 63 games, you will end up 1/2^63 or .5 raised to the 63 power.

If you evaluate that expression, you end up with an answer of 1 chance in 9,223,372,036,854,780,000. That is 1 in 9 QUINTILLION! (9.22*10^18)

Just to touch back at the PowerBall chance of winnings, you were 3 times more likely to die from a coconut falling on your head and killing you that picking the winning numbers for the PowerBall. The chance of winning the PowerBall was 1 in 175,223,510.

Compare the odds, friends! You are 52 billion times more likely to win the PowerBall than pick a bracket with 100% accuracy.

What does all this mean?

Coconuts are going to be raining down all over people!

Bring this to your classroom. What questions can we have kids investigate?

1. How many USA (population thereof) would it take to equal the odds of 1 person randomly picking the a correct bracket?
2. If chances were M&Ms, how many equivalents of Soldier's Field would be filled?

What questions can the students come up with in investigating these numbers?

The need for STEAM

I will now give fair warning...I am going to get on a soapbox for this post.

http://farm1.staticflickr.com/115/316350341_00239c8fc2_z.jpg?zz=1
 

I was reading through my Zite feed and found this post on how at the heart of every Pixar animation is a computation engine designed with the rules of geometry and physics. This just reinforces that idea to me that schools need to focus on more than STEM (Science, Technology, Engineering, Mathematics) career pathways and redefine them as STEAM (add Art) career pathways.

The POTUS has put a lot of money and policy (Race to the Top) behind the creation of these new STEM career pathways to create a new supply of trained and educated workers for these career fields. Primarily, the major need for these fields is due to the retirement of workers after the last push for these career fields from the Sputnik scare of 1957. With the idea that the Russians were able to create and successfully orbit and artificial satellite during the escalation of the Cold War, Americans had to ensure that we would be second to no one. This resulted in a huge push for more scientists, mathematicians, and engineers, and President Kennedy's decree that we would reach the moon by the end of the decade (1960s).

With it now being 2013, all of those engineers, scientists, and mathematicians have had a very successful 30+ years in the field and are now retiring in droves. As we push forward, we have to recognize that the world has radically changed since 1957.

I was very fortunate to experience a very well rounded education when I was in high school. I had the chance to explore visual as well as performance art in addition to my studies of science, math, and the humanities. I joke with people that I was much smarter when I was in high school because I was able to discuss topics of biology, chemistry, physics, history, literature, and others. Once I went to college, I chose a major and focused on the study of biology and my working knowledge of the other subjects went by the wayside. But, again, I was fortunate to still have the chance to continue a personal pursuit of vocal performance with the Varsity Men's Glee Club and an a cappella group, No Strings Attached.

It is this well rounded education that has enriched my life and my work in education. But a big part of that is that the pursuit of biology, education, and vocal performance were my passions. Educators and policy makers are pushing these STEM career pathways into 6th and 7th grade. As students move through these pathways year to year, it will become progressively difficult to alter pathways the later they go in the schooling. Should we be asking students to select a career pathway in 6th grade?

And why this push? Our test scores are low, compared to other nations. So, how do we fix this? Test more and narrow focus. What happens when a student's passions are not found within STEM? What happens when students grow and develop and their passions change?

If we are able to evolve STEM into STEAM, that would at least provide more students options and open more pathways to students who already like the STEM careers.

Case in point, a friend of mine had quite a bit of talent in the sciences, but she also had a lot of talent in drawing, which is where her passions were (as she was graduating college). Luckily enough, she had good educators around her to guide her to her career in medical illustration. Just as the Pixar article suggested, within the arts, STEM is already present, but not necessarily a conscious part of the career choice process.

Additionally, recent conversations I have had with various colleagues and friends have mentioned how their ability to express themselves, in both written and verbal formats, have allowed them to advance and collect more grant dollars than any of their subject specific trainings. Despite their lack of formal testing under NCLB, the arts need to be emphasized and encouraged.

Education needs to move full STEAM ahead.

Is your degree worth it?!

Saw this from my Twitter feed and it needed to be shared.

Too often students will pick schools or majors for the most trivial reasons. I, myself, picked my major because it was what I liked. I did not put any thought into possible career paths or potential earnings. Even the school I picked, which I LOVE, was picked because I figured that I could live with the choice and it would save my parents some money. Other times, students will pick a major simply because of the potential earnings and have no idea of the amount of work needed in order to be successful.

Hopefully, this infographic will help students find the middle ground between those two areas...
Original Source: Source

Degree Dilemma infographic

Standards...Not Nuggets

When I was in high school, I thought my biology teacher was the best teacher I ever had. He promoted deeper thinking and understanding and challenged us to apply what we were learning to new situations. While it may have been inappropriate, I heard him once say that another teacher was teaching "Gee Whiz" biology where the focus was on the interesting little nuggets that can be found in this area of science as opposed to discovering the connecting thread of all of those ideas.

When I take a look at tests that I have written, I know that I reviewed what was discussed in the unit (which was based on objectives aligned with the learning standards) and wrote questions accordingly. I attempted to have question frequency represent the importance of the topic in relation to the unit and the entire year and the time spent in class discussing and explore the topic. (One noted exception was Hardy-Weinberg equilibrium, but I told students about that ahead of time. We spent over a week learning how to solve these problems, but there were only 2 questions on the unit exam. This was mainly due to my desire to prep the students for Advanced Placement Biology and received positive feedback from both the students and the teacher about this practice...but I digress)

Recently, I was looking at a rating system for a intervention model of a grant application that our district is going to pursue. We were going to have to have the committee members look at particular characteristics and then, based on the characteristics selected, determine which intervention model is the best fit. Because I have been on various committees like this and know that some people, including myself, are a little mouthy and push their ideas onto others, I wanted to find a way to allow everyone to express their own ideas and then look at the group data to protect the integrity of everyone's voice. As I did this, I wrote some equations into the Excel spreadsheet that would, based on the characteristics selected, would determine the percent alignment with each model.

While this is an isolated event, the concept can be expanded to looking at standards and objective-based assessments as opposed to nugget-based. Previously, I wrote a post about providing students with more immediate feedback using Google forms and included some "coding" instructions. I want to expand on that here...

Before, I was writing about grading the entire assessment based on total points. If the questions are written to reflect one specific objective/topic, then the questions can be coded as such (overtly done in my example below) and the scoring can then be adjusted to reflect topic/standard specific questions and their level of mastery. Take a look at the sample below:




You can see that in the 2nd line of each question, I have included an Objective number. Using these, we can then select those questions to score based by objective and get a mastery level based on these particular questions.

What you will need to do is to develop your quiz in the Google docs and then enter the answer key as the first entry in the spreadsheet. When you look at the spreadsheet, you will see the the questions/column headers are in the first row. After the questions, I added the column headers of Objective 1 and Objective 2.

 

The grading formula for Objective 1 is seen below.

=((IF(B2=$B$2,1,0))+(IF(E2=$E$2,1,0))+(IF(F2=$F$2,1,0)))/3

 

Essentially, what the formula is telling the spreadsheet to do is if the entry in cell B2 is the same as $B$2, then give it a score of 1, if not, a score of 0. Now, the difference between B2 and $B$2 is that when you drag this formula down the spreadsheet for every entry, the B2 will change to B3, B4, B5, etc for each subsequent entry; the $ in front of the cell letter and number makes it static, and will not change with a dragging of the formula. (The same is try for C, D, and E).

 

Since we are looking for mastery of Objective 1, I made the scoring formula reflect only the assessment items that were coded for that objective. On the Google form, they are items 1, 4, & 5, which correspond to columns B, E, and F in the spreadsheet. In order to develop a mastery level, the spreadsheet will take the score for each of those items and then divide by 3 because that is the total number of items for this objective. The color coding occurs with some simple conditional formatting. If the numerical value in the Objective 1 column (H) is greater than .5, the background will become green. Likewise, if less than .5, it will become red. Because there are three items, it is impossible to get an answer of .5, which is why I made it the scoring differentiator. For this instance, we will define mastery as 67% or 2 out of 3 questions per objective. In order to get the conditional formatting, you can either right-click in the particular cell or look under the "Format" menu.

 

The same was done for the Objective 2 column (I), but the scoring formula was adjusted to reflect only the objective 2 questions (2, 3, & 6 or columns C, D, and G, respectively).

 

Once students have taken the assessment, you can drag the contents of the cells H2 and I2 down for each entry on the spreadsheet. By using the "$" in the scoring formula, only the cells without the "$" will change to match the subsequent entry lines. The conditional formatting will be dragged down to the subsequent cells also! Unfortunately, you cannot drag the contents before students have made the entry on the form. If you try the assessment, try dragging it yourself by clicking on the spreadsheet link!

 

This can be extended to more questions per objective and more objectives. This can also be expanded to include question types of "Choose all that apply" (using the Check Box option on Google forms), but this will require a little more work.

 

Explore and play.

 

If you have any questions, feel free to contact me via the blog comments or on Twitter (@misterabrams).

Change your perspective

I just got back from a training seminar on how to be a support coach for the Rising Star program. For those unaware, Rising Star is the new school improvement accountability system for the state of Illinois. The whole point of the Rising Star system is for schools and districts to look at the past and current conditions of their systems in order to make appropriate changes (backed by research) to positively impact student achievement. During this training, we discussed monitoring and sustainability of programs.

Part of the emphasis of the training is to look at the programs and steps of implementation and ask the harder questions about what is not working and why. Part of the challenge to making these changes is the dreaded phrase of "we have always done it this way". My quick response would be "Look where that got you", but that is not supportive to the change process. The entire point of Rising Star is to make changes in our past practices to make vast improvements in our schools.

Whenever we are looking to do something different, why do we often start at what we currently do. We must have made the choices to our current location because they seemed to be the best option at the time. (Sometimes it is a best option for a teacher and not necessarily the students, but that is a different conversation). If that is the case, when we look at the choices to do something different, how would we ever arrive at a different conclusion?

Sometimes it takes standing on your head or flipping your frame of reference. Case in point can be seen here:

Deeper thinking can happen when the status quo is challenged and we look for new perspectives. How's this for a change in perspective: If we want students to be more engaged in the learning process, ask them about their passions! Students can provide the adults with all sorts of ways they want to learn and it will be authentic for them. (And, teachers will get more engaged because they won't be able to pass out the same worksheet they have done for the past umpteen years...)

Deeper thinking can be activated and even framed with aspects of the common core and the next generation science standards. We learn science by doing, not by memorizing! We need to have the credo in our classes and schools that FAIL means First Attempt In Learning!

Go back to basics and when you look at the path you used to take, when you took that right before, this time, jump off of the path and go exploring!

Here are some other ways to consider flipping: http://misterabrams.blogspot.com/2012/10/flip-ordinary.html

Technology integration within the Math Common Core State Standards

From my Twitter feed, I came across a blog post from Tara Linney about integrating technology into the ELA Common Core Standards. I thought that it might be useful for mathematics also. Here it is.

Some integration will be implicit and inferred and other ideas will be very explicit. The standards are taken directly from http://CoreStandards.org. The use of algebraic thinking and models within the standards, while not explicitly stating the use of technology, commonly utilizes technology to assist students in developing or presenting their explanations. As we progress into high school, the areas for technology integration increase dramatically due to the complexity of the mathematics involved.

From the Standards for Mathematical Practice

Make sense of problems and persevere in solving them.

"Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem."

2. Reason abstractly and quantitatively

"the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved."

4. Model with mathematics
"They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas."

5. Use appropriate tools strategically
"Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software...When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts."

 

Operations and Algebraic Thinking

 

Kindergarten

CCSS.Math.Content.K.OA.A.2 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.


CCSS.Math.Content.K.OA.A.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).


CCSS.Math.Content.K.OA.A.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

First Grade
CCSS.Math.Content.1.OA.A.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.¹


CCSS.Math.Content.1.OA.A.2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Second Grade
CCSS.Math.Content.2.OA.A.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.¹


CCSS.Math.Content.2.OA.C.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.


CCSS.Math.Content.2.OA.C.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Third Grade
CCSS.Math.Content.3.OA.A.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.¹


Fourth Grade
CCSS.Math.Content.4.OA.A.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.¹


CCSS.Math.Content.4.OA.C.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

Fifth Grade

CCSS.Math.Content.5.OA.B.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

Number & Operations in Base Ten

First Grade

CCSS.Math.Content.1.NBT.C.4 Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

CCSS.Math.Content.1.NBT.C.6 Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Second Grade
CCSS.Math.Content.2.NBT.B.7 Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Third Grade
CCSS.Math.Content.3.NBT.A.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fourth Grade
CCSS.Math.Content.4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.

CCSS.Math.Content.4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

CCSS.Math.Content.4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Fifth GradeCCSS.Math.Content.5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm.

CCSS.Math.Content.5.NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

CCSS.Math.Content.5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.Number & Operations -- Fractions Third GradeCCSS.Math.Content.3.NF.A.3b Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. CCSS.Math.Content.3.NF.A.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Fourth GradeCCSS.Math.Content.4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

CCSS.Math.Content.4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.CCSS.Math.Content.4.NF.B.3b Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.CCSS.Math.Content.4.NF.B.3d Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

CCSS.Math.Content.4.NF.B.4a Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

CCSS.Math.Content.4.NF.B.4b Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

CCSS.Math.Content.4.NF.B.4c Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

CCSS.Math.Content.4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.

Fifth Grade

CCSS.Math.Content.5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

CCSS.Math.Content.5.NF.B.4a Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

CCSS.Math.Content.5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

CCSS.Math.Content.5.NF.B.7a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

CCSS.Math.Content.5.NF.B.7b Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

CCSS.Math.Content.5.NF.B.7c Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Measurement & Data

First Grade

CCSS.Math.Content.1.MD.B.3 Tell and write time in hours and half-hours using analog and digital clocks.

Second Grade

CCSS.Math.Content.2.MD.C.7 Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

Third Grade

CCSS.Math.Content.3.MD.C.7c Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.

High School

Calculate expected values and use them to solve problems

CCSS.Math.Content.HSS-MD.A.1 (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

CCSS.Math.Content.HSS-MD.A.2 (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

CCSS.Math.Content.HSS-MD.A.3 (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.

CCSS.Math.Content.HSS-MD.A.4 (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?

Use probability to evaluate outcomes of decisions

CCSS.Math.Content.HSS-MD.B.5 (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

CCSS.Math.Content.HSS-MD.B.5a Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

CCSS.Math.Content.HSS-MD.B.5b Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

CCSS.Math.Content.HSS-MD.B.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

CCSS.Math.Content.HSS-MD.B.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Geometry

Kindergarten

CCSS.Math.Content.K.G.B.5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.

CCSS.Math.Content.K.G.B.6 Compose simple shapes to form larger shapes. For example, “Can you join these two triangles with full sides touching to make a rectangle?”

Fifth Grade

CCSS.Math.Content.5.G.A.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Seventh Grade

Draw construct, and describe geometrical figures and describe the relationships between them.

CCSS.Math.Content.7.G.A.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

CCSS.Math.Content.7.G.A.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

CCSS.Math.Content.7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

CCSS.Math.Content.7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

CCSS.Math.Content.7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

CCSS.Math.Content.7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

Eighth Grade

Understand congruence and similarity using physical models, transparencies, or geometry software.

CCSS.Math.Content.8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations:

CCSS.Math.Content.8.G.A.1a Lines are taken to lines, and line segments to line segments of the same length.

CCSS.Math.Content.8.G.A.1b Angles are taken to angles of the same measure.

CCSS.Math.Content.8.G.A.1c Parallel lines are taken to parallel lines.

CCSS.Math.Content.8.G.A.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

CCSS.Math.Content.8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

CCSS.Math.Content.8.G.A.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

CCSS.Math.Content.8.G.A.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Understand and apply the Pythagorean Theorem.

CCSS.Math.Content.8.G.B.6 Explain a proof of the Pythagorean Theorem and its converse.

CCSS.Math.Content.8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

CCSS.Math.Content.8.G.B.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

CCSS.Math.Content.8.G.C.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Ratios & Proportional Relationships

Seventh Grade

CCSS.Math.Content.7.RP.A.2a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

CCSS.Math.Content.7.RP.A.2d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

The Number System

Sixth Grade

CCSS.Math.Content.6.NS.A.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?.

CCSS.Math.Content.6.NS.C.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

Eigth Grade

CCSS.Math.Content.8.NS.A.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

CCSS.Math.Content.8.NS.A.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

Expressions & Equations

Sixth Grade

CCSS.Math.Content.6.EE.C.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

Seventh Grade

CCSS.Math.Content.7.EE.B.4b Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

Eigth Grade

CCSS.Math.Content.8.EE.A.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

CCSS.Math.Content.8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

CCSS.Math.Content.8.EE.C.8a Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

CCSS.Math.Content.8.EE.C.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

Functions

Eigth Grade

Define, evaluate, and compare functions.

CCSS.Math.Content.8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.¹

CCSS.Math.Content.8.F.A.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

CCSS.Math.Content.8.F.A.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

Use functions to model relationships between quantities.

CCSS.Math.Content.8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

CCSS.Math.Content.8.F.B.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Statistics & Probability

Seventh Grade

CCSS.Math.Content.7.SP.C.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

CCSS.Math.Content.7.SP.C.7a Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.

CCSS.Math.Content.7.SP.C.7b Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?

Eigth Grade

CCSS.Math.Content.8.SP.A.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

CCSS.Math.Content.8.SP.A.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

High School Number & Quantity

CCSS.Math.Content.HSN-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.

Perform arithmetic operations with complex numbers.

CCSS.Math.Content.HSN-CN.A.1 Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.

CCSS.Math.Content.HSN-CN.A.2 Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

CCSS.Math.Content.HSN-CN.A.3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

Represent complex numbers and their operations on the complex plane.

CCSS.Math.Content.HSN-CN.B.4 (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

CCSS.Math.Content.HSN-CN.B.5 (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 i)³ = 8 because (-1 + √3 i) has modulus 2 and argument 120°.

CCSS.Math.Content.HSN-CN.B.6 (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

Use complex numbers in polynomial identities and equations.

CCSS.Math.Content.HSN-CN.C.7 Solve quadratic equations with real coefficients that have complex solutions.

CCSS.Math.Content.HSN-CN.C.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x – 2i).

CCSS.Math.Content.HSN-CN.C.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

Represent and model with vector quantities.

CCSS.Math.Content.HSN-VM.A.1 (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

CCSS.Math.Content.HSN-VM.A.2 (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

CCSS.Math.Content.HSN-VM.A.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.

Perform operations on vectors.

CCSS.Math.Content.HSN-VM.B.4 (+) Add and subtract vectors.

CCSS.Math.Content.HSN-VM.B.4a Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

CCSS.Math.Content.HSN-VM.B.4b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

CCSS.Math.Content.HSN-VM.B.4c Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

CCSS.Math.Content.HSN-VM.B.5 (+) Multiply a vector by a scalar.

CCSS.Math.Content.HSN-VM.B.5a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v_x, v_y) = (cv_x, cv_y).

CCSS.Math.Content.HSN-VM.B.5b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

Perform operations on matrices and use matrices in applications.

CCSS.Math.Content.HSN-VM.C.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

CCSS.Math.Content.HSN-VM.C.7 (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

CCSS.Math.Content.HSN-VM.C.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.

CCSS.Math.Content.HSN-VM.C.9 (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

CCSS.Math.Content.HSN-VM.C.10 (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

CCSS.Math.Content.HSN-VM.C.11 (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

CCSS.Math.Content.HSN-VM.C.12 (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.

High School: Algebra

CCSS.Math.Content.HSA-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.

CCSS.Math.Content.HSA-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

CCSS.Math.Content.HSA-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

CCSS.Math.Content.HSA-REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

CCSS.Math.Content.HSA-REI.C.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable.

CCSS.Math.Content.HSA-REI.C.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

CCSS.Math.Content.HSA-REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

CCSS.Math.Content.HSA-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions

CCSS.Math.Content.HSA-REI.D.12 Graph the solutions to 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 to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

High School: Functions

Interpret functions that arise in applications in terms of the context.

CCSS.Math.Content.HSF-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

CCSS.Math.Content.HSF-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

CCSS.Math.Content.HSF-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Analyze functions using different representations.

CCSS.Math.Content.HSF-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

CCSS.Math.Content.HSF-IF.C.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.

CCSS.Math.Content.HSF-IF.C.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

CCSS.Math.Content.HSF-IF.C.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

CCSS.Math.Content.HSF-IF.C.7d (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

CCSS.Math.Content.HSF-IF.C.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

CCSS.Math.Content.HSF-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

CCSS.Math.Content.HSF-IF.C.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

CCSS.Math.Content.HSF-IF.C.8b Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.

CCSS.Math.Content.HSF-IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

 

Build a function that models a relationship between two quantities.

CCSS.Math.Content.HSF-BF.A.1 Write a function that describes a relationship between two quantities.

CCSS.Math.Content.HSF-BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context.

CCSS.Math.Content.HSF-BF.A.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

CCSS.Math.Content.HSF-BF.A.1c (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

CCSS.Math.Content.HSF-BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Build new functions from existing functions.

CCSS.Math.Content.HSF-BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

CCSS.Math.Content.HSF-BF.B.4 Find inverse functions.

CCSS.Math.Content.HSF-BF.B.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1.

CCSS.Math.Content.HSF-BF.B.4b (+) Verify by composition that one function is the inverse of another.

CCSS.Math.Content.HSF-BF.B.4c (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.

CCSS.Math.Content.HSF-BF.B.4d (+) Produce an invertible function from a non-invertible function by restricting the domain.

CCSS.Math.Content.HSF-BF.B.5 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Construct and compare linear, quadratic, and exponential models and solve problems.

CCSS.Math.Content.HSF-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.

CCSS.Math.Content.HSF-LE.A.1a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

CCSS.Math.Content.HSF-LE.A.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

CCSS.Math.Content.HSF-LE.A.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

CCSS.Math.Content.HSF-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

CCSS.Math.Content.HSF-LE.A.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

CCSS.Math.Content.HSF-LE.A.4 For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Interpret expressions for functions in terms of the situation they model.

CCSS.Math.Content.HSF-LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context.

Extend the domain of trigonometric functions using the unit circle.

CCSS.Math.Content.HSF-TF.A.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

CCSS.Math.Content.HSF-TF.A.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

CCSS.Math.Content.HSF-TF.A.3 (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number.

CCSS.Math.Content.HSF-TF.A.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Model periodic phenomena with trigonometric functions.

CCSS.Math.Content.HSF-TF.B.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

CCSS.Math.Content.HSF-TF.B.6 (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

CCSS.Math.Content.HSF-TF.B.7 (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

Prove and apply trigonometric identities.

CCSS.Math.Content.HSF-TF.C.8 Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

CCSS.Math.Content.HSF-TF.C.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

High School: Modeling

One of the insights provided by mathematical modeling is that essentially the same mathematical or statistical structure can sometimes model seemingly different situations. Models can also shed light on the mathematical structures themselves, for example, as when a model of bacterial growth makes more vivid the explosive growth of the exponential function.

The basic modeling cycle is summarized in the diagram. It involves (1) identifying variables in the situation and selecting those that represent essential features, (2) formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables, (3) analyzing and performing operations on these relationships to draw conclusions, (4) interpreting the results of the mathematics in terms of the original situation, (5) validating the conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable, (6) reporting on the conclusions and the reasoning behind them.

High School: Geometry

CCSS.Math.Content.HSG-CO.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

CCSS.Math.Content.HSG-CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

CCSS.Math.Content.HSG-CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

CCSS.Math.Content.HSG-CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

CCSS.Math.Content.HSG-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

CCSS.Math.Content.HSG-CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

CCSS.Math.Content.HSG-CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

CCSS.Math.Content.HSG-SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor:

CCSS.Math.Content.HSG-SRT.A.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

CCSS.Math.Content.HSG-SRT.A.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

CCSS.Math.Content.HSG-SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

CCSS.Math.Content.HSG-SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

CCSS.Math.Content.HSG-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

CCSS.Math.Content.HSG-C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

CCSS.Math.Content.HSG-C.A.4 (+) Construct a tangent line from a point outside a given circle to the circle.

CCSS.Math.Content.HSG-C.B.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector..

CCSS.Math.Content.HSG-GPE.A.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

CCSS.Math.Content.HSG-GPE.A.2 Derive the equation of a parabola given a focus and directrix.

CCSS.Math.Content.HSG-GPE.A.3 (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

CCSS.Math.Content.HSG-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

CCSS.Math.Content.HSG-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

CCSS.Math.Content.HSG-MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

CCSS.Math.Content.HSG-MG.A.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

High School: Statistics & Probability

Summarize, represent, and interpret data on a single count or measurement variable

CCSS.Math.Content.HSS-ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

CCSS.Math.Content.HSS-ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

CCSS.Math.Content.HSS-ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

CCSS.Math.Content.HSS-ID.A.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Summarize, represent, and interpret data on two categorical and quantitative variables

CCSS.Math.Content.HSS-ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

CCSS.Math.Content.HSS-ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

CCSS.Math.Content.HSS-ID.B.6a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

CCSS.Math.Content.HSS-ID.B.6b Informally assess the fit of a function by plotting and analyzing residuals.

CCSS.Math.Content.HSS-ID.B.6c Fit a linear function for a scatter plot that suggests a linear association.

Interpret linear models

CCSS.Math.Content.HSS-ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

CCSS.Math.Content.HSS-ID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.

CCSS.Math.Content.HSS-ID.C.9 Distinguish between correlation and causation.

CCSS.Math.Content.HSS-IC.A.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?

CCSS.Math.Content.HSS-IC.B.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

CCSS.Math.Content.HSS-IC.B.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

CCSS.Math.Content.HSS-IC.B.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

CCSS.Math.Content.HSS-IC.B.6 Evaluate reports based on data.

CCSS.Math.Content.HSS-CP.A.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

 

CCSS.Math.Content.HSS-CP.B.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

CCSS.Math.Content.HSS-CP.B.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

CCSS.Math.Content.HSS-CP.B.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

CCSS.Math.Content.HSS-CP.B.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.