RSS
 

Posts Tagged ‘Math’

What if… we did math right?

24 Oct

What if we made developmentally sequencel goals that we want our children to reach.

what if these goals includes procedures and concepts?

what if we designed learning material to help students achieve these goals and had multiple approaches to teaching allowing the child to work with the easiest approach for their learning style.

What if the students got lots of practice and teacher feedback.

what if students could pretest themselves to see if they need more time or are ready for a test.

what if we provide more time and resources when a child isn’t yet ready instead of just having then fail a test.

what if they take a test and we spend time afterwards, helping them correct and understand what they got wrong.

what if we don’t move on to the next unit until we have a certain level of mastery with the current unit.

what if we continue to include previous material in new units so students don’t forget older topics and retain their mastery.

why don’t we focus on quality over quantity and not worry about kids memorizing formulas but instead can they correctly use and apply them.

teachers can be guides and facilitators  in the classroom, kids who are further along can help give lessons to their peers, online resources can be used to free up the traditional approach.

Assessments can contain questions that require many different types of learning, procedural, conceptual, appplications, theorical, experimental, etc.

why can’t we change how we do math? It seems so simple.

 

 

 

 
 

Quality over Quantity: A New idea for Math Education

08 Aug

“I can’t do Math,” you hear this said over and over by children and adults alike.  It is even “acceptable” to tout that “math isn’t your thing.  You don’t hear people saying, “I can’t read,” yet it is okay to almost brag that you can’t do mathematics.  Why is math such a hard subject for our country?  When and how does it fall apart?  As a math educator, I see so many solutions to our national math crisis that have just never been tried.  We always seem to just play around with the ideas of “the basics,” learning the concepts behind the mathematics (conceptual learning, new math, mathematical modeling), and procedural learning (very similar to the “basics” in many ways.)  All those things are important and we have a problem of tending to lean to one side vs. the other rather than keeping a reasonable balance between the two.  However, what I see as the biggest problem is looking at, “what is our ultimate goal?”  When I read an article that says a California College has done away with the requirement that all students must show mastery in Intermediate Algebra for college because non-STEM students don’t need math, it gets me thinking.

 

If non-STEM majors don’t need math, then do STEM majors no longer need to take literature classes and humanities classes required in the general education classes, these are not “needed” for their majors?  Why do undergraduate degrees require students to take general education classes in addition to their major focus?  We know the answer. It is the same reason why high schools require 4 English classes, 4 Math classes, 3-4 Science classes, 3-4 History classes, etc., in order to make a well rounded educated person.  Just like English, knowing math provides a level of competence for getting around in the world, it allows you to think critically, math is used in many places that kids don’t realize until they get to be an adult.  Adults who truly understand Intermediate Algebra, will be able to make more sound financial decisions in their own personal financial choices.

 

Additionally, Intermediate Algebra as a prerequisite for a college level math course, shouldn’t be too hard since Intermediate Algebra is a class that should be mastered in high school.  So, why is a high school math giving college students so much trouble that a college has to drop a high school remedial math class requirement?  This is because how we currently teach high school math is a failure.  Let’s face it, some students will struggle more with mathematical concepts and others will move on and take Calculus 3 before graduating high school.  There is nothing wrong with either student but we act like there is and we need to stop this.  We need to stop putting on kids on the same math trajectory and expecting it to work.  

 

My feeling is that the goal for graduation of high is to pass, with a B or better, Intermediate Algebra (which should replace tedious useless work with real world knowledge like understanding the Normal Distribution so you can talk intelligently about IQ scores and statistical research as well as linking concepts to real world like amortization tables for car loans and mortgages, these ideas are more important that long division of polynomials and adding rational fractions which is tedious.)  Students should be able to take the “slow path” to math if they need it where they learn the main topics in Algebra 1, some lighter topics in Algebra 2, and some of the basic ideas of Geometry (no geometric proofs).  The goal would be mastery of these topics at a B level.  Anyone graduating should be ready to prove their understanding to a college prerequisite test and be ready for a Pre-Calculus class, although, if they are not a STEM major, they may choose Statistics or Financial Math.  

 

Right now, in NC, we require students to take 4 years of math.  They start learning Algebra 1 concepts as early as middle school so that once in high school, they are already learning topics in Algebra 1, Geometry, and some starter Statistical topics.  By Math 2, they are being introduced to Trigonometry, Algebra 2, Probability, more Geometry, including proofs, and a small amount of what used to be in Pre-Calculus.  By Math 3, the students are finishing Algebra 2, finishing Geometry, taking on more topics from Pre-Calculus, and adding in more Statistical topics.  After Math 3, students must take a fourth math course.  Most non Honors students take Intro to College math, which ends up being a review of Algebra topics or Discrete Math, which goes into Probability, Statistics, and Decision Making.  If students were mastering all these topics, this would be wonderful but students are barely grasping all of this.  We need to slow it down and cut out the fourth class, cut stuff from Math 1-3, and although still require 4 full years (not semester blocks) of math for the non-Honors track but focus on QUALITY of instruction and MASTERY of learning, rather than QUANTITY of material we can “say” they were exposed to.  We will have students who learn more, are less stressed, and have a higher success rate in future math courses in college.

 

Students who are on the Honors track, can continue to be on their own schedule as they should not be slowed down.  They can meet that high school requirement while in middle school, take the “test” showing mastery and once they get to high school, they can be moving on to learning the missing pieces from Algebra 2, Geometry (with proofs), and move into modeling classes or Pre Calculus, Calculus, Statistics, and beyond.

 

Written by:

Lynne Gregorio, Ph.D. Mathematics Education

 
 

Learning Mathematics Through Achievement Learning Based Model

07 Apr

What is an Achievement Learning Based Model in Mathematics?  This is a developmental model where the student has mastery of one topic before moving onto the next topic.  In using this model, one would first need to identify your academic goals.  For most educational institutions, the academic goals is that the student learn and maintain that learning.  Currently, most school systems use a model of a set of curriculum standards, currently the Common Core standards in much of the United States for Mathematics, and students are exposed to a classroom with objectives to teach these concepts that lead to the understanding and mastery of the standards.  Students start in Kindergarten and each school year is generally broken into 4 quarters where teachers are given a pacing guide so that they can get through all the objectives needed to FINISH the standards for each grade level.  Students are given classroom assessments along the way, including possibly quarterly benchmark tests and in later grades some type of End of Year test that measures the mastery of the standards for students.

The teachers are generally not allowed much freedom in deviating from this pacing guide and required topics that they must get through since these are all prerequisites for the knowledge base students will build on during the following year.  In the early elementary years, United States students actually do fairly well overall keeping up with the pace, although, there has been much controversy over the Common Core Standards for many different reasons including political reasons, lack of testing, and pedagogical methodology.  At some point, however, students (probably almost all students at one time or another) will come across a concept, unit, or topic either in elementary school, middle school, or high school where they will get confused.  There is no time built in for teachers to realize that the class is “lost” and they need to spend longer on a specific topic.  There isn’t even the flexibility for teachers to “speed up” when topics are easy so that they can slow down later when needed.  I have seen teachers finish a unit early and just give students “free time” since they were off their pacing guide.  Teachers are not being taught about “reading” students for understanding and “how to adjust” their lessons based on individual class feedback when taking methods classes because this isn’t even allowed anymore.  Teaching is no longer an art that allows for creativity and talent and therefore all the good teachers are leaving the profession in droves.

Let’s pretend for a minute that we care about student learning as our main goal.  If we consider an Achievement Learning Based Model, we can put student achievement before our need for control, before our need for cattle car education, before our need for convenience.  An Achievement Learning Model would require more work but in the age of technology, it is so very doable.  Many schools are considering a flipped curriculum these days.  A flipped curriculum if done correctly works like this:  lessons for each unit are taped, students watch the tapes for homework and then when they come to class, they spend classtime doing active learning with a teacher available for help.  Note, this is not time when the teacher sits and grades papers or takes time off, the teacher is actively participating with the students but it allows them time to have someone help with the active learning part, working problems rather than the static part of learning, watching the lesson.

Here is how the Achievement Learning Model can be added into the flipped curriculum model.  Let’s say that you have a group of high school freshmen who are taking Common Core Math 1 or Algebra 1 (we will just refer to to it as CCM1 here.)  Students will get a goal sheet of the units they need to cover, homework needed to turn in, and assessments they need to complete.  Students will watch the lesson at home.  The next day they come into the classroom and they work on problems assigned to them.  They start with easy problems and get problems that get more difficult as they are successful.  Once they are getting enough problems correct, they move onto to the second unit.  For some students this might be one day, for other students it might take longer.  When it gets close to quiz time, the student takes a practice quiz and self corrects the quiz.  The goal is that they don’t take an assessment until they are having success with their homework and practice quizzes.  If they did well (show mastery), they take their quiz, if not, they work more problems, get more help.  Each student works at his or her own pace.  However, the teacher does oversee the pace of each student and certain requirements are placed on students who are not putting in the effort (which is different from those struggling with the content).  At school, there are after school hours in place for students to come in and continue the same “work” they would do in class.  Every student will be successful since they don’t move on until they have shown success.  The goal is 4 years of math so students “take” math every semester, where a student ends up in their knowledge base will be different for every student.   At the end of CCM1, some will have finished the course and be ready to take the final exam.  If any finish early, they will be helpers to the remaining students!  What a great way to reinforce their knowledge.  If a student does not finish, they can continue in CCM1 the next year until it is complete and move into the CCM2 whenever they finish and start there.

Bright students may have a schedule that looks like this:

Block scheduling:  (just a sample)

First Semester        Second Semester

CCM 1                            CCM2

CCM3                             Precalculus

Calc AB                         Calc BC

AP Stat                          CCM – helper

 

 

A slower student might look like this:

First Semester           Second Semester

CCM1                               CCM1

CCM1                               CCM2

CCM2                               CCM2

CCM2                               CCM3

CCM3                               CCM3

 

Each student takes math every semester, each student has mastery but they get to do things at the pace they need and they will know far more mathematics than our current model where many get D’s and forget what they have learned.   This model needs the following to be successful:

1.  A good teacher who is excellent at explaining the content on the videos is easy to follow steps and includes problems for the students to “practice” while watching the video that shows that the student watched and paid attention to the video.

2.  A good curriculum writer who can create good practice problems so that students can have sufficient practice until they reach mastery with problems starting easy and getting more difficult and have practice quizzes and tests for students to take so that they know when they are ready for the real exam and ready to move forward.

3.  Teacher education where teachers are taught how to manage this new type of classroom, facilitate appropriate groupings among students working on the same topics, “read” students so they know who knows what and who is confused, be able to delegate helper students from within the class to students who need help, to be able to provide the best use of their time during the regular “workshop” settings of daily education.

4.  Test to see at what age students would be mature enough to handle “self” learning, although it will be new and a great skill that students will be learning so it is expected that students will have a normal adjustment period despite maturity issues.

 

Written by:

 

Lynne Gregorio, Ph.D.

 
 

The problems with Common Core Math 2 – why students are failing

18 Jan

As I watched one of the students I tutor fail Common Core Math 2 for the semester, I can’t help but ask myself, what did I do wrong?  Another student, passed the class (with an A, I believe) but failed the exam (the teacher chose not to count the exam towards the course grade).  Any quality teacher who has students “fail,” should ALWAYS ask themselves, “What changes do I need to make?”

So, I pondered this question.  I looked at the type of student that each student was, the type of teacher each student had, the school each student attended, and the work we did together in our tutoring sessions.  I was able to find my answer after these considerations.

The first piece of the puzzle comes from the type of student I worked with.  One student was a student who spent many hours studying and lots of extra effort above and beyond our tutoring sessions.  I also like to rank each student with how “easy” math comes to them.  On a scale from 1 to 10, with a 1 representing a student who really struggles to grasp mathematical concepts, struggles with number sense, and just doesn’t have a logic / math brain to a 10 where the student just “gets” math without even trying, math just makes sense automatically and is like breathing, I will rank each student to provide prospective.  This student is probably around a  7.  The other student did not spend any time outside of our tutoring sessions working on math, didn’t really see doing well in math as a priority and ranks lower around a 5.  She doesn’t get totally lost but can’t seem to put the ideas together and connect them.  She also doesn’t spend time memorizing what is needed to do well.

The second piece of the puzzle comes from the school system, school and teachers these students have.  Both students are in the same school system but at different schools.  One school clearly has higher expectations than the other school and tends to ask harder questions on tests.  Neither teacher seemed “terrible” or “good.”  Neither teacher seemed to care too much about the success of their students based on my interactions / discussions with their families.

These previous two pieces certainly play a role in student success.  From me, both students got the same help from me but students need to spend outside time studying, memorizing, and practicing problems to be successful.  However, one of the biggest challenges I see is that the pace of the curriculum, especially since this school system uses block scheduling (math classes are 90 minutes a day and an entire math class is completed in 1/2 year).  If you have a student who ranks a 7 or above, they can probably handle the pace of learning Common Core 2 in one semester but for students who struggle with math (especially for those with weak math backgrounds, poor number sense, poor study habits, etc.) expecting them to be able to pass Common Core Math 2 in one semester is akin to expecting someone to become an expert on Calculus in 10 days of lessons.  There is only so fast someone can learn information and that is not being taken into consideration.  Why aren’t we offering a Common Core Math 2A and 2B class for students who need to learn at a slower pace?  They did this for Common Core Math 1 (in fact that is your only option in our county) but for Common Core Math 2, your only option is to learn the entire content in 1 semester.  For bright / math minded students, it is a good option and should remain so that these students can move ahead and take AP Calculus and AP Statistics during high school but for the average or below average student (in mathematics), we need to offer math at a slower pace.

Currently, we pass students on with a D in Common Core 1 into this fast paced Common Core 2 class.  Students with a D in Common Core 1, are not prepared to even take the content of Common Core math 2, let alone take it at the pace of 1 semester.  Many of these D students were “gifted” their D’s as I have witnessed.  I have students who can’t solve a basic linear equation on their own, couldn’t tell you the difference between linear, quadratic, and exponential equations, and couldn’t solve or graph any quadratics receive a D in the course and now I know they will be completely lost and unsuccessful in Common Core 2 because they do not have any of the prerequisite knowledge needed for success in Common Core 2.  Yet, teachers continue to “pass” students along because they can’t “fail” too many students or they will get in trouble with the administration.

We seem to forget what the goal is.  Do we want to just pass students along or do we want them to have an understanding of mathematics that makes them college ready?  If we need to slow things down, allow students more time, allow students to repeat classes, then this is what we should do.  We also continue to allow lateral entry teachers because we are short on math teachers, yet we don’t value them.  Lateral entry teachers (and many current teachers) seem to lack the skills needed to help students learn how to study mathematics, another important step for success.  Rarely do I see students come to me with a list of topics they will be covering, review sheets with problems and solutions that are representative of what they will be tested on for quizzes, tests, and finals.  If students had these materials, they could learn more effective ways to prepare for mathematics assessments and be more successful instead most of my students have no idea what to expect on their assessments and no problems that are representative of what they are supposed to know and practice right before an exam.

 
 

New ways to help students be successful in mathematics

07 Nov

Although this idea applies at any age level, I am going to direct at the age I work most closely with – high school students.  Most of my students are North Carolina students taking the new Common Core 1, 2, and next year they will start Common Core 3.  I am slowly watching as the scope and sequence that North Carolina has chosen unfolds for each level.  Common Core 1 contains many Algebra 1 topics and Common Core 2 is more heavy on Algebra 2 than on Geometry.  I find that many of my students can do the work in isolation.  By this, I mean, I teach them a topic – for example solving exponent equations that do not require logs:  3^(x+1) * 9^(2x -3) = 9^(3x+4) and once I teach the topic, they get it and are able to do it!  They can complete a full page of work and get the answers right.  The next day, students will come in with another topic, maybe finding the inverse of functions.  I teach them the rule (exchange x and y, solve for the new y).  They then do a page of those problems correctly.  Prior to the test, students get many different topics in “isolation” from their teachers that we cover and they are successful at.  However, when the test comes, they fail!  Why?

Students don’t remember which problems require which steps.  They don’t spend time memorizing what type of problems match which type of problem solving skills and they seem to lack the ability to just look at a problem and use their overall knowledge of math (a long list of other math skills they have forgotten from previous years) to be able to reason out the answer.  Many times, they don’t even know what the question is asking. ” Oh, I did a page of problems where I switched x and y and then solved for the new y, but that was called an INVERSE?”

How do we fix this?  Well, it really needs to be fixed retroactively as students need to remember all their old skills as they move on and apply the new skills.  If one of their problems has a (1/81) and they are supposed to covert that to 3 to a power in Common Core 2, but they forgot all about negative exponents they learned in Common Core 1, then they are adding double the work.  They have the new skill to learn about solving the exponent equations and they have to re-learn all their exponent rules.  Hopefully, they are just “dusty,” and it doesn’t take too much and you can remind them that fractions mean negative exponents.

Moving away from the retroactive problem, let’s just focus on the best fix we can do.  When teachers assign homework, they assign a page of all problems from the current isolated area.  Solve 15 problems of exponent equations on Monday, solve 15 problems of inverses on Tuesday, solve 15 problems of solving radical equations on Wednesday, etc.  If in addition to this, teachers each night gave students a sheet with one question from each area to solve and the “wording” of how it will be asked of them – for example, in trig, we say, “Solve the triangle.”  What does that mean?  It means find all sides and angles, well if students don’t know, they will find out before the test!  Teachers forget that the goal is to help the students learn the material BEFORE we assess them!!

So, a student will be essentially be given “baby review” sheets all along and I wouldn’t even limit the questions to just the test questions since many of these students forget everything but if they do one problem each night from a section, it will keep them fresh.  How do I complete the square?  How do I find the vertex of a quadratic when it is in standard form?  How do I solve exponential equations?  How do I factor when the leading coefficient is not 1?  How do I factor difference of perfect squares?  Math asks you to remember a lot!!  We need to show kids how to do it!  We need to help them be successful.  If we can model good study habits, when they go to college, they will use these on their own.

If you are a parent with a struggling student, begin to make up (or hire a tutor) to make up baby review sheets that ask one problem of each type of thing the student should know.  When you see they are finally solid on a topic, remove it and just add it back randomly as a check!  Good luck!

 
2 Comments

Posted in Education

 

Recipe to create Common Core Math 1, 2, and 3 for High School

17 Sep

Ingredients:

All topics in Algebra 1, Geometry, and Algebra 2

 

Directions:

1.  Dissect all topics in Algebra 1, Geometry, and Algebra 2 and break down into individual objectives.

2.  Rewrite each objective so that it reads like it is more in-depth , remove most of the algorithmic processes, and use the word “modeling” as much as possible.

Note:  If done correctly, you should now have about 200 objectives

3.  Make a chart that lists all objectives and pick different subsets of the 200 objectives and put them under labels for Common Core 1, 2, and 3.

4.  Be sure that each state and each district/county within a state has a different subset of objectives for each course so students can never successfully move between states or counties.

5.  Also be sure each course has a set of 5 disjoint groups with objectives under each group such as Algebra, Geometry, Probability,Trigonometry etc.  so that students jump from topic to topic rather than learn in a linear fashion.  Math is NOT allowed to be linear,  it should remain disjointed as much as possible.

6.  Now you have Common Core 1, Common Core 2, and Common Core 3.

Other notes for a successful recipe:

*  Try not to use textbooks

* Don’t give students any reference material to follow that relates to each topic as they do their homework every night

* Don’t align assessments with the standards or let a third party make assessments for you

* Make the objectives so difficult to follow that parents don’t have a clue what their students are doing and can’t provide support at home

 

Example of Chart of Creating Common Core

One states choices for their objectives

Another states choices for their objectives

(Examples of disconnect:  Illinois – circles are covered in CCM2, in North Carolina, circles are covered in CCM3.  Some states are covering Exponential functions in CCM1, others are waiting until CCM2.  Some are doing more Geometry in CCM1, others in CCM2, and some save most of the Geometry for CCM3!  Other states are still following Common Core but on a traditional path which means using the Common Core objectives but within the context of Algebra 1, Geometry, and Algebra 2 .)

 

This is my lovely state – they won’t give me a link but a download only so I posted their whole curriculum here.

Math I

 

The Real Number System    N-RN

Extend the properties of exponents to rational exponents.

N-RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

 

N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

 

Note: At this level, focus on fractional exponents with a numerator of 1.

 

Quantities       N-Q

Reason quantitatively and use units to solve problems.

N-Q.1Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

 

N-Q.2Define appropriate quantities for the purpose of descriptive modeling.

 

N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

 

Seeing Structure in Expressions      A-SSE

Interpret the structure of expressions.

A-SSE.1Interpret expressions that represent a quantity in terms of its context.

  1. Interpret parts of an expression, such as terms, factors, and coefficients.
  2. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

 

Note:  At this level, limit to linear expressions, exponential expressions with integer exponents and quadratic expressions.

 

A-SSE.2Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

 

Write expressions in equivalent forms to solve problems.

A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

  1. Factor a quadratic expression to reveal the zeros of the function it defines.

 

Note:  At this level, the limit is quadratic expressions of the form ax2 + bx + c.

 

 

Arithmetic with Polynomials & Rational Expressions       A-APR

Perform arithmetic operations on polynomials.

A-APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

 

Note: At this level, limit to addition and subtraction of quadratics and multiplication of linear expressions.

 

Creating Equations   A-CED

Create equations that describe numbers or relationships.

A-CED.1Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

 

Note:  At this level, focus on linear and exponential functions.

 

A-CED.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

 

Note:  At this level, focus on linear, exponential and quadratic.  Limit to situations that involve evaluating exponential functions for integer inputs. 

 

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

 

Note:  At this level, limit to linear equations and inequalities.

 

A-CED.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

 

Note:  At this level, limit to formulas that are linear in the variable of interest, or to formulas involving squared or cubed variables.

 

Reasoning with Equations & Inequalities   A-REI

Understand solving equations as a process of reasoning and explain the reasoning.

A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

 

Solve equations and inequalities in one variable.

A-REI.3Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

 

Solve systems of equations.

A-REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

 

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

 

Represent and solve equations and inequalities graphically.

A-REI.10Understand 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).

 

Note:  At this level, focus on linear and exponential equations.

 

A-REI.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.

 

Note:  At this level, focus on linear and exponential functions.

 

A-REI.12Graph 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.

 

Interpreting Functions          F-IF

Understand the concept of a function and use function notation.

F-IF.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

 

F-IF.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

 

Note:  At this level, the focus is linear and exponential functions.

 

F-IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

 

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

F-IF.4For 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.

 

Note:  At this level, focus on linear, exponential and quadratic functions; no end behavior or periodicity.

 

 

F-IF.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.

 

Note:  At this level, focus on linear and exponential functions.

 

F-IF.6Calculate 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.

 

Note:  At this level, focus on linear functions and exponential functions whose domain is a subset of the integers. 

 

Analyze functions using different representations.

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

  1. Graph linear and quadratic functions and show intercepts, maxima, and minima.
  2. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

 

Note:  At this level, for part e, focus on exponential functions only.

 

F-IF.8Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

  1. 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.

 

Note:  At this level, only factoring expressions of the form ax2 + bx +c, is expected. Completing the square is not addressed at this level.

 

  1. 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.

 

F-IF.9Compare 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.

 

Note:  At this level, focus on linear, exponential, and quadratic functions.

 

Building Functions    F-BF

Build a function that models a relationship between two quantities.

F-BF.1Write a function that describes a relationship between two quantities.

  1. Determine an explicit expression, a recursive process, or steps for calculation from a context.
  2. 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.

 

Note: At this level, limit to addition or subtraction of constant to linear, exponential or quadratic functions or addition of linear functions to linear or quadratic functions.

 

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

 

Note: At this level, formal recursive notation is not used. Instead, use of informal recursive notation (such as NEXT = NOW + 5 starting at 3) is intended.

 

Build new functions from existing functions.

F-BF.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.

 

Note: At this level, limit to vertical and horizontal translations of linear and exponential functions. Even and odd functions are not addressed.

 

Linear, Quadratic, & Exponential Models           F-LE

Construct and compare linear and exponential models and solve problems.

F-LE.1Distinguish between situations that can be modeled with linear functions and with exponential functions

  1. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
  2. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
  3. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

 

F-LE.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).

 

F-LE.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.

 

Note:  At this level, limit to linear, exponential, and quadratic functions; general polynomial functions are not addressed.

 

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

F-LE.5Interpret the parameters in a linear or exponential function in terms of a context.

 

Congruence   G-CO

Experiment with transformations in the plane.

G-CO.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.

 

Note:  At this level, distance around a circular arc is not addressed.

 

Expressing Geometric Properties with Equations G-GPE

Use coordinates to prove simple geometric theorems algebraically.

G-GPE.4 Use coordinates to prove simple geometric theorems algebraically.  For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

 

Note:Conics is not the focus at this level, therefore the last example is not appropriate here.

 

G-GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

 

G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

 

Note:  At this level, focus on finding the midpoint of a segment.

 

G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

 

Geometric Measurement & Dimension       G-GMD

Explain volume formulas and use them to solve problems.

G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.  Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

 

Note:  Informal limit arguments are not the intent at this level. 

 

G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*

 

Note:  At this level, formulas for pyramids, cones and spheres will be given.

 

Interpreting Categorical & Quantitative Data       S-ID

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

S-ID.1Represent data with plots on the real number line (dot plots, histograms, and box plots).

 

S-ID.2Use 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.

 

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

 

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

S-ID.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.

 

S-ID.6Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

  1. 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 and exponential models.
  2. Informally assess the fit of a function by plotting and analyzing residuals.

 

Note:  At this level, for part b, focus on linear models.

 

  1. Fit a linear function for a scatter plot that suggests a linear association.

 

Interpret linear models.

S-ID.7Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

 

S-ID.8Compute (using technology) and interpret the correlation coefficient of a linear fit.

 

S-ID 9Distinguish between correlation and causation.

 


Math II

 

The Real Number System    N-RN

Extend the properties of exponents to rational exponents.

N-RN.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.

 

Quantities       N-Q

Reason quantitatively and use units to solve problems.

N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

 

N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.

 

N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

 

Seeing Structure in Expressions      A-SSE

Interpret the structure of expressions.

A-SSE.1 Interpret expressions that represent a quantity in terms of its context.

  1. Interpret parts of an expression, such as terms, factors, and coefficients.
  2. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

 

Note:  At this level include polynomial expressions 

 

A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as

(x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

 

Write expressions in equivalent forms to solve problems.

A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

  1. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

 

Arithmetic with Polynomials & Rational Expressions       A-APR

Perform arithmetic operations on polynomials.

A-APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

 

Note:  At this level, add and subtract any polynomial and extend multiplication to as many as three linear expressions. 

 

 

 

Understand the relationship between zeros and factors of polynomials.

A-APR.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.

 

Note:  At this level, limit to quadratic expressions.

 

Creating Equations   A-CED

Create equations that describe numbers or relationships.

A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

 

Note:  At this level extend to quadratic and inverse variation (the simplest rational) functions and use common logs to solve exponential equations.

 

A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

 

Note:  At this level extend to simple trigonometric equations that involve right triangle trigonometry.

 

A-CED.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.

 

Note:  Extend to linear-quadratic, and linear–inverse variation (simplest rational) systems of equations. 

 

A-CED.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

 

Note:  At this level, extend to compound variation relationships.

 

Reasoning with Equations & Inequalities   A-REI

Understand solving equations as a process of reasoning and explain the reasoning.

A-REI.1Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

 

Note:  At this level, limit to factorable quadratics.

 

A-REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

 

Note:  At this level, limit to inverse variation.

 

Solve equations and inequalities in one variable.

A-REI.4 Solve quadratic equations in one variable.

b.  Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a andb.

 

Note:  At this level, limit solving quadratic equations by inspection, taking square roots, quadratic formula, and factoring when lead coefficient is one.  Writing complex solutions is not expected; however recognizing when the formula generates non-real solutions is expected.  

 

Solve systems of equations.

A-REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line

y = –3x and the circle x2 + y2 = 3.

 

Represent and solve equations and inequalities graphically.

A-REI.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).

 

Note:  At this level, extend to quadratics.

 

A-REI.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.

 

Note:  At this level, extend to quadratic functions. 

 

Interpreting Functions          F-IF

Understand the concept of a function and use function notation.

F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

 

Note:  At this level, extend to quadratic, simple power, and inverse variation functions.

 

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

F-IF.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.

 

Note:  At this level, limit to simple trigonometric functions (sine, cosine, and tangent in standard position)with angle measures of 180  or less.  Periodicity not addressed. 

 

 

F-IF.5Relate 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.

 

Note:  At this level, extend to quadratic, right triangle trigonometry, and inverse variation functions.

 

Analyze functions using different representations.

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

  1. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
  2. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

 

Note:  At this level, extend to simple trigonometric functions (sine, cosine, and tangent in standard position)

 

F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

  1. 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.

 

Note:  At this level, completing the square is still not expected.

 

F-IF.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.

 

Note:  At this level, extend to quadratic, simple power, and inverse variation functions.

 

Building Functions    F-BF

Build a function that models a relationship between two quantities.

F-BF.1 Write a function that describes a relationship between two quantities.

  1. Determine an explicit expression, a recursive process, or steps for calculation from a context.

 

Note:  Continue to allow informal recursive notation through this level.

 

  1. 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.

 

Build new functions from existing functions.

F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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.

 

Note:  At this level, extend to quadratic functions and, kf(x).

 

Congruence   G-CO

Experiment with transformations in the plane

G-CO.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).

 

G-CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

 

G-CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

 

G-CO.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.

 

Understand congruence in terms of rigid motions

G-CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

 

G-CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

 

G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

 

Prove geometric theorems

G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

 

Note:  At this level, include measures of interior angles of a triangle sum to 180° and the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length.

 

Make geometric constructions

G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

 

Similarity, Right Triangles, & Trigonometry         G-SRT

Understand similarity in terms of similarity transformations

G-SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor:

  1. 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.
  2. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

 

Define trigonometric ratios and solve problems involving right triangles

G-SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

 

G-SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.

 

G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

 

Apply trigonometry to general triangles

G-SRT.9(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

 

G-SRT.11(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

 

Expressing Geometric Properties with Equations G-GPE

Translate between the geometric description and the equation for a conic section

G-GPE.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.

 

Note:  At this level, derive the equation of the circle using the Pythagorean Theorem.

 

G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

 

Geometric Measurement and Dimension    G-GMD

Visualize relationships between two-dimensional and three-dimensional objects

G-GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

 

Modeling with Geometry      G-MG

Apply geometric concepts in modeling situations

G-MG.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).

 

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

 

G-MG.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).

 

Making Inferences & Justifying Conclusions         S-IC

Understand and evaluate random processes underlying statistical experiments

S-IC.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?

 

Make inferences and justify conclusions from sample surveys, experiments, and observational studies

S-IC.6 Evaluate reports based on data.

 

 

 

 

Conditional Probability and the Rules of Probability       S-CP

Understand independence and conditional probability and use them to interpret data

S-CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

 

S-CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

 

S-CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

 

S-CP.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.

 

S-CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

 

Use the rules of probability to compute probabilities of compound events in a uniform probability model

S-CP.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.

 

S-CP.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.

 

S-CP.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.

 

S-CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

 


Math III

 

The Real Number System    N-RN

Use properties of rational and irrational numbers.

N-RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

 

Quantities                                                                                                                                           N-Q

Reason quantitatively and use units to solve problems.

N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

 

N-Q.2Define appropriate quantities for the purpose of descriptive modeling.

 

N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

 

The Complex Number System                                                                                             N-CN

Perform arithmetic operations with complex numbers.

N-CN.1 Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a andb real.

 

N-CN.2 Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

 

Use complex numbers in polynomial identities and equations.

N-CN.7 Solve quadratic equations with real coefficients that have complex solutions.

 

N-CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

 

Seeing Structure in Expressions                                                                                          A-SSE

Interpret the structure of expressions.

A-SSE.1 Interpret expressions that represent a quantity in terms of its context.

a.  Interpret parts of an expression, such as terms, factors, and coefficients.

  1. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

 

A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

 

Write expressions in equivalent forms to solve problems.

A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

  1. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

A-SSE.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.

 

Arithmetic with Polynomials and Rational Expressions                                                    A-APR

Perform arithmetic operations on polynomials.

A-APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

 

Understand the relationship between zeros and factors of polynomials.

A-APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

 

A-APR.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.

 

Use polynomial identities to solve problems.

A-APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.

 

Rewrite rational expressions.

A-APR.6  Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

 

A-APR.7  (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

 

Note:Limit to rational expressions with constant, linear, and factorable quadratic terms.

 

Creating Equations                                                                                                               A-CEDCreate equations that describe numbers or relationships.

A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

 

A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

 

A-CED.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.

 

A-CED.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

 

 

 

Reasoning with Equations & Inequalities   A-REI

Understand solving equations as a process of reasoning and explain the reasoning.

A-REI.1Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

 

A-REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

 

Solve equations and inequalities in one variable.

A-REI.4 Solve quadratic equations in one variable.

  1. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (xp)2 = q that has the same solutions. Derive the quadratic formula from this form.
  2. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a andb.

 

Represent and solve equations and inequalities graphically.

A-REI.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).

 

A-REI.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.

 

Interpreting Functions          F-IF

Understand the concept of a function and use function notation.

F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

 

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

F-IF.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.

 

F-IF.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.

 

Analyze functions using different representations.

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

  1. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
  2. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

 

F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

  1. 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.

 

F-IF.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.

 

Building Functions                                                                                                                            F-BF

Build a function that models a relationship between two quantities.

F-BF.1 Write a function that describes a relationship between two quantities.

  1. Determine an explicit expression, a recursive process, or steps for calculation from a context.
  2. 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.

 

F-BF.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.

F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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.

 

F-BF.4 Find inverse functions.

  1. 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.

 

Linear and Exponential Models                                                                                                      F-LE

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

F-LE.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.

 

F-LE.4 For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

 

Trigonometric Functions                                                                                                                  F-TF

Extend the domain of trigonometric functions using the unit circle.

F-TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

 

F-TF.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.

 

Model periodic phenomena with trigonometric functions.

F-TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

 

Prove and apply trigonometric identities.

F-TF.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.

 

Congruence                                                                                                                           G-CO

Experiment with transformations in the plane

G-CO.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.

 

Prove geometric theorems

G-CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

 

G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

 

G-CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

 

Make geometric constructions

G-CO.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.

 

Similarity, Right Triangles, & Trigonometry                                                                     G-SRT

Understand similarity in terms of similarity transformations

G-SRT.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.

 

G-SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

 

Prove theorems involving similarity

G-SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

 

G-SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

 

Circles                                                                                                                                                            G-C

Understand and apply theorems about circles

G-C.1 Prove that all circles are similar.

 

G-C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

 

G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

 

Find arc lengths and areas of sectors of circles

G-C.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.

 

Expressing Geometric Properties with Equations G-GPE

Translate between the geometric description and the equation for a conic section

G-GPE.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.

 

G-GPE.2 Derive the equation of a parabola given a focus and directrix.

 

Modeling with Geometry      G-MG

Apply geometric concepts in modeling situations

G-MG.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).

 

Interpreting Categorical and Quantitative Data                                                                            S-ID

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

S-ID.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.

 

Making Inferences and Justifying Conclusions                                                                             S-IC

Understand and evaluate random processes underlying statistical experiments

S-IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

 

Make inferences and justify conclusions from sample surveys, experiments, and observational studies

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

 

S-IC.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.

 

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

S-IC.6 Evaluate reports based on data.

 

Using Probability to Make Decisions           S-MD

Use probability to evaluate outcomes of decisions

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

 

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

 

 
 

What should teachers know in order to teach? (Mathematics)

02 Aug

Do you ever wonder what training your child’s teacher gets to become an educator?  What do colleges require for teachers to earn their teaching credentials?  What do states require to hire a teacher into their school system?  Why do we seem to have such a small number of truly exceptional teachers?  Have you ever gone to ratemyteacher.com or ratemyprofessor.com ?  What do these sites say about our teachers?  Some report, “avoid this teacher at all costs.”  Sometimes might say, “you can get an easy A but you won’t learn anything about the subject matter.”  What do we want in a teacher?  Parents may want different things than students.  Some students may not actually care about learning the content, they only care if they get a good grade.  Some parents feel that way too.  There are some private schools where you basically pay for this option, give the school your money, they will make sure the classes are easy enough for students to get A’s and B’s.  Don’t get me wrong, there are plenty of private schools that do not work this way but I have seen a few (and I am sure there are more) that this seems to be the approach when I compare their class expectations and work to public or charter schools.  The bottom line is that “good grades” becomes a desire of parents and students and this sometimes can trump good teaching, which is unfortunate.

Taking that out of the equation for the sake of this blog, let’s assume all grades are equal.  What skill set should a teacher have?  The first answer is usually that they should know the knowledge area that they are teaching.  That seems obvious but the problem with this requirement is that it can be weighed so heavily that many states and schools will allow that to be the only qualification.  This is called “Lateral Entry” and I see it all the time in job postings.  If a person hasn’t earned a degree in mathematics or science education but they have a higher degree (masters or Ph.D.) in the field, they are allowed to join the work force as a teacher despite not having any formal education as an educator.  They may or may not have to take a state mandated test.  These tests can be in both content and pedagogy.  However, the graders of the pedagogical tests seem to accept a wide range of answers for a “pass” since it doesn’t seem to be a stumbling block for lateral entry candidates to get hired.  Besides, in general, the pedagogical tests don’t test that you know alternative approaches to teaching your subject such or effective ways to reach non-auditory learners or students with learning disabilities.  To be honest, you jot down some B.S. and your done.

Most of our teachers, however, do go through the traditional route of getting a true certification in education.  Let’s look at my own experience.  To become certified as a 9-12 math teacher, I had to meet all the requirements of a Bachelors of Science in Mathematics (which were quite rigorous at my particular college compared to others – I needed to take courses such as Calc 1, 2, 3, Differential Equations, Abstract Algebra 1 and 2, Real Analysis 1 and 2, Topology, Statistics 1, 2, plus computer programming classes, Discrete Math, and probably some others I can’t even remember.  Most schools do not require that much mathematical “theory;” for those who don’t know what that means think about your proofs from high school geometry but much more difficult.)  Additionally, I had to take a methods class.  This methods class was to prepare me for the actual teaching!  This should have been THE SINGLE MOST IMPORTANT class I took.  Why was there only ONE class that would show me HOW to teach math when clearly I can do math and all my teaching was going to be at a level far lower than anything I took for my Bachelors except maybe Calculus?  So, what did I do in that one methods class?  Well, not much.  We learned a format for how to write lesson plans.  Learning how to “format” lesson plans doesn’t really teach someone how to teach.  We were given assignments where we wrote 2 lesson plans.  That’s it, 2 lesson plans!  We explored some hands-on manipulatives such as base 10 blocks and Algebra tiles and we did some content work at the high school level to practice our lower level high school skills (we practiced from the NY Regents exams).  This is what was supposed to prepare me for teaching.  I also had to take a course on the history of mathematics, somehow that also would make me a better math teacher.  Finally, you must do student teaching.  I ended up waiting until grad school to do my student teaching.  This is your “on the job training.”  You are given the content material of the lesson you are supposed to teach, then you teach it and your cooperating teacher and college supervisor watch you and provide feedback.  In general, you can’t fail at this.  You may discover that teaching is not for you if things don’t go well.  In my case, I was lucky, since it was through grad school, we taught summer school and were the ONLY teachers (a group of us).  We were videotaped and then watched our videos and we all critiqued them together.  As a doctoral student, however, I worked as supervisor of the more traditional student teachers who took over for a classroom teacher for a semester and I provided the feedback along with the cooperating teacher so I got to experience student teaching in this fashion as well.

If you got lost or your eyes glazed over in the last paragraph, let me summarize:  there was very little done to prepare a teacher for teaching.  It wasn’t done for me and it continues to not be done for new rising teachers.  It seems that you either have the natural ability to figure out how to effectively reach students or you follow the same traditional path of how many of you were taught: students sit and listen while teacher lectures, teacher provides a lot of  basic procedural steps or possibly conceptual theory at a level that is too far above a student’s level and students get lost and teacher doesn’t even realize it.

So, what are the qualities of a good teacher?  What information should be imparted to all students learning to be teachers and all teachers taking Continuing Education Credits?  Here is my list for mathematics.  I would have different lists for other subjects although some of this can be generalized:

1.  Know your content (although sometimes those who were the strongest students in school can make the worst teachers because they don’t understand how students can’t just “see” the solutions like they can.)

2.  Know and read your students (you should know how each student is doing in your class at all times, during lectures you should be able to “read” when you have lost the class and make adjustments, this isn’t about YOU, it is about the students.)

3.  Be able to represent the material in multiple ways for visual and auditory learners (don’t make the assumption that everyone learns the way YOU do, present the same material in different ways and be sure to draw pictures and explain the pictures for visual learners.)

4.  Write down the steps for the procedures for the students (don’t assume that the student is going to be able to remember that when you complete the square that you have to take half the x term, square it, add it both sides, then create a factor of (x- 1/2term)^2 = total, take the square root of both sides, remember the plus/minus, and solve.  You can show a problem with this all worked out (sorry for those that I lost with this example but the point is to include a list of the steps so you don’t get lost!) but the students will forget how those numbers magically showed up without you helping them remember.

5.  Learn TRICKS to make things easier:  for Geometry and proofs, did you know you should start backwards with what you are trying to prove?  With factoring something like 6x^2-5x-4, learn the BOTTOMS UP approach, it is the quickest and easiest way to solve quadratics when the leading coefficient is not equal to one, there are many more TRICKS, learn them and teach them.

6.  Teach developmentally.  Start with easier problems and then build on that while adding slightly harder pieces, this is the easiest way to learn.

7.  Assign homework.  Students can’t master material without practice – don’t assign 4 problems it isn’t enough for students to learn but also don’t assign 50 problems, pick and choose the BEST problems for the student to do, don’t just day 1-50 odd, are all those GOOD problems and worth the time of the student.  Do your homework before you give them homework.

8.  Watch your tests, if students are failing, then YOU are not doing your job right and need to make adjustments.

In a perfect world, I would have students take a MINIMUM of 3 methods classes to become certified to teach.  During these classes, I would make sure all of the above were taught to students and that students had practice implementing these things.  How can we make STUDENTS better in mathematics?  The answer is simple, make our TEACHERS better.

 

Written by:  Lynne Gregorio, Ph.D.

Owner:  Apex Learning Center in Apex, NC

 

 
 

Fixing Today’s Education System: A look into the new Common Core Standards

12 Apr

Fixing Today’s Education System

Part 1:  Introduction

If you look at reports that compare the United States education to other developed countries, you will find that the U.S. is not highly ranked.  In mathematics, in particular, the U.S. is only ranked around 25th according to the Program for International Student Assessment that ranked 70 countries.  Why is our country failing our students in education?  What can we do to improve the future of education in the United States?

Although, I will address things broadly, I will put much of the focus on mathematics for two reasons.  First, it is the area that we are the worst at and it is the area that I am most qualified to address.  If you look at the most recent mathematical reform you see some general trends.  Schools were teaching math but decided that students were lacking concepts and thus changed the curriculum to a generation who learned under New Math.  New Math was supposed to do a better job at presenting concepts to students.  However, New Math seemed to fail and so the educators in high places deemed a new approach called, “Back to Basics.”  Back to Basics was an attempt at not confusing students too much with all these concepts New Math unleashed upon the children that were unsuccessful and instead encouraged students to focus on the basics of math – adding, subtracting, multiplying, and dividing.  However, after a while on this new approach, a new group of educators, yet again realized that students could do the procedural math but were again not really understanding the concepts and able to problem solve.  After Back to Basics was deemed a failure, the new push was to create a new curriculum of standards that was created by a group of math teachers from the National Council of Teachers of Mathematics who would write out specific goals that included both procedural and conceptual knowledge for students to grasp mathematics.  From the NCTM standards, states developed their own “Standard Course of Study” that would meet the NCTM standards and this was the new plan of action to make students successful in mathematics.  However, we continued to see problems with students in math and no real improvement in grades.  The educators now decided that there were too many differences across the states and again not enough conceptual understanding was being taught.  One state had one curriculum and another state had something completely different.  We needed a more cohesive curriculum across all states.  Not only that, we needed to get teachers to go back to making sure students can actually explain every math step they perform, being sure they understand the concept – in other words, more of a focus on conceptual understanding rather than procedural understanding (read New Math here.)  So, they created the Common Core Standards.  These new standards show each grade level what they should learn and give examples to those who read them about how to interpret different mathematical concepts that the standards want students to learn.  This is where we are at today.

If you look at the history and the big picture, you can see that we just keep doing the same thing.  Pushing our focus back and forth between procedural and conceptual knowledge.  Each time our plan doesn’t work, so we switch to the other focus and forget that we already tried this “idea” just in a different way.  The idea of having a common core for all states may sound “different,” but this common core has not been providing the common curriculum among schools in the same county, never mind within the same state as intended. I can only give examples from schools that I know but I would like to provide these as a way of illustrating how the idea of providing a document of “common core,” delivering it to each school provide an insufficient execution of the same content even within the same county..  I run a learning center and tutor children from many different schools.  This is the first semester our school system has offered Common Core Mathematics to students in its public schools.  Here is what I am observing.

 

School A:

Prior to common core, the course was Algebra 1 – it has since “changed” and now teachers should be implementing this new curriculum for Common Core 1.

Last Year in Algebra 1:  Students worked out of an old Algebra 1 textbook, teacher shows examples, students do homework, teacher gives tests, students are not allowed to use calculators, technology is not encouraged at all or used to teach concepts in Algebra.

This Year in Common Core 1:  NOTHING has changed, except, teachers are working at a slightly faster pace so they can add a couple new sections (worksheets) that are covered in common core during the last couple weeks of school relating to Geometry.  There is nothing different in how Common Core is being taught this year than the way they taught Algebra 1 last year.   Basically, it is the same course with 1-2 lessons of Geometry added in.

 

School B:

Prior to common core, the course was Algebra Part 1 and Part 2 – the teachers used an Algebra book and students worked through problems, took quizzes and tests often.  They used calculators as often as they wanted.  Students were often given 40 problems a night to reinforce the algebra concepts they were learning.  The topics fell in a nice sequencial pattern.  Teachers taught some technology but kids were allowed to use technology as much as they wanted to if they learned more on their own.

This year as Common Core 1:  Students are being taught out of worksheet packets that cover topics that more closely fit the Common Core topics.  The overall coverage of material is moving much slower and although students are getting taught material that better fits the curriculum, they are still not being encouraged to build concepts or practice as much as one would expect given the intent of the common core class.  However, overall School B has taken on the spirit much better than School A.  Students will often get homework with 3-4 problems to solve and this is insufficent to help them internilze the material and even work enough of a variety of different types of problems.  The glasses are very disorganzized and do a lot of discovery learning.  For some students who are smart, they may learn this way but the average student and definately the struggling student does not do well with discovery learning.  These students need examples, explanations, links and connections to things they already know, and sufficient practice on these topics.  This is not happening at this school.  Some times the students get tests and have to tell the teacher that the questions are asking things that he forgot to teach them.  There are no study guides, no direction, and not enough problems to prepare you for what is expected of you.

 You can see how different these two schools are and yet these students are taking the same course. Which is better?  Well, there are things that School A does better since it teaches math in an orderly fashion and gives enough homework for students to get the content.  However, they don’t spend time linking concepts (which the other school does).  They also don’t allow technology and the tests assume 60% calculative active, so students need to be strong in their ability to use calculators.  Overall, the kids are learning more but there areas of improvement – pulling out some wasted time on certain topics and use that time to cover other topics mentioned in the CCM1 standards.

 However, there are some things that school B does better.  They show students a lot of applications of linear, quadratic, and exponential models.  They spend a lot of time creating equations with various given information.  They ask thinking questions.  But, they do too much and the procedural knowledge gets lost and the student only ‘kind of” gets the conceptual knowledge being presented because it isn’t being linked to something they already knew.  They don’t have enough repetition and don’t do enough procedural knowledge.  The order of the material does not fit will which also impedes their ability to link key ideas for students.  They also will waste a week on a very easy topic but then rush through a hard topic in 2-3 days.  Students, overall, in school B are struggling more than in school A.

Here is another example.

 

4th grade School C:

Fourth graders in math at this school are being taught the topics stated in the common core curriculum.  Each week, they have to write journal entries that explain reasons behind what they do.  They have to be able to explain things like why a square is a rectangle but a rectangle is not a square in words.  They explore multiple ways to do different types of problems and then have to explain how they used their approach to get their answer.  The teachers ask, “why?” questions on tests and homework.  The teachers anticipate that the new End of Course tests will also be asking students open ended questions.

 

4th grade School D:

Students at this school, in the same town, are using the same homework book they used last year before common core.  They never get asked to explain any reasoning behind their work.  They simply do calculations and word problems.  The new thing the teacher added was word problems from an online web site the students log into at home.  Overall, there appears to be very little to no change in how this year’s material is taught compared to last year even though there is a whole new curriculum.

 

Another example of how much schools differ within the county goes even outside the common core concept.  Schools don’t even have the same grading rules, some schools are significantly harder than other schools, and some schools inflate grades.  All of this leads to the fact that an A, B, C – they have no meaning.  I had one student in Honors Geometry at this school that I think is a very harsh school for grading, get a C. He was very bright, he knew the concepts very well.  At any other school, he would have gotten an A.  His teachers just graded hard and gave really hard tests.  I had another student who got an A+ in Algebra 2 at one school who grades very easily and the classes in general are very easy.  It is a private school and I find this to be a trend.  It is almost as if you pay to get your child good grades.  Her knowledge in Algebra was poor but her grade was an A+.  All the children I know who struggle with grades in public school and then switch to these private schools start getting A’s and B’s.  I am not in the classroom and don’t see the tests, so I can’t judge but my point here is the overall inequity among grades in these schools.  Here is another example:  School A gives 0’s when a student doesn’t hand in work, this brings down grades significantly causing students to get poor grades.  Right down the road, School B says, “it is too hard to recover from a 0, so we will turn all 0’s to 50’s.”  Right there, is a huge difference in what a student’s score at School A and School B will be if they miss some assignments.  All of these grades are meaningless.  The only ones that count are on tests such as the AP exams or SAT’s where everyone takes the same test, however, I am not a fan on SAT questions as they don’t match up with what students actually learn in high school.  The math SAT questions are not traditional Algebra and Geometry questions, they are novel problem solving questions and some students who are good students, may just not have success on that type of test.  Additionally, the SAT verbal requires memorization of hundreds of vocabulary words to do well.  Many of these words are words that are very obscure and do not predict the success of a student. 

It isn’t just math where we see these large variations among school.  I was at an IEP meeting the other day and the first grade teacher told me how the child, a young boy, in her class needed help with writing.  I asked her about her expectations for writing in the first grade.  She told me they were writing persuasive essays and handed me a sheet that said he would need to have a topic sentence, use at least once complex sentence, have a sentence that uses some level of complex punctuation such as commas in a list of items, have descriptive words in his sentences in addition to correct spelling, punctuation, grammar, and there had to be an overall flow in the essay that showed one event following the other.  This is the very beginning of the second quarter of FIRST grade and he is a boy!  This was so developmentally inappropriate for this child to be expected to write a persuasive essay with all these expectations.  If a child of age 5 or 6 can even understand the concept of a persuasive essay, that would be an achievement!  Although, she was the most extreme example I have seen, many of the children who come to my center come in with 2’s (the grading system in our state is 1,2,3,4 and a 2 is that you are not performing at the expected level) because the expectations in writing in our state are just too high.  The children might have a chance of reaching these goals if teachers actually spent time teaching students grammar but they don’t.  Children come in expected to write complete essays with perfect sentences and with all these specific goals, however, they can’t tell me what a noun or a verb is.  Teachers, at least in my state, are just ignoring grammar altogether.  They don’t seem to think it is relevant to writing.  On top of that, the teachers have no idea how to teach writing.  They just have “expectations” that the students will do it.  So, the smart ones learn to figure it out on their own or get outside help and the others just crash and burn.  When my son was in fifth grade, the teachers were so focused on the enormous amount of social studies the state / school required they present to students that although my son couldn’t yet write a sentence, they felt that taking the time to do so was not as important as the time they needed for social studies.  I mentioned, at an IEP meeting, that he couldn’t spell either, and they told me that it didn’t matter because that is what spell checkers on computers were for.  It was appalling.  How many teachers give spelling tests to students and don’t care about the results?  When your child gets a word wrong on a spelling test, is anything ever done about it?  Do they revisit the word so that the child eventually learns to spell the word or is it just marked in a book and on they go?  How much time does the teacher put into picking out the words for a spelling list?  Again, I saw the difference between schools in this aspect.  I had my child at one school where the teacher picked completely random words that had no relationship to each other and weren’t even words that my child might use at his age.  Another school, the words were more appropriate, although still weren’t related to each other so that they helped teach a phonetic pattern to a child, although I have seen (through my center) a few good teachers give spelling lists with words that were well thought out, grouped according to sound patterns and age appropriate.  The variation, however, among spelling words is so inconsistent just as everything else.  My own daughter is very advanced in spelling and was doing 6th grade spelling words in 3rd grade.  When passed onto 4th grade, her new teacher had her repeat all the lessons she did in 3rd grade from the 6th grade spelling book (even though she got all 100’s in 3rd grade on the words), saying that review was good.  Actually, my feeling is that she was just too lazy to find lessons for her even though I offered to provide them.  So, I ask, what about the good spellers?  What about the weak spellers?  Shouldn’t we assess a child accurately (and do teachers know how to do this) and match their spelling lists to their actual level so they are learning appropriate information?

Children learn from building on what they already know.  If you jump too far ahead of what they know, they can’t make that leap and you will waste your time and their precious time teaching them things they aren’t ready for.  When children get behind in school but we fail to acknowledge that or make adjustments and just keep them with the rest of the class, they will only get further and further behind.  You can’t go from adding one digit numbers to subtraction with borrowing across zero.  You won’t make it.  You can’t take a reader who is comprehending at grade 2 and expect them to be successful in grade 3 or 4 just because you passed them along.  What does this do to our children?  The biggest problem I run across in my center is a students sense of themselves as a learner.  I call it their educational self-esteem.  When a child fails all the time, sees that 2 on their report card, or is the one who “just doesn’t get it” over and over, they begin to bring those thoughts into their sense of self.  They see themselves as a dumb person and this stays with them for life and impacts their future education because they will always figure, “they just can’t do it.”  When I taught undergraduate and graduate students, I would often ask them to write a math biographical essay about themselves.  I wanted them to tell me how they saw themselves as a learner in mathematics and what molded them to feel they way they do.  The majority of students who were non-math majors had negative math self-esteem.  They felt they weren’t good in math and never would be.  They often relayed stories of terrible events that happened such as teachers telling them they were stupid in front of the class and belittling them.  Others just got beaten down by the system of poor teaching and bad experiences.  After teaching my math classes with a “everyone can do this attitude,” and presenting material in a way that always leads to the “why couldn’t anyone have showed me this before?” question, my student’s attitude would change.  They would be shocked to learn that it was in them all along and they just needed someone to believe in them and teach in less traditional ways that allow for greater learning to take place.  We will talk about teaching techniques later on.  The point here is that all of these inappropriate expectations, all of these failing grades, and having teachers who don’t really want to be in the classroom for the right reasons provide children with negative educational self-esteem.  We need to break this cycle.

 

Next Blog:  How Do We Fix It?

Future Blogs:  Looking at the common core grade by grade

 
 

North Carolina Common Core Math 1 – what is it?

11 Dec

As the owner of The Apex Learning Center, I see students from many different schools who come in for help with mathematics.  This is the first year that North Carolina has adopted the Common Core Standards.  North Carolina chose to implement these new standards without the funds to purchase any text books that support these new standards and have provided minimal to no teacher education relating to these changes.  From what I have heard, their idea of teacher education is a top down approach.  They meet with the lead teachers once or twice with information about the changes and those teachers are to pass the information on to the rest of the classroom teachers at the school.  The individual classroom teachers get no hands on training and the lead teachers get “information” but no hands on training.

I have mentioned this before but feel it is worth mentioning again that one of the big ideas of common core mathematics is the encourage conceptual understanding in mathematics.  Mathematics has three parts:  first, procedural knowledge – the ability to perform operations and get correct answers.  This can be done at levels such as addition and multiplication facts or even taking an integral using the chain rule and quotient rule combined in Calculus.  The level doesn’t matter, it still breaks math down into a series of steps that can be followed that end with a correct answer.  Procedural mathematics is very important, all of elementary school mathematics requires procedural mathematics for the majority of its work.  People who have basic procedural knowledge of mathematics can get by in the world but will often say, “I was never good in mathematics.”  They will not see the need for mathematics in everyday life because they don’t understand how it can be used in everyday life outside of the basics of procedural arithmetic.  The second part of mathematics is a conceptual understanding.  In this instance, students understand not just how to compute, but understand why they are doing mathematics.  They will have strong number sense; they can apply arithmetic to word problems including multi-step word problems.  They understand the idea that an integral measures the area under a curve or how mean, median, and mode can all measure the concept of what is “typical” about data but provide different answers in different situations.  The  third part is when students who are able to apply mathematics to novel situations. If a student is given a problem that uses the underlying concepts and procedures of mathematics they know but they have never seen a similar type of problem solved, this student can still solve the problem.  This last type of student has reached the highest level in mathematics.

If you look at mathematics history you will see that our educators who determine “what the world should know in mathematics,” have grappled with procedural vs. conceptual understanding.  A long time ago, we had a system in place and students were not performing at a level that was good enough in mathematics.   Math educators decided that it was because the focus was too much on procedural understanding and not enough on building conceptual understanding, hence NEW MATH began.  Ask people who went through the New Math period, they were lost and confused as teachers, who were not trained to teach New Math, tried to change their behavior and teach math more conceptually.  It was a disaster and scores reflected that, so after some time, they made a new movement that was called “Back to Basics.”  BTB followed New Math and was supposed to undo the years of confusion we put our children through by just going back to a more procedural focus such as getting math facts under their belts.  However, the BTB era did not achieve the desired results either so out of that came the NCTM Standards and No Child Left Behind.  Still, we find that we fall below in scores across the board, especially in mathematics.  A new group of educators got together and determined two things were needed:  one a common set of standards across the country and a focus on conceptual understanding (hmm… anyone thinking New Math here?)  Hence, we now have Common Core.

Back to  North Carolina, which is the only state I am qualified to talk about although I am told it is the same elsewhere.  Our department of education decides to adopt Common Core.  This post focuses only on Common Core Math 1.  This is replacing Algebra 1 for students.  All public students (public and charter school) who would have taken Algebra 1 this year are now in Common Core Math 1 instead.  I asked teachers, “What is Common Core Math 1?”  The answer I got was, “It is Algebra 1 with a few things removed and a few pieces of Geometry and Statistics added.”  This was the most common answer.  All our schools – this includes middle and high schools – are teaching CCM 1 with no text books and nothing more than some information passed down from their lead teacher or their own interpretation of what they read.  So what does this translate to?  How does this COMMON (which makes me laugh because the curriculum was much more COMMON when we all taught Algebra 1 than it is now) Core Math 1 look across different schools?  Is it Algebra 1?  Should students leave CCM1 with most of the skills from Algebra 1 intact?  Should they be able to solve absolute value inequalities, solve systems of equations, factor all types of trinomials, find zeros, do linear regression, write equations of lines from 2 points, and more? Or is it okay if students just do labs that let them play around with numbers in discovery learning and if they “discover” the concepts from these labs, great, if not – well… we presented a “conceptual approach” — wasn’t THAT what common core was supposed to be about?

Let’s take 2 different schools and compare what they are doing at the half way point of CCM 1.

School Number 1, we will call SN1 for short.  They are a charter middle school.  My son attends this school so I do homework with him every night in common core and therefore am very familiar with their choice of implementation of Common Core Math 1.  He was given an old Algebra book that they have used to teach the Algebra 1 that CCM1 is replacing at the beginning of the year to work out of.  I am told, they are focusing more on conceptual understanding and this text has a lot of word problems at the end of each chapter and those problems are always assigned.  Hmmm… is conceptual understanding the same as applications?  Not sure I equate those two things.  I am also told there will be some hand outs eventually to fill in concepts that are in CCM1 that are not in the book.  I haven’t seen anything yet but they are flying through the book so at the rate they are going – maybe there will be on time.  The book has 12 chapters.  The students will have finished 7 out of 12 at the halfway point.  The book contains topics that many Algebra books would skip and has a whole chapter on Statistics so it is very full of Algebraic topics, a few of which are even seen in a typical Algebra 2 course. Students have learned to solve multi-step equations, inequalities, absolute value equations and inequalities, they have solved y=mx+b problems with the typical questions (given a point and slope or given 2 points).  They had to learn all 3 forms point slope, slope intercept, and standard form (note that I did find somewhere in the CCM 1 standards that say Standard Form would no longer be taught, but that part of the standards was lost at this school.)  They had to explain equations from graphs and they are solving systems of equations using all 3 methods.  The teacher does not allow the calculator and does very little with technology.  Next semester they will learn exponents, factoring, solving quadratics, graphing quadratics, exponential equations, arithmetic with polynomials including Algebra 2 topics here of rational polynomial addition, and statistical concepts.  I am sure by now you can see this is simply :  Algebra 1.  To be honest, I am happy that my son is learning Algebra 1 because personally I think Common Core Mathematics is going to be a disaster and I want my son to know Algebra 1, Geometry, and Algebra 2.  I wish technology was brought into the classroom as that would allow for concept building instead of the procedural focus which is how it is taught if you ignore technology.

Let’s now consider School Number 2, SN2.  This is a public high school.  I tutor a freshman student who is taking CCM1 at SN2 three times per week so I am very familiar with her work for the class.  Remember there are no CCM1 text books so her school made packets for the course.  I have never seen anything more disorganized in my life.  I am still in awe that this is their idea of CCM1.  They don’t have any problems for students to work.  The student usually gets a very small amount of homework that is nothing you would see in an Algebra book.  So, they will not be leaving the course with experience doing any of the above mentioned things from SN1.  The teachers are trying to build conceptual understanding with no procedural requirements to the class at all.  Their book is a series of lab experiments that the students are supposed to infer mathematics from.  It never directly teaches anything.  The homework never reinforces anything and is very arbitrary.  Many nights there is no homework or maybe 2 questions.  My student is getting a poor grade because it is very difficult to tutor someone when you are not inside the “brain” of the creater of this curriculum since it is so chaotic and there is no practice for the students. You don’t know what the teacher wants them to know.  The students don’t get to bring home tests, so I don’t even know what they are asked although at one point there was a question about finding slope (a procedural topic) on a test and the students pointed out to their teacher that he never taught them that.  He realized it and gave them a quick lesson and then let them retake those test questions.  He didn’t link that to anything.  I am thankful that in my student’s 8th grade curriculum, before the switch, (I tutored her then also) she had enough Algebra that she knows how to find equations of lines and other basic Algebra skills that she isn’t losing an entire semester/year.  However, next year the current 8th graders who are in CCM 8th grade won’t be so lucky since those Algebra skills have been removed and “supposedly” placed in CCM1 although at this school, they are not there.  To date, the students in SN2 have had a unit on Statistics, they can solve multi-step (but not too hard) equations, and they haven’t worked with inequalities much except to say “at least means” greater than or equal to and basic ideas such as this.  They have been studying functions in a round about way for a long time – they can name domain and range and tell if something is a function or not and find f(4) if f(x) = x + 3.  They can calculate slope and they learned NOW, NEXT commands for associating one variable with another.  This approach has done nothing  but totally confuse my student, she doesn’t see any links or connections and has no idea at all about what they are even working on.

The course material for the same course at SN2 vs. SN1 is so totally different.  Neither of them are following the spirit of the Common Core Standards.  As is usually done, things are at one end of the extreme or the other.  SN1 is teaching a typical procedural Algebra 1 class with applications.  They are not using technology and not building conceptual understanding within the context of the course.  They are not following the standard guidelines by removing things that are outdated such as switching forms in Algebra to standard form and are covering topics that are reserved for other courses when the time could be spent exploring with technology.  SN2 has gone to the other extreme, they have done away with all procedural methods and are trying to get students to understand mathematical concepts through labs and explorations with very little practice from the students, without good structure, and they are very behind.  Students in SN2 will leave CCM1 with an incredibly weak understanding of mathematics and will be far worse off than they were taking Algebra 1 at that same school last year.  Part of the problem is the curriculum and part of it is the lack of teacher training given to the teachers to implement this curriculum.  However, the biggest part at SN2 is whoever made these packets missed the boat on what the objectives of CCM1 is supposed to be and how to impart this knowledge to 14-16 year old students.

For comparison sake, I looked around at other CCM1 NC blackboard sites to see what other schools were doing without any state guidance for CCM1.  The results were very varied.  Many were like SN1 and just a repeat of Algebra 1.   Other schools seemed to be trying to do a hybrid of traditional Algebra 1 but put a great emphasis on certain topics such as exponential functions.  I did not find another school with a curriculum anything like SN2 but that doesn’t mean it isn’t out there.

In conclusion, is CCM1 just New Math all over again?  Are teachers equipped to handle teaching CCM1 at the level designed by Ph.D. educators without the years of instruction that the creaters had when coming up with the concept?  Is Common Core at all Common?  I have clearly found the answer is no and by the way this extends into the elementary school as well.  If it isn’t even Common within one state, how is it Common among all states adopting Common Core?  I would love to hear from you!

 
 

Scope & Sequence in American Mathematics

18 Sep

We do it all wrong!

Why?

1. To give students a wide breadth of mathematical knowledge
2. Because our teachers don’t know how to teach math very well
3. Because we don’t have our act together and repeat, repeat, and repeat
4. Because we feel the need to give TONS of the same type of problem
5. Because our students don’t GET MATH and we think that means we need more problems and more repetition

If we learn how to teach math better and don’t waste time with so many worksheets of the same problem and don’t confuse students with so many different mathematical topics, we could progress at a faster rate with knowledge retained by students.

The NCSCOS (North Carolina Standard Course of Study) has goals in 1) Arithmetic, 2) Geometry, 3) Statistics and Probability, and 4) Algebra and 5) Measurement. One day our teachers will be teaching factor trees, then the next they are switching to Geometry vocabulary (what are acute angles? What does it mean to be perpendicular?), the next day they are having the students do “pretend” Algebra, where we don’t actually do what would be done in Algebra, we hide it, with a missing addend instead of a variable, and then students are doing graphing and finding out how many combinations exist if I line up 4 books on a shelf, then they are working with the metric system and measure items using how many “book lengths.”

This breadth of knowledge is good but I fear that we lose so much by constantly shifting gears on our kids. They forget what they learned in the Geometry unit in grade 4 when they get to Grade 5 and we start all over again. The “Algebra” skills are not really helpful yet until the kids can grasp more advanced ideas and the uses of Algebra. Measurement, to some extent, is necessary – certainly telling time, using a basic ruler, and money. Probability and Statistics are very useful, but again, let’s finish one thing before we address a new thing.

Here is what I see as a draft of scope and sequence for elementary and middle school mathematics:

Grade K:
1. Skip counting – by 2’s, 3’s, 5’s, 10’s. Expanded skip counting by 5’s and 10’s. What if I start with 25 and now want to skip count by 10? Apply to learning to count money.
2. Tally counting and apply to telling time.
3. Adding – group by ability – some can add all 1 digit numbers here, some will need to just add low numbers +0, +1, +2.
4. Exchanging – Using base 10 blocks, teach the idea that for every 10 units you can trade in for a 10 block.
5. Modeling place value with 10’s blocks for units, 10’s, and 100’s.
6. Reading numbers to the 100’s.
7. Learning doubles: work on memorizing (using song is helpful) all the doubles.
8. Adding to 10 (10 + 3 = 13) Build the concept of place value.

Grade 1:
1. Mastery of counting money
2. Mastery of telling time
3. Adding numbers without regrouping
4. Using what was taught about regrouping in Kindergarten, expand to modeling addition with regrouping
5. Add place value to the 1000.
6. Teach addition strategies.
7. Introduce the concept of multiplication and how it applies to addition and begin study of multiplication
8. Begin unit on subtraction and teach subtraction strategies
9. Teach basic measurement

Grade 2
1. Review units on telling time, counting money, addition of numbers with and without regrouping, subtraction
2. Teach multiplication strategies
3. Teach concept of factors
4. Link multiplication to division
5. Begin the process of modeling with division
6. Do mixed word problems, teach wording of problems involving +, – , X
7. Expand place value
8. Introduce concept of fractions & decimals
9. Order decimal numbers
10. Do 2 digit by 1 digit multiplication

Grade 3
1. Solidify multiplication
2. Build to long division
3. Equivalent fractions
4. Adding fractions with like denominators
5. Using equivalent fractions, add with unlike denominators
6. Multiply fractions
7. Divide fractions
8. Introduce decimals and how they relate to fractions
8. Introduce concept of percent and how it relates to fractions and decimals
9. Convert between percents and decimals
10. Add and subtract with decimals
11. Discuss degrees (90, 180, 270, 360) – do turns with your body

Grade 4:
1. Introduce mixed numbers
2. Add & subtract with mixed numbers – use models
3. Convert mixed numbers to improper fractions
4. Multiply and divide with mixed numbers
5. Apply real world uses of percents (interest)
6. Discuss concept of variable
7. Use variable to represent unknown in math problems
8. Introduce Perimeter and Area and practice arithmetic by applying these two problems

Grade 5:
1. Discuss multiples and factors
2. Build factor trees
3. Find GCF and LCM (relate LCM to equivalent fractions)
4. Probabability and Statistics UNIT
5. Begin Pre-Algebra

Grade 6:
1. Algebra 1 with modifications (1/2 year)
2. Geometry Unit (1/2 year)

Grade 7:
1. Finish Algebra 1 curriculum (1/2 year)
2. Probability and Statistics UNIT (1/2 year)

Grade 8:
1. Advanced Algebra 1 (for students who are struggling – this is a full Algebra 1 class with topics from Algebra 2 introduced at a basic level) OR
2. GEOMETRY

Grade 9 – all students will be ready for Algebra 2 or Geometry