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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!

 
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Tutoring Common Core Mathematics 2 – the struggles for students

17 Oct

I am an educator of 25 years and have a Ph.D.  in Mathematics Education.  If you read my blog, I have been vocal about my dislike for the New Common Core curriculum.  It is very choppy and does not help students build a big picture of mathematics as they skip around from one unit to another.  For those unfamiliar with the changes to Common Core Mathematics, many of the standards from Algebra 1, Geometry, and Algebra 2 – including topics in applications of Statistics, were broken down into pieces or individual units.  Next, the creators of the curriculum went through and picked topics from across all 3 old math courses and divided them up differently into Common Core Math 1, 2, and 3.  They added a lot more verbage that describe the individual units and how they can be learned at a more conceptual level and then tried to link as many pieces as they could to mathematical modeling.

Did you ever wonder why our system used to go from Algebra 1 to Geometry and then to Algebra 2?  Why not do Algebra 1, Algebra 2, and then Geometry?  Well, a school in our district wondered the same thing, the idea was that students would forget less Algebra if they did Algebra 1 and 2 back to back and then do Geometry afterwards.  However, when they tried it, they found out the reason (probably what people found out long ago and the reason it has always been done in the old order) why students in high school do best with the order Algebra 1, Geometry, Algebra 2.  The reason is that students need time to mature into mathematics.  The higher level of mathematical thinking required in Algebra 2 was too high for many students to be successful in their sophomore year (the typical age for the average US student to take their second math course).  Geometry is much easier and gives students the chance to grow and develop before they are presented with the harder mathematical concepts learned in Algebra 2.  The school in our district did this for one year and then after finding that it was not successful, went back to the old order.

So, why do I bring that up?  Well, in Common Core 2, students are back to doing many Algebra 2 topics again in their second year of high school mathematics and we are seeing the same problem as we did when we tried to get students to take Algebra 2 in their sophomore year.  The students are not mathematically developed enough to be able process those Algebra 2 concepts.  I am currently working with some students in Common Core 2 and I want to discuss their struggles.

Common Core 2 in our district is taught in one semester on block scheduling.  This means 90 minute classes for one half of the year = an entire years class that another school might offer without block scheduling.  Therefore, the math material moves twice as fast and students can go as long as a full year between math classes.  For example, they have a class Fall Sophomore year but not again until Spring Junior Year.  Although I was given the “standards”, it was very hard to interpret what the school was actually going to do with it.  Our schools don’t use books.  My son’s school (he is taking Common Core 2 at a middle school – and therefore will have a full year since they don’t do block scheduling) have been using a an applied curriculum with outside resources that looks nothing like what the high schools (both being under the same state standards) are doing.  Other middle schools in the state have been bringing in topics that the high school students will never see since they also are not on block scheduling and it gives them much more time than the high schools.  For example, one middle school covered partitioning line segments but the high schools do not cover that.

Common Core 2 in high school has had almost no Geometry so far.  Their first unit covered dilation, translations, reflections, and rotations and during the second unit they very briefly talked about proving triangles congruent and similar.  The majority of the course has been spent on topics they did in Common Core 1, topics one would see in Algebra 2, or expanding on functions they learned about in Common Core 1.  For example, even though they fully covered how to translate, reflect, and stretch parabolas in Common Core 1 they did this again.  They are also doing it AGAIN with exponential functions.  Even though students learned how to graph quadratics in Common Core 1, they had a huge unit on graphing quadratics, finding zeros, factoring, using the quadratic formula, and finding the discriminant.  Students also learned how simply expressions with exponents including negative exponents in CCM1 yet it is repeated in CCM2.  Students learned how to simplify square roots with variables and now in CCM2, they expand it to simplifying with high roots.  They also expand by adding rational exponents and solving equations with square roots and rational exponents (an Algebra 2 topic).  Later in this unit, they will review exponential functions and look at the idea of how to write recursive functions (recursive functions are usually taught in Algebra 2 or Advanced Functions and Modeling which comes after Algebra 2).  So far, the course has had almost no resemblance to a traditional Geometry class, so it makes me ask if students are going to get any traditional Geometry teaching anymore?  I did read that “proofs” are supposed to be included in CCM3 so I wonder if CCM3 will be Geometry heavy.

Now that I have addressed the content of the course, I want to talk about student success.  I work with students who are trying to learn the material in CCM2.  The current unit with rational exponents is moving at a fast pace and requires that students have a true understanding of the “rules of mathematics” in order to be successful.  Math must be done in a specific order.  When solving a radical equation, especially one that will generate 2 answers, there are so many steps involved that any struggling student will get lost in all the steps and rules.  Unless you intuitively “get” math, these rules and required order won’t make sense.  Many kids try to memorize the steps and at a fast pace with so many rules and changes that happen from problem to problem, memorization fails.  Either you naturally understand why you are doing each step or you just can’t get there.  For so many kids, they just can’t or don’t get to that step.  Their brains are just not wired that way.  I think about it this way, I can’t whistle.  It isn’t hard for someone who can do it, they just can.  However, even though I try and try and I follow the directions everyone gives me about what to do, I am just not physically capable of whistling!  Math teachers are people who can whistle and the problem is that they don’t understand people who cannot since their brains just “get it.”  For some kids, it is a matter of not trying, not applying themselves, etc.  I see that a lot.  I also see that our curriculum and pacing is NOT helping students get where we want them to but when we have students who have brains that are just trying to memorize math as a bunch or random rules rather than seeing the big picture, we have to accommodate that by a) creating a curriculum that allows to gradually live up to the potential they are born with and b) move at a pace that gives them the time to figure out that picture.

At the current pace and with the current scope and sequence, we are just asking for failure among many students.  I leave you with this?  If our goal is to get from level 1 to 10 with students by year 12.  What is better, going slower and actually getting to 7 by year 12 or going fast and pushing to level 10 with lessons but the student is really only at a level 4 since we lost them long ago!

 
 

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).

 

 
 

Common Core Math is going to be a disaster!

10 Sep

I continue to be frustrated as we get deeper into Common Core Mathematics.  I only get to see how my county in my state is dealing with Common Core Math and I am hoping that maybe other states are doing a better job but for North Carolina, Common Core is going to leave students with a weaker understanding of mathematics and definitely a worse outlook of mathematics.  Let me tell you why.   Math is already one of those subjects that gets a bad wrap.  People get bad teachers, the subject is difficult, and for some people the abstraction of mathematics is just not the way their brains work.  My work as a teacher over the last 25 years has been to help overcome this bad wrap and within the small microcosm of my classrooms, I have.  What has been effective in doing that?  Here is the answer…

1)  You must have a curriculum that flows from simple to more complex.

2)  You must have a curriculum that has good links from one concept into the next.    Think of someone who writes poorly and jumps around from topic to topic.   Here is my example,  “My dog is the best animal in the world.  He is fluffy and soft.  I felt a soft stuffed dog at the airport.  There was a broken plane at the airport.  My car needs to go into the shop.  Cars are fun to drive.  Driving can cause accidents.  My friend was in an accident once.”    There is some connection between each sentence but not good connections that link from one thing to the next.  If my topic sentence is “My dog is the best animal in the world,” what should follow should be all about my dog and why he is the best in the world and examples to support that he is the best in the world.  This is how math needs to be for success.

3)  You must have a teacher who can provide that information to students so that they can link the new concepts to existing concepts (for all students, not just a few).

4)  Homework needs to reflect what the student is expected to know so that the student gets practice with what they will then be assessed on.  Don’t give simple homework and then ask hard questions on the test.  If you want your students to know A,B, and C.  The homework should have them practicing A,B, and C.  There should also be sufficient homework practicing A,B, and C.  Tests should never be a surprise!

5)  Assessments should be made by the teacher – we got lazy!  Since when did we start using test banks from book companies for assessments!  Assessments are just as important as teaching and yet for today’s teachers they are an afterthought!  The test should mimic what is on the homework.  All students who practiced their homework and were successful or sought help for homework, should be able to score well on tests unless there is something blocking them such as the material is not at their current developmental level.

What am I seeing in today’s Common Core High School Mathematics in North Carolina?

First, the change from Algebra 1, Geometry, Algebra 2, PreCalculus, Calculus is just BAFFLING to me.  These topics were grouped together because they all LINKED and flowed from each other.  (See #2).  Now, teachers are teaching concepts all over the place just like my little dog essay above!  Students are not linking concepts and are getting lost.  They are not putting the big picture together.  They also fail to start with simple and build to complex.  My son was given homework essentially proving congruence in triangles without anything more than a lesson on how two triangle are congruent (SSS, ASA, SAS, etc.).  In the old days, we would learn all the theorems that build around proving triangles congruent such as vertical angles congruent, reflexive property, etc. and learn how to write proofs rather than just be expected to essentially prove something out of the blue in a homework assignment.  We started with the simple and built up to the more complex.

Why are we spending all this time in Common Core Math 2 focusing on even and odd functions and complex rational functions when a student has not yet grasped the more basic concepts that are useful such as Algebra 1 and Geometry and the easier Algebra 2 concepts?  The choice in the topics and the flow in the topics in Common Core is going to create a generation of confused students.

With this jumping around approach that requires teachers to pull so many tiny pieces from all over mathematics, students never get an in-depth understanding.  The reason for cumulative mathematics is that it actually requires that you retain your previous mathematical knowledge to move on.  Teachers don’t have sufficient homework to give students for all these micro-pieces and their assessment process is pulled from someone who isn’t teaching the class but is a text book form.

I have no doubt that based on what North Carolina is doing (and I can’t speak for other states but would love to hear from others about what their states interpretation of Common Core Math is) we are headed for disaster and I am crushed that two of my 4 children are going to be causalities of this endeavor.

 
 

Common Core Math is New Math

13 Aug

I have stated this before in my blogs but after receiving a funny video and having yet again more situations fall in my lap that just get my blood boiling about Common Core, I thought I would revisit this idea with another short blog.  Let’s start with this video reminding us about New Math.

New Math is Common Core

 

For those not familiar with New Math, it was a movement of moving away from application, basic skills practice in favor of developing a deep conceptual understanding of the content.  The process became more important than the answer and providing these abstractions to children who were not ready for this and by not providing the correct amount of practice of basic skills, we lost a generation of people who neverlearned math and feel it is just a confusing mess.

“….Some of the New Math curricula were excessively formal, with little attention to basic skills or to applications of mathematics. Programs that included treatments of number bases other than base ten, as well as relatively heavy emphases on set theory, or more exotic topics, tended to confuse and alienate even the most sympathetic parents of school children. There were instances in which abstractness for its own sake was overemphasized to the point of absurdity.37 Many teachers were not well equipped to deal with the demanding content of the New Math curricula.”  (http://www.csun.edu/~vcmth00m/AHistory.html)

So, why do I feel that Common Core is the same as New Math?  Well, I can only speak for the standards and the methods that my wonderful state of North Carolina is handing out but I see an eerie similarity.  It is especially noticeable at the high school level.  It seems that this “Integrated” approach to mathematics seems to be the new “hot” item over the traditional Algebra 1, Geometry, Algebra 2 sequence.  Why?  I asked this question.  The answer was to better link concepts that relate to each other.  Hmmm… wouldn’t Algebra 1 topics best relate to topics within Algebra 1 and Geometric topics best relate to topics in Geometry?  Just seems like we already had a good system of linking similar topics.  So, moving away from that, the second reason is to provide a “Commoness” so that all states are doing the same thing.  All schools are producing students with this same deep understanding after Common Core 1, 2, and 3.  However, since some states and even individual schools did not choose to adopt these standards, we now have students taking:  Integrated Math 1,2, 3 (Common Core Style) or Algebra 1, Geometry, Algebra 2 (Still common core style), or Algebra 1, Geometry, Algebra 2 (NOT common core style).  So, what about the student who takes Common Core Math 1 and 2 at one school and learns just a little bit of Geometry in those two classes but MOST of the Geometry is withheld until Common Core 3, now that student moves onto a different school (this is common for some middle schoolers and my son may be a victim of this) and the new school doesn’t do Common Core Integrated Math but one of the other two options.  Now, he has missed most of Geometry and is taking Algebra 2 (either Common Core style or not).  He takes his SAT and they ask him about finding the area of a segment of a circle.  Well, that was done in Common Core 3 but he didn’t take that because his new school wasn’t doing the same path, he is out of luck or gets to learn all those missed concepts on his own before the SAT.

Thirdly, we haven’t yet discussed the fact that teachers are still trying to teach at this higher conceptual level with less traditional drill and practice of standard Algebraic and Geometric problems.  This is exactly what happened in New Math and we lost many students who just couldn’t grasp the concepts well enough at that higher level and without the standard drill and practice of working with exponents and solving geometric problems with parallel lines, now don’t possess those skills either.  This isn’t jut me “theorizing” it could happen.  I saw it happen last year as one school tried to teach with all investigations and a very limited amount of standard algebraic drill and practice and these students are now very weak algebra students.

My son is working from a new “investigation approach” for Common Core 2.  After going to the website to read about it, they boast how their students scored above (in one area) or the same as (in another area) other students who took a traditional 1 year Algebra class after their test students took 2 years of their Common Core sequence.  This left me scratching my head?  How can you compare two groups where one group had 1 year of instruction and the other had 2 years of instruction?  Note my comments are in blue.

“…on the Educational Testing Service’s Algebra End-of-Course Examination, students at the end of Course 2 [at the end of this second course] scored especially wellon subtests of Concepts and Processes and about the same on Skills as a national sample of students who were completing a first-year algebra course. [this other group was just completing their first year of a traditional algebra class, did they score especially well too?  didn’t say] For further discussion of the Algebra End-of-Course Examination findings.” (http://www.wmich.edu/cpmp/longitudinalstudysummary.html)

When I looked at the “further findings,” the study was never clear that it was equating students with the same number and level of math experience when comparisons were made.  It also said that the students who took the “Investigation” based class scored better on those types of problems but worse on basic skill type problems than those in a regular curriculum.  This makes obvious sense since the focus is much more applied in the “Investigation” based approach and most Algebra teachers (although this should not be the case) don’t always do enough applied problems with students although they cover the manipulations well.  When looking at SAT and ACT scores, the Investigation group did better on SAT and the traditional group did better on ACT mathematics.  With the back and forth of who was better and the lack of comparison of the same exact preparation in all cases, I don’t believe you can call this study a reliable study that in anyway shows that this “Investigation” approach provides benefit to our students.

The bottom line is that students need BOTH.  We seem to always be at one end of the spectrum or the other.  They need both conceptual understanding and procedural practice to be successful.  Every student is also different, some students will be more successful with procedural math and really have a hard time “seeing” conceptual mathematics, other students might be bored with procedural math and really be able to grasp math in a very advanced, creative way with good applications, a small amount of theory and concepts presented (these will likely be your future math majors!).

To speak for myself, I fell in love with mathematics because of Procedural Algebra.  I LOVED to manipulate algebraic equations and later Calculus derivatives and integrals.  The concepts grew for me over time but without my love for procedural mathematics, I would never have pursued higher degrees in math.

Written by:  Lynne Gregorio

Ph.D. Mathematics Education with minor in Statistics (since I loved statistical manipulation)

M.S. Mathematics with minor in Secondary Education

B.S. Mathematics with minor in Philosophy and Secondary Education

 

 

 
 

Rules for Arithmetic – What students need to know

07 Aug

With the new common core standards, students are being asked to learn a large number of math rules in a single year.  Math rules can be very daunting for students.  If you think about how they change for every little thing.  When we multiply with decimals we add up decimal points to figure out number of decimal points in the answer, when we add and subtract with decimals, we line up decimal points, when we divide with decimals we have different rules depending on whether or not the the divisor is a decimal (then we move the decimal point) or not, then we just mark the spot and divide normally.  This is just for decimals, fractions are even more difficult, especially when you deal with mixed numbers.  Adding and subtracting always have different rules than multiplying and dividing!  Almost nothing is ever just intuitive.  No we can’t take 1/2 + 1/3 and just add the tops and bottoms and say the answer is 2/5!  How are students supposed to keep all these rules straight and at the same time, we want them to conceptually understand fractions, decimals, percents, geometry, algebra, patterns, and more!

I suggest flash cards.  On one side is the type of problem such as addition of fraction with unlike denominators and the other side contains the steps for performing it or the rules.  The student reviews the flash cards and practices a problem similar to the type on the flash as he reviews that rule, this process continues until all the rules are solid.  So, what are the rules that you should make sure your child knows?  Here they are:

  1. Adding decimals
  2. Comparing decimals
  3. Subtracting decimals (include the case such as:  4.1-.27)
  4. Multiplying decimals (I have students count the decimal places, put that number in a circle off to the side, remove decimals, multiply regularly, and then put decimal in at end.)
  5. Dividing decimals when divisor is a whole number and dividend is a decimal (4.46 / 2)
  6. Dividing decimals when divisor is a decimal and dividend is a decimal and also case of whole number (2.46 / .2 or 246 / .2)
  7. Adding fractions with unlike denominators (denominators have no factors in common and denominators do have a factor in common)
  8. Finding the LCM or LCD (lowest common denominator)
  9. Reducing a fraction to simplest form
  10. Adding mixed numbers
  11. Subtracting mixed numbers (when you need to borrow and not borrow)
  12. Multiplying fractions (where you can and cannot cross cancel)
  13. Dividing fractions
  14. Multiplying mixed numbers
  15. Dividing mixed numbers
  16. Finding GCF
  17. Writing a prime factorization
  18. Evaluating a number with an exponent
  19. Order of operations (remember that multiplication and division are on the same level and addition and subtraction are on the same level)
  20. Finding multiples of a number
  21. Converting from a mixed number to an improper fraction
  22. Converting from an improper fraction to a mixed number
  23. Divisibility Rules for (2, 3, 6, 5, 9, and 10)
  24. Knowing vocabulary of Sum, product, quotient, difference
  25. Rules for rounding numbers (include rules for rounding when you have 34.399, rounding to hundredth place)
  26. Key words for problems (Each means multiply or divide; of means multiple; How many more/fewer – means subtract)
  27. Rules for scientific notation
  28. Adding positive plus a negative
  29. How to approach subtraction with integers (switch / change approach)
  30. Multiplying negative times negative
  31. Multiplying with opposite signs (pos X neg or neg X pos)
  32. How to convert fractions to percents
  33. How to convert fractions to decimals
  34. How to convert decimals to percents
  35. How to solve percent word statements (is/of = %/100)

There are also many Geometry terms that can be added.  I realize some of these are not in the correct order (finding LCM, should be in card order before adding fractions when applied).  Most of these up through #26 are appropriate for students who have finished the 5th grade.  $27-#35 include 6th grade concepts.  There are additional concepts that can be added for 6th graders that start to include rules for algebra such as combining like terms, distributive property, etc.  However, if you want your child to be a strong math student – don’t just practice random problems, have them communicate the actual rules first and study these in the same way they might study a spelling list or science vocabulary for a test.  I have made some flash cards that you can print at:  http://www.flashcardmachine.com/machine/?read_only=2512894&p=9f2s

Lynne Gregorio, Ph.D.

 
 

Alternative Ways to Teach Mathematics for Common Core – in response to video for TERC

03 Aug

What do you think about this?  This came from a site called Stopcommoncorenc.org

 

Note this link comes from a site that clearly has a negative opinion about these alternative methods.  Is she justified?  Well, let’s not jump to conclusions and say, “Yes.”  I think the answer is “almost yes.”  I am a firm believer that nothing is ever black and white.  Common Core mathematics encourages students to learn alternative ways of thinking about mathematics.  They encourage students to delve deeper into the meaning behind the computation.  They want students to know the meaning behind an algorithm at a mature level.  This is a very LOFTY goal.  In fact, after doing my dissertation, I learned that many teachers don’t even have this level of understanding.  My daughter was taking 5th grade mathematics this past year using the North Carolina Common Core curriculum.  She was required to solve all her problems three different ways and then she had to write a journal entry that explained the WHY behind what she did.  This really isn’t too far off from what is in these videos.  Of course she was also taught or allowed to use the standard algorithm, but only in addition to other less effective algorithms that she had to learn.  These less effective algorithms were meant to build her conceptual understanding.  Was it effective?  I really don’t think so.  I don’t think you need alternative algorithms to build conceptual knowledge.  I think you can just build conceptual knowledge with good teaching.

For example, I have no idea how TERC might teach adding mixed numbers but when I look at how to teach this I know that some teachers (most) would teach it by just teaching the algorithm:

2 3/4 + 1 3/4

1.  Add 3/4 + 3/4  = 6/4

2.  Convert 6/4 to    1    2/4

3.  Add the whole number pieces:  2 + 1 + the extra 1 from step 2 = 3

4.  Final answer 3    2/4  or reduce to 3  1/2

 

However, a better approach might be to demonstrate this with a concrete approach rather than starting abstractly with just numbers.

Draw 2 and 3/4 a pies and another 1 3/4 pies.  Show how you can move one piece of the second partial pie to fill up the other partial pie, giving you 3 full pies and  2/4 left over pie (or 1/2 left over pie) – final answer 3 2/4 or 3 1/2.

From here you can now relate the concrete picture to the abstract numbers.

 

This is what I mean by “good teaching” and is much better than using 3-4 additional ineffective algorithms or “discovery” learning approaches.  Discovery learning was something that I was introduced to in my methods class when getting my degree to become a teacher.  It seemed like such a good idea at the time, instead of you telling the student what the answer was or how the problem worked, you devised a lab situation where they naturally “discovered” the solution all on their own.  Since they created it, it was more active learning and hence would stay with them longer than just being told.  The idea sounds great and active learning DOES produce better retention.  The problem is that you have to actually create that situation you are looking for.  If you just give them labs that the teacher THINKS will lead them to building conceptual knowledge but instead just ends up being a lot of extra steps (see the first video example), the impact is the exact opposite.  Our local high school tried this.  They decided to teach Common Core Math 1 concepts through discovery learning.  Instead of giving lectures on exponential functions and showing relationships, demonstrating examples, and having students work problems, they created these lab experiences for these freshmen (and a few sophomores) to do.  The students that I worked with completed the labs but it was clear that they had no meaning for them.  The intent was that this was supposed to build all the concepts they needed about exponential functions when in fact, the students just found answers to questions without piecing together the big picture or linking one lab to the next.  It was an utter failure.  Discovery learning is very hard to pull off.  Not only does it require really good lab experiences but it requires the right kind of student that can learn that way.   Students have a variety of learning styles and many need to be taught by concrete methods, examples, and linking new information to things they already know.

Another important point made by these videos is that it is a waste of time to teach the student the standard algorithm because they will eventually just rely on the calculator anyway.  Is this true?  Well, it is true if we make it true.  It also NEEDS to be true for some students.  If you read my post about accommodations for students with memory problems, they will NEED to have a calculator accommodation because they lack the same ability that the non-LD student has to memorize rote facts.  For this student, I think they do need a calculator so that they can focus more on the big picture and less on things that will never happen for them.  However, what about the student who can learn their multiplication facts but just doesn’t want to put in the time.  What about the 9th grader who still breaks out the calculator for 8×3?  One might argue that we always do have technology at our disposal.  With cell phones that have calculators on them, one is almost never without the ability to use a calculator these days.  Does that mean it is okay to rely on it?  I am a strong believer in technology and feel that we should not be doing tedious problems that one would not do in the “real” world without a calculator, in class without a calculator.  However, for things that one should just “know,” that demonstrates a basic understanding of basic mathematics, yes, students should be required to do these things without a calculator unless them have a documented learning disability.  In other words, let the real world determine the appropriate use of technology in the classroom.

What about the second book that was discussed in the video?  Was it all bad?  I have had students who have used both those algorithms:  the one that uses the place value and the lattice multiplication.  Here are my thoughts on those.  First, I like the idea of showing and having a lab on how 2 digit multiplication can be done with the list of numbers that show the true place value.  This really helps a student “see” what is going on behind double digit multiplication.  In fact, there are other ways that organize it even better that a teacher can use.  We do want to teach in a way that is more than just procedural mathematics.  Not all students will grasp everything but it will help some.  However, this does not mean that the ultimate goal is to leave out the standard algorithm.  The other algorithm is used to demonstrate conceptual understanding, not to be an algorithm that students continue to use.  My student who did try to use this as their only algorithm, did make many more mistakes than students using the traditional algorithm.  As for the lattice method, there is no reason conceptually to show this method.  However, for some students, the organization (keeping things in boxes) helps them from a visual perspective and that is why a teacher might choose to show this.  The student that used this algorithm used it successfully for a long time, he is a rising 9th grader and I wouldn’t be surprised if he doesn’t still use it (I only tutor him on occasion now when he has a test but it is still his algorithm of choice.)  The point of all of this is two-fold: one, use alternative approaches IF it helps a student grasp the concept more and two, allow them to use an alternative algorithm if they can be as fast, as accurate, and as effective with it.  Give the student the choice, it is frustrating when teachers tell students they “have” to use a specific algorithm.  Once a student finds one that meets the criteria of fast, accurate, and effective, they should not be forced to practice other algorithms except in sense of in class concept building labs.

Written by:  Lynne Gregorio, Ph.D. Mathematics Education

 

 
 

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

 

 
 

Review of Khan Academy

28 Jul

Okay, I am going to head over to Khan Academy and give you a blow by blow review from how user friendly it is to how well it teaches mathematics.

When I first get on the site, it clearly directs me where to go if I want to learn Algebra.   So I will click on the word Algebra.  However, after that first click, now I am bit more confused.  There is a place that says Introduction to Algebra so let’s start there.  Now I get some choices about history of Algebra and the Why’s of Algebra.  When reading another review, it commented about the lack WHY in Khan videos.  It appears that maybe they are trying to make up for some of that.  My problem is that this information needs to be integrated into each lesson not as a separate section.  It is like this is written here for an educator.  I don’t know that I see a student accessing the “Why we do things to both sides,” but let’s take a look.  After watching all these videos in this section, I give the overall videos a thumbs up!  However, the organization gets a thumbs down!  This should be considered a regular lesson for all of Khan academy for every topic.  For each topic, they should start with what they call the “WHY,”  I call it good teaching.  From there, they can move into some more procedural practice videos, and finally student practice videos.  The organization could look like this:

  1. Solving One Step Equations Lesson – here would be their “Why Videos” Relating to that
  2. Procedural videos for solving One Step Equations – this can be what I have traditionally seen on Khan Academy, many videos of people just solving problems
  3. Practice for the student (starting with easier problems and getting more difficult)
  4. NOW – Khan would move into the next developmental step
  5. Solving Two Step Equations Lesson – the rest of their WHY videos here.
  6. Procedural Videos for students to watch (starting with easier problems and getting more difficult)
  7. Practice for student (starting with easier problems and getting more difficult)

Let’s move on and see what else Khan has to offer…

So here is where things go down hill for Khan.  They just had these nice videos that did a good example of explaining the concept of solving one and two step equations.  Following down, the next thing on that page (which seems like what I should click on since it follows those videos) is something about Yoga… I decide to skip the Yoga and click on something that has some math terms in it so I click to the third section.  Here I get a list of options of videos to watch.  Now remember, I just learned some very basic ideas of how to solve an equation using a balance scale.  After getting confused by Yoga, I see Variables Expressions and Equations.  This is sort of random lecture that doesn’t really link me into anything and makes some medium jumps and has some confusing pieces in it.  The lesson is trying to teach you to substitute in a number to a variable expression but instead of building it in a developmental approach they jump to x+y+z=5 and start letting y=2 and z=3 and solving for x.  None of it is done in an organized fashion.  Letting that go, let’s just skip to the next video, maybe it will get better…

The next one says solving inequalities and equations through substitution (I picked #3).   Well, this involves solving equations, we did some of that but we have never talked about inequalities yet….  The problem says, “If r is the number of hot dogs Joey can eat in a minute, and N is the total number of hot dogs he eats in the contest, if Joey can eat 6 4/5 hot dogs per minute, how many hot dogs does he eat in the 10 minute contest?”  They give you the equation N/10 = r.

Here is the basic explanation they give to start, “He eats N hot dogs, and the contest is 10 minutes so we divide that by 10 and we get the number of hot dogs on average he eats.  So if Joey at 6 4/5 hot dogs per minute so they are saying r is 6 4/5, so they are saying, what is N going to be?”  Hmmm… did you follow that?  Did that explanation make you UNDERSTAND anything about the problem?  Do you get it better?  If the instructor just says, well plug in r= 6 4/5 and solve for N, have you learned anything?  One suggestion he gives to solve the problem is to “just try out numbers.”  Wow, that is a great idea (sarcasm).

Summary so far:

  1. You want to learn Algebra
  2. Go to site and find Algebra page – easy
  3. Great starter videos on how to solve one and two step equations
  4. After finishing those and maybe you can solve 2x -3 = 9 (by the way, only learned how to do it with numbers that work out)
  5. You see something about Yoga
  6. You skip Yoga and go to the next lesson and find problems like this hot dog problem that explains nothing, makes little sense, and is totally out of place.

Let’s leave behind “Introduction to Algebra” as that is clearly too hard and see if plain old “Algebra” is easier.

We choose linear equations, the first choice, and what is a variable, that sounds nice and basic.  Here we go, this is a much better place to start.  The lessons here so far have been basic and sequential.  A few minor problems are noted.  For example, when teaching how to plug in to evaluate an expression:  4n^1 + 2n^0.  (Carrot here means raised to the power of).  When “teaching” that anything to the first power is equal to itself, the instructor used a variable as their example x^1=x, this is teaching at a more abstract level rather than showing things such as 4^1 = 4 and 7^1 = 7 which is concrete and more easily grasped by the beginning student who just learned about x’s 5 minutes ago.

They begin to substitute in numbers with 2 variables and include positive and negative numbers.  In their example, they end up with -10 – 15.  They write the answer is -25.  The very first comment shows the lack of a strong teacher in these videos.  The first comment says, “why is -10-15 = -25, I think it should be 5.”  She continues to say, “Wouldn’t it have be -10–15 to get -25.”  Good teachers already know the mistakes that students will make and include this information into their lectures.  If I were to make this video, I would always quickly reteach older concepts whenever possible.  In this case, I would remind students that when dealing with positive and negative numbers and subtraction, you always want to do “switch change.”  Change the minus to a + and the sign of the number after that to its opposite.  -10-15 = -10 + -15, this makes seeing the -25 much easier for students.

Going back to their user interface.  One comment asked about practice problems.  The answer was that the lessons with the stars are the practice problems.  Having this written somewhere on the site would be helpful as it wasn’t clear to me either until I read the comment feedback.  On the practice problems, I do like the hint buttons and that you can ask for several hints if you get stuck.  However, I don’t like that the practice problems are not developmental.  The ideal situation would for them to start off easy and after you get 2 easy ones right in a row without hints, you get harder problems, and so on.

In the next unit, they talk about solving inequalities.  I don’t like the use of the words, “swap the inequalities.”  Most math teachers refer to it as flipping it.  The second thing is that their first introductory problem is -.5x < 7.5.  Their first step is to multiply by -2.  Again, “magic” math!  Most kids at this point CANNOT make the leap to seeing -.5 is =-1/2 and using its inverse.  Two seems like an odd number to throw into a problem with 5’s and 7.5’s.  Don’t start with something like that when the point you are trying to make is about flipping inequality signs, why muddy the water with something so confusing as -.5 and its inverse being -2.  He also immediately introduces the idea of how to graph inequalities on a number line and use interval notation.  He just sort of does it, no explanation of the number line and a quick explanation of the interval notation.  After years and years of experience with students, I know that kids can’t just hear that or view it at that speed with such little discussion and explanation and understand the concept.  There needs to be a whole lesson just on graphing inequalities (when to use open and closed circles and why) and writing interval notation, not an after thought in this lesson.  This is also another issue with Kahn.  Some areas they go so slow and other areas that need to go slow, they whip through it so fast.  It is like they have never worked with real students of various levels before.

We are at the point where big leaps are being made in Algebra.  I was pleased with the intro to solving equations videos.  I liked the discussion about what a variable and overall most of the plugging in to equations by substitution wasn’t too bad but now that we are getting into the meat of Algebra, we will start losing students.

At this point, I randomly picked another topic to see how it was done.  I picked completing the square.  If you read the comments you get the full the picture:

  1. I don’t understand why c=22
  2. I can’t see how this video has anything to do with practice
  3. Is this Algebra 1?
  4. I still don’t get how to do this?  Why is the FOIL method on the right?
  5. I don’t get that pattern about (x+a), my teacher showed us a different way

The teacher goes through a symbolic proof.  Although this is nice for Algebra teachers, be real, the majority of kids don’t get this at this level.  What you need to do for kids is provide a written set of steps, let them know where it is used, why it is important, and situations of application.

Overall, there are some decent videos but the entire system needs an overhaul.  Here are my recommendations.

  1. Someone who is a very strong math educator and who really knows how to teach well should go through each video and make suggestions for change.
  2. The organization needs updating for ease of use.
  3. A developmental approach needs to be used for both problems in lessons and practice.
  4. Teachers should not rely on this for their lessons as it stands, the lessons are not strong enough and it is missing the benefits of dynamic teaching.

Written by:  Lynne Gregorio, Ph.D. Mathematics Education

 

 

 
 

Is Khan Academy Appropriate for a Flipped Classroom?

28 Jul

This blog post talks about the concept of using a Flipped Classroom in general and specifically educators who use Khan Academy as there lectures for the Flipped Classroom.

 

There has been so much hype about Khan Academy.  Each time I have gone onto the website, I have failed to be impressed.  I will plan a review in another post.  However, leaving those comments for that time, educators say that despite Khan’s limitations, is it still appropriate for a flipped classroom.  For those that don’t know, a flipped classroom is where students get their “lessons” at home via video and then do their homework during classroom time.  The idea is that lectures are not usually dynamic and that students need more help from a teacher during the active part of learning – when they are solving problems and doing their homework.

This brings me to a couple questions –

  1. Should teacher lectures be dynamic?
  2. Can a static lecture produce the same results in a flipped curriculum?
  3. Are teachers using this idea of a flipped curriculum properly?

 

When I took high school and college classes, many of my classes were static lectures where teachers talked, we took notes, and then were assigned homework.  If the teacher was good enough, your notes allowed you to be successful doing your homework and the next day the teacher went over the homework (all or the ones the students asked about) and moved onto the next lesson.  However, this model never worked for me as an educator.  My goal as a teacher was to make sure my students understood the content I was teaching them.  Therefore my lectures were always dynamic.  Some of this was communicated to me during my teacher education classes while some of it was common sense and instinct.  My lessons looked like this:  Teach a topic, give student problems relating to this topic to try right after the topic demonstration, communicate with students about their success and failure of those problems, teach next topic and continue.  If I just had students watching me do a whole unit on the board of the several topics I needed to cover, I would a) lose the attention of some of the class b) not know if they were understanding the topics and c) not be able to adjust the teaching to fit that classes individual learning needs.

By assuming that a static lecture (even in a flipped classroom) is the best approach, you are taking away so much from the student.  First, who is providing the lecture content?  Khan Academy, in my opinion (review coming) does a poor job of presenting static lectures.  First of all they do not build up concepts in a developmental approach.  Topics should be presented where it builds on student current knowledge.  Easier problems should come first and then more difficult problems should be added.  This helps students be successful and build procedural and conceptual knowledge.  Khan also just demonstrates information without any explanation, I call it “Magic Math.”  For example, I saw them do a lesson on solving inequalities.  They had -5x < 15.  Their first step was multiple by (-1/5).  They just said do this and moved on, doing it on both sides, switching the sign (they did mention because they divided by a negative in passing) and finished the problem.  To a student learning how to do this, knowing to multiply by (-1/5) to both sides seems like “magic.”  WHERE did that come from???  There is no 1/5 anywhere in the PROBLEM!!  There was no discussion of inverses, etc.  Even if a student caught on, do they understand why they are doing it?  Can they apply it to a novel situation?  Helping a student put these pieces together is the responsibility of the teacher.

If the teacher follows a strict definition of the “flipped classroom,” their classroom might look like this:  assign Khan Academy lectures for homework, have student come to class and work on problems in small groups or individually.  Teachers may or may not walk around and help students who get stuck.  As an example, my son’s class did this (luckily, they did their own videos rather than use Khan and the lessons were much better) however, during homework time, there was little interaction from the teacher (as reported by my son.)  When this happens the students don’t get that conceptual understanding, teachers don’t learn what their students’ know and don’t know, and the dynamic process that is so important to learning is not happening.

I believe that teaching a lesson for the first time should be dynamic and by using a flipped classroom, you lose that opportunity to make it that way.  It is even worse if the student is sent home to watch videos from a site such as Khan Academy where the lessons are mostly procedural, not developmental, sometimes have errors, and do not allow for any type of interactive process to happen.  When students come to class to do their assignments, they may be confused from poor teaching standards that are present in these lectures.  Only the brightest students will have success and we will continue to have the many problems we have with our students failing mathematics.  Teachers will need to re-explain concepts from the beginning at a more developmental approach one on one to students who got “lost” during the lectures and this defeats the idea of the flipped classroom.

If there are teacher-made videos for the flipped classroom (and I have seen some good ones), I feel the process can work better.  However, I still think that it is the teacher’s job to teach in a dynamic way and allow the student practice time at home on their own to make sure they have grasped the ability to be independent on the problems.  For example, when you drive with someone in a car and watch where you are going, often you can’t then get there on your own but if you get behind the wheel yourself, you are much more likely to remember how to get there.  The same holds true for math.  If the teacher (or fellow classmates) are holding the student’s hand through the problems during classroom homework time, the student still may be unable to do the work independently come test time.

 

Written by:  Lynne Gregorio, Ph.D. Mathematics Education

Owner:  Apex Learning Center