CAL common core standards

MATH-DUNK: The Math

The math component of Math-Dunk is something that can be varied with the needs and abilities of the students. The computational work related to sports, and Math-Dunk’s narrative backbone can be linked to any level of mathematics from arithmetic to calculus. It’s just a matter of adjusting computational problem sets to student abilities.

It is about the age of 14 that the possibility of dunking might appeal to a young male interested in basketball. At that age, the average 14 years old should be entering the 9thgrade. Though their bodies might be progressing at a typical rate, their academic skill sets, if they are African American males, will be on average, two or more years below grade level.  To compensate for their computational shortfalls, this example below covers the material that students of any background should be expected to have mastered in grades 6, 7, and 8. 

 

State of California Common Core Math Standards: Grades 6-7-8 (condensed) 

  http://www.corestandards.org/Math

 

A. Statistics

 

  1. Data set descriptions by center, spread, and shape.

  2. A measure of center for a numerical data set summarizes all of its values with a single number:

       The Mean is a measure of centrality. The Median is a measure of centrality. 

  3.  Summarizing data sets, by identifying clusters, peaks, gaps, and symmetry.

  4.  Summarizing data set context, such as; the number of observations and the units of measurement.

  5.  Generalizations about a population from a sample are valid only if the sample is representative of that population.

  6.  Random sampling to generate data sets and learn about the importance of representative samples for drawing inferences. 

  7.  Understanding that random sampling tends to produce valid representative samples.

  8.  Variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern          with reference to the context in which the data were gathered.

  9.  Informal assessment of two data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a            

       measure of variability. 

 10. Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

 11. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as    

       clustering, outliers, positive or negative association, linear association, and non-linear association.

 12. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to        the line.

 13. Investigation of patterns of association in bivariate data.

 14. Represent and analyze quantitative relationships between dependent and independent variables.

 15. Know that straight lines are widely used to model relationships between two quantitative variables. 

 16. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way        table. 

 17. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects.

 18. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. 

 19. “Where there’s smoke there’s fire.” Correlation & Causation. [Math-Dunk inclusion not part of Common Core]

 20. Spearman Rank Order correlation coefficient.[Math-Dunk inclusion not part of Common Core]

B.  Probability 

  1. The probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

  2. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. 

  3. Develop probability models by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up.
    a. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

       b. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. 

 4.   Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the    

       compound event occurs. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the    

       probability that a girl will be selected.

 5.   Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. 

 6.   Design and use a simulation to generate frequencies for compound events. For example, answer the question: If 40% of donors have type A blood,          what is the probability that it will take at least 4 donors to find one with type A blood?  

 7.   Informal introduction to Bayesian Statistics through examination of the work of David Blackwell. [Math-Dunk inclusion not      

      part of Common Core]

  C.Ratios

  1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.”

  2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. Use ratio and rate reasoning for problem solving. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

  3. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

  4. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. 

  5. Using ratios and proportionality to solve percent problems, including those involving discounts, interest, taxes, tips, and percent increase or decrease. 

 

D. Proportional Relationships

  1. Recognize and represent proportional relationships between quantities.

       a. Represent proportional relationships by equations. For example, if total cost t is proportional to the number N of items purchased at a constant              price P, the relationship between the total cost and the number of items can be expressed as t = pn.

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

       c. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities          and commissions, fees, percent increase and decrease, percent error.

  2.  Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For          example, if a person walks1/2 mile in each1/4 hour, compute the unit rate as the complex fraction ½/¼miles per hour, equivalently 2 miles per hour.

  3.  Students construct and analyze tables, such as tables of quantities that are in equivalent ratios, and they use equations (such as 3x = y) to describe            relationships between quantities.

​E. Geometry

Triangles

  1. Explain the Pythagorean Theorem and then apply it to determine unknown side lengths in right triangles in problems of two and three dimensions. 

  2. Understand congruence and similarity by constructing triangles from three measures of angles or sides noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. 

  3. Establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. 

  4. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes. 

Two Dimensional Figures

  5.  Using facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown            angle in a figure.

  6.  Understand that a two-dimensional figure is similar and/or congruent to another if the second can be obtained from the first by a sequence of rotations,        

       reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

  7.  Verifying experimentally the properties of rotations, reflections, and translations.

  8.  Calculate areas of polygons and surface areas of prisms and pyramids by decomposing them into pieces whose areas can be determined.

  9.  Perform arithmetic operations, involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order          

       [Order of Operations]. 

 10. Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing. 

 11. For example, use the formulas V = s3and A = 6s2to find the volume and surface area of a cube with sides of length s = 1/2.

 12. Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

 13. Using the formulas for the area and circumference of a circle to solve problems 

Three Dimensional Figures

 

 14. Solve problems involving area, volume and surface area of two & three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right          prisms.

 15. Relating three-dimensional figures to two-dimensional figures by examining cross-sections. Solving problems involving area, surface area, and volume of      

       two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

 16. Find the volume of a right rectangular prism. Apply the formulas V = l*w*h and V = b*h to find volumes of right rectangular prisms with fractional edge    

       lengths.

 17. Solving problems involving the area and circumference of a circle and surface area of three-dimensional objects, in preparation for work on congruence and    

       similarity. 

 18. Solving problems involving volume of cylinders, cones, and spheres. 

F. The Number System

Fractions

  1. Using the meaning of fractions, and the relationship between multiplication and division to understand why the procedures for dividing fractions make sense. 

  2. Solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3.  In general, (a/b) ÷ (c/d) = ad/bc.) 

  3. Convert a rational number to a decimal using long division. Add, subtract, multiply, and divide multi-digit decimals. 

Absolute Values & Inequalities

  4.  Positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, credits/debits,        positive/negative electric charge); use Explain the meaning of 0 in each situation. 

  5.  Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative          quantity in a real-world situation (temperature). For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.

  6.  Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. 

  7.  Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3°C > –7°C to express the fact that –3°C is          warmer than –7°C.

  8.  Reason about and solve one-variable equations and inequalities.

  9.  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a          

       statement that –3 is located to the right of –7 on a number line oriented from left to right. 

 10. The absolute value of rational numbers and the location of points in all four quadrants of the coordinate plane. Extend number line diagrams and coordinate            axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

 11. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality            true? 

 12. Write an inequality of the form x > c or x < c to represent in a real-world problem.

 13. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 (neutral) charge because its two constituents are                oppositely charged. 

 14. Use variables to represent two quantities in a real-world problem that change in relationship to one another. Analyze the relationship between the dependent            and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and              graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

​ 15. Apply properties of operations as strategies to add and subtract rational numbers.

 16. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

 17. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations,                particularly the distributive property, that leads to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational      

       numbers by describing real-world contexts.

 18. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p          and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.

 19. Solve multi-step problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically.          For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If        you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge;        this estimate can be used as a check on the exact computation.

20.  Solve equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. For example, the perimeter of a rectangle is 54 cm. Its          length is 6 cm. What is its width?

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

 22. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to            12. For example, express 36 + 8 as 4 (9 + 2).

 

Irrational Numbers

 23. Irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from fractions of integers. Among            

       irrational numbers are the ratio π of a circle's circumference to its diameter, , and the square root of two; and in fact all square roots of natural numbers, other

       than of perfect squares, are irrational. For example, the decimal representation of the number π starts with 3.14159, but no finite number of digits can    

       represent π exactly, nor does it repeat.

 24. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and

       estimate the value of expressions. For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and  

       explain how to continue on to get better approximations.

G. Expressions & Equations

  1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

  2. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent.

  3. Write, read, and evaluate expressions in which letters stand for numbers. For example, “Subtract y from 5” as 5 – y.

  4. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

  5. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5= ?

  6. Use square root and cube root symbols to represent solutions to equations of the form x2= p and x3= p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

  7. Perform operations with numbers expressed inscientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities.  For example, estimate the population of the United States as 3*10108and the population of the world as 7*10109, and determine that the world population is more than 20 times larger.

  8. Analyze and solve linear equations and pairs of simultaneous linear equations.

  9. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

  10. Analyze and solve pairs of simultaneous linear equations.

       a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of          

       intersection satisfy both equations simultaneously.

       b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. 

H. Functions - Graphing

  1. Function - a rule that assigns to each input exactly one output. 

  2. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

  3. Compare properties of two functions each represented in a different way 

       a. algebraically

       b. graphically

       c. numerically in tables 

       d. verbal description

  4.  Graph that exhibits the qualitative features of a function that has been described verbally. 

       Describe qualitatively the functional relationship between two quantities by analyzing the graph (e.g., where the function is increasing or decreasing, linear or        nonlinear). 

  5.  Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line.

​  6.  Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation        y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

  7. Graph proportional relationships and understand the unit rate (for change in X there is a corresponding change in Y) as a measure of the steepness of the        

      related line, called the slope.

  8. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r)    

      wherer is the unit rate. 

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

 10. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to      

       determine which of two moving objects has greater speed.

 11. Use the Pythagorean Theorem to find the distance between two points in a coordinate system. 

 12. Solve problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points        with the same first coordinate or the same second coordinate. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the      

       coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

 13. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first      

       coordinate or the same second coordinate. 

 14. Present examples of a function that is not linear: function A = s2giving the area of a square as a function of its side length is not linear because its graph      

       contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

For any questions, please contact David Trotman :

415-298-8979

1519 O'farrell St. San Francisco, CA 94115