WHAT THEY NEED TO KNOW
In high school, students will use a deeper understanding of mathematics to solve real-world problems. While in elementary and middle school, the math skills children need to know are organized by grade level, in high school they are grouped by concepts — such as algebra, functions, geometry — that students will tackle in various courses. These concepts build on what students learned in grade eight and move toward greater depth of knowledge and skills throughout high school.
Tests: New York City high-school teachers have already begun incorporating the Common Core math standards in classes. However, the Regents exam required for graduation, now known as Integrated Algebra, will not be updated to reflect the new curriculum until 2014, state officials say.
The Common Core high-school math standards cover these main areas:
* Number and Quantity: The Real Number System, Quantities, the Complex Number System, Vector and Matrix Quantities
* Algebra: Seeing Structure in Expressions, Arithmetic with Polynomials and Rational Expressions, Creating Equations, Reasoning with Equations and Inequalities
* Functions: Interpreting Functions, Building Functions, Linear, Quadratic, and Exponential Models, Trigonometric Functions
* Geometry: Congruence, Similarity, Right Triangles, and Trigonometry, Circles, Expressing Geometric Properties with Equations, Geometric Measurement and Dimensions, Modeling with Geometry
* Modeling: Real-world applications of mathematics from all categories
* Statistics and Probability: Interpreting Categorical and Quantitative Data, Making Inferences and Justifying Conclusions, Conditional Probability and the Rules of Probability, Using Probability to Make Decisions
Here’s a snapshot of work students will be doing:
* Create and solve equations (mathematical statements that use letters to represent unknown numbers, such as 2x-6y+z=14) with two or more variables to describe numbers or relationships
* Build an understanding of rational numbers (such as ¾) to include rational expressions (such as 3/(x-4))
* Use the structure of an expression to identify ways to rewrite it. For example, recognizing that x8-y8 is the difference between two squares and can also be written (x4)2-(y4)2
* Add, subtract, and multiply polynomials (an expression with multiple terms such as 5xy2+2xy-7)
* Interpret the slope of a line as the rate of change in two variables and the intercept as the constant term in a linear model
* Build and analyze functions that describe relationships between quantities, and use function notation (for example, f(x) denotes the output of f corresponding to the input of x)
* Represent and perform operations with complex numbers* (such as 3+5i, where i is an imaginary number and i 2 = -1)
* Understand the rules of probability, and use them to interpret data and evaluate the outcomes of decisions
* Distinguish between correlation and causation
* Interpret quantitative and categorical data
* Understand and prove geometric theorems (mathematical statements whose truth can be proven on the basis of previously proven or accepted statements)
* Use algebraic reasoning to prove geometric theorems
* Apply geometric concepts to model real-life situations
* Complex numbers are used in many scientific fields, including engineering and quantum physics
Examples of how students will develop and apply an understanding of structure and patterns in algebraic expressions:
* Interpret the structure of an expression
* Use the structure of an expression to identify ways to rewrite it. For example, x4-y4 = (x2)2-(y2)2
* Interpret one or more parts of an expression individually. For example, interpret 6+(x-2)2 as the sum of a constant and the square of x-2
* Solve quadratic equations (which include the square of a variable, such as 5×2-3x+3=0)
* Factor a quadratic expression to reveal the zeros of the function it defines
* Use the properties of exponents to transform and evaluate expressions. For example, interpret (82/3)2=(81/3)4 =24=16
* Write expressions in equivalent forms to solve problems
* Derive the formula for the sum of a finite geometric series, and use the formula to solve problems. For example, 3, 12, 48, 192 is a finite series where the ratio between each term is 4; 12/3=48/12=192/48=4
* Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. In grade eight, students solved real-world problems involving rates and discount, focusing on the computation needed to determine the final price. In high school, the emphasis is not about merely computing the final price but rather about using the structure of the answer to make a general argument.
Sample problem
Judy is working at a retail store over summer break. A customer buys a $50 shirt that is on sale for 20% off. Judy computes the discount, then adds sales tax of 10%, and tells the customer how much he owes. The customer insists that Judy first add the sales tax and then apply the discount. He is convinced that this way he will save more money because the discount amount will be larger.
a. Is the customer right?
b. Does the answer to part (a) depend on the numbers used or would it work for any percentage discount and any sales-tax percentage? Find a convincing argument using algebraic expressions and/or diagrams for this more general scenario.
Judy’s solution:
After the 20% discount, the shirt will cost 80% of the original price. 0.80($50) = $40 The tax will be 10% of this reduced price. 0.10($40) = $4 The final cost will be the reduced price plus tax. $40+$4 = $44 The equation for finding this answer is $50 (0.80)(1.10) = $44 Customer’s solution:
Before the 20% discount, the shirt cost $50. The tax will be 10% of this price. 0.10($50) = $5 The cost before the discount would be price plus tax. $50+$5 = $55 After the 20% discount, the shirt will cost 80% of this price. 0.80($55) = $44 The equation for finding this answer is $50 (1.10)(0.80) = $44 In this problem, students apply their understanding that changing the order of quantities in a multiplication problem doesn’t matter (known as the commutative property of multiplication). Students also show that given the structure of the equation used to find the answers, the answer would apply to any given combination of price, discount, and tax. For example, if we let P represent the original price, s represent the sale percentage, and t represent the tax percentage, students see that they can generalize the results.
Judy: P (1-s/100)(1+ t/100)
Customer: P (1+ t/100)(1 – s/100)
Examples of how students connect functions, algebra, and modeling to describe relationships between quantities
* Understand and use function notation (for example, f(x) denotes the output of f corresponding to the input of x)
* Interpret functions in terms of the context
* Calculate and interpret the average rate of change of a function presented in a graph or table over a given interval
* Graph functions symbolically and show key features of the graph, by hand or using technology (such as graphing calculators and computer programs) for more complicated cases
* Write a function defined by an expression in different but equivalent forms
* For a function that models a relationship between two quantities, interpret key features of graphs and tables, including intercepts, intervals where the function is increasing or decreasing, relative maximums and minimums, etc.
* Construct, compare, and apply linear, quadratic, and exponential models to solve problems to promote fluency with functions representing proportional relationships, students begin by interpreting function notation in context. For example, if h is a function that relates Shea’s height in inches to her age in years, then h(8)=50 means “When Shea is eight years old, she is 50 inches tall.”
Sample problem
The figure shows the graph of T, the temperature (in degrees Fahrenheit) over one particular 20-hour period as a function of time t.
a. Estimate T(14).
b. If t=0 corresponds to midnight, interpret what we mean by T(14) in words.
c. From the graph, estimate the highest temperature during this 20-hour period.
d. If Anya wants to go for a two-hour hike and return before the temperature is over 80 degrees, when should she leave?
Solution:
In this task, T(14) means that 14 hours after midnight, the temperature is a little less than 90 degrees Fahrenheit; T(14) is 2:00 p.m. The highest temperature on the graph is about 90 degrees. The temperature was decreasing between 4:00 p.m. and 8:00 p.m. It might have continued to decrease after that, but there is no information about the temperature after 8:00 p.m. If Anya wants to go for a two-hour hike and return before the temperature is over 80 degrees, then she should start her hike before 8:00 a.m.
Note: This is a straight-forward assessment task of reading and interpreting graphs. It requires an understanding of function notation and reinforces the idea that when a variable represents time, t = 0 is chosen as an arbitrary point in time and positive times are interpreted as times that happen after that point.
Sample problem
You work for a small business that sells bicycles, tricycles, and tandem bikes. Bicycles have one seat, two pedals and two wheels. Tricycles have one seat, two pedals, and three wheels. Tandem bikes have two seats, four pedals and two wheels.
1. On Monday, you counted 48 tricycle wheels. How many tricycles were in the shop? Write an algebraic equation that shows the relationship between the number of wheels (w) and the number of tricycles (t).
2. On Wednesday, there were no tandem bikes in the shop. There were only bicycles and tricycles. There are a total of 24 seats and 61 wheels in the shop. How many bicycles and how many triangles are in the shop? Show how you figured it out using algebra.
3. A month later, there are a different number of bicycles, tricycles tandem bikes in the shop. There are a total of 144 front steering handlebars, 378 pedals, and 320 wheels. How many bicycles, tricycles and tandem bikes are in the shop? Explain your solution.
Solution:
1. 16. Equation: t = w/3
2. 11 bicycles and 13 tricycles. Show work such as:
t: number of tricycles and b: number of bicycles
24 = t + b and 61 = 3t + 2b
so 61 = 3(24 – b) + 2b and b = 11 and t = 13
3. 67 bicycles, 32 tricycle, 45 tandem bikes.
Explanation: t: number of tricycles, b: number of bicycles, n: number tandem bikes
144 = t + b + n 378 = 2t + 2b + 4n 320 = 3t + 2b + 2n
(320 = 3t + 2b + 2n) — (288 =2t + 2b + 2n) results in t = 32 Substituting for t and using equations 1 & 2, (112 = b + n) – (314 = 2b + 4n) results in n = 45 Substituting for t and n in first equation: 144 = 32 + b + 45 results in b = 67