Mathematics Common Core Standards: Algebra
Algebra is essential knowledge for a high school student trying to meet Common Core Standards. Algebra is a branch of mathematics that uses letters and other symbols in equations to represent unknown values and then uses what is known to figure out what is unknown.
For example, if 5x = 40 (that is, 5 multiplied by x equals 40), you know that x = 8 because given what we’ve been told — the known information — 5 is the only number that makes the number sentence true (5 × 8 = 40).
Structure in expressions
A major emphasis is on interpretation of the parts of an expression, such as coefficients and terms. In math, a coefficient is the number place before a variable (so in 4x, 4 is the coefficient and x is the variable). A term can be a single number, a variable, or a coefficient and variable together.
Understanding the interaction of coefficients and variables results in students being able to rewrite the expression in different ways, which requires an understanding of each part of the expression and of how all of the parts of an expression interact given the rules of math operations.
Expressions are numbers, symbols, and operators (+, –, ×, and ÷) grouped together to show value. Expressions differ from equations, which employ the use of an equal sign (=) to show that the values on either side of it are equal or to demonstrate the value of a variable.
Practice translating written or spoken expressions into numerical expressions for use in calculations. For example, you can write the statement “2 less than 5 times a number” as the expression 5x – 2. If x = 7, then 5(7) – 2 = 35 – 2 = 33.
Polynomials and rational expressions
Students start to work with polynomials, which are expressions that have more than one variable. They use addition, subtraction, multiplication, and division with polynomials.
Students are also introduced to the concept of factoring to simplify expressions and solve problems. Factoring involves finding values that multiplied together result in the expression; for example, the expression 5x – 5 can be factored as 5(x – 1) the same way that 14 can be factored as 2 × 7.
Write out a polynomial and have your child name the parts, as in the following example:
4x2 + 5x – 3
This is a trinomial because it has three expressions linked together with operators. It’s easy to get confused and count the x and x2 as separate entities, but in this problem they’re part of the coefficients beside them. The operators (addition and subtraction signs in this problem) separate the parts of this trinomial.
The number 4 is the leading coefficient, x is a variable, 2 is an exponent, and 2 is also the highest power in the equation. The 2 at the end is a constant.
Solve this problem, which involves a polynomial: If the area of a rectangle is expressed as x2 + 7x + 12 and the length of one side is x + 4, what is the length of the other side?
To solve this problem, factor x2 + 7x + 12 as (x + 3)(x + 4), so the length of the other side is (x + 3).
Students use equations to describe the relationships that exist between variables, including solving equations that are representative of real-world situations. The use of modeling (the application of mathematical concepts to practical situations) is a significant aspect of these standards.
Relationships between variables involve the interaction between variables and coefficients. For example, in 3x = y, the value of y is dependent on the value of x. In other words, as the value of x increases or decreases, so does the value of y.
Build an equation to solve a problem involving a real-world scenario. For example, imagine that a farmer wants to build a rectangular pen for his animals. He has 200 feet of fencing materials, and he needs one side of the rectangle to be 30 feet long. How long do the other sides need to be?
Start by drawing a rectangle and labeling the two shorter sides “30 feet.” Label the two longer sides “x.”
Represented in the form of an equation, you write: x + x + 30 + 30 = 200. After combining like terms, you’re left with 2x + 60 = 200. Subtract 60 from both sides of the equation, and you get 2x = 140. Divide both sides by 2, and you have your answer: Each of the longer sides of the rectangle is 70 feet.
Reason with equations and inequalities
Students solve equations by finding accurate solutions, practicing the skill of substituting numbers for variables to ensure accuracy. Checking the accuracy of an answer, including how reasonable an answer is in the context of a problem, does more to build problem-solving skills than merely having a student use an algorithm to solve equations. An algorithm is a step-by-step procedure to solve a problem.
Checking answer accuracy is an essential skill. When solving for a variable in an equation, such as 3x + 5 = 35, students can plug in their answer for x to see if they’re right. So after solving for x and finding that x equals 10, they plug in 10 for x and do the math: 3(10) + 5 = 35. If both sides are equal, the problem is correct.
Checking the reasonableness of an answer involves making a logical determination of whether the answer is reasonable given the context of a math problem. For example, if an object is thrown upward in the air, how long does it take for the object to hit the ground, given that s(t) = t2 – 2t + 35, with t representing time measured in seconds?
So, t can equal –7 seconds or 5 seconds. Is –7 seconds a reasonable answer? Of course not!