HA - Heart of Algebra Module Overview

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Heart of Algebra

image of wordcloud in heart shape Algebra is the language of much of high school mathematics, and it is also an important prerequisite for advanced mathematics and postsecondary education in many subjects. The redesigned SAT focuses strongly on algebra and recognizes in particular the essentials of the subject that are most essential for success in college and careers. Heart of Algebra will assess students' ability to analyze, fluently solve, and create linear equations and inequalities. Students will also be expected to analyze and fluently solve equations and systems of equations using multiple techniques.

To assess full command of the material, these problems will vary significantly in form and appearance. Problems may be straight forward fluency exercises or may pose challenges of strategy or understanding, such as interpreting the interplay between graphical and algebraic representations or solving as a process of reasoning. Students will be required to demonstrate both procedural skill and a deeper understanding of the concepts that undergird linear equations and functions to successfully exhibit a command of the Heart of Algebra.

Mastering linear equations and functions has clear benefits to students. The ability to use linear equations to model scenarios and to represent unknown quantities is powerful across the curriculum in the postsecondary classroom as well as in the workplace. Further, linear equations and functions remain the bedrock upon which much of advanced mathematics is built. Consider, for example, that derivatives in calculus are used to approximate curves by straight lines and to approximate nonlinear functions by linear ones. Without a strong foundation in the core of algebra, much of this advanced work remains inaccessible.  

Essential Questions

  • What is the difference between a constant and a variable?  
  • What is a linear equation and how can it be solved?
  • What is a linear inequality and how can it be solved?
  • How does solving an equation or inequality in one variable differ from solving one with two variables?
  • How can you write a linear equation to model a given situation?
  • How can a system of equations have zero, one, or infinitely many solutions?

Key Terms

 1. Constant - A value that does not change, any number.

 2. Coefficient - the number multiplied or divided with the variable, for example, 2x + y = 3.   The   coefficient of the x term is 2 and the coefficient of the y term is 1.

 3. Consistent Dependent -  Lines which intersect at all points (the same line).

 4. Consistent Independent - Lines that have one intersection point.

 5. Dependent Variable - The variable whose answer is found in an equation after the independent variable has a value input into the equation.

 6. Domain - The x values that may be input into an equation to yield the y values, the range.

 7. Equation - A mathematical statement stating two expressions have the same value. Any number sentence using an = symbol meaning equality, both sides of the math equation are the same value when evaluated.     5 = 2 + 3.

 8. Inconsistent Lines -   Lines that are parallel, thus having no intersection point.

 9. Independent Variable - The variable in the equation that controls the value of the dependent variable. The independent variable has a substitution value chosen and then the dependent variable is evaluated.

10. Inequality - A mathematical statement indicating that two quantities are not strictly equal, but may have a relationship with or without equality. The mathematical inequality statement will use the symbols:

≤      Less than or equal to, indicates that the math expression on the left is smaller or equal to the math expression on the right;  5 ≤ 6 + 2 or 5 ≤ 6 -1.

≥      Greater than or equal to, indicates the math expression on the left is  greater or equal to the math expression on the right: 10 ≥ 6 - 4 or 10 ≥ 3 + 7.

<      Less than, indicates the math expression on the left is smaller or equal to the math expression on the right;  5 < 6.  

>      Greater than, indicates the math expression on the left is greater or equal to the math expression on the right;   7 > 2.

≠      Not equal to, indicates the math expression on the left is not equal to the math expression on the right;  -3 ≠ 3.

11. Linear Equation - An algebraic equation, such as y = 2x + 7 , in which the highest degree term in the variable or variables is of the first degree (the variables have an understood power of 1:  y^1 = 2x^1 + 7 where the exponents are the understood 1, the first degree and not written. The graph of such an equation is a straight line if there are two variables.

12. Range - The y values evaluated by inputting the domain, the x values, into an equation.

13. Slope - Intercept Form - An equation used to represent any straight line, y = mx + b, where m is the slope (the incline of the line) and b is the y-intercept, where the line crosses the y-axis.

14. System of Equations - two equations that have a relationship together, either parallel or intersecting at one point, or intersecting at all points.

15. Variable - Any letter or combination of letters used to represent a number value in an expression or an equation. Each variable has a meaning in the context of the problem (see Slope-Intercept Form for an example); a placeholder for a value in an equation.

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