8.3 The student will a) estimate and determine the two consecutive integers between which a square root lies; and b) determine both the positive and negative square roots of a given perfect square. Understanding the Standard Essential Knowledge and Skills A perfect square is a whole number whose square root is an integer. The square root of a given number is any number which, when multiplied times itself, equals the given number. Both the positive and negative roots of whole numbers, except zero, can be determined. The square root of zero is zero. The value is neither positive nor negative. Zero (a whole number) is a perfect square. The positive and negative square root of any whole number other than a perfect square lies between two consecutive integers (e.g., lies between 7 and 8 since 72 = 49 and 82 = 64; lies between -4 and since (-4)2 = 16 and (-3)2 = 9). The symbol may be used to represent a positive (principal) root and - may be used to represent a negative root. The square root of a whole number that is not a perfect square is an irrational number (e.g., is an irrational number). An irrational number cannot be expressed exactly as a fraction where b does not equal 0. Square root symbols may be used to represent solutions to equations of the form x2 = p. Examples may include: If x2 = 36, then x is = 6 or =-6. If x2 = 5, then x is or −. Students can use grid paper and estimation to determine what is needed to build a perfect square. The square root of a positive number is usually defined as the side length of a square with the area equal to the given number. If it is not a perfect square, the area provides a means for estimation. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Estimate and identify the two consecutive integers between which the positive or negative square root of a given number lies. Numbers are limited to natural numbers from 1 to 400. (a) Determine the positive or negative square root of a given perfect square from 1 to 400. (b)
8.4 The student will solve practical problems involving consumer applications. Understanding the Standard Essential Knowledge and Skills Rational numbers may be expressed as whole numbers, integers, fractions, percents, and numbers written in scientific notation. Practical problems may include, but are not limited to, those related to economics, sports, science, social science, transportation, and health. Some examples include problems involving the amount of a pay check per month, commissions, fees, the discount price on a product, temperature, simple interest, sales tax and installment buying. A percent is a ratio with a denominator of 100. Reconciling an account is a process used to verify that two sets of records (usually the balances of two accounts) are in agreement. Reconciliation is used to ensure that the balance of an account matches the actual amount of money deposited and/or withdrawn from the account. A discount is a percent of the original price. The discount price is the original price minus the discount. Simple interest (I) for a number of years is determined by finding the product of the principal (p), the annual rate of interest (r), and the number of years (t) of the loan or investment using the formula I = prt. The total value of an investment is equal to the sum of the original investment and the interest earned. The total cost of a loan is equal to the sum of the original cost and the interest paid. Percent increase and percent decrease are both percents of change measuring the percent a quantity increases or decreases. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Solve practical problems involving consumer applications by using proportional reasoning and computation procedures for rational numbers. Reconcile an account balance given a statement with five or fewer transactions. Compute a discount or markup and the resulting sale price for one discount or markup. Compute the sales tax or tip and resulting total. Compute the simple interest and new balance earned in an investment or on a loan given the principal amount, interest rate, and time period in years. Compute the percent increase or decrease found in a practical situation.
change in each y-value compared to the corresponding change in the x-value. slope = m = = = −3 The slope, m, and y-intercept of a linear function can be determined given the graph of the function. Example: Given the graph of the linear function, determine the slope and y-intercept. Given the graph of a linear function, the y-intercept is found by determining where the line intersects the y-axis. The y-intercept would be 2 or located at the point (0, 2). The slope can be found by determining the change in each y-value compared to the change in each x-value. Here, we could use slope triangles to help visualize this: slope = m = = = −3 Graphing a linear function given an equation can be addressed using different methods. One method involves determining a table of ordered pairs by substituting into the equation values for one variable and solving for the other variable, plotting the ordered pairs in the coordinate plane, and connecting the points to form a straight line. Another method involves using slope triangles to determine points on the line. Example: Graph the linear function whose equation is y = 5x − 1. In order to graph the linear function, we can create a table of values by substituting arbitrary values for x to determining coordinating values for y: function using verbal descriptions, tables, equations, and graphs. (e).
graph. Not all values for x could be represented and connecting the points would not be appropriate. Example: The function y = −2.5x + 20 represents the number of gallons of water (y) remaining in a 20-gallon tank being drained for x number of minutes. The domain in this function would be continuous. There would be an x-value representing any point in time until the tank is drained so connecting the points to form a straight line would be appropriate (Note: the context of the problem limits the values that x can represent to positive values, since time cannot be negative.). Functions can be represented as ordered pairs, tables, graphs, equations, physical models, or in words. Any given relationship can be represented using multiple representations. The equation y = mx + b defines a linear function whose graph (solution) is a straight line. The equation of a linear function can be determined given the slope, m, and the y-intercept, b. Verbal descriptions of practical situations that can be modeled by a linear function can also be represented using an equation. Example: Write the equation of a linear function whose slope is and y-intercept is −4, or located at the point (0, −4). The equation of this line can be found by substituting the values for the slope, m = , and the y-intercept, b = −4, into the general form of a linear function y = mx + b. Thus, the equation would be y = x – 4. Example: John charges a $30 flat fee to trouble shoot a personal watercraft that is not working properly and $50 per hour needed for any repairs. Write a linear function that represents the total cost, y of a personal watercraft repair, based on the number of hours, x, needed to repair it. Assume that there is no additional charge for parts. In this practical situation, the y-intercept, b, would be $30, to represent the initial flat fee to trouble shoot the watercraft. The slope, m, would be $50, since that would represent the rate per hour. The equation to represent this situation would be y = 50x + 30.