Unit of Study
❑Topical X Skills-based ❑ Thematic
In this unit of study, students will apply the properties of exponents. They will represent very small or very large numbers in scientific notation, perform operations and learn how to interpret when “E” appears on the calculator. Students will provide examples of linear equations with one, infinitely many, or no solution. Students will understand that real numbers are rational or irrational; will place them on the number line and compare them.
Common Core Essential State Standards
Domains: Expressions and Equations (8.EE), Number System (8.NS)
Clusters: Work with radicals and integer exponents.
Analyze and solve linear equations and pairs of simultaneous linear equations.
Know that there are numbers that are not rational, and approximate them by
8.EE.1 KNOW and APPLY the properties of integer exponents to GENERATE equivalent numerical expressions. For example: 32 × 3–5 = 3–3 = 1/33 = 1/27.
8.EE.2 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.
8.EE.3 USE numbers expressed in the form of a single digit times an integer power of 10 to ESTIMATE very large or very small quantities, and to EXPRESS how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.
8.EE.4 PERFORM operations with numbers expressed in scientific 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 (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
8.EE.7 SOLVE linear equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
b. Solve linear equations with rational number coefficients, including equation whose solutions require expanding expressions using the distributive property and collecting like terms.
8.NS.1 KNOW that numbers that are not rational are called irrational. UNDERSTAND informally that every number has a decimal expansion; for rational numbers SHOW that the decimal expansion repeats eventually, and CONVERT a decimal expansion which repeats eventually into a rational number.
8.NS.2 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 (e.g., π2). 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
Standards for Mathematical Practice
(students need to know)
(students need to be able to do)
· Properties of exponents
· I can use properties of exponents to simplify expressions
· Solve and explain x2 = p
· Solve and explain x3 = p
· I can solve and explain equations in the form of x2 = p and x3 = p
· Numbers represented in scientific notation
· Comparison of quantities written in scientific form
· I can represent very small or very large numbers in scientific notation.
· I can compare quantities written in scientific notation.
· Computation with numbers written in scientific notation and decimal form
· Interpret and read “E” when it appears on the calculator
· I can compare and compute numbers in scientific notation and decimal form.
· I can interpret how to read answer when the “E” appears con the calculator.
· Examples of linear equations with one, infinitely many, or no solution
· Equations that include rational number coefficients, expanding expressions, and combining like terms
· I can provide examples of linear equations in one variable with one, infinitely many or no solutions.
· I can solve equations that include rational number coefficients, expanding expressions, and combining like terms.
· Rational or irrational numbers
· I can classify numbers as rational or irrational and explain why including that:
· decimals that repeat or terminate in 0s are rational.
· all fractions are rational.
· square roots of positive numbers are irrational, unless the radicand (the number inside the square root sign) is a perfect square.
· cube roots are irrational, unless the radicand is a perfect cube.
· π and e are special numbers that are irrational.
· Use of knowledge of square roots of perfect squares to estimate value of other square roots.
· Order rational numbers
· Location of irrational numbers on number line
· I can use the knowledge of square roots of perfect squares to estimate value of other square roots.
· I can order rational and irrational numbers on number line.
· I can find approximate location of irrational numbers on a number line.
Corresponding Big Ideas
· How can you apply properties of exponents to simplify expressions?
· Students will apply the following laws of exponents to write equivalent expressions for a given expression:
o am × an = am+n
o (am)n = am×n
o a–1 = 1/a and a–m = 1/am
· How can you solve equations of the form x2 = p?
· How can you solve equations of the form x3 = p?
· Students will solve equations in the form x2 = p, where p is a positive rational number and represent the solution using the square root symbol, and can explain when and why the solution must include the ± (plus or minus) symbol.
· Students will solve equations in the form x3 = p, where p is a positive rational number and represent the solution using the cube root symbol. The student knows that the cube root of a positive number is positive, and the cube root of a negative number is negative.
· How can you represent very large or very small numbers in scientific notation?
· How can you compare two quantities written in scientific notation?
· Students will represent very large and very small quantities/measurements in scientific notation. This includes:
o Converting a given number into scientific notation.
o Converting from scientific notation to decimal form.
o Knowing that a number written in scientific notation with 10 to a negative power represents a number between 0 and 1.
· Students will compare two quantities written in scientific notation and can reason how many times bigger/smaller one is than the other without having to convert each number back into decimal form.
· How can you solve problems where both decimal and scientific notation are used?
· How can you interpret the answer when “E” appears on the calculator?
· Students will solve problems that involve very large or very small quantities, including problems where both decimal and scientific notation are used.
· Students will interpret how to read the answer when the “E” appears when using a calculator to perform a calculation in which the answer will be a very large or very small number.
· How can you provide examples of linear equations with one, infinitely many or no solution?
· How can you solve equations that include rational number coefficients, expanding expressions, and combining like terms?
· Students will provide an example of a linear equation in one variable that has:
§ exactly one solution (e.g., x + 3 = 7)
§ infinitely many solutions (e.g., 2x + 4 = x + 2)
§ no solutions (e.g. x + 5 = x + 3)
· Students will solve linear equations (i.e., find the value of the variable that satisfies the equation) that include:
§ rational number coefficients (e.g., )
§ expanding expressions (e.g., 3(x + 5) = 18
§ combining like terms (e.g., 5x – 3 = 2x + 12)
· How can you classify numbers as rational or irrational and explain?
· Students will classify numbers as rational or irrational and explain why.
· How can you use knowledge of square roots of perfect squares to estimate the value of other squares?
· How can you order and compare rational or irrational from least to greatest?
· How can you approximate the location of irrational numbers on the number line?
· Students will estimate value of square root of non perfect squares.
· Students will order rational and irrational numbers.
· Students will approximate location of irrational numbers on the number line.
laws of exponents, power, perfect squares, perfect cubes, root, square root, cube root, scientific notation, standard form of a number.
intersecting, parallel lines, coefficient, distributive property, like terms
real numbers, irrational numbers, rational numbers, integers, whole
numbers, natural numbers, radical, radicand, square roots, perfect squares, cube roots, terminating decimals, repeating decimals, truncate
8.EE.1 - 4
Define and give examples of vocabulary and expressions specific to this standard (properties, integer exponents, positive, negative, equivalent numerical expressions, raising to a power, square root, cube root, squaring, cubing, rational, irrational, inverse operations, scientific notation, distributive property, rational numbers, variables, equality, equation, solution, like terms, constant, value, linear equation, expanding expressions, coefficients, etc.)
SWBAT use sentence frames to explain why some equations have no solution. (Where variables cancel out, constants are not equal.)
SWBAT compare equations which have one solution (true equality) and multiple solutions.
SWBAT explain the properties of integer exponents to a partner.
SWBAT write a paragraph to explain how to interpret numbers written in scientific notation.
SWBAT explain to a group how to use laws of exponents to multiply or divide numbers written in scientific notation.
SWBAT write a paragraph to describe the relationship of square roots and squares and of cube roots and cubes.
SWBAT use sequence words and phrases to explain the step-by-step process of solving linear equations, including those with rational coefficients, expanding expressions using distributive property and collecting like terms.
SWBAT transform equations into simpler forms until an equivalent equation results and explain the process.
SWBAT transform equations into simpler forms until an equivalent equation results and explain the process
Language Learning Strategies
SWBAT interpret how the graph of the solution of a linear equation represents the coordinates of the point where the two lines would intersect.
SWBAT explain why the graph of the solution of a linear equation represents a pair of parallel lines.
SWBAT interpret the graph of an equation with infinitely many solutions (same line).
Information and Technology Standards
8.TT.1.2 Use appropriate technology tools and other resources to organize information (e.g. graphic organizers, databases, spreadsheets, and desktop publishing).
8.RP.1.1 Implement a project-based activity collaboratively.
8.RP.1.2 Implement a project-based activity independently.
Instructional Resources and Materials
NCDPI Realigned Resources
Connected Math 2 Series
Partners in Math Materials
RCDay 2/20/12: Ponzi Pyramid Scheme Task