ALEX Classroom Resource

  

Equivalent Expressions Using Exponents

  Classroom Resource Information  

Title:

 Equivalent Expressions Using Exponents

URL:

 https://aptv.pbslearningmedia.org/resource/mgbh-math-ee-8exp/equivalent-expressions-using-exponents/

Content Source:

 PBS
Type: Interactive/Game

Overview:

Students will apply their critical thinking skills to learn about multiplication and division of exponents. This interactive exercise focuses on positive and negative exponents and combining exponents in an effort to help students recognize patterns and determine a rule.

This resource is part of the Math at the Core: Middle School collection.

Content Standard(s):
Mathematics
MA2015 (2016)
Grade: 9-12
Algebra I
32 ) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8]

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. [F-IF8a]

b. Use the properties of exponents to interpret expressions for exponential functions. [F-IF8b]

Example: Identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, and y = (1.2)t/10, and classify them as representing exponential growth and decay.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.F.HS.32- Identify the y-intercept of a linear equation in the form of y=mx+b as (0,b).
Mathematics
MA2015 (2016)
Grade: 9-12
Algebra II with Trigonometry
13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

Example: See x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).
Mathematics
MA2019 (2019)
Grade: 7
Accelerated
14. Develop and apply properties of integer exponents to generate equivalent numerical and algebraic expressions. [Grade 8, 3]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Use their understanding of exponents as repeated multiplication to create equivalent expressions and justify integer exponent properties.
Teacher Vocabulary:
  • Integer Exponent
Knowledge:
Students know:
  • That whole number exponents indicate repeated multiplication of the base number and that these exponents indicate the actual number of factors being produced.
Skills:
Students are able to:
  • Develop integer exponent operations in order to generate equivalent expressions through addition, multiplication, division and raising a power by another power with expressions containing integer exponents.
Understanding:
Students understand that:
  • Just as whole number exponents represent repeated multiplication, negative integer exponents represent repeated division by the base number.
  • The exponent can be translated (visually. i.e. listing out the factors) to represent the exact number of factors being repeated so that the use of integer exponent operations ("rules") can be proven/make sense.
Diverse Learning Needs:
Mathematics
MA2019 (2019)
Grade: 8
3. Develop and apply properties of integer exponents to generate equivalent numerical and algebraic expressions.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Use their understanding of exponents as repeated multiplication to create equivalent expressions and justify integer exponent properties.
Teacher Vocabulary:
  • Integer Exponent
Knowledge:
Students know:
  • that whole number exponents indicate repeated multiplication of the base number and that these exponents indicate the actual number of factors being produced.
Skills:
Students are able to:
  • Develop integer exponent operations in order to generate equivalent expressions through addition, multiplication, division and raising a power by another power with expressions containing integer exponents.
Understanding:
Students understand that:
  • just as whole number exponents represent repeated multiplication, negative integer exponents represent repeated division by the base number.
  • The exponent can be translated (visually, listing out the factors) to represent the exact number of factors being repeated so that the use of integer exponent operations ("rules") can be proven/make sense.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.8.3.1: Define exponent, power, coefficient, integers, equivalent, and numerical expression.
M.8.3.2: Restate negative exponents as positive exponents in the form 1/x2 .
M.8.3.3: Restate zero exponents as 1 (x0 = 1).
M.8.3.4: Recognize to add exponents when multiplying terms with like bases (Property of product of powers).
M.8.3.5: Recognize to subtract exponents when dividing terms with like bases (Property of quotient of powers).
M.8.3.6: Compute a numerical expression with positive exponents.
M.8.3.7: Restate exponential numbers as repeated multiplication.
M.8.3.8: Compute problems with adding and subtracting integers.

Prior Knowledge Skills:
  • Define exponent, numerical expression, algebraic expression, variable, base, power, square of a number, and cube of a number.
  • Compute a numerical expression with exponents, with or without a calculator.
  • Restate exponential numbers as repeated multiplication.
  • Choose the correct value to replace each variable in the expression (Substitution).
  • Calculate the multiplication of single or multi-digit whole numbers, with or without a calculator.
  • Define integers, positive and negative numbers.
  • Demonstrate the location of positive and negative numbers on a vertical and horizontal number line.
  • Give examples of positive and negative numbers to represent quantities having opposite directions in real-world contexts.
  • Discuss the measure of centering of 0 in relationship to positive and negative numbers.
  • Discover that the opposite of the opposite of a number is the number itself.
  • Show on a number line that numbers that are equal distance from 0 and on opposite sides of 0 have opposite signs.
Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
5. Use the structure of an expression to identify ways to rewrite it.

Example: See x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Make sense of algebraic expressions by identifying structures within the expression which allow them to rewrite it in useful and more efficient ways.
Teacher Vocabulary:
  • Terms
  • Linear expressions
  • Equivalent expressions
  • Difference of two squares
  • Factor
  • Difference of squares
Knowledge:
Students know:
  • Algebraic properties.
  • When one form of an algebraic expression is more useful than an equivalent form of that same expression.
Skills:
Students are able to:
  • Use algebraic properties to produce equivalent forms of the same expression by recognizing underlying mathematical structures.
Understanding:
Students understand that:
  • Generating equivalent algebraic expressions facilitates the investigation of more complex algebraic expressions.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.5.1: Define equivalent expressions.
ALGI.5.2: Rewrite an exponential expression in an alternative way.
ALGI.5.3: Rewrite a quadratic expression in an alternative way.
ALGI.5.4: Rewrite a linear expression in an alternative form.
ALGI.5.5: Understand that rewriting an expression in different forms in a problem context can shed light on the problem.
ALGI.5.6: Recall properties of exponents.

Prior Knowledge Skills:
    li>Give examples of the properties of operations including distributive, commutative, and associative.
  • Recall how to find the greatest common factor.
  • Combine like terms of a given expression.
  • Recognize the property demonstrated in a given expression.
  • Simplify expressions with parentheses (Ex. 5(4 + x) = 20 + 5x).
  • Simplify an expression by dividing by the greatest common factor (Ex. 18x + 6y= 6(3x + y).
  • Define linear expression, rational, coefficient, and rational coefficient.
    Example: See x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2
  • y2)(x2 + y2).
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.11.5 Solve simple algebraic equations using real-world scenarios with one variable using multiplication or division.

Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
24. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a linear or exponential function,
  • Create a sequence from the functions and examine the results to demonstrate that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals.
  • Use slope-intercept form of a linear function and the general definition of exponential functions to justify through algebraic rearrangements that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals.

  • Given a contextual situation modeled by functions, determine if the change in the output per unit interval is a constant being added or multiplied to a previous output, and appropriately label the function as linear, exponential, or neither.
Teacher Vocabulary:
  • Linear functions
  • Exponential functions
  • Constant rate of change
  • Constant percent rate of change
  • Intervals
  • Percentage of growth
  • Percentage of decay
Knowledge:
Students know:
  • Key components of linear and exponential functions.
  • Properties of operations and equality
Skills:
Students are able to:
  • Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity changes at a constant rate per unit interval relative to another (linear).
  • Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity grows or decays by a constant percent rate per unit interval relative to another (exponential).
Understanding:
Students understand that:
  • Linear functions have a constant value added per unit interval, and exponential functions have a constant value multiplied per unit interval.
  • Distinguishing key features of and categorizing functions facilitates mathematical modeling and aids in problem resolution.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.24.1: Define linear function and exponential function.
ALGI.24.2: Distinguish between graphs of a line and an exponential function.
ALGI.24.3: Identify the graph of an exponential function.
ALGI.24.4: Identify the graph of a line.
ALGI.24.5: Plot points on a coordinate plane from a given table of values. a.
ALGI.24.6: Divide each y-value in a table of values by its successive y-value to determine if the quotients are the same, to prove an exponential function.
ALGI.24.7: Subtract each y-value in a table of values by its successive y-value to determine if the differences are the same, to prove a linear function.
ALGI.24.8: Apply rules for adding, subtracting, multiplying, and dividing integers. b.
ALGI.24.9: Define constant rate of change as slope.
ALGI.24.10: Subtract each y-value in a table of values by its successive y-value to determine if the differences are the same, to prove a linear function.
ALGI.24.11: Recognize the calculated difference is the constant rate of change.
ALGI.24.12: Apply rules for adding, subtracting, multiplying, and dividing integers. c.
ALGI.24.13: Define exponential growth and decay.
ALGI.24.14: Divide each y-value in a table of values by its successive y-value to determine if the quotients are the same, to prove an exponential function.
ALGI.24.15: Apply the rules of multiplication and division of integers.

Prior Knowledge Skills:
  • Recognize ordered pairs.
  • Identify ordered pairs.
  • Recognize linear equations.
  • Recall how to solve problems using the distributive property.
  • Define linear and nonlinear functions, slope, and y-intercept.
  • Analyze the graph to determine the rate of change.
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.12.24 Given a simple linear function on a graph, select the model that represents an increase by equal amounts over equal intervals.
Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
25. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a contextual situation shown by a graph, a description of a relationship, or two input-output pairs,
  • Create a linear or exponential function that models the situation.
  • Create arithmetic and geometric sequences from the given situation.
  • Justify the equality of the sequences and the functions mathematically and in terms of the original sequence.
Teacher Vocabulary:
  • Arithmetic and geometric sequences
  • Arithmetic sequence
  • Geometric sequence
  • Exponential function
Knowledge:
Students know:
  • That linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
  • Properties of arithmetic and geometric sequences.
Skills:
Students are able to:
  • Accurately recognize relationships within data and use that relationship to create a linear or exponential function to model the data of a contextual situation.
Understanding:
Students understand that:
  • Linear and exponential functions may be used to model data that is presented as a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
  • Linear functions have a constant value added per unit interval, and exponential functions have a constant value multiplied per unit interval.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.25.1: Define linear function and exponential function.
ALGI.25.2: Define arithmetic sequence, geometric sequence, and input-output pairs.
ALGI.25.3: Define sequences and recursively-defined sequences.
ALGI.25.4: Recognize that sequences are functions whose domain is the set of all positive integers and zero.
ALGI.25.5: Given a chart, write an equation of a line.
ALGI.25.6: Given a graph, write an equation of a line.
ALGI.25.7: Given two ordered pairs, write an equation of a line.

Prior Knowledge Skills:
  • Given a function, create a rule.
  • Recognize numeric patterns.
  • Recall how to complete input/output tables.
  • Demonstrate how to plot points on a Cartesian plane using ordered pairs.
  • Define function, ordered pairs, input, output.
  • Graph a linear equation given the slope-intercept form of an equation.
  • Graph a function given the slope-intercept form of an equation.
  • Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
  • Generate the slope of a line using given ordered pairs.
  • Recall the rules for multiplying integers.
  • Define quotient, divisor, and integer.
  • Solve addition and subtraction of multi-digit whole numbers.
  • Solve addition and subtraction of multi-digit decimal numbers (emphasis on alignment).
  • Recall basic multiplication and division facts.
  • Solve multiplication problems involving multi-digit whole numbers and decimal numbers.
  • Solve division problems involving multi-digit whole numbers and decimal numbers.
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.12.24 Given a simple linear function on a graph, select the model that represents an increase by equal amounts over equal intervals.
Tags: arithmetic sequence, equivalent, exponential function, exponents, expression, geometric sequence, graph, input, linear function, output, table
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This resource is free for students and teachers to use. 

  This resource provided by:  
Author: Hannah Bradley
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