ALEX Classroom Resource

  

Grade 8 Mathematics Module 7, Topic A: Square and Cube Roots

  Classroom Resource Information  

Title:

 Grade 8 Mathematics Module 7, Topic A: Square and Cube Roots

URL:

 https://www.engageny.org/resource/grade-8-mathematics-module-7-topic-overview

Content Source:

 EngageNY
Type: Lesson/Unit Plan

Overview:

In Module 7, Topic A, students learn the notation related to roots (8.EE.A.2). The definition for irrational numbers relies on students’ understanding of rational numbers, that is, students know that rational numbers are points on a number line (6.NS.C.6) and that every quotient of integers (with a non-zero divisor) is a rational number (7.NS.A.2). Then irrational numbers are numbers that can be placed in their approximate positions on a number line and not expressed as a quotient of integers. Though the term “irrational” is not introduced until Topic B, students learn that irrational numbers exist and are different from rational numbers. Students learn to find positive square roots and cube roots of expressions and know that there is only one such number (8.EE.A.2). Topic A includes some extension work on simplifying perfect square factors of radicals in preparation for Algebra I.

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 8
1. Define the real number system as composed of rational and irrational numbers.

a. Explain that every number has a decimal expansion; for rational numbers, the decimal expansion repeats or terminates.

b. Convert a decimal expansion that repeats into a rational number.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Provide an example of both rational and irrational numbers in ratio form as well as the decimal expansion taken from the quotient of that ratio.
  • Convert a repeating decimal into a rational number.
Teacher Vocabulary:
  • Real Number System
  • Ratio
  • Rational Number
  • Irrational Number
Knowledge:
Students: know that any ratio a/b, where b is not equal to zero, has a quotient attained by dividing a by b.
  • know that the real number system contains natural numbers, whole numbers, integers, rational, and irrational numbers.
  • know that every real number has a decimal expansion that is repeating, terminating, or is non-repeating and non-terminating.
    • Skills:
    Students are able to:
    • define the real number system by giving its components.
    • Explain the difference between rational and irrational numbers. specifically how their decimal expansions differ.
    • Convert a ratio into its decimal expansion and take a decimal expansion back to ratio form.
    Understanding:
    Students understand that:
    • all real numbers are either rational or irrational and
    • Every real number has a decimal expansion that repeats, terminates, or is both non-repeating and non-terminating.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.1.1: Define expanding decimals, terminating decimals, rational number, and irrational number.
    M.8.1.2: Identify and give examples of rational numbers.
    M.8.1.3: Demonstrate how to convert fractions to decimals.
    M.8.1.4: Recall steps for division of fractions.

    Prior Knowledge Skills:
    • Define rational number.
    • Plot pairs of integers and/or rational numbers on a coordinate plane.
    • Arrange integers and /or rational numbers on a horizontal or vertical number line.
    • Locate the position of integers and/or rational numbers on a horizontal or vertical number line.
    • Identify a rational number as a point on the number line.
    • Recognize place value of whole numbers and decimals.
    • Give examples of rational numbers.
    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.1 Add and subtract fractions with like denominators (e.g. halves, thirds, fourths, tenths).
    M.AAS.8.1a Add and subtract decimals to the hundredths place.
    M.AAS.8.1b Convert a fraction with a denominator of 100 to a decimal.
    Mathematics
    MA2019 (2019)
    Grade: 8
    2. Locate rational approximations of irrational numbers on a number line, compare their sizes, and estimate the values of the irrational numbers.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Estimate the value of an irrational number and use that estimate to compare an irrational number to other numbers and to place irrational numbers on a number line alongside or between rational numbers.
    Teacher Vocabulary:
    • Rational
    • Irrational
    Knowledge:
    Students know:
    • the difference between a rational and an irrational number.
    • That real numbers and their decimal expansions can be approximated using a common place value to compare those expansions.
    Skills:
    Students know: the difference between a rational and an irrational number.
  • That real numbers and their decimal expansions can be approximated using a common place value to compare those expansions.
    • Understanding:
    Students understand that:
    • an estimation of the value of an irrational number can be used to compare an irrational number to other numbers and to place them on a number line.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.2.1: Define square root, expressions, and approximations.
    M.8.2.2: Identify properties of exponents.
    M.8.2.3: Recall how to compare numbers.
    M.8.2.4: Identify perfect squares and square roots.
    M.8.2.5: Demonstrate how to locate points on a vertical or horizontal number line.
    M.8.2.6: Recall how to estimate.

    Prior Knowledge Skills:
    • Define equivalent, simplify, term, distributive property, associative property of addition and multiplication, and the commutative property of addition and multiplication.
    • Simplify expressions with parentheses (Ex. 5(4 + x) = 20 + 5x).
    • Combine terms that are alike of a given expression.
    • Recognize the property demonstrated in a given expression.
    • Discuss various strategies for solving real-world and mathematical problems.
    • Recall steps for solving fractional problems.
    • Identify properties of operations for addition and multiplication.
    • Recall the rules for multiplication and division of rational numbers.
    • Recall the rules for addition and subtraction of rational numbers.
    • Demonstrate the location of positive and negative numbers on a vertical and horizontal number line.
    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.2 Compare quantities represented as decimals in real-world examples to the hundredths place.

    Mathematics
    MA2019 (2019)
    Grade: 8
    4. Use square root and cube root symbols to represent solutions to equations.

    a. Evaluate square roots of perfect squares (less than or equal to 225) and cube roots of perfect cubes (less than or equal to 1000).

    b. Explain that the square root of a non-perfect square is irrational.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Evaluate expressions involving squared and cubed numbers.
    • Solve equations with radicals with a square or cube root solution.
    Teacher Vocabulary:
    • Radical
    • Square Root
    • Cube Root
    Knowledge:
    Students know:
    • that the square root of a non-perfect square is an irrational number.
    • Equations can potentially have two solutions.
    • how to identify a perfect square/cube.
    Skills:
    Students are able to:
    • define a perfect square/cube.
    • Evaluate radical expressions representing square and cube roots.
    • Solve equations with a squared or cubed variable.
    Understanding:
    Students understand that:
    • there is an inverse relationship between squares and cubes and their roots.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.4.1: Define square root, cube root, inverse, perfect square, perfect cube, and irrational number.
    M.8.4.2: Recognize the inverse operation of squaring a number is square root and the inverse of cubing a number is cube root.
    M.8.4.3: Restate exponential numbers as repeated multiplication.
    M.8.4.4: Calculate the multiplication of single or multi-digit whole numbers.
    M.8.4.5: Recognize rational and irrational numbers.

    Prior Knowledge Skills:
    • Restate exponential numbers as repeated multiplication.
    • Define rational number.
    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.4 Calculate the square of numbers 1 through 10.
    Tags: cube root, decimal, expressions, irrational, number line, rational, square root
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    Comments

    There are five lessons in this topic.

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    Author: Hannah Bradley
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