ALEX Resources

   View Standards     Standard(s): [MA2019] REG-8 (8) 4 :

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.

[MA2019] REG-8 (8) 4 :

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.

This lesson will develop the knowledge of squared and cubed numbers. The students will know when to use the square root and cube root to solve an equation. The students will memorize perfect squares and some cube roots. The answers will be left in radical form. Finally, the students will be able to identify the radicals as rational or irrational.

This lesson results from the ALEX Resource Gap Project.

   View Standards     Standard(s): [MA2019] ACC-7 (7) 10 :

10. 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 in a pattern or terminates.

b. Convert a decimal expansion that repeats in a pattern into a rational number. [Grade 8, 1]

[MA2019] ACC-7 (7) 15 :

15. 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. [Grade 8, 4]

[MA2019] REG-8 (8) 1 :

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.

[MA2019] REG-8 (8) 4 :

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.

Visualize a strategy for approximating the square roots of nonperfect squares through modeling. This video focuses on a ratio, the number of extra tiles over the number of tiles for the next square that will give you a fractional approximation of the square root of a number. This video was submitted through the Innovation Math Challenge, a contest open to professional and nonprofessional producers. This resource is part of the Math at the Core: Middle School collection.

   View Standards     Standard(s): [MA2019] REG-8 (8) 1 :

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.

[MA2019] REG-8 (8) 2 :

2. Locate rational approximations of irrational numbers on a number line, compare their sizes, and estimate the values of the irrational numbers.

[MA2019] REG-8 (8) 4 :

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.

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.

   View Standards     Standard(s): [MA2019] REG-8 (8) 1 :

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.

[MA2019] REG-8 (8) 2 :

2. Locate rational approximations of irrational numbers on a number line, compare their sizes, and estimate the values of the irrational numbers.

[MA2019] REG-8 (8) 4 :

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.

In Module 7, Topic B, students learn that to get the decimal expansion of a number (8.NS.A.1), they must develop a deeper understanding of the long division algorithm learned in Grades 6 and 7 (6.NS.B.2, 7.NS.A.2d). Students show that the decimal expansion for rational numbers repeats eventually, in some cases with zeros, and they can convert the decimal form of a number into a fraction (8.NS.A.2). Students learn a procedure to get the approximate decimal expansion of numbers like the square root of 2 and the square root of 5 and compare the size of these irrational numbers using their rational approximations. At this point, students learn that the definition of an irrational number is a number that is not equal to a rational number (8.NS.A.1). In the past, irrational numbers may have been described as numbers that are infinite decimals that cannot be expressed as a fraction, like the number pi. This may have led to confusion about irrational numbers because until now, students did not know how to write repeating decimals as fractions and further, students frequently approximated pi using 22/7 leading to more confusion about the definition of irrational numbers. Defining irrational numbers as those that are not equal to rational numbers provides an important guidepost for students’ knowledge of numbers. Students learn that an irrational number is something quite different than other numbers they have studied before. They are infinite decimals that can only be expressed by a decimal approximation. Now that students know that irrational numbers can be approximated, they extend their knowledge of the number line gained in Grade 6 (6.NS.C.6) to include being able to position irrational numbers on a line diagram in their approximate locations (8.NS.A.2).