Students discover that all repeating decimals are rational numbers because they correspond to a fraction, and both represent the same location on the number line. This idea is extended to include irrational numbers, and distinguishing between rational and irrational numbers. Students extend their understanding of square roots to include estimating irrational square roots, finding common cube roots, and writing and solving simple equations involving square and cube roots.
Key Concepts:
 Rational numbers have a decimal expansion that repeats either with 0s or a set of repeating digits. The decimal form of rational numbers has the same value and names the same point on the number line as the fraction form of the number.
 Because we can find find the area of squares given the side length, we should be able to find the side length of a square given the area. This requires finding a number which, multiplied by itself, gives the area. Some of those numbers cannot be expressed as the ratio of integers, and are called "irrational" numbers.
 Irrational numbers can be measures of lengths, and therefore can be located a distance from 0 on the number line.
Prior Knowledge Needed:
 Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons; (Grade 5, Unit 6; 5.NBT.A.3b)
 Understand a rational number as a point on the number line; (Grade 6, Unit 6; 6.NS.C.6)
 Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats; Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities; (Grade 7, Unit 1; 7.NS.A.2d; 7.EE.B.4)
 Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms; (Grade 7, Unit 11; 7.G.B.6)
Units on the Horizon:
 Pythagorean theorem (Grade 8, Unit 5)
 Exponents and scientific notation (Grade 8, Unit 12)
Lessons

Lesson objective: Understand that rational numbers have a decimal expansion that repeats either with 0s or a set of repeating digits. The decimal form of rational numbers has the same value and names the same point on the number line as the fraction for...

Lesson objective: Apply understanding of rational numbers as distances to solve a problem. This lesson provides an opportunity for students to apply their knowledge and understanding of converting and adding rational numbers to a reallife situation. St...

Lesson objective: Understand that all repeating decimals have a fraction equivalent. Students bring prior knowledge of converting decimals into fractions from 5.NBT.A.3b. This prior knowledge is extended to repeating decimals as students examine multipl...

Lesson objective: Practice writing repeating decimals in fraction form. This lesson helps to build procedural skill with writing repeating decimals as fractions with an integer numerator and an integer denominator. A number line is used here because it ...

Lesson objective: Understand that the square roots of some numbers are irrational, but can still be approximated. Students bring prior knowledge of finding side lengths of squares from 7.G.B.6. This prior knowledge is extended to include irrational squa...

Lesson objective: Solve for rational and irrational square roots and cube roots. This lesson helps to build procedural skill with finding square roots and cube roots. Visual models and tables are used here to build a conceptual understanding of roots, a...

Lesson objective: Apply understanding of irrational square roots to solve a reallife problem. This lesson provides an opportunity for students to apply their knowledge and understanding of estimating irrational square roots to a reallife situation. St...

Lesson objective: Extend understanding that irrational numbers represent real distances by reasoning about the locations of numerical and variable expressions on a number line. Students bring prior knowledge of irrational numbers such as \(\mathrm\pi\) ...

Lesson objective: Practice estimating irrational square roots and order irrational numbers on a number line. This lesson helps to build fluency with identifying the whole numbers between which an irrational square root falls, and comparing the values of...

Lesson objective: Apply understanding of irrational numbers as actual distances to solve a realworld problem. This lesson provides an opportunity for students to apply their knowledge and understanding of irrational numbers to a reallife situation. St...