In Grade 8, students get more comfortable with the use of rational and irrational numbers to meet Common Core Standards. A rational number is one that can be expressed as a simple fraction. An irrational number has no fractional equivalent; for example, the value of pi (3.1415926535897…) can’t be expressed as 3 along with a fraction. Students are also introduced to several new concepts and skills, including the following:

• The relationship between exponents and radicals: While an exponent tells you how many times to multiply a number by itself, a radical, also referred to as a root, tells you how many times to divide a number by itself; for example, the square root of 4 is 2, because 2 × 2 = 4. The cube root of 27 is 3, because 3 × 3 × 3 = 27.

• Functions: Functions are rules that define the output for any given input; for example y = x + 2 is a rule that defines the value of y in terms of the value of x. If you know that x is 3, then you know that y is 5 because the rule says that 2 must be added to any input.

• Analyzing two- and three-dimensional objects: Students use distance, angle, and similarity to analyze shapes. They’re also introduced to the Pythagorean theorem: the rule that in a right triangle (a triangle with a 90 degree angle), the square of the hypotenuse (longest side) is equal to the sum of the squares of the two other sides.

Significant emphasis is placed on the skills that prepare students for high-school algebra.

## The number system

In Grade 8, students discover the difference between rational and irrational numbers and are asked to put irrational numbers on a number line as accurately as possible to the nearest rational numbers. This enables students to compare the values of multiple irrational numbers to rational numbers.

Review the difference between rational and irrational numbers and then move on to placing these numbers on a number line. Find the approximate value of an irrational number in relation to rational numbers on the number line by rounding to a specified digit (you can start simple with whole numbers and then move on to rounding to the tenths, hundredths, and thousandths place).

For example, if you ask your child to round to the tenths place, pi can be rounded to 3.1, and the square root of 2 can be rounded to 1.4. Place the answers on a number line so she can see the relationship between the irrational number and its relative placement on a number line.

## Expressions and equations

Students use expressions and equations with exponents (for example, the number 2 in 42) and radicals () when using square roots and cube roots or solving problems written in scientific notation. A significant aspect of this domain in Grade 8 is solving linear equations (an equation that results in a straight line when graphed), including the use of graphing.

Practice graphing linear equations. Create a simple coordinate plane and remember to make sure that the equation results in graphing a straight line. For the most part, you can write linear equations by avoiding the use of exponents or radicals.

## Functions

Functions play a big role in Grade 8. Students get comfortable with how functions work by representing them with numbers, in tables, and on graphs.

Practice writing various functions and talking through the input and output. For example, f(x) = x + 1 indicates that the function is adding the number 1 to any value substituted for x. So if the number 2 takes the place of x, the output is 3. If the input is 5, the output is 6.

## Geometry

Students look at geometric shapes and determine whether they’re congruent (the same size and shape) using various movements, tools, and methods. They use the Pythagorean theorem (a2 + b2 = c2) to find the length of unknown sides of right triangles and explore its application in real-world settings.

Draw several right triangles and label two of the three sides with a number. Challenge your child to use the Pythagorean theorem to find the length of the side that’s missing a value.

## Statistics and probability

Sets of data that include two variables (bivariate) require students to explore different ways of interpreting data, specifically using scatterplots. Students interpret and explain information gathered and use it to draw conclusions.