Test II Mathematics Study Guide for the GACE

Understanding the Base 10 System

An understanding of the base 10 system is essential as students solve problems with both whole numbers and decimals. The base 10 system is our everyday number system where integers 0-9 take on different values based on their place value. When introducing the base 10 system, students often use manipulatives like base 10 blocks or straws to build understanding. For example, with straws, a straw would equal 1, 10 straws bundled together would equal 10, and 10 bundles of straws would equal 100.

Place Value

Place value is the value of each digit in a number, depending on its position in that number. For example, in the number 413, the value of the 4 is 400, the value of the 1 is 10, and the value of the 3 is three. This can also be shown by decomposing the number like this: 413 = 400 + 10 + 3.

Foundations of Place Value

Place value depends on the integer’s order of placement on one side of a decimal point. Each place is ten times larger than the place to its right. Numbers to the left of a decimal point have the following values:

• Ones (1)
• Tens (10)
• Hundreds (100)
• Thousands (1,000)
• Ten thousands (10,000)
• Hundred thousands (100,000)

Using Place Value in Operations

Students should be able to generalize their understanding of place value to solve problems with and compare numbers. This understanding begins with comparing and solving problems with the same number of digits. Students progress from solving problems that have only one digit (ones’ place).

Using in Multi-Digit Operations

Students progress to using place value to solve problems with multiple digits and varied numbers of digits. A strategy that may help students with this concept is rounding. Rounding is the process of changing a number to make it simpler to operate with. Usually, a particular place value is chosen as the point to which the student simplifies. Additionally, numbers 5 and up are usually rounded “up” and numbers 4 and below are usually rounded “down.” For example, rounding 568 to the nearest ten would result in 570, because 8 rounds up to 10.

Using with Decimals

Like the numbers to the left of the decimal point, the numbers to the right of the decimal point are also each 10 times larger than the number in the place to their right. Numbers to the right of the decimal point have the following values:

• Tenths (.1)
• Hundreths (.01)
• Thousandths (.001)
• Ten thousandths (.0001)
• Hundred thousandths (.00001)

Fractions

A fraction is a quantity that is not equal to one whole number. Fractions can also be written as decimals. Some fractions are more than one, and are written as improper fractions.An example of an improper fraction is\(\frac{6}{5}\). This can be decomposed into a mixed number.

A mixed number is a whole number and a fraction together. For example, \(\frac{6}{5}\)can be decomposed to a whole number (1) and a new fraction (\(\frac{1}{5}\)). The resulting mixed number is \(1 \frac{1}{5}\).

Understanding Fractions

A fraction is made up of a numerator and a denominator. The denominator is the number on the bottom and represents the number of equal parts which a quantity is being split into. A numerator is the number on top and represents the number of parts being included.

For example, \(\frac{1}{6}\) means that one out of 6 equal parts is being considered. Early learners can use visual fraction models to build an understanding of fractions. A visual fraction model shows the denominator as a shape broken into equal parts, and the number in the numerator filled in. For the fraction \(\frac{1}{6}\), a shape is broken up into \(6\) equal parts, with \(1\) part shaded in to represent the numerator. Once students have an understanding of what a fraction is, they can begin to find equivalent fractions, reduce fractions, and use fractions in operations.

Here are \(2\) such models for the fraction\(\frac{1}{6}\):

Equivalent Fractions

An equivalent fraction is a fraction that has the same value, but uses a different numerator and denominator. This can be done by multiplying or dividing both the numerator and denominator by the same number.

For example, \(\frac{1}{2}\) is equivalent to\(\frac{2}{4}\). This was found by multiplying both the numerator and denominator by 2.

\[\frac{1}{2} \times \frac{2}{2} = \frac{2}{4}\]

This process is often used when adding and subtracting fractions. In order to add and subtract fractions, both denominators must be the same. This process is called finding the common denominator. Another simple way to do this is to multiply the denominators together to arrive at a common number.

Reducing Fractions

Reducing a fraction means that a student simplifies the numbers in the fraction to their lowest possible terms. This is done by finding the greatest common factor, or the highest number that divides equally into both numbers. For example, the fraction \(\frac{4}{20}\) can be reduced because both \(4\) and \(20\) are divisible by \(2\) and \(4\). Since \(4\) is the greatest common factor, we can divide both numbers by \(4\) and get the resulting simplified fraction:\(\frac{1}{5}\).

\[\frac{4 \div 4}{20 \div 4} = \frac{1}{5}\]

Using Multiplication and Division with Fractions and Whole Numbers

When students have a strong understanding of theories and practice of multiplication and division with whole numbers, they can apply this knowledge, combined with their knowledge of fractions, to multiply and divide fractions.

Multiplying Fractions

To multiply a fraction, students simply multiply the numerators to find the new numerator and multiply the denominators to find the new denominator. Reducing the fractions can happen before or after the numbers are multiplied.

Dividing Fractions

Dividing fractions is a similar process to multiplying fractions, but there is one extra step. Students must use the reciprocal of the second fraction. The reciprocal is the inverse, or “flipped” form of the fraction. In the example \(\frac{3}{4}\) \(\div\)\(\frac{6}{7}\), the first step is to “flip” the second fraction by switching the numerator and denominator and then multiplying the new fractions. The new problem would be\(\frac{3}{4} \times \frac{7}{6}\).

Solve Real-World Problems Using Fractions

The use of fractions and problems with fractions is a skill that is used in everyday life. While we often think of typical situations like dividing up food or objects, fractions are also used in other mathematical situations like telling time (\(15\) minutes is \(\frac{1}{4}\)of an hour) or counting money (\(50\) cents is half of a dollar).