Test II Mathematics Study Guide for the GACE

Measurement and Data

Measurement refers to the process of finding a number that represents the size or amount of something. A unit of measurement represents “one” unit of the standard of measurement for that situation. The standard of measurement is most often pre-determined. Some examples are minutes to measure time, centimeters to measure length, and pounds to measure weight. Students understand and apply knowledge of measurement, measurable attributes, and mathematical reasoning to identify, describe, and compare measurements.

In the beginning stages of measurement, students often use non-standard units. Non-standard units of measurement help students understand the concept of measurement without having to use standard tools like a ruler or scale. For example, a student could use paper clips or hand spans to measure their desk, or a balance to compare the weights of two items. Later, students use standard units, which are more traditionally and officially used among older students and adults to measure.

Measurable Attributes

One object may have many measurable attributes. For example, a dog could be measured for length, weight, and height, but length and height would use a different standard of measurement than weight.

Identify

Students should be able to identify the measurable attributes of a specific object and understand that one object can have more than one measurable attribute.

Describe

Students can describe an object based on its attributes. Beginning students use words like tall, short, long, heavy, and light without numerical measurements. As students become comfortable using both standard and non-standard units of measure, they give specific numerical measurements to show measurable attributes.

Compare

Students can compare objects based on one attribute at a time. They use appropriate vocabulary like:

“____ is taller than ____” and “____ weighs 2 pounds less than ____.”

For example, they can compare the weight of two dogs, and then compare the length of two dogs.

Measuring Length

Students use multiple units to measure length. When beginning to measure length, students often use non-standard units of measurement. These can be things like paper clips, cubes, or shoes. Eventually, students use standard tools like a ruler to measure the length of objects using standard units.

Standard units of length in most of the world are metric, like centimeters, meters, and kilometers. The United States often uses inches, feet, and miles from the standard system of measurement. Students use addition and subtraction to solve problems in which they must take multiple measurements to find the total or compare lengths.

Measuring Perimeter and Area

Perimeter is the sum of all sides of shape. Area is the measurement of the surface of or space inside a shape. Again, when students begin to measure these things, they often use nonstandard units like cubes to measure the outside (perimeter) or tell how many fit inside a shape (area). Later, they use standard units and equations.

Measuring Angles

An angle is a figure formed by two rays that meet at a common endpoint. Angles can be measured using a protractor. Students are able to identify types of angles such as:

• Right: a 90-degree angle
• Obtuse: an angle between 90 and 180 degrees
• Acute: an angle between 0 and 90 degrees

Measuring Volume

Volume is a measure of the space inside a 3-dimensional shape and is measured in cubic units. The basic formula for the volume of a rectangular solid is length x width x height.

Students can relate volume to multiplication and division by using measurements and mathematical reasoning to find the volume or, when the volume is known, find a missing measurement.

Data

Data is a collection of facts. These facts can be collected in many ways and can be qualitative (descriptive) or quantitative (numerical).

Representing Data

Data can be represented or displayed through methods like charts and graphs. A graph is a pictorial representation or diagram of data that displays data in an organized way. These visual representations aid students in comparing, analyzing, and interpreting the collection of facts. Some examples of graphs are:

• Bar graph: uses bars of different heights, and often different colors, to represent data. It is usually used to compare quantities.
• Line graph: uses plot points to show a change in quantity over time.
• Plot graph: Usually has a number line at the bottom, with check marks or tallies above each number to show quantity.

Interpreting Data

Using data and visual representations, students ask and answer questions about the data at hand. This often requires knowledge of counting, cardinality, and operations.

Geometry

Geometry is the mathematical study of space, shapes, lines, and angles. Students may begin their understanding of geometry by drawing shapes, then identifying their features, and eventually adding on measurement and analysis. Geometry is used in many real-life scenarios such as building and planning space.

Shapes

There are 2 types of shapes: two-dimensional (2D) shapes and three-dimensional (3D) shapes. 2D shapes are two-dimensional, or flat. They lie flat on a plane and are measured by width and height. Some examples are squares, circles, and triangles. 3D shapes are three-dimensional and take up physical space. Examples of these include cubes, prisms, and spheres.

Attributes of Shapes

Students are able to tell about shapes based on both defining and non-defining attributes. Defining attributes of shapes are ones that are necessary to define a shape like angles, sides, and faces. Non-defining attributes are ones that are not necessary to define a shape, like color, patterns, or overall size.

For example, defining attributes of a square would be that it has 4 equal-sided and 4 right angles. Non-defining attributes might be that it is small and red.

Classifying Shapes

Students can classify shapes by the properties of their lines and angles. A line is an object that extends in both directions with no known end. A line segment is one part of a line that has two endpoints. A ray has one endpoint and extends without a known end in the other direction. When two rays share an endpoint, an angle is formed.

The Coordinate Plane

The coordinate plane, also known as the Cartesian Plane, is a 2 dimensional plane with an x and y axis used for plotting and graphing points. The vertical axis is called the y-axis and the horizontal axis is called the x-axis. The lines run perpendicular and their intersection point is zero. On the x axis, numbers to the right of the y axis intersection are positive and numbers to the left are negative. On the y axis, numbers above the x axis intersection are positive and numbers below are negative. The plane is divided into four quadrants:

• Quadrant 1 is the quadrant on the top right. All points are positive.
• Quadrant 2 is on the top left. The y axis is positive and the x axis is negative.
• Quadrant 3 is on the bottom left. All points are negative.
• Quadrant 4 is on the bottom right. The y axis is negative and the x axis is positive.

Graphing Points

Coordinates on a plane are given in the format (a,b) and tell the distance from the origin point on both the x axis and the y axis. The x axis is stated first. For example, the coordinate (-5, 7) would be located at the point that is -5 from the origin on the x axis and 7 from the origin on the y axis. The coordinate is graphed at the meeting point of those two locations. This particular example is in Quadrant 2.