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### Descriptive Statements:

- Demonstrate knowledge of properties of numbers and number systems, operations, place value, rounding, comparing and ordering numbers, and equivalent representations of numbers.
- Use a variety of models to represent numbers and operations.
- Demonstrate knowledge of prime and composite numbers, divisibility rules, least common multiple, and greatest common factor.
- Solve problems involving integers, rational numbers, fractions, decimals, ratios, proportions, percent, exponents, and scientific notation.
- Apply knowledge of basic concepts of probability, including the use of simulations and counting procedures to estimate probabilities.
- Demonstrate knowledge of computation, including the use of mental math and estimation.

### Sample Item:

A student has a bag containing 96 hard candies and wants to give an equal number of them
to each of 6 friends. Which of the following mathematical operations would be most useful
for the student to use when determining how many candies each friend should receive?

- addition
- division
- subtraction
- multiplication

Correct Response and Explanation (Show Correct ResponseHide Correct Response)

**B.** This question requires the examinee to demonstrate knowledge of
operations. The operation of division can be defined as partitioning a certain number
of items into a specified number of groups in order to determine how many items each
group contains.

### Descriptive Statements:

- Demonstrate knowledge of mathematical reasoning and proofs.
- Apply knowledge of various strategies and procedures used in problem-solving situations.
- Translate between verbal descriptions and mathematical language and symbols to express quantitative relationships and to solve problems.
- Apply knowledge of a variety of diagrams, models, charts, manipulatives, and other tools used to represent mathematical concepts and real-world situations.
- Apply knowledge of statistical measures (e.g., mean, median, mode, range, frequency distribution) to describe and analyze data.
- Apply knowledge of data interpretation and of methods for displaying data in a variety of formats.

### Sample Item:

A member of a local environmental group uses a spreadsheet to track the amount of paper,
plastic, glass, and aluminum a community recycles each month, as shown below. The
spreadsheet software can automatically create several types of graphs from the data.
Which of the following types of graphs would be most appropriate for visually representing the
percentages of the various recycled materials relative to the total amount recycled?

blank |
A |
B |
C |

1 |
**Recycled Material** |
**Number of Pounds** |
**Percentage of Total** |

2
| Paper |
1380 |
69% |

3
| Plastic |
280 |
14% |

4
| Glass |
240 |
12% |

5
| Aluminum |
100 |
5% |

- circle graph
- line graph
- bar graph
- scatter plot

Correct Response and Explanation (Show Correct ResponseHide Correct Response)

**A.** This question requires the examinee to apply knowledge of graphs to
represent real-world situations. A circle graph is used to compare the parts of a whole
and the area of each sector is proportional to the fraction or percentage of the
area of the circle that it represents.

### Descriptive Statements:

- Recognize patterns in numbers, shapes, and data and ways to use variables, expressions, equations, and inequalities to communicate quantitative relationships.
- Apply knowledge of patterns to model real-world situations and make predictions.
- Recognize types and properties of functions.
- Use algebraic concepts to solve equations and real-world problems.

### Sample Item:

[0, 1, 5, 14, 30, 55, . . . ]

Which of the following numbers would appear next in the patterned sequence above?

- 64
- 81
- 91
- 121

Correct Response and Explanation (Show Correct ResponseHide Correct Response)

**C.** This question requires the examinee to recognize patterns in numbers.
In the pattern given, the differences between the successive terms in the pattern are
1, 4, 9, 16, and 25, which are the squares of the first 5 integers. To determine the
next term, the square of 6, or 36, must be added to 55 to get 91.

### Descriptive Statements:

- Recognize types and properties of lines, angles, and two- and three-dimensional shapes, including symmetry, congruence, and similarity.
- Solve problems involving perimeter, area, volume, geometric transformations, measurement, scale, and coordinate systems.
- Use geometric concepts to solve real-world problems.
- Identify and use appropriate measurement units, tools, and measurement techniques in various situations.
- Convert measurements within the metric and customary systems.

### Sample Item:

Two rectangles, A and B, are similar. Rectangle A has a length of 100 centimeters and a
width of 50 centimeters. If the area of rectangle B is 200 square centimeters, what is
its perimeter?

- 40 centimeters
- 60 centimeters
- 80 centimeters
- 120 centimeters

Correct Response and Explanation (Show Correct ResponseHide Correct Response)

**B.** This question requires the examinee to solve problems involving
perimeter. If rectangles A and B are similar, the length and width of rectangle B must
be proportional to the length and width of rectangle A. In addition, the product of the
length and width of rectangle B must equal 200 square centimeters. If the length of
rectangle B is 20 centimeters and its width is 10 centimeters, both of these conditions
are met. Therefore, the equation representing the perimeter of rectangle B is
20 + 10 + 20 + 10 = 60.