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High School — Mathematics
Number, Number Sense and Operations

When students encounter a number like 25, those who have good number sense will realize this number is a multiple of 5, that it is 25% of 100, and that it is a perfect square. This type of understanding is necessary for students to grasp mathematical concepts. As students pursue estimation and computation with operations, they will increasingly rely on their sense of numbers.

Having a strong sense of numbers involves understanding the types of numbers in the real number system. For example, students should be able to distinguish a given number as rational or irrational. Number sense also includes the ability to express equivalent numbers in a variety of forms, such as expressing one million or one billionth in scientific notation.

Operating on numbers and discussing those operations is also vital to a student's mathematical understanding. Students should be able to discuss properties of operations and determine when they hold true. Finally students should be able to use estimation and computation to solve problems and to justify their answers.

Even though students will not receive a score for the Mathematical Processes standard on the Ohio Graduation Test (OGT), it is still an important part of the curriculum. Content and processes should be taught in tandem. To better understand Number, Number Sense and Operations, click on the dropdown menu above and select Mathematical Processes.

The content in this Teaching Tool is based largely on the Ohio Mathematics Content Standards and Benchmarks and includes released items from the OGT. Additionally, these materials are aligned with the National Council of Teachers of Mathematics (NCTM) standards . While various activities are suggested for working with students, this Teaching Tool is designed to complement a rigorous, research-based curriculum, not to substitute for one.



Number, Number Sense and Operations


1. Number, Number Sense and Operations

Click on the following benchmarks for more information and for links to annotated OGT items.

a.

Benchmark A: Use scientific notation to express large numbers and numbers less than one.

Scientific notation is a way to express very large and very small numbers using shorthand. Numbers written in scientific notation are written as the product of a power of 10 and a decimal number greater than or equal to 1 and less than 10. Here are examples of some large and small numbers written in scientific notation.

Standard NotationScientific Notation
4,300,000
27,690
0.0065
0.000000341


It is important for students to be familiar with scientific notation since they will be encountering numbers written in this form in their science and social science classes. Students should also become familiar with how calculators display scientific notation and should be able to use technology to solve problems using scientific notation.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

b.

Benchmark B: Identify subsets of the real number system.

At this level students should know the different sets of real numbers: natural numbers, whole numbers, integers, rational and irrational numbers. They should understand that each is a subset of the real number system and that some are subsets of each other. For example, the natural numbers are a subset of whole numbers. Rational and irrational numbers are both subsets of the real number system, so if we combine the rational and irrational numbers, we have the entire set of real numbers. When given a real number, students should be able to identify the correct subset(s) for the number.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

c.

Benchmark C: Apply properties of operations and the real number system, and justify when they hold true for a set of numbers.

This benchmark focuses on students' ability to understand and use such mathematical properties as identity, closure, commutative, associative and inverse properties. Students should be able to use operations on different subsets of real numbers and predict the results. For example, students should be able to discuss such questions as: "If I multiply two irrational numbers, do I get an irrational number?" and "What is the opposite (inverse) operation of finding the square root?"

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

d.

Benchmark D: Connect physical, verbal and symbolic representations of integers, rational numbers and irrational numbers.

At this level students need to expand their understanding of integers and both rational and irrational numbers. For example, students can understand the irrational number as the length of the hypotenuse of an isosceles right triangle with legs equal to 1. Students should also be able to explain what it means to find the square root, cube root or nth root of a number.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

e.

Benchmark E: Compare, order and determine equivalent forms of real numbers.

Students should be able to compare numbers that are decimals, fractions, irrational and integers. For example, students might be asked to order from least to greatest.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

f.

Benchmark F: Explain the effects of operations on the magnitude of quantities.

Students should already be able to add, subtract, multiply and divide integers and explain the effects of these operations. At this level students also should be able to explain what happens when we raise a quantity to a power or find a square root. For example, students should be able to explain what it means to raise 2 to the fifth power.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

g.

Benchmark G: Estimate, compute and solve problems involving real numbers, including ratio, proportion and percent, and explain solutions.

Students should be able to use ratios, proportions and percents in various problem solving situations. Students should be able to use real numbers fluently to make estimates, perform calculations and to compare estimates and actual answers. Here is a brief explanation of ratio, proportion, and percent.

  • A ratio is the relationship between two quantities expressed as a quotient of two numbers or as two numbers separated by a colon. For example, if your friend has twice as many apples as you have, we could express this relationship as a ratio or 2:1. A number that can be written as a rational number is a rational number.
  • A proportion is an equation showing that two ratios are equal. For example, since and are equivalent, we can write the proportion . When we have a proportion, we can always calculate the "missing" number if only three numbers are known. For example, given the proportion , we can solve for x.
  • Students should think of percents as an amount "out of 100." We encounter percents while shopping, in the news, in politics and in sports.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

h.

Benchmark H: Find the square root of perfect squares, and approximate the square root of non-perfect squares.

Students at this level should understand that squaring and taking the square root are inverse operations. They should be able to use this relationship to simplify and approximate square roots. For example, they can simplify because . Students should also be able to approximate square roots of numbers using their relationship to the square roots of perfect squares. For example, students should be able to determine that is between 11 and 12 because 130 is greater than 121 ( ) but less than 144 ( ).

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

i.

Benchmark I: Estimate, compute and solve problems involving scientific notation, square roots and numbers with integer exponents.

Students should understand scientific notation, square roots, and integer exponents. Their background with scientific notation should include performing operations and comparing numbers in this form. They should be able to use the order of operations to simplify expressions with exponents and roots and estimate nth roots between consecutive integers.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

 

ABOUT THE MATH
  • Rational Numbers

    Rational numbers are any numbers that can be written in the form , where a and b are integers and b is not equal to zero. Examples are . The first example is the repeating decimal 3 which can be expressed as . The second number is already expressed as a fraction with integers in the numerator and denominator. The third example is rational not because it is currently a fraction, but because you can write it in the equivalent form which is now a fraction with integer values in the top and bottom. One way for students to decide whether a number is rational is to check whether the decimal equivalent terminates or repeats.

  • Irrational Numbers

    Irrational numbers are all the real numbers that are not rational. Irrational numbers cannot be written as a ratio of two integers. The decimal form of the number never terminates and never repeats. Common examples of irrational numbers are square roots that do not contain perfect squares inside the radical, such as or or the value of .

  • Comparing Subsets of Real Numbers

    As students identify the subsets of real numbers, it's helpful for them to create a hierarchy that ranks the subsets. The diagram below shows the hierarchy of real numbers.

    The Venn diagram is also helpful for students to understand the different types of real numbers.

  • Closure Property

    When we combine an operation with a subset of numbers we can ask, "Is this operation closed on this set of numbers?" A set of numbers is closed under an operation if, for any two numbers from a set of numbers, we can perform the operation on the two numbers and the result is a number in our original set of numbers. For example, the natural numbers (integers greater than 0) are closed under addition because we can add any two natural numbers e.g. 1 + 1 = 2 or 1 + 2 = 3 and will get another natural number. However, the natural numbers are not closed under subtraction. As a counterexample, we can pick the numbers 1 and 2. Consider, for example, the equation 1 - 2 = -1. This results in an answer of -1, which is not a natural number.

  • Commutative and Associative Properties

    The commutative property applies for some operations when the order of the numbers can be changed without affecting the result. For example, 2 + 3 gives the same result as 3 + 2. Since this will be true for any two real numbers added together, the commutative property holds for addition. For subtraction, 2 - 3 does not give the same result as 3 - 2; This counterexample demonstrates that the commutative property does not hold for subtraction.

    The associative property says that the result for a single operation does not change due to the grouping. For example, illustrates the associative property of multiplication. However, does not equal , illustrating that the associative property does not hold for subtraction.

  • Simplifying Square Roots

    Answers involving square roots should be written in simplest form. For example, if the length of a hypotenuse is , this answer must be simplified. A square root in simplest form has no perfect square factors (other than 1) under the radical sign. Here is an example of one approach to simplifying square roots:

    Simplify .

    • When simplifying square roots, first check to see if the number under the radical sign, the radicand, is a perfect square. In this example, the radicand, 72, is not a perfect square.
    • Next, we want to find the factors of the radicand. The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.
    • Look at the list of factors and find the largest perfect square. Thirty-six is the largest perfect square that is a factor of 72.
    • We can now rewrite the expression.
    • Now simplify:
    • is in simplest form because the number under the radicand, 2, has no perfect square factors other than 1.

  • Estimating Square Roots

    Students should be able to approximate the value of a square root without a calculator using what they know about perfect squares. For example, we know that is between 6 and 7 because 41 is between the perfect squares, 36 () and 49 ().



Strategies

Help With Fundamentals

Listed here are a few of the difficulties students might have with this topic, along with a few suggestions of how to address these obstacles. Once you have identified a particular difficulty, find a strategy that will help students overcome the problem.



Additional Instruction and Practice

If your students need additional instruction and practice, you can supplement your instruction with activities. Here are a couple of activities you might want to try.

Activity 1

When learning about proportions, it is helpful for students to use concrete models. Have students compare a photograph with an enlargement. After determining the ratio of the original to the enlargement, have students determine the lengths of objects in the enlargement based on the length of the same object in the original.

Activity 2

In order to increase students' understanding of irrational numbers, have them create right triangles with integer lengths for the legs. Then have them find the length of the hypotenuse by the Pythagorean theorem and use technology to approximate the square root. Finally, have them measure the hypotenuse to see if their estimates were reasonable.

Activity 3

Have students search the Internet to find numbers that would require the use of scientific notation. If the number isn't in scientific notation, have students express it in scientific notation. If it already is in scientific notation, have students convert it to expanded form.



Advanced Work

Imaginary Numbers

After exploring real numbers, students can study the imaginary numbers. Imaginary numbers are the square roots of negative numbers. Together with the real numbers they form the set of complex numbers.

Problem 1

What are all solutions to the equation over the set of imaginary numbers?

Method

This question asks us to find all solutions to the equation instead of just the real solutions. Therefore, we need to consider those answers that are the square roots of negative numbers. To solve the equation, we solve for .

Then we take the square root of both sides.

Usually is rewritten as . Therefore, our solution is .


Answer

Answer Explanation

Extension

When students are familiar with complex numbers, they are ready to understand how to visualize these numbers on the complex plane. Students should be able to graph complex numbers on this plane.