## B1. Number Sense

### Specific Expectations

By the end of Grade 8, students will:

#### Rational and Irrational Numbers

B1.1

represent and compare very large and very small numbers, including through the use of scientific notation, and describe various ways they are used in everyday life

- The numerals 0 to 9 are used to form numbers. They are referred to as the digits in a number and each digit corresponds to a place value. A billion is “a thousand millions”, and a trillion is “a thousand billions” or “a million millions”. After the trillions period come quadrillions, quintillions, sextillions, septillions, octillions, and so on. Each period is 1000 times the preceding one.
- Reading numbers involves interpreting them as a quantity when they are expressed in words, in standard notation, in expanded notation, or in scientific notation. Large numbers may be expressed as a decimal number with the unit expressed in words. For example, 36.24 trillion is equivalent to 36 240 000 000 000 = 3.624 × 10
^{13}. - When a number is expressed in scientific notation, there is only one non-zero digit to the left of the decimal point. Thus, 36.24 × 10
^{12}is not in scientific notation because there are two digits to the left of the decimal point. In scientific notation, 36 240 000 000 000 is written as 3.624 × 10^{13}. - In words, 37 020 005 205 is written and said as “thirty-seven billion twenty million five thousand two hundred five”. Sometimes an approximation to a large number is used to describe a quantity. For example, the number 37 020 005 205 may be rounded to 37 billion or 37.02 billion, depending on the amount of precision needed.
- Understanding the magnitude of a large number may be done by comparing it to other numbers and quantities. For example:
- One million seconds is around 11.5 days.
- One billion seconds is around 32 years.
- One trillion seconds is around 32 000 years.

- A number greater than 1 that is written in scientific notation can be written in standard notation by multiplying the decimal number by ten the number of times indicated by its exponent. For example, for 3.2 × 10
^{5}, 3.2 is multiplied by ten, five times. The result is 320 000. - A number written in standard notation can be written in scientific notation. For a number greater than 1, a decimal point is positioned so that the first non-zero digit is to the left of the decimal point, and then the exponent for the base ten is determined by counting the number of times that decimal number needs to be multiplied by 10 to produce that number in standard notation. For example, 156 000 000 000 = 1.56 × 10
^{11}. - Very small numbers refer to numbers between 0 and 1. The closer the number is to zero the smaller the number is. These numbers can also be written in scientific notation. A negative exponent is used to indicate that the decimal number needs to be divided by 10 that many times. For example, for 5.2 × 10
^{-8}, 5.2 is divided by 10 eight times to become^{ }0.000000052. - To write a small number in scientific notation, the decimal point is positioned so that the first non-zero digit is to the left of the decimal point, and then the exponent is determined by counting the number of times that decimal number needs to be divided by 10 to produce that number in standard notation. For example, 0.0034 = 3.4 × 10
^{-3}. - Numbers expressed in scientific notation can be compared by considering the number of times the decimal number is multiplied or divided by ten. The more times it is multiplied by ten, the greater the number. The more times it is divided by ten, the smaller the number.

*Note*

- Every strand of mathematics relies on numbers.
- Some numbers have cultural significance.
- Real-life contexts can provide an understanding of the magnitude of large and small numbers.
- The number 1 in scientific notation is 1 × 10
^{0}. - The exponent on the base ten, in scientific notation, indicates the number of times the decimal number is multiplied or divided by ten, not how many zeros need to be included for a number to be written in standard notation.
- When inputting numbers electronically, the “^” sign is used for exponents; for example, 10
^{6}would be entered as 10^6.

B1.2

describe, compare, and order numbers in the real number system (rational and irrational numbers), separately and in combination, in various contexts

- Real numbers are a set of numbers that contain all rational and irrational numbers.
- Rational numbers are those that can be expressed in the form , where
*a*and*b*are integers; for example, −, 3.12, , −7, 0, 205, , −32.5. - Fractions (positive and negative) are rational numbers. Any fraction can be expressed as a decimal number that either terminates or repeats.
- Fractions can be written in a horizontal format (e.g., 1/2 or ½) as well as stacked format (e.g., ).
- Whole numbers are rational numbers since any whole number can be expressed as a fraction (e.g., 5 = ).
- Integers (whole numbers and their opposites) are rational numbers since any integer can be expressed as a fraction (e.g., −4 = −, +8 = +).
- Irrational numbers are numbers that cannot be expressed as a fraction. Examples of irrational numbers include decimal numbers that never repeat or terminate (e.g., 3.12122122212222…), pi (π), and square roots of non-perfect squares (e.g., ).
- Rational and irrational numbers can be represented as points on a number line to show their relative distance from zero.
- The farther a number is to the right of zero on a horizontal number line, the greater the number.
- The farther a number is to the left of zero on a horizontal number line, the lesser the number.
- There are an infinite number of numbers in the real number system.

*Note*

- Since Grade 1, students have been working with whole numbers. The set of whole numbers (W) is a subset of integers (I), which are a subset of rational numbers (Q).
- Since Grade 1, students have been working with positive fractions, which are rational numbers. Negative fractions are introduced in Grades 7 and 8 as students represent, compare, and order negative fractions. Students will perform operations with negative fractions in secondary school because they are still developing the skills in Grade 8 to perform operations with integers.
- In Grade 7, students were introduced to pi, which is an irrational number. They may have worked with approximations of pi (3.14 or ), which are rational numbers. In Grade 8, students are introduced to other types of irrational numbers.
- In Grade 8, the focus is supporting students in making connections among the different number systems and the way they have been building knowledge through the years about the real number system.

B1.3

estimate and calculate square roots, in various contexts

- The inverse of squaring a number is to take its square root.
- Each positive number has two possible square roots. For example, the square roots of 9 are +3 and −3 because (+3)(+3) = 9 and (−3)(−3) = 9.
- The symbol means the positive square root. The symbol ± means both the positive and the negative square root.
- Depending on the context, only the positive square root may be appropriate. For example, given the area of a square, the length of its side is determined by taking the square root of the area. Since the side is a dimension, it makes sense to determine only the positive square root.
- Square roots of non-perfect squares are irrational and are left in radical form (e.g., ) or approximated to a decimal number.
- Estimating the square roots of non-perfect squares involves identifying the two perfect squares that are closest to it. For example, is between and , so the first step is to determine the square root of those perfect squares, that is, = 7 and = 8. The next step is to estimate a value that is close to the closest square root. Since 60 is closer to 64 than to 49, then can be estimated as 7.8.

*Note*

- Solving for a length using the Pythagorean theorem involves applying squares and square roots of numbers.
- A spatial interpretation of a square number is to think of the area of a square with side length the square root of the area (
*side*×*side*or*s*).^{2} - If the area of a square is 9, its side length is or 3.
- 9 is a perfect square.

- If the area of a square is 5, its side length is .
- 5 is an imperfect square and its square root is an irrational number, with a decimal that never repeats or terminates.

- Perfect squares can be calculated. Imperfect squares can only be estimated. Calculators give approximations of all square roots of non-perfect square numbers.

#### Fractions, Decimals, and Percents

B1.4

use fractions, decimal numbers, and percents, including percents of more than 100% or less than 1%, interchangeably and flexibly to solve a variety of problems

- Converting between fractions, decimals, and percents often makes calculations and comparisons easier to understand and carry out.
- Fractions, decimals, and percents all describe relationships to a whole. While fractions may use any number as a denominator, decimal units are in powers of ten (tenths, hundredths, and so on) and percents express a rate out of 100 (“per cent” means “per hundred”).
- Relationships of quantities relative to a whole can be expressed as a fraction, a decimal number, and a percent. The choice of using a fraction, decimal number, or a percent can vary depending on the context of a problem.
- When fractions are considered as a quotient, the numerator is divided by a denominator and the result is a decimal representation that can be converted to a percent.
- To convert a percent to a fraction, it can first be represented out of 100 and then an equivalent proper fraction or mixed number can be made. For example, 104.6% = = = = .
- Some decimal numbers when converted to a percent result in whole number percents, and others result in decimal percents (e.g., 0.15 = 15%, 0.642 = 64.2%, 3.425 = 342.5%).
- Percents can be whole number percents (e.g, 32%, 168%) or decimal percents (0.5%, 43.6%, 108.75%). Percents can be understood as decimal hundredths.
- Percents can be composed from other percents. A 15% discount combines a 10% discount and a 5% discount. A 13% tax adds 10% and another 3% (3 × 1%).
- To convert a percent to a decimal number, the percent is divided by 100 (e.g., 35.4% = 0.354, 0.1% = 0.001).
- There are three types of problems that involve percents – determining the percent a quantity represents relative to a whole; finding the percent of a number; and finding a number given the percent.
- Common benchmark fractions, decimals, and percents include:
- 150% = = 1.50
- 100% = 1 = 1.00
- 75% = = 0.75
- 50% = = 0.50
- 25% = = 0.25
- 20% = = 0.20
- 10% = = 0.10
- 5% = = 0.05
- 1% = = 0.01
- 0.1% = = 0.001

- Unit fraction conversions can be scaled to determine non-unit conversions. For example:
- = 0.25 = 25%, so = 0.125 = 12.5% (half of one fourth).
- = 0.125 = 12.5%, so = 0.375 = 37.5 (three times one eighth).

*Note*

- Sometimes when working with percents, students may work with complex fractions in which a decimal number is the numerator. An equivalent proper fraction or mixed number can be made by multiplying both the numerator and denominator by the appropriate number of tens.
- More than one strategy can be used to solve problems involving percents. For example, a coat is on sale for 25% off. The cost of the coat can be determined by finding 25% of the original price and then subtracting that discount value from the original price. Another strategy could be to determine 75% of the original price.