## B1. Number Sense

### Specific Expectations

#### Rational Numbers

B1.1

read and represent whole numbers up to and including one million, using appropriate tools and strategies, and describe various ways they are used in everyday life

- Reading numbers involves interpreting them as a quantity when they are expressed in words, in standard notation, or in expanded notation.
- 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. For example, in the number 945 107, the digit 9 represents 9 hundred thousands, the digit 4 represents 4 ten thousands, the digit 5 represents 5 thousands, the digit 1 represents 1 hundred, the digit 0 represents 0 tens, and the digit 7 represents 7 ones.
- There are patterns to the way numbers are formed. Each place value column, or period, repeats the 0 to 9 counting sequence.
- Any quantity, no matter how great, can be described in terms of its place value. For example, 1500 may be said as fifteen hundred or one thousand five hundred.
- A number can be represented in expanded form (e.g., 634 187 = 600 000 + 30 000 + 4000 + 100 + 80 + 7, or 6 × 100 000 + 3 × 10 000 + 4 × 1000 + 1 × 100 + 8 × 10 + 7) to show place value relationships.
- Numbers are used throughout the day, in various ways and contexts. Most often numbers describe and compare quantities. They express magnitude, and provide a way to answer questions such as “how much?” and “how much more?”.

*Note*

- Every strand of mathematics relies on numbers.
- Numbers may have cultural significance.
- Seeing how a quantity relates to
*other*quantities supports students in developing an understanding of the magnitude or “how muchness” of a number. - There are patterns in the place value system that support the reading, writing, saying, and understanding of numbers and that suggest important ways for numbers to be composed and decomposed.
- The
*place*(or position) of a digit determines its*value*(*place value*)*.*The 5 in 511, for example, has a value of 500, not 5. - A zero in a column indicates that there are no groups of that size in the number. It serves as a placeholder, holding the other digits in their correct “place”.
- The value of the columns increases by a constant “times 10” multiplicative pattern. For example, as the digit 5 shifts to the left, from 5000 to 50 000, the digit’s value becomes 10 times as great. As it shifts to the right, from 5000 to 500, its value becomes one tenth as great.
- To find the value of a digit in a number, the value of the digit is multiplied by the value of its place. For example, in the number 52 036, the 5 represents 50 000 (5 × 10 000) and the 2 represents 2000 (2 × 1000).
- Expanded notation represents the value of each digit separately, as an expression. Using expanded form, 7287 is written as 7287 = 7000 + 200 + 80 +7, or 7 × 1000 + 2 × 100 + 8 × 10 + 7 × 1.
- Each period – thousands, millions, billions, trillions – is 1000 times the previous period.

- The
- A “hundreds-tens-ones” pattern repeats within each period (ones, thousands, millions, billions, and so on), and each period is 1000 times the one preceding it. Exposure to these patterns, and the names of these periods, also satisfies a natural curiosity around “big numbers” and could lead to conversations about periods beyond millions (billions, trillions, quadrillions, and so on).

- The number “five hundred eight thousand thirty-seven” is written as “508 037” and not “508 1000 37” (as if being spelled out with numbers). Listening for the period name (508 thousand), and the hundreds-tens-ones pattern that precedes the period, gives structure to the number and signals where a digit belongs. If there are no groups of a particular place value, 0 is used to describe that amount, holding the other digits in their correct place.
- Large numbers are difficult to visualize. Making connections to real-life contexts helps with this, as does comparing large numbers to other numbers using proportional reasoning. For example, a small city might have a population of around 100 000, and 1 000 000 would be 10 of these cities.

B1.2

read and represent integers, using a variety of tools and strategies, including horizontal and vertical number lines

- Integers are whole numbers and their opposites.
- Zero is neither negative nor positive.
- On a horizontal number line, positive integers are displayed to the right of zero and negative integers are displayed to the left of zero.
- On a vertical number line, positive integers are displayed above the zero and negative integers are displayed below the zero.
- Integers can be represented as points on a number line, or as vectors that shows magnitude and direction. The integer −5 can be shown as a point positioned 5 units to the left of zero or 5 units below zero. The integer −5 can also be shown as a vector with its tail positioned at zero and its head at −5 on the number line, to show that it has a length of 5 units and is moving in the negative direction.
- Each integer has an opposite, and both are an equal distance from zero. For example, −4 and +4 are opposite integers and both are 4 units from zero.
- Zero can be represented with pairs of opposite integers. For example, (+3) and (−3) = 0.
- Integers measure "whole things" relative to a reference point. For example, 1 degree Celsius is used to measure temperature. Zero degrees is freezing (reference point). The temperature +10
^{° }C is ten degrees above freezing. The temperature −10^{° }C is ten degrees below freezing.

*Note*

- Engaging with everyday examples of negative integers (e.g., temperature, elevators going up and down, sea level, underground parking lots, golf scores, plus/minus in hockey, saving and spending money, depositing and withdrawing money from a bank account, walking forward and backwards) helps build familiarity and a context for understanding numbers less than zero.
- Pairs of integers such as (+2) and (−2) are sometimes called "zero pairs".
- The Cartesian plane (see Spatial Sense, E1.3) uses both horizontal and vertical integer number lines to plot locations, and negative rotations to describe clockwise turns (see Spatial Sense,

B1.3

compare and order integers, decimal numbers, and fractions, separately and in combination, in various contexts

- Numbers with the same units can be compared directly (e.g., 72.5 cm
^{2}compared to 62.4 cm^{2}). - Sometimes numbers without the same unit can be compared, such as 6.2 kilometres and 6.2 metres. Knowing that the unit
*kilometre*is greater than the unit*metre*can allow one to infer that 6.2 kilometres is greater than 6.2 metres. - Sometimes numbers without the same unit may need to be rewritten with the same unit in order to be compared. For example, 1.2 metres and 360 centimetres can be compared as 120 centimetres and 360 centimetres. Thus, 360 centimetres is greater than 1.2 metres.
- Whole numbers (zero and positive integers) and decimal numbers can be compared and ordered according to their place value.
- Benchmark numbers can be used to compare quantities. For example, is greater than and 0.25 is less than , so is greater than 0.25.
- If two fractions have the same denominator, then the numerators can be compared. In this case the numerator with the greater value is the greater fraction because the number of parts considered is greater (e.g., ).
- If two fractions have the same numerators, then the denominators can be compared. In this case the denominator with the greater value is the smaller fraction because the size of each partition of the whole is smaller (e.g., ).
- Having more digits does not necessarily mean that a number is greater. For example, −7528 has four digits but it is less than +3 because −7528 is less than zero and +3 is greater than zero.
- Any positive number is greater than any negative number.
- When comparing positive numbers, the greater number is the number with the greater magnitude. On a horizontal number line, the greater number is the farthest to the right of zero. On a vertical number line, the greater number is the farthest above zero.
- When comparing negative integers, the least number is the negative integer with the greater magnitude. On a horizontal number line, the lesser number is the farthest to the left of zero. On a vertical number line, the lesser number is the farthest below zero.
- Numbers can be ordered in ascending order – from least to greatest – or can be ordered in descending order – from greatest to least.

*Note*

- Comparing numbers helps with understanding and describing their order and magnitude.
- Visual models like the number line – particularly if they are proportional – can be used to order numbers; show the relative magnitude of numbers; and highlight equivalences among fractions, decimals, and whole numbers.
- The absolute value of a number is its distance from zero, or its magnitude. Both −5 and +5 have an absolute value of 5, because both are 5 units from zero on the number line.

#### Fractions, Decimals, and Percents

B1.4

read, represent, compare, and order decimal numbers up to thousandths, in various contexts

- The place value of the first position to the right of the decimal point is tenths. The second position to the right of the decimal point is hundredths. The third position to the right of the decimal point is thousandths.
- Decimal numbers can be less than one (e.g., 0.654) or greater than one (e.g., 24.723).
- The one whole needs to be shown or explicitly indicated when decimal numbers are represented visually since their representation is relative to the whole.
- Decimal numbers can be represented as a composition or decomposition of numbers according to their place value. For example, decimals can be written in expanded notation 3.628 = 3 + 0.6 + 0.02 + 0.008, or 3 × 1 + 6 × 0.1 + 2 × 0.01 + 8 × 0.001.
- Decimal numbers can be compared by their place value. For example, when comparing 0.8250 and 0.845, the greatest place value where the numbers differ is compared. For this example, 2 hundredths (from 0.825) and 4 hundredths (from 0.845) are compared. Since 4 hundredths is greater than 2 hundredths, 0.845 is greater than 0.825.
- Numbers can be ordered in ascending order – from least to greatest or can be ordered in descending order – from greatest to least.

*Note*

- Between any two consecutive whole numbers are decimal thousandths. For example, the number 3.628 describes a quantity between 3 and 4 and, more precisely, between 3.6 and 3.7 and, even more precisely, between 3.62 and 3.63.
- Decimals are sometimes called decimal fractions because they represent fractions with denominators of 10, 100, 1000, and so on. Decimal place value columns are added to describe smaller partitions. Decimals, like fractions, have a numerator and a denominator; however, with decimals, only the numerator is visible. The denominator (or unit) is “hidden” within the place value convention.
- Decimals can be composed and decomposed like whole numbers. Expanded notation shows place value subdivisions (e.g., 3.628 = 3 + 0.6 + 0.02 + 0.008, or 3 × 1 + 6 × 0.1 + 2 × 0.01 + 8 × 0.001).
- The decimal point indicates the location of the unit. The unit is always to the left of the decimal point. There is symmetry around the unit column, so tens are matched by tenths, and hundreds are matched by hundredths. Note that the symmetry does not revolve around the
*decimal*, so there is no “oneth”:

- Between any two places in the base ten system, there is a constant 10 : 1 ratio, and this is true for decimals as well. As a digit shifts one space to the right it becomes one tenth as great and if it shifts two spaces to the right it becomes one hundredth as great. So, 0.005 is one tenth as great as 0.05, one hundredth as great as 0.5, and one thousandth as great as 5. This also means that 5 is 1000 times as great as 0.005.
- As with whole numbers, a zero in a decimal indicates that there are no groups of that size in the number:
- 5.007 means that there are 5 wholes, 0 tenths, 0 hundredths, and 7 thousandths.
- 5.100 means that there are 5 wholes, 1 tenth, 0 hundredths, and 0 thousandths.
- 5.1 (five and one tenth), 5.10 (5 and 10 hundredths), and 5.100 (5 and 100 thousandths) are all equivalent (although writing zero in the tenths and hundredths position can indicate the precision of a measurement; for example, the race was won by 5.00 seconds and the winning time was 19.29 seconds). Writing zero in the tenths, hundredths, and thousandths position can indicate the precision of a measurement (e.g., baseball batting averages are given to the nearest thousandths).

- Decimals are read in a variety of ways in everyday life. Decimals like 2.5 are commonly read as two point five; in math, the term pi (π) is commonly approximated as three point one four; the decimal in baseball averages is typically ignored. However, to reinforce the decimal’s connection to fractions, and to make visible its place value denominator, it is recommended that decimals be read as their fraction equivalent. So, 2.573 is read as “2 and 573 thousandths.”

B1.5

round decimal numbers, both terminating and repeating, to the nearest tenth, hundredth, or whole number, as applicable, in various contexts

- Rounding makes a number simpler to work with and is often used when estimating computations, measuring, and making quick comparisons.
- A decimal number is rounded to the nearest hundredth, tenth, or whole number based on which hundredth, tenth or whole number it is closest to. If it is the same distance, it is usually rounded up. However, depending on context it may be rounded down.

*Note*

- Decimal numbers that terminate are like 3.5, 46.27, and 0.625.
- Decimal numbers that repeat are like 3.555555… and can be represented using the symbol with a dot above the repeating digit, (e.g., ). If a string of digits repeats, a bar can be shown above the string, or dots above the first and last digits (e.g., 3.546754675467 is written as , or ).
- Rounding involves making decisions about what level of precision is needed and is often used in measurement. How close a rounded number is to the actual amount depends on the unit it is being rounded to: the larger the unit, the broader the approximation; the smaller the unit, the more precise. Whether a number is rounded up or down depends on the context.

B1.6

describe relationships and show equivalences among fractions and decimal numbers up to thousandths, using appropriate tools and drawings, in various contexts

- Any fraction can become a decimal number by treating the fraction as a quotient (e.g., = 8 ÷ 5 = 1.6).
- Some fractions as quotients produce a repeating decimal. For example, = 1 ÷ 3 = 0.333… or = 1 ÷ 7 = . When decimal numbers are rounded they become approximations of the fraction.
- If a fraction can be expressed in an equivalent form with a denominator of tenths, hundredths, thousandths, and so on, it can also be expressed as an equivalent decimal. For example, because can be expressed as , it can also be expressed as 0.25.
- A terminating decimal can be expressed in an equivalent fraction form. For example, 0.625 = , which can be expressed as other equivalent fractions, or or .
- Any whole number can be expressed as a fraction and as a decimal number. For example, 3 = = 3.0.

*Note*

- Decimals are how place value represents fractions and are sometimes called decimal fractions. While fractions may use any number as a denominator, decimals have denominators (units) that are based on a system of tens (tenths, hundredths, and so on).