## B1. Number Sense

### Specific Expectations

#### Whole Numbers

B1.1

read, represent, compose, and decompose whole numbers up to and including 100 000, 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, in expanded notation, or on a number line.
- 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, with the number 45 107, 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 period repeats the 0 to 9 counting sequence. Any quantity, no matter how great, can be described in terms of its place value.
- A number can be represented in expanded form as 34 187 = 30 000 + 4000 + 100 + 80 + 7, or as 3 × 10 000 + 4 × 1000 + 1 × 100 + 3 × 10 + 7, to show place value relationships.
- Numbers can be composed and decomposed in various ways, including by place value.
- Numbers are composed when two or more numbers are combined to create a larger number. For example, the numbers 100 and 2 can be composed to make the sum 102 or the product 200.
- Numbers can be decomposed as a sum of numbers. For example, 53 125 can be decomposed into 50 000 and 3000 and 100 and 25.
- Numbers can be decomposed into their factors. For example, 81 can be decomposed into the factors 1, 3, 9, 27, and 81.
- 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 helps in understanding the magnitude, or “how muchness”, of a number.
- Closed number lines with appropriate scales can be used to represent numbers as a position on a number line or as a distance from zero. Depending on the number, estimation may be needed to represent it on a number line.
- Partial number lines can be used to show the position of a number relative to other numbers.
- Composing and decomposing numbers in a variety of ways can support students in becoming flexible with their mental math strategies.
- Open number lines can be used to show the composition or decomposition of large numbers without drawing the number line to scale.
- It is important for students to understand key aspects of place value. For example:
- The order of the digits makes a difference. The number 21 385 describes a different quantity than 82 153.
- The
*place*(or position) of a digit determines its*value*(*place value*)*.*The 5 in 51 981, for example, has a value of 50 000, not 5. To determine the value of a digit in a number, multiply the value of the digit by the value of its place. For example, in the number 15 236, the 5 represents 5000 (5 × 1000), and the 2 represents 200 (2 × 100). - Expanded notation represents the values of each digit separately, as a sum. Using expanded form, 7287 is written 7287 = 7000 + 200 + 80 + 7 or 7 × 1000 + 2 × 100 + 8 × 10 + 7 × 1.
- A zero in a column indicates that there are no groups of that size in the number. It serves as a placeholder and holds the other digits in their correct “place”. For example, 189 means 1 hundred, 8 tens, and 9 ones, but 1089 means 1 thousand, 0 hundreds, 8 tens, and 9 ones.
- The value of the digits in each of the positions follows a “times 10” multiplicative pattern, when moving right to left. For example, 5000 is 10 times greater than 500, and 500 is 10 times greater than 50. Conversely, a “divide by ten” pattern is observed when moving left to right. For example, 500 is 10 times smaller than 5000, and 50 is 10 times smaller than 500.
- Going from left to right, a “hundreds-tens-ones” pattern repeats within each period (units, thousands, millions, billions, and so on). Exposure to this larger pattern and the names of the periods – into millions and beyond – satisfies a natural curiosity around “big numbers”, although students at this grade do not need to work beyond 100 000.

- The number “seventy-eight thousand thirty-seven” is written as “78 037”, and not “78 000 37” (as if being spelled out with numbers). Listening for the period name (seventy-eight
*thousand*) gives structure to the number and signals where a digit belongs. If there are no groups of that place value in a number, 0 is used to describe that amount, holding the other digits in their correct place.

B1.2

compare and order whole numbers up to and including 100 000, in various contexts

- Numbers are compared and ordered according to their “how muchness” or magnitude.
- Numbers with the same units can be compared directly (e.g., 72 cm
^{2}compared to 62 cm^{2}). Numbers that do not show a unit are assumed to have units of ones (e.g., 75 and 12 are considered as 75 ones and 12 ones). - Sometimes numbers without the same unit can be compared, such as 6200 kilometres and 6200 metres. Knowing that the unit
*kilometre*is greater than the unit*metre*, and knowing that 6200 kilometres is greater than 6200 metres can allow one to infer that 6200 kilometres is a greater distance than 6200 metres. - Sometimes numbers without the same unit may need to be rewritten to have the same unit in order to be compared. For example, 12 metres and 360 centimetres can be compared as 1200 centimetres and 360 centimetres. Therefore, 12 metres is greater than 360 centimetres.
- Benchmark numbers can be used to compare quantities. For example, 41 320 is less than 50 000 and 62 000 is greater than 50 000, so 41 320 is less than 62 000.
- Numbers can be compared by their place value. For example, when comparing 82 150 and 84 150, the greatest place value where the numbers differ is compared. For this example, 2 thousands (from 82 150) and 4 thousands (from 84 150) are compared. Since 4 thousand is greater than 2 thousand, then 84 150 is greater than 82 150.
- Numbers can be ordered in ascending order – from least to greatest – or can be ordered in descending order – from greatest to least.

*Note*

- Numbers can be compared proportionally. For example, 100 000 is 10 times greater than 10 000; it is also 100 times greater than 1000. It would take 1000 hundred-dollar bills to make $100 000.
- Depending on the context of the problem, numbers can be compared additively or multiplicatively.

#### Fractions, Decimals, and Percents

B1.3

represent equivalent fractions from halves to twelfths, including improper fractions and mixed numbers, using appropriate tools, in various contexts

- Equivalent fractions describe the same relationship or quantity.
- When working with fractions as a quotient, equivalent fractions are ones that have the same result when the numerators are divided by the denominators.
- When working with fractions as a part of a whole, the partitions of the fraction can be split or merged to create equivalent fractions. The whole remains the same size.
- When working with fractions as a comparison, the ratios between the numerator and the denominator of equivalent fractions are equal.

*Note*

- Models and tools can be used to develop understanding of equivalent fractions. For example:
- Fraction strips or other partitioned models, such as fraction circles, can be used to create the same area as the original fraction using split or merged partitions.
- Strips of paper can be folded to show the splitting of partitions to create equivalence.
- A double number line or a ratio table can be used to show equivalent fractions based on different scales.

- A fraction is a number that conveys a relationship between two quantities.
- A fraction can represent a quotient (division):
- It shows the relationship between the number of wholes (numerator) and the number of partitions the whole is being divided into (denominator).
- For example, 3 granola bars (3 wholes) are shared equally with 4 people (number of partitions), which can be expressed as .

- A fraction can represent a part of a whole:
- It shows the relationship between the number of parts selected (numerator) and the total number of parts in one whole (denominator).
- For example, if 1 granola bar (1 whole) is partitioned into 4 pieces (partitions), each piece is one fourth () of the granola bar. Two pieces are 2 one fourths () of the granola bar, three pieces are 3 one fourths () of the granola bar, and four pieces are four one fourths () of the granola bar.

- A fraction can represent a comparison:
- It shows the relationship between two parts of the same whole. The numerator is one part and the denominator is the other part.
- For example, a bag has 3 red beads and 2 yellow beads. The fraction represents that there are two thirds as many yellow beads as red beads. The fraction , which is as a mixed number, represents that there are 1 and one half times more red beads than yellow beads.

- A fraction can represent an operator:
- When considering fractions as an operator, the fraction increases or decreases a quantity by a factor.
- For example, in the case of of a granola bar, of $100, or of a rectangle, the fraction reduces the original quantity to its original size.

B1.4

compare and order fractions from halves to twelfths, including improper fractions and mixed numbers, in various contexts

- When working with fractions as parts of a whole, the fractions are compared to the same whole.
- Fractions can be compared spatially by using models to represent the fractions. If an area model is chosen, then the areas that the fractions represent are compared. If a linear model is chosen, then the lengths that the fractions represent are compared.
- 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., < ).
- Fractions can be compared by using the benchmark of "half"and considering each fraction relative to it. For example, is greater than because is greater than one half and is less than one half.
- Fractions can be ordered in ascending order – least to greatest – or in descending order – greatest to least.

*Note:*

- The choice of model used to compare fractions may be influenced by the context of the problem. For example:
- a linear model may be chosen when the problem is dealing with comparing things involving length, like lengths of a ribbon or distances.
- an area model may be chosen when the problem is dealing with comparing the area of two-dimensional shapes, like a garden or a flag.

B1.5

read, represent, compare, and order decimal numbers up to hundredths, 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.
- Decimal numbers can be less than one (e.g., 0.65) or greater than one (e.g., 24.72).
- 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 compared and ordered by identifying the size of the decimal number visually relative to 1 whole. Using knowledge of fractions (e.g., ), or thinking about money (e.g., $2.50 is more than $2.05), are helpful strategies when comparing decimal numbers.

*Note*

- Between any two consecutive whole numbers are other numbers. Decimals are how the base ten number system shows these “in-between” numbers. For example, the number 3.62 describes a quantity between 3 and 4 and, more precisely, between 3.6 and 3.7.
- Decimals are sometimes called
*decimal fractions*because they represent fractions with denominators of 10, 100, 1000, and so on. The first decimal place represents tenths, the second represents hundredths, and so on. Columns can be added indefinitely to describe smaller and smaller partitions. Decimals, like fractions, have what could be considered a numerator (a count of units) and a denominator (the value of the unit); however, with decimals, only the numerator is visible. The denominator (or unit) is “hidden” within the place value convention. - 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
*ones*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. If 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, 0.05 is one tenth as great as 0.5 and one hundredth as great as 5. It also means that 5 is 100 times as great as 0.05, in the same way that there are 100 nickels ($0.05) in $5.00.
- As with whole numbers, a zero in a decimal indicates that there are no groups of that size in the number:
- 5.07 means 5 ones, 0 tenths, 7 hundredths.
- 5.10 means 5 ones, 1 tenth, 0 hundredths.
- 5.1 (five and one tenth) and 5.10 (5 and 10 hundredths) are 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).

- 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; and decimals used in numbered lists function merely as labels, like in a numbered list. 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, 2.57 should be read as “2 and 57 hundredths”. - Decimals can be compared and ordered like any other numbers, including fractions. Like fractions, decimals describe an amount that is relative to the whole.
- Many of the tools that are used to represent whole numbers can also be used to represent decimal numbers. It is important to emphasize 1 whole to recognize the representation in tenths and hundredths and not as wholes. For example, a base ten rod that was used to represent 10 ones can be used to represent 1 whole that is partitioned into tenths, and a base ten flat that was used to represent 100 ones can be used to represent 1 whole that is partitioned into hundredths.

B1.6

round decimal numbers to the nearest tenth, in various contexts

- Rounding makes a number simpler to work with and is often used when estimating computations, measuring, and making quick comparisons.
- Rounding compares a number to a given reference point – is it closer to this or to that? For example, is 1.75 closer to 1 or to 2? Is 1.84 closer to 1.8 or to 1.9?
- Rounding 56.23 to the nearest tenth becomes 56.2, since 56.23 is closer to 56.2 than 56.3 (it is three hundredths away from 56.2 versus seven hundredths away from 56.3).
- Rounding 56.28 to the nearest tenth becomes 56.3, since 56.28 is closer to 56.3 than 56.2.
- If a decimal hundredth is exactly between two decimal tenths, the convention is to round up, unless the context suggests differently (e.g., 56.25 is rounded to 56.3.)

- In the absence of a context, numbers are typically rounded around the midpoint.

*Note*

- As with whole numbers, rounding decimal numbers involves making decisions about the level of precision needed. Whether a number is rounded up or down depends on the context and whether an overestimate or an underestimate is preferred.

B1.7

describe relationships and show equivalences among fractions, decimal numbers up to hundredths, and whole number percents, using appropriate tools and drawings, in various contexts

- 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 (“percent” means “per hundred”). For both decimals and percents, the “denominator” (the value of the unit or the divisor) is hidden within the convention itself (i.e., the place value convention and the percent sign).
- Percent is a special rate, “per 100”, and can be represented with the symbol %. The whole is partitioned into 100 equal parts. Each part is one percent, or 1%, of the whole.
- The unit fraction expressed as a quotient is 1 ÷ 100 and the result is 0.01, which is read as one hundredth. This unit fraction and its decimal equivalent are equal to 1%.
- Any fraction can be expressed as a fraction with a denominator of 100.
- A decimal hundredth can be rewritten as a whole number percent (e.g., 0.56 = 56%).
- If a fraction or decimal number can be expressed as a hundredth, it can be expressed as a whole number percent. For example, is equivalent to , and 0.8 is equivalent to 0.80, and they are both equivalent to 80%.

- Common benchmark percentages include:
- 1% = = 0.01
- 10%, 20%, 30%, … = , , , … = 0.1, 0.2, 0.3, …
- 20% = or = 0.2
- 25% = = 0.25
- 50% = = 0.5
- 75% = = 0.75
- 100% = 1 = 1.00

- A percent can be greater than 100% (e.g., 150% = = 1.50).
- Some fractions are easier than others to express with a denominator of 100.