## B1. Number Sense

### Specific Expectations

#### Whole Numbers

B1.1

read, represent, compose, and decompose whole numbers up to and including 10 000, 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, or represented using physical objects or diagrams.
- 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 4107, the digit 4 represents 4 thousands, the digit 1 represents 1 hundred, the digit 0 represents 0 tens, and the digit 7 represents 7 ones.
- There are patterns in the way numbers are formed. Each place value column 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 (e.g., 4187 = 4000 + 100 + 80 + 7 or 4 × 1000 + 1 × 100 + 8 × 10 + 7 × 1) 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, 1300, 200, and 6 combine to make 1506.
- Numbers are decomposed when they are represented as a composition of two or more smaller numbers. For example, 5125 can be decomposed into 5000 and 100 and 25.
- Tools may be used when representing numbers. For example, 1362 may be represented as the sum of 136 ten-dollar bills and 1 toonie or 13 base ten flats, 6 base ten rods, and 2 base ten units.
- 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.
- When a number is decomposed and then recomposed, the quantity is unchanged. This is the conservation principle.
- There are non-standard but equivalent ways to decompose a number using place value, based on understanding the relationships between the place values. For example, 587 could be decomposed as 58 tens and 7 ones or decomposed as 50 tens and 87 ones, and so on.
- Composing and decomposing numbers in a variety of ways can support students in becoming flexible with their mental math strategies.
- 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.
- Open number lines can be used to show the composition of large numbers without drawing them 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 385 describes a different quantity than the number 853.
- 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. 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 5236, the 5 represents 5000 (5 × 1000) and the 2 represents 200 (2 × 100). - 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”. 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. For example, 500 is 10 times greater than 50, 50 is 10 times greater than 5, and 5 is 10 times greater than 0.5.

- Going from left to right, a “hundreds-tens-ones” pattern repeats within each period (ones, 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 thousands.

B1.2

compare and order whole numbers up to and including 10 000, 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., 7645 kilometres compared to 6250 kilometres). 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 625 kilometres and 75 metres. Knowing that the unit "kilometres" is greater than the unit "metres", and knowing that 625 is greater than 75, one can infer that 625 kilometres is a greater distance than 75 metres.
- Benchmark numbers can be used to compare quantities. For example, 4132 is less than 5000 and 6200 is greater than 5000, so 4132 is less than 6200.
- Numbers can be compared by their place value. For example, when comparing 8250 and 8450, the greatest place value where the numbers differ is compared. For this example, 2 hundreds (from 8250) and 4 hundreds (from 8450) are compared. Since 4 hundreds is greater than 2 hundreds, 8450 is greater than 8250.
- Numbers can be ordered in ascending order – from least to greatest – or can be ordered in descending order – from greatest to least.

*Note*

- An understanding of place value enables whole numbers to be compared and ordered. There is a stable order to how numbers are sequenced, and patterns exist within this sequence that make it possible to predict the order of numbers and make comparisons.

B1.3

round whole numbers to the nearest ten, hundred, or thousand, in various contexts

- Rounding numbers is often done to estimate a quantity or measure, estimate the results of a computation, and make quick comparisons.
- Rounding involves making decisions about what level of precision is needed, and is used often in measurement. How close a rounded number is to the actual amount depends on the unit it is being rounded to. The result of rounding a number to the nearest ten is closer to the original number than the result of rounding the same number to the nearest hundred. Similarly, the result of rounding a number rounded to the nearest hundred is closer to the original number than the result of rounding the same number to the nearest thousand. 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. For example, when paying by cash in a store, the amount owing is rounded to the nearest five cents (or nickel).
- In the absence of a context, numbers are typically rounded on a midpoint. This approach visualizes the amount that is halfway between two units and determines whether a number is closer to one unit than the other.
- Rounding 1237 to the nearest 10 becomes 1240, since 1237 is closer to 1240 than 1230.
- Rounding 1237 to the nearest 100 becomes 1200, since 1237 is closer to 1200 than 1300.
- Rounding 1237 to the nearest 1000 becomes 1000, since 1237 is closer to 1000 than 2000.
- If a number is exactly on the midpoint, convention rounds the number up (unless the context suggests differently). So, 1235 rounded to the nearest 10 becomes 1240.

#### Fractions and Decimals

B1.4

represent fractions from halves to tenths using drawings, tools, and standard fractional notation, and explain the meanings of the denominator and the numerator

- A fraction is a number that tells us about the 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 three 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 decrease 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.

*Note*

- A fraction is a number that can tell us information about the relationship between two quantities. These two quantities are expressed as parts and wholes in different ways, depending on the way the fraction is used.
- as a quotient (3 ÷ 4): 3 represents three wholes divided into 4 equal parts (wholes to parts relationship).
- as a part of a whole: 3 is representing the number of parts selected from a whole that has been partitioned into 4 equal parts (parts to a whole relationship).
- as a comparison: 3 parts of a whole compared to 4 parts of the same whole (parts to parts of the same whole relationship).

- A fraction is an operator when one interprets a fraction relative to a whole. For example, each person gets three fourths of a granola bar, or three one fourths of the area is shaded.

B1.5

use drawings and models to represent, compare, and order fractions representing the individual portions that result from two different fair-share scenarios involving any combination of 2, 3, 4, 5, 6, 8, and 10 sharers

- Fair sharing means that quantities are shared equally. For a whole to be shared equally, it must be partitioned in such a way that each sharer receives the same amount.
- Fair-share or equal-share problems can be represented using various models. The choice of model may be influenced by the context of the problem. For example,
- A set model may be chosen when the problem is dealing with objects such as beads or sticker books. The whole may be the entire set or each item in the set.
- A linear model may be chosen when the problem is dealing with things involving length, like the length of a ribbon or the distance between two points.
- An area model may be chosen when the problem is dealing with two-dimensional shapes like a garden plot or a flag.

- Fractions that are based on the same whole can be compared by representing them using various tools and models. For example, if an area model is chosen, then the area that the fractions represent are compared. If a linear model is chosen, then the lengths that the fractions represent are compared.
- Ordering fractions requires an analysis of the fractional representations. For example, when using an area model, the greater fraction covers the most area. If using a linear model, the fraction with the larger length is the greater fraction.

*Note*

- Words can have multiple meanings. It is important to be aware that in many situations, fair does not mean equal, and equal is not equitable. Educators should clarify how they are using the term “fair share” and ensure that students understand that in the math context fair means equal and the intent behind such math problems is to find equal amounts.
- Fair-share or equal-share problems provide a natural context for students to encounter fractions and division. Present these problems in the way that students will best connect to.
- Different modes and tools can be used to represent fractions:
- Set models include collections of objects (e.g., beads in a bag, stickers in a sticker book), where each object is considered an equal part of the set. The attributes of the set (e.g., colour, size, shape) may or not be considered. Either each item in the set can be considered one whole or the entire set can be considered as the whole, depending on the context of the problem. If the entire set is the whole it will be important that the tool used can be easily partitioned. For example, concrete pattern blocks are difficult to partition; however, paper pattern blocks could be cut.
- Linear models include number lines, the length of relational rods, and line segments. It is important for students to know the difference between of a line segment and as a position on a number line. Three fourths of a line segment treats the fraction as an operator and the whole is represented by the entire length of the line segment; for example, if the whole line segment represent 8 apples, then would be positioned at the 6. Three fourths as a position on a number line treats the fraction as a part-whole relationship where the number 1 on the number line is 1 whole, the number 2 on the number line is 2 wholes, and so on. So as a position, is located three fourths of the way from 0 to 1.
- Other measurement models include area, volume, capacity, and mass. Area is the most common model used with shapes like rectangles and circles. Circles are difficult to partition when the fractions are not halves, fourths, or eighths, so providing models of partitioned circles is imperative. Making connections to the analog clock may also be helpful; for example, past the hour.

B1.6

count to 10 by halves, thirds, fourths, fifths, sixths, eighths, and tenths, with and without the use of tools

- To count by a fractional amount is to count by a unit fraction. For example, when counting by the unit fraction one third, the sequence is: 1 one third, 2 one thirds, 3 one thirds, and so on. Counting by unit fractions can reinforce that the numerator is actually counting units. A fractional count equivalent to the unit fraction makes one whole (e.g., 3 one thirds).
- A fractional count can exceed one whole. For example, 5 one thirds means that there is 1 whole (or 3 one thirds) and an additional 2 one thirds.
- The numerator of a fraction shows the count of units (the denominator).

*Note*

- Counting by the unit fraction with a visual representation can reinforce the relationship between the numerator and the denominator as parts of the whole. Fractions can describe amounts greater than 1 whole.
- The fewer partitions of a whole, the smaller the number of counts needed to make a whole. For example, it takes 3 counts of one third to make a whole, whereas it takes 5 counts of one fifth to make a whole.
- When the numerator is greater than the denominator (e.g., ), the fraction is called
*improper*and can be written as a mixed number (in this case, ). Understanding counts can support understanding the relationship between improper fractions and mixed numbers. - Counting of unit fractions is implicitly the addition of unit fractions.

B1.7

read, represent, compare, and order decimal tenths, in various contexts

- The place value of the first position to the right of the decimal point is tenths.
- Decimal tenths can be found in numbers less than 1 (e.g., 0.6) or more than 1 (e.g., 24.7).
- When representing a decimal tenth, the whole should also be indicated .
- Decimal tenths can be compared and ordered by visually identifying the size of the decimal number relative to 1 whole.
- Between any two consecutive whole numbers are other numbers. Decimal numbers are the way that the base ten number system shows these “in-between” numbers. For example, the number 3.6 describes a quantity between 3 and 4.
- As with whole numbers, a zero in a decimal indicates that there are no groups of that size in the number. So, 5.0 means there are 0 tenths. It is important that students understand that 5 and 5.0 represent the same amount and are equivalent.
- Writing zero in the tenths position can be an indication of the precision of a measurement (e.g., the length was exactly 5.0 cm, versus a measurement that may have been rounded to the nearest ones, such as 5 cm).
- Decimals are read in a variety of ways in everyday life. Decimals like 2.5 are commonly read as two point five; the decimal in baseball averages is typically ignored (e.g., a player hitting an average of 0.300 is said to be “hitting 300”). To reinforce the decimal’s connection to fractions, and to make evident its place value, it is highly recommended that decimals be read as their fraction equivalent. So, 2.5 should be read as “2 and 5 tenths”. The word "and" is used to separate the whole-number part of the number and the decimal part of the number.
- Many tools that are used to represent whole numbers can be used to represent decimal numbers. It is important that 1 whole be emphasized to see the representation in tenths and not as wholes. For example, a base ten rod or a ten frame that was used to represent 10 wholes can be used to represent 1 whole that is partitioned into tenths.

B1.8

round decimal numbers to the nearest whole number, in various contexts

- Rounding numbers is often done to estimate a quantity or a measure, to estimate the results of a computation, and to estimate a comparison.
- A decimal number rounded to the nearest whole number means rounding the number to the nearest one; for example, is 1.7 closer to 1 or 2?
- Decimal tenths are rounded based on the closer distance between two whole numbers. For example:
- 56.2 is rounded to 56, because it is two tenths from 56 as opposed to eight tenths to 57.

- If a decimal tenth is exactly between two whole numbers, the convention is to round up, unless the context suggests differently – in some circumstances, it might be better to round down.

*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.

B1.9

describe relationships and show equivalences among fractions and decimal tenths, in various contexts

- The fraction as a quotient is 1 ÷ 10 and the result is 0.1, which is read as one tenth.
- A count of decimal tenths is the same as a count of unit fractions of one tenth and can be expressed in decimal notation (i.e., 0.1 (1 one tenth), 0.2 (2 one tenths), 0.3 (3 one tenths), and so on).
- A count of 10 one tenths makes 1 whole and can be expressed in decimal notation (1.0).
- A count by tenths can be greater than 1 whole. For example, 15 tenths is 1 whole and 5 tenths and can be expressed in decimal notation as 1.5.