• Reading numbers involves interpreting them as a quantity when they are expressed in words or numerals, or represented using physical quantities 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 25, the digit 2 represents 2 tens and the digit 5 represents 5 ones.
• Sometimes numbers used every day do not represent a quantity. For example:
  • Postal codes and license plates are made up of the numerals 0 to 9 and letters.
  • Addresses are assigned numbers in order.
  • Numbers on sports jerseys organize the players on a team.
  • Numbers can rank positions, such as finishing 3rd in a race.
• Sometimes numbers used every day describe quantities (e.g., 12 turtles). 
• Recognizing small quantities without counting (subitizing) is helpful for working with numbers.
• Numbers can be represented in a variety of ways including the use of counts such as tallies, position, or distance on a number line, in words, using money, and using mathematical learning tools such as ten frames.

Note

• Every strand in the mathematics curriculum relies on numbers.
• Numbers may have cultural significance.
• Subitizing can help lay the foundation for work with place value, addition and subtraction, and estimation. For example, students looking at the number 32 can visualize 3 ten frames and 2 more; students can see a representation of 7 as 3 + 4 (see B1.2).
• Subitizing is easier when objects are organized (e.g., dots on a die or a domino) than when they are unorganized (i.e., objects randomly positioned).
• Sometimes a quantity is recognized all at once (perceptual subitizing).
• Sometimes a quantity is recognized as smaller quantities that can be added together (conceptual subitizing).

• Numbers are composed when two or more numbers are combined to create a larger number. For example, twenty and five are composed to make twenty-five.
• Numbers are decomposed when they are taken apart to make two or more smaller numbers that represent the same quantity. For example, 25 can be represented as two 10s and one 5.

Note

• When a number is decomposed and then recomposed, the quantity is unchanged. This is the conservation principle.
• Numbers can be decomposed by their place value.
• Composing and decomposing numbers in a variety of ways can support students in becoming flexible with their mental math strategies for addition and subtraction.
• Certain tools are helpful for showing the composition and decomposition of numbers. For example:
  • Ten frames can show how numbers compose to make 10 or decompose into groups of 10.
  • Rekenreks can show how numbers are composed as groups of 5s and 10s or decomposed into 5s and 10s.
  • Coins and bills can show how numbers are composed and decomposed according to their values.
  • Number lines can be used to show how numbers are composed or decomposed using different combinations of “jumps”.

• Numbers are compared and ordered according to their “how muchness”.
• Numbers with the same units can be compared directly. For example, 5 cents and 20 cents, 12 birds and 16 birds. Numbers that do not show a unit are assumed to have units of ones (e.g., 5 and 12 are considered as 5 ones and 12 ones).
• Numbers can be ordered in ascending order – from least to greatest – or can be ordered in descending order – from greatest to least.

Note

• The "how muchness" of a number is its magnitude.
• There is a stable order to how numbers are sequenced, and patterns exist within this sequence that make it possible to predict the order and make comparisons.
• The sequence from 1 to 19 has fewer patterns than sequences involving greater numbers and so requires a lot of practice to consolidate.
• The “decades” that follow the teens pick up on the 1 to 9 pattern. Within each decade, the 1 to 9 sequence is repeated. After 9 comes the next decade. The pattern of naming the decade is not always overt in English. For example, 30 means “three tens”, but this connection may not be noticed by hearing the word “thirty”.
• Number lines and hundreds charts model the sequence of numbers and can be used to uncover patterns.

• Estimation is used to approximate quantities that are too great to subitize.
• Different strategies can be used to estimate the quantity in a collection. For example, a portion of the collection can be subitized and then that amount visualized to count the remainder of the collection.
• Although there are many different ways to count a collection (see B1.5), if the count is carried out correctly, the count will always be the same.

Note

• Estimating collections involves unitizing, for example, into groups of 5, and then counting by those units (skip counting by 5s).
• Estimation skills are important for determining the reasonableness of calculations and in developing a sense of measurement.

• The count of objects does not change, regardless of how the objects are arranged (e.g., close together or far apart) or in what order they are counted (order irrelevance).
• Counting objects may involve counting an entire collection or counting the quantity of objects that satisfy certain attributes.
• Objects can be counted individually or in groups of equal quantities. The skip count is based on the number of objects in the equal groups.
• Each object in a collection must be touched or included in the count only once and matched to the number being said (one-to-one-correspondence).
• The numbers in the counting sequence must be said once, and always in the standard order (stable order).
• The last number said during a count describes how many there are in the whole collection (cardinality), including when groups are combined to solve an addition problem.

Note

• The counting principles are: one-to-one correspondence, stable order, conservation principle, order irrelevance, and cardinality.
• When skip counting groups of objects of the same quantity, the unit of skip count is the number of objects in each group. For example, when each group has two objects, the counter should count by twos.
• When skip counting a set of objects that leaves remainders or leftovers, the leftovers must still be counted for the total to be accurate. For example, when counting a collection of 37 by 5s, the 2 left over need to be counted individually and added to 35.
• Skip counting is an efficient way to count larger collections, and it also helps build basic facts and mental math strategies and establishes a strong foundation for multiplication and division.
• Counts can be tracked using tally marks. An application of this is identified in the Data strand; see D1.2.

• Fair-sharing or equal-sharing means that quantities are shared equally. For a whole to be shared equally, it must be partitioned so that each sharer receives the same amount.

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 encountering fractions and division. Present these problems in the way that students will best connect to.
• Whole numbers and fractions are used to describe fair-share or equal-share amounts. For example, 5 containers of playdough shared between 2 people means that each person receives 2 containers and half of another container. Or each person could receive 5 halves, depending on the sharing strategy used.
• Fractions have specific names. In Grade 1, students should be introduced to the terminology of “half/halves” and “fourth/fourths”.

• When something is shared fairly, or equally as two pieces, each piece is 1 one half of the original amount. Two one halves make up a whole.
• When something is shared fairly, or equally as four pieces, each piece is 1 one fourth of the original amount. Four one fourths make up a whole.
• If the original amount is shared as two pieces or four pieces, the fractions one half and two fourths are equivalent.
• A half of a half is a fourth.
• If something is cut in half, it is not possible for one person to get “the big half” while the other person gets “the small half”. If something is cut in half, both pieces are exactly equal. If there is a “big half”, then it isn’t a half.

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.
• Different fractions can describe the same amount as long as they are based on the same whole.
• The size of the whole matters. If 1 one half and 1 one fourth are based on the same whole, then 1 one half is twice as big as 1 one fourth. But if a small sticky note is cut into halves, and a big piece of chart paper is cut into fourths, then the 1 one fourth of the chart paper is bigger than the 1 one half of the sticky note.
• The fair-share problems that students engage in for learning around B1.6 will provide the opportunity to notice that 1 one half and 2 one fourths are the same amount.
• Students in this grade are not expected to write fractions symbolically; they should write “half”, not “”.

• When one whole is shared equally by a number of sharers, the number of sharers determines the size of each individual portion and is reflected in how that portion is named. For example, if a whole is equally shared among eight people, the whole has been split into eighths, and each part is one eighth of the whole. One eighth is a unit fraction, and there are 8 one eighths in a whole.
• The size of the whole matters. When comparing fractions as numbers, it is assumed they refer to the same-sized whole. Without a common whole, it is quite possible for one fourth to be larger than one half.
• Sharing a whole equally among more sharers creates smaller shares; conversely, sharing a whole equally among fewer sharers creates larger shares. So, for example, 1 one fourth is larger than 1 one fifth, when taken from the same whole or set.