B1. Number Sense
Specific Expectations
Whole Numbers
- representing whole numbers:
- in words (e.g., one hundred fifty-eight)
- using base ten materials (e.g., 1 hundred, 5 tens, 8 ones):
- as numerals in standard notation (e.g., 158)
- in expanded form (e.g., 100 + 50 + 8)
- on a number line (e.g., 43):
- as the distance (magnitude) from zero on a number line (e.g., 67):
- composing whole numbers:
- composing 150 using quarters:
- 25¢, 50¢, 75¢, 100¢, 125¢, 150¢ (six quarters in all)
- composing using relational rods (e.g., 124 can be composed using 12 tens and 2 twos):
- composing 150 using quarters:
- decomposing whole numbers:
7 | 25 | 50 |
6 and 1 5 and 2 4 and 3 3 and 4 2 and 5 1 and 6 |
2 tens and 5 ones 1 ten and 15 ones 25 ones |
5 tens 4 tens and 10 ones 10 fives 5 tens 25 + 25 |
- 117 in expanded form:
- 100 + 10 + 7
- 130 on a number line:
- 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 107, 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 decade 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., 187 = 100 + 80 + 7 or 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, 30, 20, and 5 are composed to make 55.
- Numbers are decomposed when they are represented as a composition of two or more smaller numbers. For example, 125 can be represented as 100 and 25; or 50, 50, 20, and 5.
- 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 in the mathematics curriculum relies on numbers.
- Numbers may have cultural significance.
- 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, 187 could be decomposed as 18 tens and 7 ones or decomposed as 10 tens and 87 ones, and so on.
- Composing and decomposing numbers in a variety of ways can support students in becoming flexible with 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 using 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”.
- Breaking down numbers and quantities into smaller parts (decomposing) and reassembling them in new ways (composing) highlights relationships between numbers and builds strong number sense.
- Composing and decomposing numbers is also useful when doing a calculation or making a comparison.
- As students build quantities to 200 concretely, they should use both written words and numerals to describe the quantity so that they can make connections among the representations.
Have students look for numbers up to 200 in the world around them and use these examples to describe and compare quantities and measurements. Have them use other real-life examples to show what numbers and quantities up to 200 look like. Ask questions such as “How many?” and “How many more?” to engage students in the world of numbers.
Have students create base ten materials by bundling craft sticks into groups of tens and hundreds. For example, they could show 158 as 1 hundred-stick bundle made up of 10 ten-stick bundles, 5 ten-stick bundles, and 8 single sticks. Have them write the quantity as a number and in expanded form (100 + 50 + 8) to show the decomposition. Support students in decomposing and composing the quantity in more than one way by unbundling and re-bundling the sticks (e.g., as 15 ten-stick bundles and 8 single sticks; as 13 ten-stick bundles and 28 single sticks), and have them record their combinations using number sentences (equations). Ensure that students understand that no matter how the sticks are bundled, the total quantity remains the same.
Have students use an open number line to show various ways to reach 189 and record the sequences of jumps using number sentences. Emphasize that each of these ends at the same point – 189 – and that there are many different ways to make the same quantity.
As students become familiar with number lines and develop an understanding of the various ways they can partition a number, have them create clues to determine Which Number Am I?. For example:
- My number has 1 hundred, 6 tens, and 6 ones.
- My number has 16 tens and 16 ones.
- My number has three digits, is less than 130, has a zero in the tens place, and has 9 ones.
Have students visualize these amounts and use models such as the number line or base ten materials to verify their predictions. Challenge students to create clues that have exactly two possible answers (e.g., I am an odd number that is less than 125, with 12 tens).
B1.2
compare and order whole numbers up to and including 200, in various contexts
- comparing whole numbers:
- using digit values:
- 105 and 107:
- the hundreds and the tens digits are the same, so compare the ones digits
- since the digit 7 is greater than 5, 107 is greater than 105
- 105 and 107:
- using benchmark numbers like 0, 50, 100, 150, and 200:
- 132 and 130: 130 is closer to 100 than is 132, so 130 is less than 132
- 147 and 155: 147 is less than 150 and 155 is greater than 150, so 155 is greater than 147
- using digit values:
- ordering whole numbers:
- arranging numbers in ascending order from least to greatest:
- 32, 98, 107, 120, 143
- arranging numbers in descending order from greatest to least:
- 143, 120, 107, 98, 32
- arranging numbers in ascending order from least to greatest:
- Numbers are compared and ordered according to their “how muchness” or magnitude.
- Numbers with the same units can be compared directly (e.g., 145 minutes compared to 62 minutes). 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).
- Benchmark numbers can be used to compare quantities. For example, 32 is less than 50 and 62 is greater than 50, so 32 is less than 62.
- Numbers can be compared by their place value. For example, 200 is greater than 20 because the digit 2 in 200 represents 2 hundreds and the 2 in 20 represents 2 tens; one hundred is greater than one ten.
- Numbers can be ordered in ascending order – from least to greatest – or they can be ordered in descending order – from greatest to least.
Note
- Moving between concrete (counting objects and sets) and abstract (symbolic and place value) representations of a quantity builds intuition and understanding of numbers.
- Understanding place value enables any number 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 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 to describe the number of tens in a number. This pattern is not always overt in English. For example, 30 means “three tens”, but this connection may not be noticed by only hearing the word “thirty”.
- Within each decade, the 1 to 9 sequence is repeated. After 9 comes the next decade. After 9 decades comes the next hundred.
- The 1 to 9 sequence names each hundred. Within each hundred, the decade sequence and the 1 to 99 sequences are repeated.
- Number lines and hundred charts model the sequence of numbers and can be used to uncover patterns.
Give students any two digits (using cards from 0 to 9), and have them make the two possible two-digit numbers. Ask them which of the two numbers is greater and which is lesser. Have them use place-value language or other number relationships to explain how they decided to order the digits. Ground the conversation in concrete materials to ensure that students understand the significance of the two different place-value columns.
Create a physical number line using a clothesline. On the clothesline, use clothespins to hang benchmark numbers proportionally from 0 to 200 (e.g., 0, 50, 100, 150, 200). Give students a number card (any number from 0 to 200), and have them use the benchmark numbers to decide where to position it on the clothesline. Ask them to explain their reasoning.
The clothesline task from Sample Task 2 can be extended to a partner or trio game. Each player takes five cards from a deck of cards numbered from 0 to 200 and organizes their cards in front of them, face up, with numbers arranged from least to greatest. One after another, players pick up a card from the remaining deck and place it in the correct order with respect to their other cards. The game continues until a player gets three consecutive numbers. Support students in comparing the numbers on their cards and discussing their strategies for ordering.
Create a 0 to 200 chart, leaving “blanks” in the chart that students must fill in. Have them explain how they figured out the missing numbers, and, as they do, highlight the patterns in the number system.
B1.3
estimate the number of objects in collections of up to 200 and verify their estimates by counting
- collections:
- unorganized collection of different types of objects:
- tulips in the garden
- cars in the parking lot
- Estimation is used to approximate large quantities and develops a sense of magnitude.
- Different strategies can be used to estimate the quantity in a collection. For example, a small portion of the collection can be counted, and then used to visually skip count the rest of the collection.
- The greater the number of objects in the skip count, the fewer the number of counts are needed.
- Although there are different ways to count a collection (see B1.4), if the count is carried out correctly, the count will always be the same.
Note
- Estimation strategies often build on “unitizing” an amount (e.g., “I know this amount is 10”) and visually repeating the unit (e.g., by skip counting by 10s) until the whole is filled or matched. Unitizing is an important building block for place value, multiplication, measurement, and proportional reasoning.
Fill identical containers with varying amounts of the same countable object, up to 200 objects in each container. For example, several mason jars could be filled with different numbers of centicubes. Place a label with the actual quantity on one or more of the containers so that students have a benchmark for their estimates. Have students use the benchmark container(s) and other spatial strategies to estimate the amounts in the mystery containers and explain their reasoning.
Fill identical containers with the same number of different-size objects (e.g., one container has 154 connecting cubes and another has 154 centicubes). Place a label with the actual quantity on one container, and have students estimate the quantity of objects in the other. Have students record their estimates and verify by skip counting (see B1.4). As students come to notice that the containers have the same quantity of objects, ask them to explain how this could be. Emphasize that a quantity can look very different depending on what is being counted. To connect this to measurement work, use the language of units (e.g., the size of the unit changes, but the quantity is the same).
Develop estimation tasks by filling different-shape containers with an equal number of same-size (congruent) objects (e.g., fill both a tall, thin container and a short, wide container with 135 centicubes). Have students estimate the quantities in each and skip count to check their estimates. Support students in recognizing that the shape of the container also affects what a certain quantity looks like. Discuss the different strategies that students used, for example, unitizing a group of 10 and visually repeating that unit to estimate how many groups of 10 it takes to fill the container.
B1.4
count to 200, including by 20s, 25s, and 50s, using a variety of tools and strategies
- counting experiences using tools:
- using play money:
- by 1s using loonies
- by 2s using toonies
- by 5s using nickels or $5 bills
- by 10s using dimes or $10 bills
- by 20s using $20 bills
- by 50s using $50 bills
- using a grid, hundred chart, or number line to show counts from a starting number, for example, showing counting by 10:
- using play money:
- using an open number line to show hops or to skip count:
- counting forward by 20s, 25s, and 50s, starting at 0:
- The count of objects does not change, regardless of how the objects are arranged (e.g., close together or far apart).
- Counting usually has a purpose, such as determining how many are in a collection, how long before something will happen, or to compare quantities and amounts.
- Counting objects may involve counting an entire collection or counting the quantity of objects that satisfy certain attributes.
- A count can start from zero or from any other starting number.
- The unit of skip count is identified as the number of objects in a group. For example, when counting by twos, each group has two objects.
- Counting can involve a combination of skip counts and single counts.
Note
- 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 number of objects must remain the same, regardless of how they are arranged, whether they are close together or spread far apart (conservation principle).
- The objects can be counted in any order, and the starting point does not affect how many there are (order irrelevance).
- The last number said during a count describes how many there are in the whole collection. It does not describe only the last object (cardinality).
- When all objects are not accounted for by using a skip count then the remaining objects are counted on either individually or by another type of skip count. For example, when counting a collection of 137 objects by 5s, the 2 left is counted on either by 1s or by 2s.
- Counting by ones up to and over 100 reinforces the concept that the 0 to 9 and decade sequence that appeared in the first hundred repeats in every hundred.
- Skip counting is an efficient way to count collections, and it also helps build basic facts and mental math strategies and establishes a strong foundation for multiplication and division.
To ensure that students can count up to and over 100, have them count by 1s from 96 to 120. Listen for students who do not yet recognize that the decades pattern repeats in every hundred and instead believe that the next number after 109 is 200 (e.g., the next big number that they know). Support students in developing their understanding by using a calculator to display the counting sequence (e.g., by repeatedly pressing the equal key: 96 + 1 = = = = ...). Before each tap of the equal sign, have students predict the next number in the sequence. Look for other opportunities, such as the page numbers in a book, to reinforce the patterns that exist in every hundred and in every decade.
Create a two-hundred chart, and have students count orally by 20s, 25s, and 50s. Discuss patterns that they notice in each sequence that might make the skips easier to recall.
Ask students what coins they might have if they had 175¢ in total, or what bills they might have if they had $135. Ask them to represent the money in a drawing and explain how they could use skip counting to count it.
- even numbers as matching pairs:
- 4 and 6 have matching pairs (even):
- 5 and 7 have one left over (odd):
- A whole number is even if it can be shared into two equal-sized groups or many groups of 2 without a remainder.
- A whole number is odd if it cannot be shared into two equal-sized groups or into many groups of 2 without a remainder.
Note
- There are patterns in the number system that can be used to identify a whole number as even or odd. For example, if a whole number with more than one digit ends in an even number, it is even.
Have students describe things that come in pairs. For example, animals commonly have two eyes and two ears; eggs come in cartons of six pairs; socks are sold in pairs. Use the idea of pairs to introduce the term even (i.e., a number is even if it can be split into sets of 2 [pairs]). Then have students consider things that do not come in even arrangements. For example, there are five fingers on a hand; many animals have one tail; tricycles have three wheels. Have students create a collage of objects with even and odd groupings. Have them explain how they know which objects represent even numbers and which represent odd numbers.
To strengthen their understanding of even and odd numbers, have students identify the even and odd numbers on a two-hundred chart and identify common characteristics of each. Have students think about other numbers in real-life contexts (e.g., their street address; the number of their favourite hockey player; number of siblings; their age) and discuss whether each of those numbers is even or odd.
Fractions
B1.6
use drawings to represent, solve, and compare the results of fair-share problems that involve sharing up to 10 items among 2, 3, 4, and 6 sharers, including problems that result in whole numbers, mixed numbers, and fractional amounts
- possible drawings for equal-sharing problems:
- 10 carrot sticks are shared equally:
- between 2 students:
- 10 carrot sticks are shared equally:
- between 3 students:
- between 6 students:
- 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 students to encounter 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, 4 pieces of ribbon shared between 3 people means that each person receives 1 whole ribbon and 1 one third of another ribbon.
- When assigning these types of problems, start with scenarios where there is a remainder of 1. As students become adept at solving these problems, introduce scenarios where there is a remainder of 2 that needs to be shared equally.
- Fractions have specific names. In Grade 2, students should be using the terminology of “halves”, “fourths”, and “thirds”.
Pose (or have students pose) a variety of equal-sharing problems that involve 2, 3, and 4 sharers. In general, difficulty increases depending on the number of sharers (e.g., 2, then 4, then 3) and on whether the solution results in a mixed number with a unit fraction (e.g., 3 and one fourth); a mixed number with a non-unit fraction (e.g., 3 and two thirds or 3 and 2 one thirds); or a non-unit fraction (e.g., two thirds). The following are examples that increase in complexity:
- If 9 blocks of clay are shared equally between 4 students, how many blocks does each student get? (Each student gets 2 and one fourth blocks of clay.)
- If 3 friends want to share 4 submarine sandwiches so that each person gets the same amount, how much does each friend get? (Each friend gets 1 and one third submarine sandwiches.)
- There are 8 pieces of ribbon to be shared by 3 students for their craft. If they share the ribbon equally, how much ribbon does each student get? (Each student gets 2 and two thirds pieces of ribbon, or 2 and 2 one thirds pieces of ribbon).
- 6 friends have 8 granola bars. If they share the granola bars equally, how many granola bars does each friend get? (Each friend gets 1 and two sixths granola bars each or 1 and 2 one sixths granola bars.)
Have students model each situation with a drawing, and support those who would benefit from acting out the situation using strips of paper. Ask students how they decided to share the pieces that were left over. Introduce fraction names (halves, fourths, thirds, sixths), and support students in understanding the language and conventions of mixed numbers and fractions; for example, 1$$\frac{1}{3}$$ is 1 and one third. Students should not be expected to write fractions or mixed numbers using fraction notation, but teachers may model this convention, making connections to the representations when appropriate.
B1.7
recognize that one third and two sixths of the same whole are equal, in fair-sharing contexts
- equal-sharing context showing that one third and two sixths are equal:
- 3 students share 4 four pieces of paper:
Sharing with Thirds | Sharing with Sixths |
- When something is shared fairly, or equally as three pieces, each piece is 1 one third of the original amount. Three one thirds make up a whole.
- When something is shared fairly, or equally as six pieces, each piece is 1 one sixth of the original amount. Six one sixths make up a whole.
- If the original amount is shared as three pieces or six pieces, the fractions 1 one third and 2 one sixths (two sixths) are equivalent, and 2 one thirds (two thirds) and 4 one sixths (four sixths) are equivalent.
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.
- Fair-share problems involving six sharers that result in remainders (see B1.6) provide a natural opportunity to recognize that 1 one third and 2 one sixths (two sixths) are equal.
Create and pose equal-sharing situations, such as the following, that lead students to equivalent answers (i.e., one third and two sixths):
- 6 friends are equally sharing 8 apples. How much does each friend get? (Each friend gets 1 and one third apples, or 1 and two sixths apples.)
- 3 friends equally share a cheese pizza. (Each friend gets one third of the cheese pizza). The 3 friends also equally share a pepperoni pizza of the same size that is cut into 6 equal pieces. (Each friend gets two sixths of the pepperoni pizza). Do the friends get more cheese or more pepperoni pizza? (Each friend gets two sixths of the pepperoni pizza, or one third of the pepperoni pizza, which is the same amount as the cheese pizza.)
Encourage students to act out each situation using pieces of paper, or model it with a drawing, and use the language of mixed numbers and fractions (one third, one sixth, two sixths) to describe the result. As students share their solutions, have them determine whether one third and two sixths of the apple or pizza represent the same amount. Support the conversation by having them use their paper models or drawings to explain their thinking.