B1. Number Sense
Specific Expectations
Whole Numbers
B1.1
read and represent whole numbers up to and including 50, and describe various ways they are used in everyday life
- numbers used in everyday life:
- numbers related to a board game
- road signs – speed, distance to a location, exit numbers
- numbers on licence plates
- calendars
- phone numbers
- house numbers
- age
- money
- numbers on a phone or remote control
- numbers on a bank machine
- shoe size
- numbers on a clock
- numbers on a menu
- representing whole numbers:
- in words (e.g., thirty-six)
- as numerals in standard notation (e.g., 36)
- using tools (e.g., rekenrek, ten frames, connecting cubes, tallies):
36 | |
- 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).
Have students look for numbers in the world around them, including in their classroom. Have them identify instances where numbers are used to describe “how many” (a quantity) and where numbers are used as a label or to describe the sequence in which something happens.
Have students play a matching set game where they recognize and match different representations of the same number. For example, they might create sets of matching cards, with each set featuring a single quantity depicted in numbers, words, number expressions (e.g., 2 + 3), and drawings (e.g., of a ten frame, a dot card, a number line).
Have students represent relevant real-life examples of what 50 looks like in different contexts. Have them describe, for example, where they might observe 50 people and compare that to 50 pencils. Support students in noticing that, while 50 people fill a room, 50 pencils can fit in a pencil case. Through providing students with different experiences of visualizing 50, reinforce the idea that although the objects being counted are different, the amount or quantity is the same. Ask students to select their own number and record two different contexts for it.
Dot cards use familiar arrangements to emphasize number relationships and help students develop subitizing skills. To enable students to practise subitizing in larger group settings, create a set of “dot plates” (arrangements of dots on paper plates created using stickers or a bingo dabber) and use them as flash cards for a minds-on activity. Hold up a plate for up to 3 seconds, then hide it again and ask students how many dots they saw and how they knew the number. Students might also use dot plates with their peers to practise their basic facts to 10.
- composing whole numbers:
- 28 composed using relational rods:
- 28 ones:
- 28 composed using relational rods:
- 2 tens and 1 eight:
- 5 fives and 1 three:
- 14 twos:
- decomposing whole numbers:
- 45 decomposed into nine groups of 5:
- 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”.
Have students use number lines, rekenreks, and ten frames to create and represent different ways a number can be composed and decomposed. Use addition and subtraction equations to show how the same ideas can be expressed symbolically. For example, a jump of 3 from 25, landing on 28, can be written as 25 + 3 = 28. 28 is decomposed into 25 and 3.
Have students use money (coins or bills) to show different ways to make a certain quantity. For example, ask them to show 40¢ in three different ways. Represent the different combinations using an equation, and emphasize that the count remains constant regardless of how a quantity is counted or arranged.
B1.3
compare and order whole numbers up to and including 50, in various contexts
- using a number line and connecting cubes:
- 15, 23, 33, and 40 arranged in order on a number line:
- 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.
Put the following numbers on a hundred chart: 13, 23, 32, 43. Then show students a different set of four numbers: 14, 24, 33, 44. Finish with 15, 25, 34, 45. Ask students what they notice that could help them make comparisons. For example, they might notice that the number is in a lower row if the tens digit is more, or that the column for a number is based on its ones digit and not its tens digit. Connect these numerical conversations to concrete quantities to make visible the different values of the ones columns and tens rows (e.g., in 23, 24, and 25, the 2 means there are two tens).
Ask students to compare the same number used to count different units. For example, have them compare five nickels, five dimes, and five loonies, and ask if these have the same money value. Support students in understanding that the value of a unit affects the value of a quantity.
B1.4
estimate the number of objects in collections of up to 50, and verify their estimates by counting
- collections:
- a collection of objects that are the same size but different colours:
- a collection of objects with mixed shapes, sizes, and orientations:
- a collection of objects that are all the same shape, size, and colour:
- a collection of 13 shells, arranged as two rows of five and three more
- a collection of 28 beads, arranged as five rows of five and three more
- a group of students lined up in a row
- a group of students at the back of the classroom
- a group of students spread out across five tables
- games such as mancala and dominoes
- 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.
Show students a container of small manipulatives (e.g., connecting cubes, centicubes, two-colour counters, coloured beads, pieces of ribbons, shells). Tell them that you want to scoop out, for example, 22 objects from the container. Take a scoop and have students estimate whether you scooped more than or fewer than 22 and count to verify. Invite them to choose a new target number and try this for themselves. As they become more capable of estimating with one manipulative, change the unit (the object) and support them in noticing how that affects the estimate.
Many students are not aware of the importance or relevance of estimating. They think of it as a way of getting to the exact answer, so they often change their estimate after they have done a close count. Providing students with many daily opportunities to practise estimating will help them improve their skills in this area and develop an appreciation for how useful estimating can be. For example, ask students to estimate:
- the number of buttons in the classroom. Ask: “Are there more than 20? Are there more than 50?”
- the number of days until the end of the year. Ask: “What number is it less than? What number is it more than?”
- the number of rooms in the school. Ask: “Are there more than five? Are there fewer than 20?”
- the number of teachers in the school. Ask: “Are there more than eight? Are there fewer than 20?”
- the number of basketballs in the gym. Ask: “Are there more than 10? Are there fewer than 25?”
B1.5
count to 50 by 1s, 2s, 5s, and 10s, using a variety of tools and strategies
- counting by 1s and 5s using tally marks:
- counting by 5s using skips on a number line:
- counting by 10s using beads on a rekenrek:
- counting by 10s using a bead string:
- counting by 2s on a hundred chart:
- 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.
Provide contexts and tools that authentically encourage counting by 1s (e.g., number of days, individual cubes), counting by 2s (e.g., pairs of objects such as shoes, 2-unit rods), counting by 5s (e.g., value of nickels, tally marks), and counting by 10s (e.g., value of dimes, ten frames, rows on a rekenrek).
Have students count collections in different arrangements, using different counting strategies, to internalize the counting principles.
- Show students a collection of 17 objects arranged as two rows of eight and one more. Have them predict how many objects there are just by looking (visualize), then count the collection by ones to check their prediction (verify). Ask students how many objects they think there will be if they count the collection by 2s, and listen for those who believe that the count could change (not observing the conservation principle). Have them count the collection by 2s, paying attention to those who are not yet confident about the counting sequence into and through the teens, and to those who might assign the single counter a 2 instead of a 1. If students can count the collection by 2s correctly, support them in noticing that the count remains the same, regardless of the counting strategy. To check if students realize the count can never change, ask how many objects they think there would be if they counted by 5s, and repeat the process as required. Listen for students who recognize that the count will always be the same “because that’s how many there are”.
- Show students a set of 18 objects arranged in a row. Ask how many objects there are, and have them count the collection to check. Then have them count the collection starting at someplace in the middle. Watch for students who think that collections can only be counted from left to right and are unsure what to do when they come to the end of the row and some objects are still uncounted. Support them in understanding that a collection can be counted in any order, as long as each object is counted once and only once.
To support students in understanding and practising skip-counting skills, have them:
- use a hundred chart to reinforce understanding of patterns and strengthen automaticity by skip-counting sequences, both forward and backwards
- decide what coins they could use to skip-count to 50 and use number lines to model and count the jumps
- plan how many notebooks will be needed for a class if each student gets two notebooks; discuss how skip counting could lead to a solution (change the situation to support skip-counting by 5s and 10s)
- imagine that each student takes five grapes from a bag; if there are 45 grapes, how many students get grapes (ask them to represent the solution using addition and/or subtraction equations)?
Fractions
B1.6
use drawings to represent and solve fair-share problems that involve 2 and 4 sharers, respectively, and have remainders of 1 or 2
- possible drawings for equal-sharing problems:
- 5 containers of modelling clay are shared between 2 children:
- 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”.
Create and pose equal-sharing situations that involve 2 or 4 sharers and have a result with a remainder of 1 or 2, such as the following:
- 6 granola bars are to be shared by 4 friends. How many granola bars does each friend get?
- 7 melon slices are to be shared by 2 friends. How many slices does each friend get?
- 4 students want to share 6 pieces of ribbon to decorate cards they have made. How many pieces of ribbon does each student get?
Encourage students to act out the situation using pieces of paper, or model it with a simple drawing. Introduce the language of mixed numbers and fractions (e.g., 1 and one half; 2 and 2 one fourths or 2 and two fourths) to describe the result. Ask students how they decided to share the remainders equally, and introduce fraction names (e.g., halves, fourths) to describe the remainders.
B1.7
recognize that one half and two fourths of the same whole are equal, in fair-sharing contexts
- equal-sharing context showing that one half and two fourths are equal:
- 6 pieces of string are to be shared by 4 students to make bracelets. How many pieces of string does each student get?
- In this situation, students are supported in recognizing that for both solutions, the students will get the same amount of string, that is, 1 and one half is the same as 1 and 2 one fourths, or 1 and two fourths.
- 6 pieces of string are to be shared by 4 students to make bracelets. How many pieces of string does each student get?
- 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 “$$\frac{1}{2}$$”.
Create and pose equal-sharing situations with 4 sharers, where students can arrive at equivalent answers (e.g., by using one half or two fourths). For example:
- 10 carrot sticks are to be shared by 4 friends. How many carrot sticks does each friend get?
- 4 Elders want to share 6 pieces of bannock. How many pieces of bannock does each Elder get?
Encourage students to act out the situation using pieces of paper, or model it with a simple drawing, and use the language of mixed numbers and fractions (e.g., 1 and one half, 1 and two fourths) to describe the result. As students share their solutions, have them determine whether answers with one half and those with two fourths represent the same amount. Support the conversation by having them use their drawings and models to explain their thinking.
B1.8
use drawings to compare and order unit fractions representing the individual portions that result when a whole is shared by different numbers of sharers, up to a maximum of 10
- comparing and ordering unit fractions:
- same-size pan of lasagna shared equally among two, four, and five people:
- drummers sharing a drum:
- 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.
Have students compare the results of sharing the same amount with different numbers of sharers and use these experiences to compare and order unit fractions. The following are examples of this type of equal-sharing situation:
- Who gets more melon, the friend equally sharing a melon with seven other friends or the friend equally sharing the same-size melon with eight other friends?
- The baker at Sweet Treats Bakery has just made two desserts in two different but identical rectangular pans. The baker cuts the tres leches cake in one pan into six equal-size pieces and the baklava in the other pan into eight equal-size pieces. Which pieces are bigger, the cake or the baklava? How do you know?
- The teacher has two sheets of construction paper that are exactly the same size and is preparing for a craft activity to make pinwheels. The teacher cuts the blue sheet of construction paper into four equal-size pieces and the red sheet into six equal-size pieces. Each pinwheel uses one of the pieces cut from the blue construction paper and one of the pieces cut from the red construction paper. Is the pinwheel more blue or more red? How do you know?
Support students in using the language of unit fractions (e.g., one sixth, one eighth, one fourth) to describe the size of each portion. Have them order the unit fractions from largest to smallest. Draw out the idea that more shares means smaller pieces and emphasize that this idea is helpful in understanding and comparing fractions.