B1. Number Sense
Specific Expectations
Rational Numbers
B1.1
represent and compare whole numbers up to and including one billion, including in expanded form using powers of ten, and describe various ways they are used in everyday life
- representing whole numbers:
- in words (e.g., 19 365 489 = nineteen million three hundred sixty-five thousand four hundred eighty-nine)
- as numerals in standard notation (e.g., 32 935 172)
- in expanded form (e.g., 7 259 289 = 7 000 000 + 200 000 + 50 000 + 9000 + 200 + 80 + 9 or 7 millions + 2 hundred thousands + 5 ten thousands + 9 thousands + 2 hundreds + 8 tens + 9 ones)
- representing numbers in expanded form using powers of 10:
- 345 = (3 × 102) + (4 × 101) + (5 × 1)
- 4.2 million = (4 × 106) + (2 × 105)
- 6 230 085 207 = (6 × 109) + (2 × 108) + (3 × 107) + (8 × 104) + (5 × 103) + (2 × 102) + (7 × 1)
- 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 876 345 107, the digit 8 represents 8 hundred millions, the digit 7 represents 7 ten millions, the digit 6 represents 6 millions, the digit 3 represents 3 hundred thousands, 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.
- Reading numbers involves interpreting them as a quantity when they are expressed in words, in standard notation, or in expanded notation using powers of ten. Large numbers may be expressed as a decimal number with the unit expressed in words. For example, 36.2 million is equivalent to 36 200 000 = 36.2 × 106.
- Expanded notation with powers of ten shows a number as an expression by using addition, multiplication, and exponents. The number “three hundred seven million, twenty thousand, and fifty”, can be expressed as 307 020 050 and 3 × 108 + 7 × 106 + 2 × 104 + 5 × 10.
- Numbers without units identified are assumed to be based on ones.
- Numbers can be written in terms of another number. For example:
- 1 billion is 1000 millions.
- 1 billion millimetres is equal to 1000 kilometres.
- 1 billion seconds is about 32 years.
- Sometimes an approximation of a large number is used to describe a quantity. For example, the number 7 238 025 may be rounded to 7 million, or 7.2 million or 7.24 million, depending on the amount of precision needed.
- Numbers can be compared by their place value or they can be compared using proportional reasoning. For example, 1 billion is 1000 times greater than 1 million.
Note
- Every strand of mathematics relies on numbers.
- Some numbers have cultural significance.
- Real-life contexts can provide an understanding of the magnitude of large numbers. For example:
- 1 billion seconds is about 32 years.
- When they are closest to each other, Earth and Saturn are 1.2 billion kilometres apart.
- Given that Earth’s population is 7.5 billion (and counting), if you are “one in a million”, there are 7500 people just like you.
- There are patterns in the place value system that help people read, write, say, and understand numbers, and that suggest important ways for numbers to be composed and decomposed.
- 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.
- 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”.
- The value of the columns increases and decreases by powers of ten. With each shift to the left, a digit’s value increases by a power of 10 (i.e., its value is ten times as great). With each shift to the right, a digit’s value decreases by a power of 10 (i.e., its value is one tenth as great).
- To find the value of a digit in a number, the value of a digit is multiplied by the value of its place.
- Each period – thousands, millions, billions, trillions – is 1000 times as great as the previous period. Periods increase in powers of 1000 (103).
- When inputting numbers electronically, the “^” sign is used for exponents; for example, 106 would be entered as 10^6.
Have students represent numbers up to 1 000 000 000 in different ways. For example, they might:
- look for ways that numbers in the millions are communicated in the media, and interpret large numbers that are written with decimals (e.g., 1.2 million)
- determine the number of $100 bills needed to make $1 000 000 000
- calculate how many times the population of Canada would fit into 1 000 000 000
- research the distances between planets, solar systems, and other celestial objects
- research other real-life examples that demonstrate the quantity of 1 000 000 and share them with the class
To further extend their awareness of large numbers in real-world contexts, have students gather statistics that involve large numbers, such as world and local populations, the salaries of celebrities, or data about relevant social issues. Have them compare current data to historical data and describe how it has changed over time, using both absolute and relative comparisons. Ask them to describe numbers in a variety of ways and estimate the amounts as powers or as decimal representations (e.g., 27.5 million). Have them share the information with the class.
Fermi tasks are estimation tasks that proceed from an initial assumption or estimation. For example, if we wanted to know how many hairs are on a human head, we could start from an initial estimate of 50 hairs/1 cm2 and then estimate how many square centimetres of the scalp have hair. Have students try these Fermi-style tasks to get a sense of the size of 1 000 000 000:
- Open any book in the room. Estimate the number of words on a page. Use this estimate to estimate the total number of words in the book. How many copies of this book would be needed to make 1 000 000 000 words?
- Count out loud for 1 minute, recording or having another student record how many numbers you can say in one 1 minute. Use this data to estimate how long it would take you to count to 1 000 000 000. It may be useful to ask: “How long would it take you to count to 1 000 000?” first, and then scale that up by 1000.
- Determine if a human heart will beat 1 000 000 000 times in a lifetime. Measure your heartbeat by finding your pulse at your wrist, and formulate an answer to this question by using your knowledge of mathematics.
To visualize 1 000 000 000 in comparison to 1 000 000, consider a metre stick as a number line that spans from 0 to 1 000 000 000. Have students engage in the following task.
- If one end of this metre stick represents 1 000 000 000, and the other represents 0, where would 1 000 000 fall?
Have students write down their prediction and share it with peers. Support students as they refine their prediction by counting (e.g., if they think 1 000 000 is halfway to 1 000 000 000 – a common misconception – guide them to see that, in that case, the end of the metre stick would only represent 2 000 000). Through conversation, guide students to see that if 1000 millions make 1 000 000 000, then 1 million is $$\frac{1}{1000}$$ of 1 000 000 000 on the metre stick, or $$\frac{1}{1000}$$ of a metre. Support students in recognizing that, if 1 metre represents 1 000 000 000, the 1 millimetre mark represents 1 000 000. 1 000 000 000 is 1000 times as great as 1 000 000.
B1.2
identify and represent perfect squares, and determine their square roots, in various contexts
- perfect squares and their square roots:
- 1, 4, and 9 are perfect squares:
- Any whole number multiplied by itself produces a square number, or a perfect square, and can be represented as a power with an exponent of 2. For example, 9 is a square number because 3 × 3 = 9, or 32.
- The inverse of squaring a number is to take its square root. The square root of 9 ($$\sqrt{9}$$) is 3.
- A perfect square can be represented as a square with an area equal to the value of the perfect square. The side length of a perfect square is the square root of its area. In general, the area (A) of a square is side (s) × side (s), A = s2.
Note
- Perfect squares are often referred to as square numbers.
- Students should become familiar with the common perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144) and their associated square roots.
- Squares and square roots are inverse operations.
Have students use square tiles or grid paper to show the first few square numbers. Explain that a square tile has an area of 1 × 1, 1 square unit, or 1 unit squared (unit2). The next square is the 2 × 2 square, which has an area of 4 square units, and so on. Have students create a list of perfect squares to 20 × 20 using known facts, mental math, and a calculator. For each square, have them represent the area as a number sentence (e.g., 7 × 7 = 49), and a power (e.g., 72), and describe the side length as a square root (e.g., $$\sqrt{49}$$). Support students in using the terms power, base, exponent, square root, and perfect square by creating a visual reference that illustrates the appropriate terms.
Discuss the difference between multiplication and exponentiation by having students illustrate why 72 is not the same as 7 × 2 (a common misunderstanding). Throughout the year, use exponents and square roots to describe numbers whenever possible in order to reinforce the meaning and purpose of this operation.
Show students a square labelled with its area of 400 cm2. Ask them to determine the side lengths of this square. Use this context to model the square root symbol and connect square roots to the side length of a square. Have students use a calculator (including the exponent key or power function if it is available) to extend their list from Sample Task 1 to include other perfect squares. Have them use the square root function on a calculator to determine the square root of other perfect squares (e.g., common perfect squares such as 625 and 10 000; less common perfect squares such as 2025).
B1.3
read, represent, compare, and order rational numbers, including positive and negative fractions and decimal numbers to thousandths, in various contexts
- a rational number can be:
- a decimal
- $$1.\dot3$$ or $$1.\overline{3}$$ (repeating)
- 0.25 (terminating)
- any number that can be expressed as a fraction using two integers, with a denominator that is not zero, for example:
- a decimal
5 = $$\frac{5}{1}$$ | 0 = $$\frac{0}{4}$$ | –5 = –$$\frac{5}{1}$$ | 2$$\frac{1}{3}$$ = $$\frac{7}{3}$$ | –$$\frac{3}{8}$$ |
- representing a fraction as a decimal number:
- dividing the numerator by the denominator:
- $$\frac{3}{4}$$ = 3 ÷ 4 = 0.75
- using repeating decimals:
- $$\frac{1}{3}$$ = 0.3333... = $$0.\dot3$$ or $$0.\overline{3}$$
- dividing the numerator by the denominator:
- representing a decimal number as a fraction:
- 0.75 = $$\frac{75}{100}$$ = $$\frac{3}{4}$$
- Rational numbers are any numbers that can be expressed as $$\frac{a}{b}$$, where a and b are integers, and b ≠ 0. Examples of rational numbers include: $$\frac{-5}{4}$$, $$\frac{3}{6}$$, −7, 0, 205, 45.328, $$6.\dot4$$, −32.5.
- Fractions (positive and negative) are rational numbers. Any fraction can be expressed as a decimal number that either terminates or repeats.
- Whole numbers are rational numbers since any whole number can be expressed as a fraction (e.g., 5 = $$\frac{5}{1}$$).
- Integers (whole numbers and their opposites) are rational numbers since any integer can be expressed as a fraction (e.g., −4 = $$\frac{-4}{1}$$, +8 = $$\frac{8}{1}$$).
- Rational numbers can be represented as points on a number line to show their relative distance from zero.
- The farther a rational number is to the right of zero on a horizontal number line, the greater the number.
- The farther a rational number is to the left of zero on a horizontal number line, the lesser the number.
- Fractions can be written in a horizontal format (e.g., 1/2 or ½) as well as stacked format (e.g.,$$\frac{1}{2}$$).
Note
- There are an infinite number of rational numbers.
- Whole numbers, integers, positive fractions, and positive decimal numbers can be represented using concrete tools.
- Negative fractions and negative decimal numbers can be represented using a number line.
- Negative fractions have the same magnitude as their corresponding positive fraction. The positive and negative signs indicate their relative position to zero. One way of comparing negative fractions is to rewrite them as decimal numbers (e.g., $$\frac{−4}{5}$$ = −0.8).
- Negative decimal numbers have the same magnitude as their corresponding positive decimal numbers. The positive and negative signs indicate their relative position to zero.
Complete the table so that each column shows the same value represented in different ways:
As a fraction |
$$\frac{1}{10}$$ |
$$\frac{1}{1000}$$ | ||
As a decimal | 0.001 | |||
Using words | one hundredth | one thousandth |
Build a number line clothesline (see Grade 5: B1.2, Sample Task 1), and mark the endpoints as −10 and +10. Give students a series of cards that fall between these numbers (positive and negative integers, fractions, mixed numbers, and decimals), and have them use clothespins to hang the cards in the correct order. Discuss where zero belongs, and highlight the symmetries between positive and negative numbers. As students hang their cards, have them read the numbers aloud and discuss strategies for ordering the cards.
Change the endpoints of the clothesline in Sample Task 2 to focus on different sections of the number line. Zoom in to order fractions and decimals and emphasize the infinite density of the number system. Zoom out to order very large integers (positive and negative, including numbers expressed as powers), and emphasize the infinite vastness of the number system. This concept is illustrated in the following clip:
As the endpoints change, support the use of numerical, proportional, and spatial reasoning to determine relative distances on the number line. For example, if the clothesline is labelled from 0 to 1 000 000, guide students to reason that, since 1000 is $$\frac{1}{1000}$$of 1 000 000, it should be placed at a distance of $$\frac{1}{1000}$$of the total length of the clothesline.
Fractions, Decimals, and Percents
B1.4
use equivalent fractions to simplify fractions, when appropriate, in various contexts
- identifying equivalent fractions on a multiplication chart:
- identifying equivalent fractions for $$\frac{2}{3}$$:
- focus on the × 2 and × 3 rows (either horizontally or vertically)
- read along the two rows to find equivalent fractions, such as $$\frac{4}{6}$$ is equivalent to $$\frac{8}{12}$$:
- identifying equivalent fractions for $$\frac{2}{3}$$:
- fractions written in lowest or simplest terms:
- $$\frac{8}{12}$$ is not in lowest terms because 8 and 12 have factors other than 1 in common (i.e., 2 and 4, with 4 being the greatest common factor)
- $$\frac{2}{3}$$ is in lowest terms because the only factor that 2 and 3 have in common is 1
- Equivalent positive fractions that represent parts of a whole are created by either partitioning or merging partitions.
- A fraction is simplified (in lowest terms) when the numerator (count) and the denominator (unit) have no common whole number factor other than 1 (e.g., $$\frac{3}{5}$$ is in lowest terms, $$\frac{4}{6}$$ is not in lowest terms because both the numerator and denominator have a common factor of 2).
- Multiplication and division facts are used to create equivalent fractions and reduce fractions to their lowest terms.
- All unit fractions are in lowest terms.
Note
- Positive and negative fractions can represent quotients. Fractions are equivalent when the results of the numerator divided by the denominator are the same.
- Creating equivalent fractions is used to add and subtract fractions that represent parts of a whole when their units (denominators) are different.
- When performing addition, subtraction, multiplication, and division involving fractions, the results are commonly expressed in lowest terms.
- Sometimes, when working with fractions, a fraction may become a complex fraction in which the numerator and/or the denominator are decimal numbers. To express these fractions in lowest terms, both the numerator and the denominator are multiplied by the appropriate power of ten.
Have students engage in problems that involve equivalent ratios, such as the following:
- Three-fourths of the students in a class have brown hair. What is the fewest number of students that could be in this class?
- If the class has more than 25 students, what is the minimum possible number of brown-haired students, given the equivalent ratio to the question above?
- If the class is in a lecture hall, and has 340 students in it, how many brown-haired students would there be?
Record student responses in a ratio table, such as the one below, and use this to describe the ratios as fractions, including equivalent fractions. Identify the simplest form as the least number of students in the class (i.e., where the numerator and the denominator are integers that cannot be further reduced). Guide students to see the multiplicative relationships between and within the fractions.
Brown hair | 3 | 6 | 24 | |||||
Non-brown hair | 1 | 2 | 8 | |||||
Total | 4 | 8 | 32 |
Have students consider the fraction $$\frac{30}{36}$$. Ask them to list the common factors of 30 and 36. Have them use these factors to determine other equivalent fractions, and ultimately its simplest form. Highlight the role of the greatest common factor (see B2.6) in simplifying a fraction.
Have students simplify other fractions that are not in lowest terms, such as $$\frac{8}{12}$$, $$\frac{9}{12}$$, $$\frac{6}{10}$$, $$\frac{4}{6}$$, .... Support the concept of equivalent fractions and lowest terms or simplest form through the use of fraction strips and other concrete representations. Formalize the idea that both the numerator and the denominator can be multiplied or divided by the same number to create equivalent fractions, and that dividing the numerator and denominator by their greatest common factor simplifies a fraction.
B1.5
generate fractions and decimal numbers between any two quantities
- generating fractions between two fractions with the same numerator:
- identifying a fraction between $$\frac{2}{5}$$ and $$\frac{2}{3}$$ using a stacked area model:
- "I can see that $$\frac{3}{5}$$falls between $$\frac{2}{5}$$ and $$\frac{2}{3}$$."
- "By adding fourths to my model, I can see that $$\frac{2}{4}$$ falls between $$\frac{2}{5}$$ and $$\frac{2}{3}$$."
- identifying a fraction between $$\frac{2}{5}$$ and $$\frac{2}{3}$$ using a stacked area model:
- generating fractions between two fractions with the same denominator:
- identifying a fraction between $$\frac{1}{3}$$ and $$\frac{2}{3}$$ using number lines:
- "I can make equivalent fractions using sixths. I know that $$\frac{3}{6}$$ is between $$\frac{2}{6}$$ and $$\frac{4}{6}$$."
- identifying a fraction between $$\frac{1}{3}$$ and $$\frac{2}{3}$$ using number lines:
- generating more than one fraction between two fractions:
- identifying four fractions between $$\frac{1}{3}$$ and $$\frac{2}{3}$$ using number lines:
- "If I use the equivalent fractions $$\frac{5}{15}$$ and $$\frac{10}{15}$$, I can name the additional fifteenths in between them."
- identifying four fractions between $$\frac{1}{3}$$ and $$\frac{2}{3}$$ using number lines:
- generating decimals between decimals:
- using the midpoint (mean) between 24.65 and 32.5:
- determining a decimal number between 3.14 and 3.145:
- “I can re-represent 3.14 as 3.140 and then identify a number between it and 3.145.”
- There are an infinite number of decimal numbers that fall between any two decimal numbers. The place values of the decimal numbers need to be compared to ensure that the number generated does indeed fall between the two numbers.
- The number that falls exactly between any two numbers can be determined by taking the average of the two numbers.
- To determine a fraction between any two fractions, equivalent fractions must be created so that the two fractions have the same denominator in order to do the comparison.
Note
- The number system is infinitely dense. Between any two rational numbers are other rational numbers.
Have students identify decimals that fall between 24.5 and 24.6. As they generate decimals such as 24.51 and 24.52, have them find decimals that fall between those two decimals and explain their thinking. Highlight the infinite density of numbers, and draw out the role that each new place value has in increasing the precision of a number.
Have students determine fractions that fall between $$\frac{1}{4}$$ and $$\frac{1}{5}$$. Have them use visual models, such as relational rods and fraction strips, to justify their thinking. Draw out the role that equivalent fractions can play in comparing these fractions (e.g., $$\frac{5}{20}$$ and $$\frac{4}{20}$$) and generating in-between fractions (e.g., $$\frac{9}{40}$$, $$\frac{8}{40}$$, $$\frac{7}{40}$$). Students may also make connections to the decimal and percent equivalents to $$\frac{1}{4}$$ and $$\frac{1}{5}$$ (i.e., 0.25 or 25% and 0.20 or 20%). If not, point them out, and have students draw on the learning from Sample Task 1 to generate more examples. Make connections between finding in-between decimals and finding in-between fractions.
Give students the following Mystery Decimals prompts, and have them use number lines, fraction strips, and their knowledge of equivalence to reason out possible answers. To extend this learning, have students create their own Mystery Decimals prompts.
- The decimal is:
- Clue #1: greater than $$\frac{1}{8}$$
- Clue #2: less than $$\frac{1}{5}$$
- Clue #3: a multiple of $$\frac{1}{20}$$
- 0.15
- The decimal is:
- Clue #1: between $$\frac{2}{5}$$ and $$\frac{3}{5}$$
- Clue #2: greater than $$\frac{1}{2}$$
- Clue #3: a multiple of 0.11
- 0.55
- The decimal is:
- Clue #1: less than $$\frac{7}{8}$$
- Clue #2: greater than $$\frac{3}{4}$$
- Clue #3: a multiple of 0.17
- 0.85
B1.6
round decimal numbers to the nearest tenth, hundredth, or whole number, as applicable, in various contexts
- rounding decimal numbers:
- 35.674 to the nearest hundredth is 35.67
- 35.674 to the nearest tenth is 35.7
- 35.674 to the nearest whole number is 36
- Rounding makes a number simpler to work with and is often used when estimating computations, measuring, and making quick comparisons.
- A decimal number is rounded to the nearest thousandth, hundredth, tenth, or whole number based on which hundredth, tenth, or whole number it is closest to. If it is the same distance, it is usually rounded up. However, depending on context, it may be rounded down.
- Rounding involves making decisions about what level of precision is needed and is used in measurement, as well as in statistics. How close a rounded number is to the actual amount depends on the unit it is being rounded to: the larger the unit, the broader the estimate; the smaller the unit, the more precise.
Note
- Some decimal numbers do not terminate or repeat. For example, the decimal representation for pi. When working with circles, the decimal representation of pi is usually rounded to the nearest hundredth (3.14).
Have students consider real-life examples that require rounding. For example, have students round local gas prices, such as 127.9 cents/litre, to the nearest cent. Since pennies are no longer minted and most cash registers now round to the nearest nickel when customers pay in cash, have students round prices such as $4.51, $178.85, $999.91, and $62.78 to the nearest nickel. Have them consider a situation such as the following.
- Eight friends went out for lunch together. The bill came to $89 and they decided to split it evenly among them. When they used a calculator to figure out how much each person owed ($89 ÷ 8), it showed 11.125. How much do they each owe? How much would they pay if they wanted to add a 15% tip?
After students have had a chance to work on the questions, discuss the meaning of numbers like 11.125 in dollars and cents and the strategies that students used for rounding. Have them share their strategies for mentally calculating a 15% tip and the operations they needed to arrive at the total.
B1.7
convert between fractions, decimal numbers, and percents, in various contexts
- converting fractions to decimal numbers to percents:
- represent the fraction with a denominator of 100, if possible:
- $$\frac{3}{4}$$ = $$\frac{75}{100}$$ = 0.75 = 75%
- otherwise, divide the fraction and multiply by 100:
- $$\frac{2}{3}$$
= 2 ÷ 3
= 0.66 × 100 = 66.7 %
- $$\frac{2}{3}$$
- represent the fraction with a denominator of 100, if possible:
- showing equivalence from a decimal to a fraction:
- 0.125 = $$\frac{1}{8}$$
- showing equivalence from a percent to a decimal to a fraction:
- 0.2% = 0.002 = $$\frac{2}{1000}$$
- Converting between fractions, decimals, and percents often makes calculations and comparisons easier to understand and carry out.
- Relationships of quantities relative to a whole can be expressed as a fraction, a decimal number, and a percent. Percents can be greater than 100%.
- Some fractions can be converted to a percent by creating an equivalent fraction with a denominator of 100.
- When fractions are considered as a quotient, the numerator is divided by a denominator and the result is a decimal representation that can be converted to a percent.
- Some decimal numbers when converted to a percent result in a whole number percent (e.g., 0.6 = 60%, 0.42 = 42%).
- Some decimal numbers when converted to a percent result in a decimal number percent (e.g., 0.423 = 42.3%).
- The relationship of multiplying and dividing whole numbers and decimal numbers by 100 is used to convert between decimal numbers and percents.
- Percents can be understood as decimal hundredths.
- Any percent can be represented as a fraction with a denominator of 100. An equivalent fraction can be created expressed in lowest terms.
Note
- Unit fraction conversions can be scaled to determine non-unit conversions (e.g., one fifth = 0.2, so four fifths is 0.2 × 4 = 0.8). (See B2.2.)
- Common benchmark fractions, decimals, and percents include:
- $$\frac{1}{2}$$ = 0.5 = 0.50 = 50%
- $$\frac{1}{4}$$= 0.25 = 25%
- $$\frac{1}{5}$$= 0.2 = 0.20 = 20%
- $$\frac{1}{8}$$ = 0.125 = 12.5%
- $$\frac{1}{10}$$ = 0.1 = 0.10 = 10%
Have students analyse the flag below (or something similar) and determine the area of each colour as a fraction, a decimal, and a percent. The flag may be presented with an overlaid hundred grid, some other grid, or nothing.
Have students share their strategies for determining the relative areas of the colours, and for representing these areas as fractions, decimals, and percents.
Have students create their own fraction flag, using at least four colours and at least five different regions. Challenge them to represent several benchmark fractions in their design (both unit and non-unit fractions), and discuss the strategies used to partition the flag. Once they are finished, have them trade designs and determine the fraction, decimal, and percent representations for each region.