B1. Number Sense
Specific Expectations
Rational Numbers
B1.1
read and represent whole numbers up to and including one million, using appropriate tools and strategies, and describe various ways they are used in everyday life
- representing whole numbers:
- in words (e.g., three hundred sixty-five thousand four hundred eighty-nine)
- as numerals in standard notation (e.g., 187 498)
- in expanded form (e.g., 527 910 = 500 000 + 20 000 + 7000 + 900 + 10 + 0 or 5 hundred thousands + 2 ten thousands + 7 thousands + 9 hundreds + 10 tens + 0 ones)
- in expanded notation with place value expressed as multiples of ten [e.g., 661 281 = (6 × 10 × 10 × 10 × 10 × 10) + (6 × 10 × 10 × 10 × 10) + (1 × 10 × 10 × 10) + (2 × 10 × 10) + (8 × 10) + (1 × 1)]
- in expanded notation [e.g., 397 177 = (3 × 100 000) + (9 × 10 000) + (7 × 1000) + (1 × 100) + (7 × 10) + (7 × 1)]
- expressed in different units (e.g., 263 949 = 2639 hundreds and 49 ones)
- Reading numbers involves interpreting them as a quantity when they are expressed in words, in standard notation, or in expanded notation.
- 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 945 107, the digit 9 represents 9 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.
- There are patterns to the way numbers are formed. Each place value column, or period, repeats the 0 to 9 counting sequence.
- Any quantity, no matter how great, can be described in terms of its place value. For example, 1500 may be said as fifteen hundred or one thousand five hundred.
- A number can be represented in expanded form (e.g., 634 187 = 600 000 + 30 000 + 4000 + 100 + 80 + 7, or 6 × 100 000 + 3 × 10 000 + 4 × 1000 + 1 × 100 + 8 × 10 + 7) to show place value relationships.
- 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.
- Numbers may have cultural significance.
- Seeing how a quantity relates to other quantities supports students in developing an understanding of the magnitude or “how muchness” of a number.
- There are patterns in the place value system that support the reading, writing, saying, and understanding of 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 by a constant “times 10” multiplicative pattern. For example, as the digit 5 shifts to the left, from 5000 to 50 000, the digit’s value becomes 10 times as great. As it shifts to the right, from 5000 to 500, its value becomes one tenth as great.
- To find the value of a digit in a number, the value of the digit is multiplied by the value of its place. For example, in the number 52 036, the 5 represents 50 000 (5 × 10 000) and the 2 represents 2000 (2 × 1000).
- Expanded notation represents the value of each digit separately, as an expression. Using expanded form, 7287 is written as 7287 = 7000 + 200 + 80 +7, or 7 × 1000 + 2 × 100 + 8 × 10 + 7 × 1.
- Each period – thousands, millions, billions, trillions – is 1000 times the previous period.
- A “hundreds-tens-ones” pattern repeats within each period (ones, thousands, millions, billions, and so on), and each period is 1000 times the one preceding it. Exposure to these patterns, and the names of these periods, also satisfies a natural curiosity around “big numbers” and could lead to conversations about periods beyond millions (billions, trillions, quadrillions, and so on).
- The number “five hundred eight thousand thirty-seven” is written as “508 037” and not “508 1000 37” (as if being spelled out with numbers). Listening for the period name (508 thousand), and the hundreds-tens-ones pattern that precedes the period, gives structure to the number and signals where a digit belongs. If there are no groups of a particular place value, 0 is used to describe that amount, holding the other digits in their correct place.
- Large numbers are difficult to visualize. Making connections to real-life contexts helps with this, as does comparing large numbers to other numbers using proportional reasoning. For example, a small city might have a population of around 100 000, and 1 000 000 would be 10 of these cities.
Have students represent 1 000 000 in different ways. For example, students might:
- determine how many thousands cubes would be needed to make 1 000 000 and consider the dimensions of a base ten block for 1 000 000
- determine the number of $100 bills needed to make $1 million
- figure out how many hockey arenas of people would equal 1 000 000
- provide examples for lengths of 1 000 000 km, 1 000 000 cm, and 1 000 000 mm, and for times of 1 million seconds
- research other real-life examples that demonstrate the quantity of 1 000 000 and share them with the class
Having visual models and well-known benchmark comparisons help students understand exactly how big a million is.
Have students read numbers with zeros in them, such as 509 608. Discuss the meaning of the zero(s) in the number. Emphasize the important role that zero plays as a placeholder, and how it holds the other digits in their correct place. To engage students in considering the role of zero, have them diagnose a common misconception by posing a question such as the following:
- Someone wrote the number five hundred nine thousand six hundred eight as 509 000 608. What place-value idea do you think they’re missing? How might you help them read and write numbers correctly?
Support students in recognizing that the value of the place includes the period name “thousands”. Guide them to see that mistakenly adding the zeros pushes the digits into other columns and creates a number that is actually five hundred nine million six hundred eight.
Emphasize the need to listen for the ones-tens-hundreds pattern when a number is read, and the use of the period name (e.g., thousands) to locate where the digits fall in the place-value chart.
Have students think of a metre stick as a number line from 0 to 1 000 000. Ask them to predict where 1000 falls on this number line. Have them write down their prediction and share it with a partner. Have them refine their prediction with their partner by counting using a metre stick. If they think 1000 is halfway to 1 000 000 – a common misconception – they should soon recognize that this would mean there are only 2 thousands in 1 000 000, so they will need to revise their thinking. As students refine their predictions, guide them to see that, if 1000 thousands make 1 000 000, then 1000 is $$\frac{1}{1 000}$$ of 1 000 000 or, on this number line, $$\frac{1}{1 000}$$ of a metre. Support students in recognizing that, if a metre represents 1 000 000, the 1-millimetre mark represents 1000.
B1.2
read and represent integers, using a variety of tools and strategies, including horizontal and vertical number lines
- representing integers with two-colour counters:
- assign a colour to represent positive and another to represent negative. For example, a red counter could represent (−1) and a blue counter could represent (+1). Similarly, a purple counter could represent (+1) and an orange counter could represent (−1). The choice of colour is arbitrary, not mathematical, and can be based on the materials that are readily available.
- zero pair: sum of +1 and −1:
- −4:
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With Zero Pairs |
- 2 (understood as positive 2):
- representing integers on a number line:
- −6, 0, and 3 as positions on a number line:
- −5 as a magnitude of 5, going in a negative direction from zero:
- −5 as a magnitude of 5, going in a negative direction from 4 to −1:
- 4 as a magnitude of 4, going in a positive direction from −2 to 2 on a vertical number line:
- Integers are whole numbers and their opposites.
- Zero is neither negative nor positive.
- On a horizontal number line, positive integers are displayed to the right of zero and negative integers are displayed to the left of zero.
- On a vertical number line, positive integers are displayed above the zero and negative integers are displayed below the zero.
- Integers can be represented as points on a number line, or as vectors that shows magnitude and direction. The integer −5 can be shown as a point positioned 5 units to the left of zero or 5 units below zero. The integer −5 can also be shown as a vector with its tail positioned at zero and its head at −5 on the number line, to show that it has a length of 5 units and is moving in the negative direction.
- Each integer has an opposite, and both are an equal distance from zero. For example, −4 and +4 are opposite integers and both are 4 units from zero.
- Zero can be represented with pairs of opposite integers. For example, (+3) and (−3) = 0.
- Integers measure "whole things" relative to a reference point. For example, 1 degree Celsius is used to measure temperature. Zero degrees is freezing (reference point). The temperature +10° C is ten degrees above freezing. The temperature −10° C is ten degrees below freezing.
Note
- Engaging with everyday examples of negative integers (e.g., temperature, elevators going up and down, sea level, underground parking lots, golf scores, plus/minus in hockey, saving and spending money, depositing and withdrawing money from a bank account, walking forward and backwards) helps build familiarity and a context for understanding numbers less than zero.
- Pairs of integers such as (+2) and (−2) are sometimes called "zero pairs".
- The Cartesian plane (see Spatial Sense, E1.3) uses both horizontal and vertical integer number lines to plot locations, and negative rotations to describe clockwise turns (see Spatial Sense, E1.4). Both are mathematical contexts for using and understanding positive and negative integers.
In a whole-class discussion, find out and record what students know about integers. Make a list of integers they know. For example, they might tell you that −20°C in the winter is very cold, or that the record high temperature on Earth is about +57°C. During this discussion, ask students what zero represents in the integer context they gave you (e.g., 0°C represents the temperature at which water freezes). Record their integers on both a vertical and a horizontal number line.
Have students draw their own personal horizontal and vertical number lines in their notebooks to represent positive and negative integers. Ask them to record integers that they encounter in real-world contexts (e.g., going up and down an elevator, golf scores) and to plot examples on their number lines.
Have students use a variety of models to represent integers as a positive or negative amount, direction, or change. For example:
- Create a physical integer number line on the classroom floor. Put a line of tape on the floor that represents zero as the starting point, and place smaller pieces of tape to represent positive and negative integers. Have the whole class stand on the line and have them take steps forward or backwards to represent +1 or −1. Give instructions such as: “Take two steps backwards. What integer did you land on?” For example, if a student is on 1 and moves two steps backwards, the student will land on −1. Support students in seeing that a step forward (+1) combined with a step backwards (−1) results in a change of 0 since they are back to their starting point. Have students describe combinations of moves that result in a given change, such as −3 (e.g., 3 steps forward and 6 steps backwards). If possible, students could apply this learning by programming robots to travel forward and backwards on the integer line and recording the robots' movements.
- Have students use two-colour tiles (or integer tiles) to represent positive and negative integers. Have them choose one colour to represent negative integers and the other colour to represent positive integers (e.g., red for negative and yellow for positive). Have them use the tiles to model quantities such as −4 (four red tiles) or +2 (two yellow tiles). Support them in seeing how combinations of integers can also represent an amount, similar to the number line activity above (e.g., 3 red tiles and 3 yellow tiles represent zero).
B1.3
compare and order integers, decimal numbers, and fractions, separately and in combination, in various contexts
- using benchmarks on a number line:
- using position relative to zero on a number line:
- the least number in the list is the farthest to the left or below zero
- the greatest number in the list is the farthest to the right or above zero
- Numbers with the same units can be compared directly (e.g., 72.5 cm2 compared to 62.4 cm2).
- Sometimes numbers without the same unit can be compared, such as 6.2 kilometres and 6.2 metres. Knowing that the unit kilometre is greater than the unit metre can allow one to infer that 6.2 kilometres is greater than 6.2 metres.
- Sometimes numbers without the same unit may need to be rewritten with the same unit in order to be compared. For example, 1.2 metres and 360 centimetres can be compared as 120 centimetres and 360 centimetres. Thus, 360 centimetres is greater than 1.2 metres.
- Whole numbers (zero and positive integers) and decimal numbers can be compared and ordered according to their place value.
- Benchmark numbers can be used to compare quantities. For example, $$\frac{5}{6}$$ is greater than $$\frac{1}{2}$$ and 0.25 is less than $$\frac{1}{2}$$, so $$\frac{5}{6}$$ is greater than 0.25.
- If two fractions have the same denominator, then the numerators can be compared. In this case the numerator with the greater value is the greater fraction because the number of parts considered is greater (e.g., $$\frac{2}{3}>\frac{1}{3}$$).
- If two fractions have the same numerators, then the denominators can be compared. In this case the denominator with the greater value is the smaller fraction because the size of each partition of the whole is smaller (e.g., $$\frac{5}{6}<\frac{5}{3}$$).
- Having more digits does not necessarily mean that a number is greater. For example, −7528 has four digits but it is less than +3 because −7528 is less than zero and +3 is greater than zero.
- Any positive number is greater than any negative number.
- When comparing positive numbers, the greater number is the number with the greater magnitude. On a horizontal number line, the greater number is the farthest to the right of zero. On a vertical number line, the greater number is the farthest above zero.
- When comparing negative integers, the least number is the negative integer with the greater magnitude. On a horizontal number line, the lesser number is the farthest to the left of zero. On a vertical number line, the lesser number is the farthest below zero.
- Numbers can be ordered in ascending order – from least to greatest – or can be ordered in descending order – from greatest to least.
Note
- Comparing numbers helps with understanding and describing their order and magnitude.
- Visual models like the number line – particularly if they are proportional – can be used to order numbers; show the relative magnitude of numbers; and highlight equivalences among fractions, decimals, and whole numbers.
- The absolute value of a number is its distance from zero, or its magnitude. Both −5 and +5 have an absolute value of 5, because both are 5 units from zero on the number line.
Give students a set of integers to put in order from least to greatest, both on a number line and as a list. Support those who might ignore the integer sign or be distracted by the number of digits. Reinforce the idea that every integer (except zero) has an opposite, which is the same distance from zero but on the other side.
Create a physical number line using a clothesline. On the clothesline, use a clothespin to hang 0 as a benchmark number close to the centre, and hang another number (e.g., +10) as a card to the right of zero. Give students a series of cards with positive integers, decimals, and fractions (including improper fractions and mixed numbers), and have them hang these on the number line. Once students have negotiated where positive numbers should hang, give them the symmetrically opposite integers (and optionally the accompanying negative decimals, negative mixed numbers, and negative fractions), and have them decide where these are located. (Note that, while ordering negative fractions and decimals becomes required learning in Grade 7, it can be helpful to preview this concept in Grade 6 to avoid misconceptions and emphasize larger symmetrical patterns in the number system.) Draw out the concepts that zero is the centre of the symmetry (reflection) and that the absolute value for opposite integers is the same; for example, the distance from 0 to +3 is the same as the distance from 0 to −3. Highlight that all negative numbers fall to the left of zero and all positive numbers, including positive fractions and decimals, fall to the right of zero. Similarly, on a vertical number line, all positive numbers lie above zero and all negative numbers lie below zero.
Fractions, Decimals, and Percents
B1.4
read, represent, compare, and order decimal numbers up to thousandths, in various contexts
- representing decimal numbers:
- in words (e.g., four and twenty-five thousandths)
- in standard notation (e.g., 7.635)
- in expanded form (e.g., 3.725 = 3 + 0.7 + 0.02 + 0.005)
- in expanded notation (e.g., −2.127 = (−2 × 1) + (1 × 0.1) + (2 × 0.01) + (7 × 0.001))
- in expanded notation using fractions (e.g., 62.378 = (6 × 10) + (2 × 1) + (3 × $$\frac{1}{10}$$) + (7 × $$\frac{1}{100}$$) + (8 × $$\frac{1}{1 000}$$))
- expressed in different units (e.g., 9.147 = 9147 thousandths, 7.045 = 70 tenths and 45 thousandths)
- comparing and ordering decimal hundredths:
- 3, 3.2, 3.02, 3.002:
- represent all numbers to thousandths: 3.000, 3.200, 3.020, 3.002
- arrange in ascending order: 3.000, 3.002, 3.020, 3.200
- 3, 3.2, 3.02, 3.002:
- The place value of the first position to the right of the decimal point is tenths. The second position to the right of the decimal point is hundredths. The third position to the right of the decimal point is thousandths.
- Decimal numbers can be less than one (e.g., 0.654) or greater than one (e.g., 24.723).
- The one whole needs to be shown or explicitly indicated when decimal numbers are represented visually since their representation is relative to the whole.
- Decimal numbers can be represented as a composition or decomposition of numbers according to their place value. For example, decimals can be written in expanded notation 3.628 = 3 + 0.6 + 0.02 + 0.008, or 3 × 1 + 6 × 0.1 + 2 × 0.01 + 8 × 0.001.
- Decimal numbers can be compared by their place value. For example, when comparing 0.8250 and 0.845, the greatest place value where the numbers differ is compared. For this example, 2 hundredths (from 0.825) and 4 hundredths (from 0.845) are compared. Since 4 hundredths is greater than 2 hundredths, 0.845 is greater than 0.825.
- Numbers can be ordered in ascending order – from least to greatest or can be ordered in descending order – from greatest to least.
Note
- Between any two consecutive whole numbers are decimal thousandths. For example, the number 3.628 describes a quantity between 3 and 4 and, more precisely, between 3.6 and 3.7 and, even more precisely, between 3.62 and 3.63.
- Decimals are sometimes called decimal fractions because they represent fractions with denominators of 10, 100, 1000, and so on. Decimal place value columns are added to describe smaller partitions. Decimals, like fractions, have a numerator and a denominator; however, with decimals, only the numerator is visible. The denominator (or unit) is “hidden” within the place value convention.
- Decimals can be composed and decomposed like whole numbers. Expanded notation shows place value subdivisions (e.g., 3.628 = 3 + 0.6 + 0.02 + 0.008, or 3 × 1 + 6 × 0.1 + 2 × 0.01 + 8 × 0.001).
- The decimal point indicates the location of the unit. The unit is always to the left of the decimal point. There is symmetry around the unit column, so tens are matched by tenths, and hundreds are matched by hundredths. Note that the symmetry does not revolve around the decimal, so there is no “oneth”:
- Between any two places in the base ten system, there is a constant 10 : 1 ratio, and this is true for decimals as well. As a digit shifts one space to the right it becomes one tenth as great and if it shifts two spaces to the right it becomes one hundredth as great. So, 0.005 is one tenth as great as 0.05, one hundredth as great as 0.5, and one thousandth as great as 5. This also means that 5 is 1000 times as great as 0.005.
- As with whole numbers, a zero in a decimal indicates that there are no groups of that size in the number:
- 5.007 means that there are 5 wholes, 0 tenths, 0 hundredths, and 7 thousandths.
- 5.100 means that there are 5 wholes, 1 tenth, 0 hundredths, and 0 thousandths.
- 5.1 (five and one tenth), 5.10 (5 and 10 hundredths), and 5.100 (5 and 100 thousandths) are all equivalent (although writing zero in the tenths and hundredths position can indicate the precision of a measurement; for example, the race was won by 5.00 seconds and the winning time was 19.29 seconds). Writing zero in the tenths, hundredths, and thousandths position can indicate the precision of a measurement (e.g., baseball batting averages are given to the nearest thousandths).
- Decimals are read in a variety of ways in everyday life. Decimals like 2.5 are commonly read as two point five; in math, the term pi (π) is commonly approximated as three point one four; the decimal in baseball averages is typically ignored. However, to reinforce the decimal’s connection to fractions, and to make visible its place value denominator, it is recommended that decimals be read as their fraction equivalent. So, 2.573 is read as “2 and 573 thousandths.”
Have students cut out several 10 × 10 grids and use them to represent decimal numbers such as 1.111 (i.e., 1 + 0.1 + 0.01 + 0.001). Establish that the grid represents the whole (1), and have them use place value to reason about the amount that should be shaded to represent each digit. For example, since a column or row is a tenth of the grid, one column or row would be shaded to represent 0.1. As students move to thousandths, they begin to recognize that there are no subdivisions on the grid to show this amount.
Support students as they draw on their fraction sense (i.e., the relative size of tenths, hundredths, and thousandths) and their understanding of place value (i.e., that each place-value column represents a value that is one tenth the value of the column to its left) to subdivide a hundredth square into 10 further sections to create thousandths. Have them shade one of these thousandths to show 0.001.
To draw out understanding of the relationships between thousandths and the whole, ask students how many of these newly created subdivisions there are in the whole. As students come to see that there are 1000 of these small units, support them in making the connection between 1000 small subdivisions, the fraction unit “thousandths”, and the decimal column by the same name.
To ensure that students have a strong understanding of the meaning of decimals, have them engage with the following tasks, individually at first, and then in conversation with peers and the class:
- Ask students what they would enter on a calculator to generate the number sequence 7, 0.7, 0.07, 0.007. Support them in recognizing that each number is one tenth the previous one and that dividing by 10 shifts a digit to the right. To extend their thinking, have students see if they can create this sequence using multiplication (i.e., × 0.1) and discuss why multiplying 7 by 0.1 (which is the same as dividing 7 by 10) produces a smaller number.
- Provide students with a series of numbers, such as those below, and have them determine which is the greatest in each set and explain their thinking:
- 4328 or 434 or 48
- 43.6 or 4.25 or 345
- 8.3 or 8.257 or 8.45
- 5.008 or 5.09 or 5.7
Listen to students as they explain the reason behind their choice, particularly those who have made an incorrect choice. Encourage them to use their number sense, and guide those who need support to recognize that having more digits after the decimal point does not mean a greater number.
Present the following situation:
- Paper clips come in small boxes of 100 and large boxes of 1000.
- A teacher has a container with 1350 loose paper clips.
Ask students to consider the following:
- Student A says that this is the same as 1.35 large boxes of paper clips.
- Student B says that this is the same as 13.5 small boxes of paper clips.
Can both students be correct? Explain your reasoning.
Present the following situation:
- Books from the library need to be put back on the shelves according to their Dewey decimal number, with smaller codes first.
Ask students to consider the following:
- A student says that code 599.3 should go before 599.234 because 3 is smaller than 234. Her friend disagrees. Who do you think is correct and why? Explain your reasoning
B1.5
round decimal numbers, both terminating and repeating, to the nearest tenth, hundredth, or whole number, as applicable, in various contexts
• terminating decimals:
- finite number of digits:
- 4, 4.5, 4.05, 4.005
• repeating decimals:
- infinite number of repeats of one or more digits:
- 3.333 333… = $$3.\dot{3}$$ or $$3.\overline{3}$$
- 0.121 212 12… = $$0.\overline{12}$$
- 0.425 674 256 7… = $$0.\overline{425 67}$$
- 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 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.
Note
- Decimal numbers that terminate are like 3.5, 46.27, and 0.625.
- Decimal numbers that repeat are like 3.555555… and can be represented using the symbol with a dot above the repeating digit, (e.g., $$3.\dot5$$). If a string of digits repeats, a bar can be shown above the string, or dots above the first and last digits (e.g., 3.546754675467 is written as $$3.\overline{5467}$$, or $$3.\dot546\dot7$$).
- Rounding involves making decisions about what level of precision is needed and is often used in measurement. 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 approximation; the smaller the unit, the more precise. Whether a number is rounded up or down depends on the context.
Adapt the clothesline number line (see B1.3, Sample Task 2) to focus on rounding decimals. Using a clothesline with the cards 5 and 6 pinned at opposite ends, support students in subdividing and labelling the space by tenths (5.1, 5.2, 5.3, and so on), and have them use these benchmarks to approximate where 5.879 and 5.777 would fall and explain their thinking.
Zoom in on the number line by having students change the endpoint benchmarks to 5.7 and 5.9. Discuss where to pin additional benchmark numbers (e.g., 5.75, 5.8, and 5.85) and use these to round 5.879 and 5.777 to the nearest hundredth.
B1.6
describe relationships and show equivalences among fractions and decimal numbers up to thousandths, using appropriate tools and drawings, in various contexts
- showing equivalence from decimal numbers to fractions:
- 0.3 = $$\frac{3}{10}$$
- 0.36 = $$\frac{36}{100}$$
- 0.152 = $$\frac{152}{1000}$$
- showing equivalence from fractions to decimal numbers up to thousandths:
- converting known representations:
- $$\frac{1}{2}$$ = 0.5
- $$\frac{1}{4}$$ = 0.25
- $$\frac{1}{3}$$ = $$0.\dot{3}$$ or $$0.\overline{3}$$
- $$\frac{3}{4}$$ = 0.75
- dividing the numerator by the denominator:
- $$\frac{5}{8}$$ = 5 ÷ 8 = 0.625
- converting known representations:
- Any fraction can become a decimal number by treating the fraction as a quotient (e.g., $$\frac{8}{5}$$ = 8 ÷ 5 = 1.6).
- Some fractions as quotients produce a repeating decimal. For example,$$\frac{1}{3}$$ = 1 ÷ 3 = 0.333… or $$\frac{1}{7}$$= 1 ÷ 7 = $$0.\overline{142857}$$. When decimal numbers are rounded they become approximations of the fraction.
- If a fraction can be expressed in an equivalent form with a denominator of tenths, hundredths, thousandths, and so on, it can also be expressed as an equivalent decimal. For example, because $$\frac{1}{4}$$ can be expressed as $$\frac{25}{100}$$, it can also be expressed as 0.25.
- A terminating decimal can be expressed in an equivalent fraction form. For example, 0.625 = $$\frac{625}{1000}$$, which can be expressed as other equivalent fractions, $$\frac{125}{200}$$ or $$\frac{25}{40}$$ or $$\frac{5}{8}$$ .
- Any whole number can be expressed as a fraction and as a decimal number. For example, 3 = $$\frac{3}{1}$$ = 3.0.
Note
- Decimals are how place value represents fractions and are sometimes called decimal fractions. While fractions may use any number as a denominator, decimals have denominators (units) that are based on a system of tens (tenths, hundredths, and so on).
Use the clothesline number line (see B1.3, Sample Task 2) to focus on equivalences between fractions and decimals. Create cards with fractions, decimals, and division expressions, and have students order these cards on the clothesline. Have them pin equivalent expressions (fractions, decimals, and division statements) together. Have them discuss how they might deal with repeating decimals and their fraction and division equivalents. Have them use calculators to create additional equivalent trios, and have them identify trios that produce terminating or repeating decimals.
Have students play a concentration or memory game with a variety of cards that include decimal, fraction, and model representations for a number, for example, 0.50, $$\frac{1}{2}$$, and a picture of a ten frame that is half full. Include cards with commonly used decimals and fractions, such as one tenth, one fifth, one fourth, three fourths, one third, and two thirds).