B1. Number Sense
Specific Expectations
Whole Numbers
B1.1
read, represent, compose, and decompose whole numbers up to and including 10 000, using appropriate tools and strategies, and describe various ways they are used in everyday life
- representing whole numbers:
- in words (e.g., eight thousand six hundred twenty-one)
- as numerals in standard notation (e.g., 8621)
- in expanded form (e.g., 8000 + 600 + 20 + 1)
- as a position on a number line:
- as the distance (magnitude) from zero on a number line:
- using base ten materials:
- 8621 can be most efficiently represented as 8 cubes, 6 flats, 2 rods, and 1 small cube (unit):
- composing whole numbers:
- 8621 as hops on an open number line:
- decomposing whole numbers:
- 8621 using place value:
- 8 thousands + 6 hundreds + 2 tens + 1 one
- 7 thousands + 16 hundreds + 2 tens + 1 one
- 8 thousands + 5 hundreds + 12 tens + 1 one
- 8 thousands + 6 hundreds + 1 ten + 11 ones
- 8621 using place value:
- 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 4107, the digit 4 represents 4 thousands, 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 place value column 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., 4187 = 4000 + 100 + 80 + 7 or 4 × 1000 + 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, 1300, 200, and 6 combine to make 1506.
- Numbers are decomposed when they are represented as a composition of two or more smaller numbers. For example, 5125 can be decomposed into 5000 and 100 and 25.
- Tools may be used when representing numbers. For example, 1362 may be represented as the sum of 136 ten-dollar bills and 1 toonie or 13 base ten flats, 6 base ten rods, and 2 base ten units.
- 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.
- 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, 587 could be decomposed as 58 tens and 7 ones or decomposed as 50 tens and 87 ones, and so on.
- Composing and decomposing numbers in a variety of ways can support students in becoming flexible with their mental math strategies.
- Closed number lines with appropriate scales can be used to represent numbers as a position on a number line or as a distance from zero. Depending on the number, estimation may be needed to represent it on a number line.
- Partial number lines can be used to show the position of a number relative to other numbers.
- Open number lines can be used to show the composition of large numbers without drawing them to scale.
- It is important for students to understand key aspects of place value. For example:
- The order of the digits makes a difference. The number 385 describes a different quantity than the number 853.
- 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. To determine the value of a digit in a number, multiply the value of the digit by the value of its place. For example, in the number 5236, the 5 represents 5000 (5 × 1000) and the 2 represents 200 (2 × 100).
- 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”. For example, 189 means 1 hundred, 8 tens, and 9 ones, but 1089 means 1 thousand, 0 hundreds, 8 tens, and 9 ones.
- The value of the digits in each of the positions follows a “times 10” multiplicative pattern. For example, 500 is 10 times greater than 50, 50 is 10 times greater than 5, and 5 is 10 times greater than 0.5.
- Going from left to right, a “hundreds-tens-ones” pattern repeats within each period (ones, thousands, millions, billions, and so on). Exposure to this larger pattern and the names of the periods – into millions and beyond – satisfies a natural curiosity around “big numbers”, although students at this grade do not need to work beyond thousands.
Have students show 10 000 in different ways (note that this number could be embedded in a context appropriate to students). If using base ten materials, have students create 10 000 using large cubes (thousand blocks) and then determine how many hundred flats or ten rods they would need. Have them create 10 000 using combinations of these blocks and record their various arrangements using number sentences.
Have students identify real-life examples of quantities that are around 10 000 (e.g., populations, stadium capacities, fish populations in a local lake). Have them provide examples of distances that are 10 000 kilometres, 10 000 metres, 10 000 centimetres, or 10 000 millimetres and share their ideas.
B1.2
compare and order whole numbers up to and including 10 000, in various contexts
- comparing whole numbers:
- using digit value:
- 9315 and 9327:
- 9315 = 9000 + 300 + 10 + 5
- 9327 = 9000 + 300 + 20 + 7
- the thousands and hundreds digits are the same and the tens are different, so compare the tens
- since the digit 1 is less than the digit 2 (10 < 20), 9315 is less than 9327
- 9315 and 9327:
- using other numbers as benchmarks:
- Is 4670 or 4580 closer to 4600?
- 4670 is 70 more than 4600 and 4580 is 20 less than 4600, so 4580 is closer to 4600 than 4670
- Is 4670 or 4580 closer to 4600?
- using digit value:
- ordering whole numbers:
- arranging numbers in ascending order from least to greatest, e.g., 324, 982, 1603, 4597
- arranging numbers in descending order from greatest to least, e.g., 4597, 1603, 982, 324
- Numbers are compared and ordered according to their “how muchness” or magnitude.
- Numbers with the same units can be compared directly (e.g., 7645 kilometres compared to 6250 kilometres). 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).
- Sometimes numbers without the same unit can be compared, such as 625 kilometres and 75 metres. Knowing that the unit "kilometres" is greater than the unit "metres", and knowing that 625 is greater than 75, one can infer that 625 kilometres is a greater distance than 75 metres.
- Benchmark numbers can be used to compare quantities. For example, 4132 is less than 5000 and 6200 is greater than 5000, so 4132 is less than 6200.
- Numbers can be compared by their place value. For example, when comparing 8250 and 8450, the greatest place value where the numbers differ is compared. For this example, 2 hundreds (from 8250) and 4 hundreds (from 8450) are compared. Since 4 hundreds is greater than 2 hundreds, 8450 is greater than 8250.
- Numbers can be ordered in ascending order – from least to greatest – or can be ordered in descending order – from greatest to least.
Note
- An understanding of place value enables whole numbers 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 of numbers and make comparisons.
Create a number line clothesline, and use clothespins to hang benchmark numbers from 0 to 10 000 (e.g., 0, 5000, 10 000 – or additional cards that count by thousands). To support students in considering numbers that are closer together, place numbers such as 4100 and 5100 at either end.
Have students draw a number card (any number from 0 to 10 000) and pin it in what they think is the right position. Have them share their reasoning and make connections to the idea that rounding numbers can be useful when making estimates (see also B1.3). Support students in using spatial and proportional reasoning to partition the number line into increasingly smaller parts to enhance precision. Decreasing or eliminating the benchmark numbers increases the complexity of this sort of open number line task. This task can also be done in partners or small groups using paper number lines at desks.
Have students compare numbers like 2340, 2042, and 2403 to decide which is the greatest, and explain their reasoning. For contexts involving numbers to 10 000, have students order the populations of local small towns (populations under 10 000), the capacity of local sports and entertainment venues, the distances walked for a year-long charity walkathon, or other interesting contexts.
B1.3
round whole numbers to the nearest ten, hundred, or thousand, in various contexts
- rounding in various contexts:
- population of a town to the nearest thousand
- number of books in a library to the nearest hundred
- distance around the running track to the nearest 10 metres
- Rounding numbers is often done to estimate a quantity or measure, estimate the results of a computation, and make quick comparisons.
- Rounding involves making decisions about what level of precision is needed, and is used often in measurement. How close a rounded number is to the actual amount depends on the unit it is being rounded to. The result of rounding a number to the nearest ten is closer to the original number than the result of rounding the same number to the nearest hundred. Similarly, the result of rounding a number rounded to the nearest hundred is closer to the original number than the result of rounding the same number to the nearest thousand. 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. For example, when paying by cash in a store, the amount owing is rounded to the nearest five cents (or nickel).
- In the absence of a context, numbers are typically rounded on a midpoint. This approach visualizes the amount that is halfway between two units and determines whether a number is closer to one unit than the other.
- Rounding 1237 to the nearest 10 becomes 1240, since 1237 is closer to 1240 than 1230.
- Rounding 1237 to the nearest 100 becomes 1200, since 1237 is closer to 1200 than 1300.
- Rounding 1237 to the nearest 1000 becomes 1000, since 1237 is closer to 1000 than 2000.
- If a number is exactly on the midpoint, convention rounds the number up (unless the context suggests differently). So, 1235 rounded to the nearest 10 becomes 1240.
Adapt the number line clothesline from B1.2, Sample Task 1 into a permanent classroom fixture. Throughout the year, label the clothesline at multiples of 10, multiples of 100, or multiples of 1000 (for rounding benchmarks). Place a clothespin at the halfway mark of each benchmark section. Students decide whether their number card is on the right side or the left side of the halfway mark and explain how they know. They then round their number to the nearest ten, hundred, or thousand. Have students also use desk-sized open number lines (blank number lines where the markings and numbers can be added in pencil) and work with partners to practise rounding on similar tasks using numbers up to 10 000.
Have students consider situations when it might be appropriate to round to the nearest 10, 100, or 1000. For example, they could connect rounding numbers to rounding measurements and ask when (and why) people might round to the nearest centimetre, metre, or kilometre (or any other measurement unit).
Students might also consider situations where it is better to round up, round down, or compensate by rounding up sometimes and rounding down other times. Invite students to practise their rounding and mental math skills when they go grocery shopping with a parent and determine how close they can get to the actual total of the grocery bill without going over.
Fractions and Decimals
B1.4
represent fractions from halves to tenths using drawings, tools, and standard fractional notation, and explain the meanings of the denominator and the numerator
- representing one tenth using fraction tiles:
- A fraction is a number that tells us about the relationship between two quantities.
- A fraction can represent a quotient (division).
- It shows the relationship between the number of wholes (numerator) and the number of partitions the whole is being divided into (denominator).
- For example, 3 granola bars (3 wholes) are shared equally with 4 people (number of partitions), which can be expressed as $$\frac{3}{4}$$.
- A fraction can represent a part of a whole.
- It shows the relationship between the number of parts selected (numerator) and the total number of parts in one whole (denominator).
- For example, if 1 granola bar (1 whole) is partitioned into 4 pieces (partitions), each piece is one fourth ($$\frac{1}{4}$$) of the granola bar. Two pieces are 2 one fourths ($$\frac{2}{4}$$) of the granola bar, three pieces are three one fourths ($$\frac{3}{4}$$) of the granola bar, and four pieces are four one fourths ($$\frac{4}{4}$$) of the granola bar.
- A fraction can represent a comparison.
- It shows the relationship between two parts of the same whole. The numerator is one part and the denominator is the other part.
- For example, a bag has 3 red beads and 2 yellow beads. The fraction $$\frac{2}{3}$$ represents that there are two thirds as many yellow beads as red beads. The fraction $$\frac{3}{2}$$, which is $$1 \frac{1}{2}$$ as a mixed number, represents that there are 1 and one half times more red beads than yellow beads.
- A fraction can represent an operator.
- When considering fractions as an operator, the fraction increases or decrease by a factor.
- For example, in the case of $$\frac{3}{4}$$ of a granola bar, $$\frac{3}{4}$$ of $100, or $$\frac{3}{4}$$ of a rectangle, the fraction reduces the original quantity to $$\frac{3}{4}$$ its original size.
Note
- A fraction is a number that can tell us information about the relationship between two quantities. These two quantities are expressed as parts and wholes in different ways, depending on the way the fraction is used.
- $$\frac{3}{4}$$ as a quotient (3 ÷ 4): 3 represents three wholes divided into 4 equal parts (wholes to parts relationship).
- $$\frac{3}{4}$$ as a part of a whole: 3 is representing the number of parts selected from a whole that has been partitioned into 4 equal parts (parts to a whole relationship).
- $$\frac{3}{4}$$ as a comparison: 3 parts of a whole compared to 4 parts of the same whole (parts to parts of the same whole relationship).
- A fraction is an operator when one interprets a fraction relative to a whole. For example, each person gets three fourths of a granola bar, or three one fourths of the area is shaded.
To build understanding of fractions as part of a whole, have students create a fraction kit by partitioning strips of paper into different-size fractional parts. Give each student 10 strips of paper (and have more on hand), with each strip being the same length, but preferably a different colour. Have them fold one strip of paper in halves, another in thirds, another in fourths, and so on, up to tenths, and let the unfolded strip represent a whole. On each strip, have them mark each section with a unit fraction (e.g., on the halves strip, mark the sections as $$\frac{1}{2}$$ and $$\frac{1}{2}$$; on the thirds strip mark the sections as $$\frac{1}{3}$$, $$\frac{1}{3}$$, and $$\frac{1}{3}$$; and so on).
Note: Some unit fractions, such as, $$\frac{1}{3}$$, $$\frac{1}{7}$$, $$\frac{1}{9}$$, and $$\frac{1}{10}$$, will be harder to create than others, so it is advisable to break the task up, perhaps even over several days (e.g., do whole, halves, fourths, and eighths on the first day; thirds, sixths, and ninths on another day; fifths and tenths on another day; and sevenths on the last day). It is advisable to have models for ninths, fifths, tenths, and sevenths (see BLM4: B1.4, Fraction Strips Model) that students can use when folding their own strips. Take time to support students in seeing connections between units, such as “How can I use my thirds to help make my ninths?”.
To build understanding of the way different fractions compare to each other, have students stack the fraction strips that they made in Sample Task 1 to create a fraction tower, ordered by largest to smallest unit fraction (i.e., from the whole strip to the strip partitioned into tenths), as illustrated in BLM4.1: Fraction Strip Tower. Support students in connecting the denominator to the size of the unit and the numerator to the count of unit fractions.
Adapt the number clothesline (from Sample Tasks in B1.2 and B1.3) to enable students to represent the relative locations of fractions. Have them place randomly selected fraction cards (both unit fractions and non-unit fraction cards) from 0 to 1 in order, using their fraction tower as a reference to verify the order. Use the number line to make observations about the numerators and the denominators (e.g., the larger the denominator, the smaller the part).
Draw on the equal-sharing work students do in B1.5 to make connections to fractions as division. Have students, by themselves or with a partner, represent the following situation:
- If 3 sandwiches are shared equally between 4 friends, how much of a sandwich will each friend have?
In a debrief of the activity, have students share their representations to solve the problem, which might include:
Support students in writing the accompanying division sentence 3 ÷ 4 = $$\frac{3}{4}$$ to match the actions in the situation. Guide them to notice the connection between the fraction, the division sentence, and the meaning of the numerator and the denominator. After repeated experiences with solving equal-sharing situations, guide students to notice that the numerator represents the amount being divided (the dividend), the denominator represents the number by which that amount is divided (divisor), and the fraction describes the quotient.
Pose problems where students must use pattern blocks to represent different fractional amounts, such as one fourth, for different wholes. For example:
- If the yellow hexagon is the whole, describe the quantity that the green, red, and blue blocks represent (the green triangle is $$\frac{1}{6}$$ of the yellow hexagon, the red trapezoid is $$\frac{1}{2}$$ of the yellow hexagon, and the blue rhombus is $$\frac{1}{3}$$ of the yellow hexagon).
- If the red trapezoid is the whole, describe the quantity that the yellow hexagon and the blue rhombus represent (the yellow hexagon is two whole red trapezoids, and the blue rhombus is $$\frac{2}{3}$$ of a red trapezoid).
- If the blue rhombus is the whole, describe the quantity that the red trapezoid represents (the red trapezoid is 1$$\frac{1}{2}$$ or $$\frac{3}{2}$$ of the blue rhombus). Support students in identifying the green triangle as the unit fraction ($$\frac{1}{2}$$ of the rhombus), and have them count the unit fractions needed to cover the area of the red trapezoid.
Draw out the concept that a fraction is always relative to the whole and that as the whole changes, so does the unit fraction. Connect the meaning of the denominator to the unit, and the numerator to the count.
For additional experience with representing fractional quantities, have students compare different relational rods, where the length of one rod is described in terms of another (e.g., “If the dark green rod is 1 unit, how many units is the purple rod?”).
B1.5
use drawings and models to represent, compare, and order fractions representing the individual portions that result from two different fair-share scenarios involving any combination of 2, 3, 4, 5, 6, 8, and 10 sharers
- comparing fractions using an area model:
- scenarios where the wholes are the same size:
One Third of a Hexagon | One Half of a Hexagon | One Sixth of a Hexagon |
- scenarios where the wholes are different sizes:
One Third of a Hexagon | One Third of a Trapezoid |
- Fair sharing means that quantities are shared equally. For a whole to be shared equally, it must be partitioned in such a way that each sharer receives the same amount.
- Fair-share or equal-share problems can be represented using various models. The choice of model may be influenced by the context of the problem. For example,
- A set model may be chosen when the problem is dealing with objects such as beads or sticker books. The whole may be the entire set or each item in the set.
- A linear model may be chosen when the problem is dealing with things involving length, like the length of a ribbon or the distance between two points.
- An area model may be chosen when the problem is dealing with two-dimensional shapes like a garden plot or a flag.
- Fractions that are based on the same whole can be compared by representing them using various tools and models. For example, if an area model is chosen, then the area that the fractions represent are compared. If a linear model is chosen, then the lengths that the fractions represent are compared.
- Ordering fractions requires an analysis of the fractional representations. For example, when using an area model, the greater fraction covers the most area. If using a linear model, the fraction with the larger length is the greater fraction.
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.
- Different modes and tools can be used to represent fractions:
- Set models include collections of objects (e.g., beads in a bag, stickers in a sticker book), where each object is considered an equal part of the set. The attributes of the set (e.g., colour, size, shape) may or not be considered. Either each item in the set can be considered one whole or the entire set can be considered as the whole, depending on the context of the problem. If the entire set is the whole it will be important that the tool used can be easily partitioned. For example, concrete pattern blocks are difficult to partition; however, paper pattern blocks could be cut.
- Linear models include number lines, the length of relational rods, and line segments. It is important for students to know the difference between $$\frac{3}{4}$$ of a line segment and $$\frac{3}{4}$$ as a position on a number line. Three fourths of a line segment treats the fraction as an operator and the whole is represented by the entire length of the line segment; for example, if the whole line segment represent 8 apples, then $$\frac{3}{4}$$ would be positioned at the 6. Three fourths as a position on a number line treats the fraction as a part-whole relationship where the number 1 on the number line is 1 whole, the number 2 on the number line is 2 wholes, and so on. So as a position, $$\frac{3}{4}$$ is located three fourths of the way from 0 to 1.
- Other measurement models include area, volume, capacity, and mass. Area is the most common model used with shapes like rectangles and circles. Circles are difficult to partition when the fractions are not halves, fourths, or eighths, so providing models of partitioned circles is imperative. Making connections to the analog clock may also be helpful; for example, $$\frac{1}{4}$$ past the hour.
To emphasize the importance of the whole when comparing fractions, have students draw a rectangle and shade in one half of the area. Have them draw another rectangle where one fourth of its area is larger than the first rectangle’s half. Support students in recognizing that one can only be certain that a half is larger than a fourth if they are both fractions of the same-size whole.
Have students compare the results from two different equal-sharing situations, such as the following:
- Comparing equivalent situations:
- 4 friends are sitting at a table sharing 2 apples equally. At another table, 2 friends are sharing 1 apple equally. All the apples are the same size. Who gets more apple?
- One group has 3 friends sharing 2 bags of popcorn. Another group has 6 friends sharing 4 bags of popcorn. The bags of popcorn are the same size. If the groups share their bags of popcorn equally, do the friends in the first group or the second group get more popcorn?
- Comparing unequal situations:
- Who gets more modelling clay, 6 students sharing 2 cans of modelling clay equally or 2 students sharing 1 can of modelling clay equally?
- Who gets more water, 4 friends sharing 2 jugs of water or 13 friends sharing 5 jugs of water?
- Which plant grows faster, a plant that grows 2 cm in 8 days or a plant that grows 6 cm in 12 days?
- Who gets more tea, 6 people sharing 2 pots of tea or 9 people sharing 6 pots of tea?
Have students model these situations with drawings, counters, or the fraction kit that they developed in B1.4, Sample Task 1, to get a sense of the relationships within both situations. Support students in using benchmark fractions to make a comparison (e.g., is this more or less than a half?).
Move through these situations as students are ready. Not all of these problems will be appropriate for all students; however, working with these types of comparison problems strengthens proportional reasoning and an understanding of fractions.
Adapt the Number Line Clothesline from the B1.2 and B1.3 sample tasks for use with fractions. Pin or tape benchmark cards of 0, $$\frac{1}{2}$$, and 1 in the appropriate spaces along the clothesline. Distribute cards with fourths ($$\frac{1}{4}$$, $$\frac{2}{4}$$, $$\frac{3}{4}$$, $$\frac{4}{4}$$). Have students place them on the clothesline and explain their reasoning. Continue with eighths, thirds, sixths, fifths, and tenths. Use a different colour of card for each set of denominators to make them easily distinguishable. Discuss equivalent fractions and the roles of the denominator and the numerator in placing the cards. Build this number line over time, and/or develop desk-size versions for students to use. Eventually, extend the number line to 10, or focus on a part of the 0 to 10 number line.
B1.6
count to 10 by halves, thirds, fourths, fifths, sixths, eighths, and tenths, with and without the use of tools
- counting by fourths:
- To count by a fractional amount is to count by a unit fraction. For example, when counting by the unit fraction one third, the sequence is: 1 one third, 2 one thirds, 3 one thirds, and so on. Counting by unit fractions can reinforce that the numerator is actually counting units. A fractional count equivalent to the unit fraction makes one whole (e.g., 3 one thirds).
- A fractional count can exceed one whole. For example, 5 one thirds means that there is 1 whole (or 3 one thirds) and an additional 2 one thirds.
- The numerator of a fraction shows the count of units (the denominator).
Note
- Counting by the unit fraction with a visual representation can reinforce the relationship between the numerator and the denominator as parts of the whole. Fractions can describe amounts greater than 1 whole.
- The fewer partitions of a whole, the smaller the number of counts needed to make a whole. For example, it takes 3 counts of one third to make a whole, whereas it takes 5 counts of one fifth to make a whole.
- When the numerator is greater than the denominator (e.g., $$\frac{5}{3}$$), the fraction is called improper and can be written as a mixed number (in this case, $$1 \frac{2}{3}$$). Understanding counts can support understanding the relationship between improper fractions and mixed numbers.
- Counting of unit fractions is implicitly the addition of unit fractions.
Have students play Buzz with unit fractions. The teacher or the students choose a unit fraction by which to count (for example, counting to 10 by one tenths). Students take turns counting up by unit fractions – 1 one tenth, 2 one tenths, 3 one tenths, and so on (or, alternatively, one tenth, two tenths, three tenths, and so on). Each time a student says a unit fraction that makes a whole, that student stands up and states the quantity as both a fraction and as a whole: “BUZZ! 10 one tenths (10 tenths), or one.”
Ask students, “How many $$\frac{1}{2}$$ pieces are in 10?” (20). Ask again using other unit fractions up to $$\frac{1}{10}$$. Look for patterns that connect fractions to multiplication and division. Next, give a mixed number such as 3$$\frac{1}{4}$$, and discuss how students would determine how many one fourths are in 3$$\frac{1}{4}$$. Finally, ask students to describe why numbers such as 3$$\frac{1}{4}$$ and $$\frac{13}{4}$$ or 13 fourths represent the same amount.
To reinforce connections to unit fractions, have students consider a fraction such as $$\frac{4}{5}$$. Ask them what the 4 and the 5 represent in this fraction, and have them model the fraction using a strip of paper. Have them explain how the fraction, the model, and the number expressions ( $$\frac{1}{5}$$ + $$\frac{1}{5}$$ + $$\frac{1}{5}$$ + $$\frac{1}{5}$$; 4 × $$\frac{1}{5}$$ ) are all related and identify the numerator with the count of unit fractions.
B1.7
read, represent, compare, and order decimal tenths, in various contexts
representing six tenths:
- using relational rods:
- as a position on a number line:
- as the distance (magnitude) from zero on a number line:
- using ten frames, where the frame represents one whole:
- The place value of the first position to the right of the decimal point is tenths.
- Decimal tenths can be found in numbers less than 1 (e.g., 0.6) or more than 1 (e.g., 24.7).
- When representing a decimal tenth, the whole should also be indicated .
- Decimal tenths can be compared and ordered by visually identifying the size of the decimal number relative to 1 whole.
- Between any two consecutive whole numbers are other numbers. Decimal numbers are the way that the base ten number system shows these “in-between” numbers. For example, the number 3.6 describes a quantity between 3 and 4.
- As with whole numbers, a zero in a decimal indicates that there are no groups of that size in the number. So, 5.0 means there are 0 tenths. It is important that students understand that 5 and 5.0 represent the same amount and are equivalent.
- Writing zero in the tenths position can be an indication of the precision of a measurement (e.g., the length was exactly 5.0 cm, versus a measurement that may have been rounded to the nearest ones, such as 5 cm).
- Decimals are read in a variety of ways in everyday life. Decimals like 2.5 are commonly read as two point five; the decimal in baseball averages is typically ignored (e.g., a player hitting an average of 0.300 is said to be “hitting 300”). To reinforce the decimal’s connection to fractions, and to make evident its place value, it is highly recommended that decimals be read as their fraction equivalent. So, 2.5 should be read as “2 and 5 tenths”. The word "and" is used to separate the whole-number part of the number and the decimal part of the number.
- Many tools that are used to represent whole numbers can be used to represent decimal numbers. It is important that 1 whole be emphasized to see the representation in tenths and not as wholes. For example, a base ten rod or a ten frame that was used to represent 10 wholes can be used to represent 1 whole that is partitioned into tenths.
Calculators can be very helpful for making place-value patterns visible.
- Have students enter 10 000 on a calculator and repeatedly press the equal key to repeatedly divide by 10. For example, enter [10 000], press [÷ 10] and [=]; continue to press [=] and watch the 1 move to the next column to the right each time. Draw out the fact that 1 divided by 10 is 0.1, and make the connection to fractions, where one whole split into 10 parts is $$\frac{10}{10}$$ and where 0.1 represents one of those tenths ($$\frac{1}{10}$$) (see B1.4). Read 0.1 as one tenth to make this connection explicit. Support students in seeing that the “÷ 10” pattern seen in whole-number place value is simply being extended to decimal place value. Repeat this with other numbers, such as 40 000, and have students explain why 4 ÷ 10 = 0.4.
- Have students enter 0.1 on the calculator and multiply it by 10. Draw out the fact that 0.1 × 10 is 1, and that 10 tenths makes 1 whole. Support students in making the connection to fractions (see B1.6). As students continue multiplying by 10, and move from 0.1 to 10 000, support them in noticing that each column to the left has a value that is 10 times as great as the one to its right and that this continues on both sides of the decimal point.
- Have students enter 0.1 on a calculator and add 0.1. Then ask them to repeatedly add a tenth using the equal key (i.e., count by 0.1; note the connection to B1.6). Have them write down the number they predict will next appear with each tap of the [=] key. Have them read each decimal as a fraction (e.g., one tenth, two tenths) as it appears on the display. Support students who believe that after 0.9 will come 0.10 by making connections to counting by fraction tenths (e.g., after nine tenths comes ten tenths, which equals one; see B1.6). Have students continue the count by 0.1 up to 10 or beyond to reinforce this pattern.
This task also has strong connections to B2.3, where students look for place-value patterns when multiplying by 10, 100, and 1000 and dividing by 10.
To build understanding of comparing and ordering decimals, adapt the Number Line Clothesline activity for decimals. Place benchmark cards of 0 and 3 at either end of the clothesline. Have students decide where 2.5 should go and explain their thinking. Draw out the idea that decimals (and fractions and whole numbers) occupy their own unique space on a number line, and that decimals (like fractions) can describe amounts between whole numbers. Listen for and support students who might believe that decimal numbers fall below zero (i.e., that decimals represent negative integers). To ensure that students recognize that decimals can fall between any two numbers, adjust this task so the benchmark cards are larger numbers, for example 27 and 30, and have students decide where they would place cards such as 28.5.
If the size of the whole changes, the size of the part changes as well, even though the decimal describing the part remains the same. To strengthen students’ relational thinking, have them change the unit values of different place-value blocks. For example, have them think of the thousands block as the whole or one and figure out which block represents 0.1 of it. Change the block and have students consider the flat as the whole or the rod as the whole and determine which block represents one tenth. Discuss what one tenth of the unit cube would look like. Record student thinking as sentences (e.g., $$\frac{1}{10}$$ or 0.1 of 1000 is 100; $$\frac{1}{10}$$ or 0.1 of 100 is 10).
Provide pairs of students with a set of cards showing different representations of decimal numbers (i.e., numerical and pictorial) and ask them to share the cards equally between them in face-down piles. Then have them play Greatest Number or Least Number. At the same time, both students turn over a card from their pile. The student with the greater or lesser decimal number, depending on the version being played, wins the two cards. If both cards represent the same decimal number, each player keeps their own card. Have students play until one player has all the cards or until the time allocated for the activity has elapsed.
B1.8
round decimal numbers to the nearest whole number, in various contexts
- rounding in various contexts:
- length or distance to the nearest tenth of a metre or kilometre
- mass to the nearest tenth of a kilogram
- time to the nearest tenth of a second in a race
- Rounding numbers is often done to estimate a quantity or a measure, to estimate the results of a computation, and to estimate a comparison.
- A decimal number rounded to the nearest whole number means rounding the number to the nearest one; for example, is 1.7 closer to 1 or 2?
- Decimal tenths are rounded based on the closer distance between two whole numbers. For example:
- 56.2 is rounded to 56, because it is two tenths from 56 as opposed to eight tenths to 57.
- If a decimal tenth is exactly between two whole numbers, the convention is to round up, unless the context suggests differently – in some circumstances, it might be better to round down.
Note
- As with whole numbers, rounding decimal numbers involves making decisions about the level of precision needed. Whether a number is rounded up or down depends on the context and whether an overestimate or an underestimate is preferred.
Use the Decimal Clothesline task from B1.7, Sample Task 2, and have students decide whether 2.7 is closer to 2 or 3. Support students in splitting the space into tenths – either physically or mentally – and discuss why 2.5 is the halfway point. Have them draw on their number sense to explain why 2.8 is two and eight tenths, and have them use their spatial sense to notice that 2.8 is closer to 3. Remind students of the convention that if a number is “halfway”, the number is generally rounded up. Have students use desk-size open number lines to visually locate and round decimals that fall between any two whole numbers from 0 to 10 000, and to support other work with decimals. Add contexts appropriate to the students in the class to make work with decimals more concrete (e.g., travelling from home to the Elders’ Lodge, which is 1.6 kilometres away).
Have students make connections between decimal number lines and a metre stick (e.g., one tenth of a centimetre is a millimetre). Link one tenth of a metre to the decimetre and describe the relationship between metric prefixes using the language of tenths (see also E2.2). Apply rounding techniques to measurement contexts, and discuss when it might be appropriate to round a measurement up or down (e.g., when is an overestimate better than an underestimate and vice versa?). Reinforce the idea that the space between any two numbers can be subdivided into decimal parts.
B1.9
describe relationships and show equivalences among fractions and decimal tenths, in various contexts
- showing equivalence between one half and five tenths:
Half or Five Tenths of a Ten Frame | Half or Five Tenths of a Circle |
- The fraction $$\frac{1}{10}$$ as a quotient is 1 ÷ 10 and the result is 0.1, which is read as one tenth.
- A count of decimal tenths is the same as a count of unit fractions of one tenth and can be expressed in decimal notation (i.e., 0.1 (1 one tenth), 0.2 (2 one tenths), 0.3 (3 one tenths), and so on).
- A count of 10 one tenths makes 1 whole and can be expressed in decimal notation (1.0).
- A count by tenths can be greater than 1 whole. For example, 15 tenths is 1 whole and 5 tenths and can be expressed in decimal notation as 1.5.
To support students in recognizing the relationship between decimal tenths and fraction tenths, and to reinforce equivalences among other fractions, have them use their fraction kits created for B1.4, Sample Task 1, to identify, represent, and show any combination of equivalent fractions and decimals, such as $$\frac{1}{2}$$ = $$\frac{2}{4}$$ = $$\frac{3}{6}$$ =$$\frac{4}{8}$$ = $$\frac{5}{10}$$, illustrated in BLM4: B1.9, Equivalent Fractions.
Develop a shared list to keep as a reference in the classroom.