• identity property:
  • addition and subtraction:
    • 4.365 + 0 = 4.365
    • 4.365 − 0 = 4.365
  • multiplication and division:
    • 38 × 1 = 38
    • 38 ÷ 1 = 38
• commutative property:
  • addition:
    • 3.781 + 5.23 = 5.23 + 3.781 = 9.011
  • multiplication:
    • 12 × 38 = 38 × 12 = 456
• associative property:
  • addition:
    • (3.509 + 5.02) + 2.001 = 3.509 + (5.02 + 2.001) = 10.53
  • multiplication:
    • (1.2 × 10) × 1.2 = 1.2 × (10 × 1.2) = 14.4
• distributive property:
  • multiplication over addition:
    • 15 × 27.3
      = (15 × 20) + (15 × 7) + (15 × 0.3)
      = 300 + 105 + 4.5
      = 409.5
• order of operations:
  • BEDMAS is an acronym to help students remember the correct order to perform the operations in an expression:
    • B – brackets
    • E – exponents
    • D/M or M/D – division and multiplication in the order they appear in the expression
    • A/S or S/A – addition and subtraction in the order they appear in the expression

• Properties of operations are helpful for carrying out calculations.
  • The identity property: a + 0 = a, a – 0 = a, $$\frac{a}{1}$$ = a.
  • The commutative property: a + b = b + a, a × b = b × a.
  • The associative property: (a + b) + c = a + (b + c), (a × b) × c = a × (b × c).
  • The distributive property: a × b = (c + d) × b = (c × b) + (d × b).
• The commutative, associative, and identity properties can be applied for any type of number.
• The order of operations needs to be followed when given a numerical expression that involves multiple operations. Any calculations in brackets are done first. Secondly, any numbers expressed as a power (exponents) are evaluated. Thirdly, multiplication and division are done in the order they appear, from left to right. Lastly, addition and subtraction are done in the order they appear, from left to right.
• Multi-step problems may involve working with a combination of whole numbers, decimal numbers, and fractions.
• Multi-step problems may involve working with a combination of relationships, including ratios, rates, and percents.
• There may be more than one way to solve a multi-step problem.

Note

• This expectation supports most expectations in the Number strand and is applied throughout the grade. Whether working with numbers or operations, recognizing and applying properties and relationships builds a strong foundation for doing mathematics.
• Solving problems with more than one operation involve similar processes to solving problems with a single operation. For both types of problems:
  • Identify the actions and quantities in a problem and what is known and unknown.
  • Represent the actions and quantities with a diagram (physically or mentally).
  • Choose the operations(s) that match the actions to write the equation.
  • Solve by using the diagram (counting) or using the equation (calculating).
• In multi-operation problems, sometimes known as two-step problems, there is often an ultimate question (asking for the final answer or result being sought), and a hidden question (a step or calculation that must be taken to get to the final result). Identifying both questions is an important part of solving these types of problems.
• The actions in a situation inform the choice of operation. The same operation can describe different situations.
  • Does the situation involve changing (joining, separating), combining, or comparing? Then it can be represented with addition or subtraction.
  • Does the situation involve equal groups (or rates), ratio comparisons (scaling), or arrays? Then it can be represented with multiplication or division.
• Representing a situation as an equation is often helpful in solving it.
• The same situation can be represented with different operations. Each operation has an “inverse” operation – an opposite that “undoes” it. The inverse operation can be used to rewrite an equation to make it easier to calculate, or to check whether a calculation is true.
  • The inverse of addition is subtraction, and the inverse of subtraction is addition. So, for example, $$\frac{1}{2}$$ + ? = $$\frac{3}{4}$$ can be rewritten as $$\frac{3}{4}$$ − $$\frac{1}{2}$$ = ?.
  • The inverse of multiplication is division, and the inverse of division is multiplication. So, for example, $$\frac{1}{2}$$ × ? =  $$\frac{3}{8}$$can be rewritten as $$\frac{3}{8}$$ ÷ $$\frac{1}{2}$$ = ?.

• commonly used percents, fractions, and decimal equivalents:
  • 100% = 1
  • 10% = $$\frac{1}{10}$$ = 0.1
  • 1% = $$\frac{1}{100}$$ = 0.01
  • 5% = $$\frac{5}{100}$$ = $$\frac{1}{20}$$ = 0.05
  • 25% = $$\frac{25}{100}$$ = $$\frac{1}{4}$$ = 0.25
  • 50% = $$\frac{50}{100}$$ = $$\frac{1}{2}$$ = 0.5
  • 75% = $$\frac{75}{100}$$ = $$\frac{3}{4}$$ = 0.75
  • 200% = 2

• Certain equivalent representations of percents, fractions, and decimals are more commonly used to do mental calculations than others.
• Since 1% is 1 hundredth ($$\frac{1}{100}$$ or 0.01) of an amount, then any percent can be determined by scaling it up or down. 
• Both 1% and 10% ($$\frac{1}{10}$$ or 0.1) of an amount can be calculated mentally by visualizing how the digits of a number change their place value.
• Five percent (5% = $$\frac{5}{100}$$  = 0.05) is commonly used to do mental calculations since it is half of ten percent.
• Any percent can be created as a composition of 1%, 5%, and 10%.
• Since 100% of an amount is the amount, then 200% is twice the amount.
• Any fraction can be used as an operator; however, there are certain fractions that are more common than others. For example:
  • One half of an amount ($$\frac{1}{2}$$ = 50% = 0.5).
  • One fourth of an amount, since it is half of a half ($$\frac{1}{4}$$ = 0.25 = 25%).
  • Three fourths of an amount, since it is triple one-fourth ($$\frac{3}{4}$$ = 0.75 = 75%).

Note

• For more about understanding the equivalence between percents, fractions, and decimals, see B1.7.
• Common benchmark fractions, decimals, and percents include:
  • $$\frac{1}{2}$$ = 0.50 = 50%
  • $$\frac{1}{4}$$ = 0.25 = 25%
  • $$\frac{1}{5}$$ = 0.20 = 0.2 = 20%
  • $$\frac{1}{8}$$ = 0.125 = 12.5%
  • $$\frac{1}{10}$$ = 0.1 = 0.10 = 10%
• Unit fraction conversion can be scaled to determine non-unit conversions. For example:
  • 1 one fifth ($$\frac{1}{5}$$) = 0.2, so 4 one fifths ($$\frac{4}{5}$$) = 0.2 × 4 = 0.8.
  • 1 one fifth ($$\frac{1}{5}$$) = 20%, so 4 one fifths ($$\frac{4}{5}$$) = 20% × 4 = 80%.

To practise decimals and percents, have students create a deck of “trio cards”, where each card has an equivalent fraction, decimal, or percent on it. Have them shuffle the cards and use them to play a game. It could be a matching game, like Concentration; or it could be a game like High Score where cards are turned over and the high card takes both (in the event of a tie – two equivalent representations – the next card is turned over and the winner takes both rounds of cards). Have students discuss strategies for converting between the different representations.

• increasing a number by a percentage of its initial value:
  • increasing 72 by 50%:
    • 72 is 100% of its initial value
    • The new amount is 150% of this initial value
    • 50% of 72:
      • Find half of 72:
      • 72 × $$\frac{1}{2}$$ = 36
    • 150% of 72:
      • Add 36 to the initial value:
      • 72 + 36 = 108
• decreasing a number by a percentage of its initial value:
  • decreasing 72 by 5%:
    • 72 is 100% of its initial value
    • The new amount is 95% of this initial value
  • 5% of 72:
    • Find 10% of 72 and then halve it:
    •  7.2 ÷ 2 = 3.6.
  • 95% of 72:
    • Subtract 3.6 from the initial value:
    • 72 − 3.6 =
    • 72 − 4 + 0.4 = 68.4 (using compensation)

• Calculating whole number percents is a frequently used skill in daily life (e.g., when determining sales tax, discounts, or gratuities).
• Percents can be composed from other percents. A 15% discount combines a 10% discount and a 5% discount. A 13% tax adds 10% and another 3% (3 × 1%).
• Visuals are helpful for understanding (and communicating) whether a situation describes a percent increase or decrease or a percent of the whole. Unclear language can obscure the intended meaning.
• Finding a percent of a number is scaling an amount down or up. For example, finding 10% of a number is the same as scaling that number to $$\frac{1}{10}$$ of its size, as illustrated below. On a calculator, 10% of $50 = 0.10 × 50 = 5.

• Decreasing by a given percent means the percent is subtracted from the whole. So, a 10% decrease will be 90% ($$\frac{9}{10}$$) of the whole, as illustrated below. On a calculator, decreasing 50 by 10% = 50 − 10% or 50 − (0.10 × 50) = 45.

• Increasing by a given percent means the percent is added to the whole. So, a 10% increase is 110% ($$1 \frac{1}{10}$$) of the whole, as illustrated below. On a calculator, increasing 50 by 10% = 50 + 10%  or 50 + (0.10 × 50) = 55.

Note

• Mental math is not always quicker than paper and pencil strategies, but speed is not its goal. The value of mental math is in its portability and flexibility, since it does not require a calculator or paper and pencil. Practising mental math strategies also deepens understanding of numbers and operations.
• Estimation is a useful mental strategy when either an exact answer is not needed or there is insufficient time to work out a calculation.

• representing addition with integers:
  • using different models:
    • assign a colour to represent positive and another to represent negative. For example, a red arrow could represent (−1) and a blue arrow 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 Principle: (+3) + (−3) = 0

• representing subtraction with integers:
  • using different models:

• When adding and subtracting integers, it is important to pay close attention to all of the elements of the statement. For example:
  • (+4) + (−3) may be interpreted as combining positive four and negative three.
  • 4 + (−3) may be interpreted as adding negative three to positive four.
  • (−4) − (+3) may be interpreted as determining the difference between negative four and positive three by comparing them.
  • (−4) − 3 may be interpreted as taking away positive three from negative four.
  • 4 − 3 may be interpreted as taking away positive three from positive four.
  • −4 − 3 may be interpreted as taking away positive three from negative four. 
• The order in which integers are written in an addition statement does not matter because the commutative property holds true (e.g., −5 + 3 = 3 + (−5)). It is important to note that the sign directly in front of the number belongs to the number.
• The order in which integers are written in a subtraction statement does matter because the commutative property does not hold true. For example, (−5) − (+3) = −8 and (+3) − (−5) = +8; they do not produce the same result.
• Addition and subtraction are inverse operations; therefore, a subtraction expression can be rewritten as an addition expression by adding its opposite (e.g., (−5) − (+ 3) = (−5) + (−3) and 2 − (−4) = 2 + (+4)).
• When two positive integers are added together, the result is positive. This can be visualized on a number line as:
  • two vectors moving in a positive direction (right or up);
  • a vector moving in a positive direction from a positive starting position.
• When two negative integers are added together, the result is negative. This can be visualized on a number line as:
  • two vectors moving in a negative direction (left or down);
  • a vector moving in a negative direction from a negative starting position.
• When a positive and a negative integer are added together, the result is negative if the absolute value of the negative integer is greater than the absolute value of the positive integer. This can be visualized on a number line as:
  • one vector moving in a positive direction and the other vector with a greater magnitude moving in a negative direction (the sign of the resultant vector is negative);
  • a vector moving in a negative direction from a positive starting position with the head of the vector to the left (or below) zero;
  • a vector moving in a positive direction from a negative starting position with the head of the vector to the left (or below) zero.
• When a positive and a negative integer are added together, the result is positive if the absolute value of the positive integer is greater than the absolute value of the negative integer. This can be visualized on a number line as:
  • one vector moving in a negative direction and the other vector with a greater magnitude moving in a positive direction (the sign of the resultant vector is positive);
  • a vector moving in a positive direction from a negative starting position with the head of the vector to the right (or above) zero;
  • a vector moving in a negative direction from a positive starting position and the head of the vector to the right (or above) zero.
• If the two integers added together have the same sign, then their magnitudes are added together.
• If the two integers added together have different signs, then their magnitude is determined by taking the absolute difference between them.
• “Zero pairs” are the sum of a positive and a negative number that results in zero.
• Depending on the models and the integers that are involved in a subtraction, zero pairs may need to be introduced in order to act out the situation. For example, if the situation involves taking away a negative amount but only positive amounts are shown, then adding zero pairs will allow for the negative amount to be removed.
• If the situation involves comparing two integers, the two integers can be represented as positions on a number line to determine the distance between them (magnitude). The order in which the subtraction statement is written is important in determining the sign. The sign is determined by the direction of the movement from the point represented by the integer after the minus sign (subtrahend) to the point represented by the integer in front of the minus sign (minuend). For example:
  • For 10 − (+2) = +8, the distance between positive 10 and positive 2 is 8; the movement from positive 2 to 10 is in a positive direction.
  • For (+2) − (+10) = −8, the distance between positive 2 and positive 10 is 8; the movement from positive 10 to positive 2 is in a negative direction.
  • For (2) − (−10) = +12, the distance between positive 2 and negative 10 is 12; the movement from negative 10 to positive 2 is in a positive direction.

Note

• Familiar real-world contexts for negative and positive integers (e.g., temperature, elevators going up and down, parking garages, sea level, golf scores, plus/minus in hockey, gaining and losing money, walking forward and backwards) provide a starting point for understanding adding and subtracting with integers.
• Situations involving addition and subtraction can be modelled using tools such as a number line and integer tiles.
• When writing an equation, integers are often placed inside brackets and the equation written as (+3) − (−2) = (+5). If an integer sign is not included, the number is considered positive, so it is also true that 3 − (−2) = 5. These conventions help reduce confusion between the number and the operation.
• Change can be represented by a positive or negative integer (e.g., rise of 4 expressed as +4, drop of 4 expressed as −4).
• A quantity relative to zero can be represented by a positive or negative integer (e.g., temperature is 3 degrees, temperature is −5 degrees).
• The integers in a situation may be interpreted as changes or as quantities. For example, if the temperature outside drops 5 degrees and then 3 degrees, this may be expressed as the addition of two drops [(−5) + (−3)] or as a subtraction of 3 degrees (−5 − 3). Both statements result in the same answer (−8), meaning the temperature decreased by 8 degrees.
• Modelling addition and subtraction expressions can help with the interpretation of the result relative to the context.

• adding fractions:
  • $$\frac{2}{3}$$ + $$1\frac{3}{4}$$
    • "I think this is going to be a little bit more than 2." 
    • "To make this easier to work with, I'm going to write the mixed number as two fractions."
  • $$\frac{2}{3}$$ + 1 + $$\frac{3}{4}$$
    • $$\frac{2}{3}$$ + $$\frac{4}{4}$$ + $$\frac{3}{4}$$ 
      • "Now I can add the fourths together."
  • $$\frac{2}{3}$$ + $$\frac{7}{4}$$
    • "I have to add $$\frac{2}{3}$$ and $$\frac{7}{4}$$. I think that 12 is a good common denominator for the equivalent fractions."
  • $$\frac{2}{3}$$ × $$\frac{4}{4}$$ = $$\frac{8}{12}$$ and $$\frac{7}{4}$$ × $$\frac{3}{3}$$ = $$\frac{21}{12}$$
    • = $$\frac{8}{12}$$ + $$\frac{21}{12}$$ = $$\frac{29}{12}$$ = $$2\frac{5}{12}$$
• subtracting fractions:
  • $$1\frac{3}{4}$$ − $$\frac{2}{3}$$
    • "I think this is going to be a little bit more than 1."
    • "To make this easier to work with, I'm going to write the mixed number as two fractions." 
    • 1 + $$\frac{3}{4}$$ − $$\frac{2}{3}$$
      • = $$\frac{4}{4}$$ + $$\frac{3}{4}$$ − $$\frac{2}{3}$$
        • "I can add the fourths together and then subtract $$\frac{2}{3}$$."
    • $$\frac{7}{4}$$ − $$\frac{2}{3}$$
      • "I think that 12 is a good common denominator for my equivalent fractions."
    • $$\frac{7}{4}$$ × $$\frac{3}{3}$$ = $$\frac{21}{12}$$ and $$\frac{2}{3}$$ × $$\frac{4}{4}$$ = $$\frac{8}{12}$$
      • $$\frac{21}{12}$$ − $$\frac{8}{12}$$ =  $$\frac{13}{12}$$ = $$1\frac{1}{12}$$

• The addition and subtraction of fractions with the same denominator can be modelled on the same number line. Each one whole on the number line can be partitioned by the number of units indicated by the denominator. For example:
  • To model $$\frac{3}{4}$$ + $$\frac{2}{4}$$ = $$\frac{5}{4}$$ , the number line is partitioned into fourths. Three fourths can be represented as a point and then a vector can be drawn from that point to the right for a distance of two fourths of a unit. The head of the vector is at the point five fourths.
  • To model $$\frac{7}{3}$$ − $$\frac{2}{3}$$ = $$\frac{5}{3}$$, the number line is partitioned into thirds. Seven thirds and two thirds are represented as points. The distance between the two points is five thirds.
• Strategies to add and subtract fractions with unlike denominators depend on the types of fractions that are given. For example:
  • Mental math can be used to create wholes (ones). For example, for $$\frac{1}{2}$$ + $$\frac{3}{4}$$ = $$1 \frac{1}{4}$$, knowing that three fourths is composed of one half and one fourth, the two halves are combined to make one, and then one fourth is added on.
• Equivalent fractions may be created so that both fractions have a common denominator (e.g., $$\frac{2}{3}$$ + $$\frac{1}{2}$$ can be scaled so that both have a denominator of 6, which results in the equivalent expression $$\frac{4}{6}$$ + $$\frac{3}{6}$$). This can be modelled using a double number line.

Note

• Fractions are commonly added and subtracted in everyday life, particularly when using imperial units (inches, feet, pounds, cups, teaspoons). Imperial units are commonly used in construction and cooking.
• Only common units can be added or subtracted, whether adding or subtracting whole numbers, decimals, or fractions. Adding fractions with like denominators is the same as adding anything with like units:
  • 3 apples and 2 apples are 5 apples.
  • 3 fourths and 2 fourths are 5 fourths.
• The numerator in a fraction represents the count of unit fractions. The denominator represents what is being counted (the unit). To add or subtract fractions is to change the total count of units. This is why only the numerator is added or subtracted.
• There are helpful ways to visualize the addition and subtraction of fractions. Drawings, fraction strips, clock models, and rulers in imperial units can be used to generate equivalent fractions and model how these common units can be combined or separated.
• The three types of addition and subtraction situations (see B2.1) also apply to fractions.

• greatest common factor (GCF) for 24 and 56:
  • making a list of common factors:
    • 24: 2, 3, 4, 6, 8, 12
    • 56: 2, 4, 7, 8, 14, 28
    • Common factors: 2, 4, 8
      • GCF: 8
  • using a factor tree:
    • expressing each number as a product of its prime factors:

• identifying the factors they have in common and determining their product:
  • $$\sf \small \enclose{circle}[mathcolor=CornflowerBlue]{\color{black} 2 × 2 × 2}{\color{black} ×\space3 = 24}$$
  • $$\sf \small \enclose{circle}[mathcolor=CornflowerBlue]{\color{black}2 × 2 × 2} {\color{black} ×\space7 = 56}$$
  • $$\sf \small \enclose{circle}[mathcolor=CornflowerBlue]{\color{black}2 × 2 × 2} {\color{black}= 8}$$
  • The GCF for 24 and 56 is 8.
• lowest common multiple (LCM) for 4, 6, and 8:
  • creating a sequential list of multiples for each of the numbers and then identifying the first number that is the same in all the lists:
    • $$\sf \small {\color{black} 4: 4, 8, 12, 16, 20,} \enclose{circle}[mathcolor=CornflowerBlue]{\color{black} 24}{\color{black}, 28, 32, … }$$
    • $$\sf \small {\color{black} 6: 6, 12, 18,} \enclose{circle}[mathcolor=CornflowerBlue]{\color{black} 24}{\color{black}, 30, 36, … }$$
    • $$\sf \small {\color{black} 8: 8, 16,} \enclose{circle}[mathcolor=CornflowerBlue]{\color{black} 24}{\color{black}, 32, 49, … }$$
      • $$\sf \small {\color{black} LCM: 24}$$

• A number can be written in terms of its factors. For example, the factors of 6 are 1, 2, 3, and 6.
• One is a factor for all numbers. Some numbers only have 1 and themselves as factors, and they are called prime numbers (e.g., 3, 5).
• To determine the common factors among two or more numbers, factors are listed and then the common factors are identified, including the greatest one they have in common. For example:
  • The factors of 6 are {1, 2, 3, 6}.
  • The factors of 12 are {1, 2, 3, 4, 6, 12}.
  • The common factors of 6 and 12 are {1, 2, 3, 6}.
  • The greatest common factor of 6 and 12 is 6.
• The multiples of a number are the multiplication facts related to that number (e.g., the multiplication facts for 2 are 2, 4, 6, 8, 10, 12, …).
• To determine the lowest common multiple among two or more numbers, multiples are listed and then the common multiples are identified, including the lowest one they have in common. For example:
  • The multiples of 3 are {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 32, 36 …} and they are all divisible by 3.
  • The multiples of 4 are {4, 8, 12, 16, 20, 24, 28, 32, 36 …} and they are all divisible by 4.
  • The common multiples of 3 and 4 are {12, 24, 36 …}.
  • The lowest common multiple of 3 and 4 is 12.
• The lowest common multiple of a set of numbers is the smallest whole number that divides evenly into all numbers in the set.

Note

• Knowing the greatest common factor among numbers can help with reducing the number of steps to simplify a fraction into its lowest terms. In this case the greatest common factor is being determined for the numerator and the denominator.
• Knowing the lowest common multiple among numbers can help with creating equivalent fractions in order to add or subtract fractions with a common denominator. In this case the lowest common multiple is being determined for all the denominators.

• exponential notation:
  • 2 × 2 × 2 × 2 × 2 = 2⁵:

• Exponentiation is a fifth number operation, like addition, subtraction, multiplication, and division. It is written as bⁿ  and involves two numbers, where b is the base and n is the exponent or power. 
• Exponential notation signifies the multiplication of factors that are all the same, often referred to as repeat multiplication, and known as a power.
• The power has two components: the base and the exponent. The base is the factor that is being repeated, and the exponent states the number of those factors and is written as a superscript. For example: 
  • 5² has a base of 5, an exponent of 2, and represents 5 × 5.
  • 10⁵ has a base of 10, an exponent of 5, and represents 10 × 10 × 10 × 10 × 10.
• To evaluate a power means to determine the result. Often the power would be rewritten as a product to determine its result. For example, 2⁴ = 2 × 2 × 2 × 2 = 16.
• Powers are used to express very large and very small numbers. They are also used to describe very rapid growth (such as doubling) that increases over time. 
• Any number can be written as a power with an exponent 1(e.g., 5 = 5¹).

Note

• The term "power of 10" means the base is 10.
• When a number is expressed in expanded form, the place value is written as a power of ten, which means the base is 10. The exponent is dependent on the place value. For example, 500 = 5 × 10² and 5000 = 5 × 10³.
• Exponential notation can also apply to variables, such as in a formula. For example, in the formula for the area of a circle,A = πr², the r² means r × r.
• Using patterns can help with understanding the relationship between the exponents of the same base.

Have students use a calculator or a spreadsheet to determine the powers of 2 and record them in a list, expressing them as repeated multiplication and as a total: 2¹ = 2, 2² = 2 × 2 = 4, 2³ = 2 × 2 × 2 = 8, …. Draw students’ attention to the rate of growth that occurs during exponentiation. Use this context as the basis for posing a question:

• If you could have $10 million now, or be given an amount equal to one nickel doubled every day for the next 30 days, which would you pick?

Have students share their choice and their reasoning. Discuss how this problem could be represented as a power and as repeated multiplication. Guide them to see that for the second choice:

• on day 1, the amount is 0.05 
• on day 2, that amount is doubled to 0.05 × 2 (or 0.05 × 2¹)
• on day 3, the day 2 amount is doubled to 0.05 × 2 × 2 (or 0.05 × 2²), and so on.

Have students predict the amount on Day 30 and then use a calculator or spreadsheet to determine the answer. Support them in using a calculator to show 0.05 × 2²⁹ by using repeated multiplication (i.e., 2 × 0.05 = = = =) or the exponent key (i.e., [0.05] [×] [2] [x^y] [29] [=]). Have them write their exponentiation as a power (2²⁹) and as a number (i.e., 0.05 × 2²⁹ = 26 843 545.60. Ask them to express the number in expanded form using powers of 10 (see B1.1).

• multiplying a fraction by a fraction using an area model:
  • $$\frac{1}{3}$$ × $$\frac{1}{5}$$ = $$\frac{1}{15}$$ 

• dividing a fraction by a fraction using a number line:
  • $$\frac{3}{4}$$ ÷ $$\frac{1}{2}$$ = $$1 \frac{1}{2}$$

• dividing a fraction by a fraction using an area model:
• $$\frac{3}{4}$$ ÷ $$\frac{1}{2}$$ = $$1 \frac{1}{2}$$

• The multiplication and division of two fractions can be interpreted based on the different ways fractions are used: as a quotient, as parts of a whole, as a comparison (ratio), and as an operator. 
• The multiplication of two fractions as operators can be modelled as follows:
  • For $$\frac{1}{2}$$ × 1, the fraction one half as an operator can visually be shown as one half of a rectangle.

• Therefore $$\frac{1}{2}$$ × $$\frac{1}{2}$$, is one half of the one half of a rectangle. The result is one fourth of a rectangle.

• Division of fractions can be interpreted in two ways:
  • 4 ÷ $$\frac{1}{2}$$  = ? can be interpreted as “How many one halves are in four?”

• Two one halves make 1, so eight one halves make 4. Therefore, 4 ÷ $$\frac{1}{2}$$  = 8.

• 4 ÷ $$\frac{1}{2}$$ = ? can also be interpreted as “If 4 is one half of a number, what is the number?”

• Since 4 is one half of a number, the other one half is also 4. Therefore, 4 ÷ $$\frac{1}{2}$$ = 8

• Division of a fraction by its unit fraction (e.g., $$\frac{5}{8}$$ ÷ $$\frac{1}{8}$$) can be interpreted as "How many counts of the unit are in the fraction (i.e., how many one eighths are in five eighths)?" The result is the number of counts (e.g., there are 5 counts of one eighth).
• Dividing a fraction by a fraction with the same denominator (e.g., $$\frac{6}{8}$$ ÷ $$\frac{2}{8}$$)  can be interpreted as “How many divisors are in the dividend?” In the fraction strip below, notice there are three counts of two eighths that are in six eighths. Similar to the division of a fraction by its unit fraction, the result is the count.

• Sometimes the division of a fraction by a fraction with the same denominator has a fractional result. For example: $$\frac{5}{8}$$ ÷ $$\frac{2}{8}$$.

• Notice there are 2 two eighths in five eighths, and then $$\frac{1}{2}$$ of another two eighths. 
• Therefore, $$\frac{5}{8}$$ ÷ $$\frac{2}{8}$$ = $$2 \frac{1}{2}$$.

Note

• When multiplying a fraction by a fraction using the area of a rectangle, first the rectangle is partitioned horizontally or vertically into the same number of sections as one of the denominators. Next, the region represented by that fraction is shaded to show that fraction of a rectangle. Next, the shaded section of the rectangle is partitioned in the other direction into the same number of sections as the denominator of the second fraction. Now it is possible to identify the portion of the shaded area that is represented by that fraction.
• Any whole number can be written as a fraction with one as its denominator. A whole number divided by a fraction can be used to support students in understanding the two ways division can be interpreted. If context is given, usually only one or the other way is needed. Dividing a whole number by a fraction also helps with making connections to thinking about division of a fraction as the multiplication of its reciprocal.
• In general, dividing fractions with the same denominator can be determined by dividing the numerators and dividing the denominators.
• Multiplying fractions follows a developmental progression that may be helpful in structuring tasks for this grade:
  • A fraction by a whole number (e.g., 5 × $$\frac{3}{8}$$; 5 groups of $$\frac{3}{8}$$).
  • A whole number by a fraction (e.g., $$\frac{3}{4}$$ × 24; multiplication as scaling).
  • A fraction by a fraction, no partitioning (e.g., $$\frac{1}{3}$$ × $$\frac{3}{4}$$; multiplication as scaling).
  • A fraction by a fraction, with partitioning (e.g., $$\frac{2}{3}$$ × $$\frac{3}{4}$$; $$\frac{2}{3}$$ × $$\frac{4}{5}$$).
• Dividing fractions also follows a developmental progression that may be helpful in structuring a task for this grade:
  • A whole number divided by a whole number (e.g., 8 ÷ 3).
  • A fraction divided by a whole number (e.g., $$\frac{3}{4}$$ ÷ 2).
  • A whole number divided by a unit fraction (e.g., 5 ÷ $$\frac{1}{3}$$).
  • A whole number divided by a fraction (e.g., 5 ÷ $$\frac{2}{3}$$).
  • A fraction divided by a unit fraction (e.g., $$\frac{7}{8}$$ ÷ $$\frac{1}{8}$$).
  • A fraction divided by a fraction, with the same denominator and a result that is a whole number (e.g., $$\frac{4}{5}$$ ÷ $$\frac{2}{5}$$).
  • A fraction divided by a fraction, with the same denominator and a result that is a fractional amount (e.g., $$\frac{3}{4}$$ ÷ $$\frac{1}{2}$$ ; $$\frac{1}{2}$$ ÷ $$\frac{3}{4}$$).

• multiplying decimal numbers using an area model:
  • 6.3 × 2.8:
    • decompose the numbers into their whole and decimal parts and create appropriate sections in a rectangle
    • determine the product of the numbers describing the length and width of each section, that is, the area of each section, and then calculate the sum of the products:

• multiplying decimal numbers using an algorithm:
  • 7.1 × 8.5:
    • “I know that 7 × 8 = 56 and that 8 × 8 = 64, so the answer must be a number in between.”

• “Knowing that my answer must be between 56 and 64, it must be 60.35.”
• dividing decimals by creating whole numbers:
  • 15.4 ÷ 0.04:
    • multiplying both numbers by the same amount:
    • 15.4 ÷ 0.04
      = (15.4 × 100) ÷ (0.04 × 100)
      = 1540 ÷ 4
      = 385

• Any decimal number multiplied by one is that decimal number.
• One tenth × one tenth results in a hundredths product (0.1 × 0.1 = 0.01), similar to 10 × 10 = 100.
• One tenth × one hundredth results in a thousandths product (0.1 × 0.01 = 0.001), similar to 10 × 100 = 1000.
• One hundredth × one hundredth results in a ten thousandths product (0.01 × 0.01 = 0.0001), similar to 100 × 100 = 10 000.
• A strategy to multiply decimal numbers is to decompose them as a product of whole numbers with tenths, hundredths, or thousandths and then apply the associative property. For example:
  • 23.5 × 0.03

          = 235 × 0.1 × 3 × 0.01

          = 705 × 0.001

          = 0.705

• Sometimes a combination of words and numbers may be helpful, such as:
  • 23.5 × 0.03

          = 235 tenths x 3 hundredths

          = 705 thousandths or 0.705

• The area model can be used to multiply decimal numbers. The decimal numbers can represent the dimensions of a rectangle. Each dimension can be decomposed into its place value, and then the area of each of the sections formed can be determined. For example:
  • 23.5 × 0.3 can be decomposed as 23 and 5 tenths, and 3 tenths. The partitions that result are 23 by 0.3 (6.9), and 5 tenths by 3 tenths (0.15). The total area is 6.9 + 0.15 = 7.05.

• Standard multiplication algorithms for whole numbers can also be applied to decimal numbers. As with whole numbers, these algorithms add partial products to create a total. For example, with 23.5 × 0.3, the partial products are formed by multiplying each digit according to its place value by three tenths.

• To divide decimal numbers, an equivalent division statement with a whole number divisor can be used since the results will be the same. For example: 70.5 ÷ 0.5 = 705 ÷ 5 when both the dividend and the divisor are scaled by 10, and 705 ÷ 5 = 141. In some cases, mental calculations can be used to determine the result and at other times the standard algorithm may be applied.
• Estimating a product or quotient prior to a calculation helps in assessing whether a calculation is reasonable.

Note

• Support students in making connections between the area model and the standard algorithms for multiplication.
• Depending on the context, the multiplication of a decimal number may be relative to a scale factor, a measurement, or a partial group.
• Division by decimal amounts disrupts the notion that division “makes numbers smaller”. The question “How many tenths can be made from 3?” (i.e., 3 ÷ 0.1) will produce an answer that is larger than three – in fact, it will be ten times as large. As with fractions and measurement, the smaller the unit, the greater the count.

Have students play the Target Game in pairs, sharing one calculator. Player 1 enters any number on the calculator. Player 2 has to multiply this number by another to try to get as close as possible to the target number of 100. Player 1 then multiplies the new number by another number to get nearer to 100. The player who gets closest to 100 after a given number of rounds is the winner.

• proportional situations:
  • the length of the diameter is twice the length of the radius of a circle
  • conversions between metric units (e.g., km to cm)
  • constant rates (e.g., driving at a steady speed for a duration of time)
  • quantities that are related multiplicatively
  • equivalent ratios (e.g., 1 : 2 is equivalent to 2 : 4 and 3 : 6) 
  • equivalent fractions (e.g., $$\frac{1}{2}$$, $$\frac{2}{4}$$, $$\frac{3}{6}$$)
• non-proportional situations:
  • the relationship between the diagonal and the perimeter of a rectangle
  • fluctuating rates (e.g., running at different speeds in a race)
  • a growing pattern that increases from one term to the next by different values
  • a shrinking pattern that decreases from one term to the next by different values

• A proportional relationship is when two variables change at the same rate. For example, depositing $5 into a savings account every month is a proportional situation because the relationship between months and money is constant: $5 per month. Note that the change is additive ($5 more per month), but the relationship is multiplicative ($5 per month):

• A non-proportional relationship is when two variables do not change at the same rate. For example, a deposit of $5 one month and $2 the next is not proportional because the growth is not constant. The line on a graph would be jagged, not straight.
• Tables and graphs are helpful for seeing proportional (or non-proportional) relationships. 
• A proportion is a statement that equates two proportions (ratios, rates): $$\frac{a}{b}$$ = $$\frac{c}{d}$$. There are four ways that the proportion can be written for it to hold true. For example, 3 km for every 5 hours and 6 km for every 10 hours can be expressed as:
  • $$\frac{3}{5}$$ = $$\frac{6}{10}$$ or
  • $$\frac{3}{6}$$ = $$\frac{5}{10}$$ or
  • $$\frac{5}{3}$$ = $$\frac{10}{6}$$ or
  • $$\frac{6}{3}$$ = $$\frac{10}{5}$$.
• Problems involving proportional relationships can be solved in a variety of ways, including using a table of values, a graph, a ratio table, a proportion, and scale factors.

Note

• Problems that involve proportions with whole numbers provide an opportunity to apply mental calculations that use multiplication and division facts. For example, to solve for m, in a proportion like $$\frac{m}{9}$$ = $$\frac{12}{54}$$, one could determine the multiple of 9 that gives 54 and then use that to divide 12 by to find m.
• If two quantities change at the same rate, the quantities are proportional.

Have students decide if the pricing on two products is proportional or non-proportional:

• A six-pack of pens costs $4.59. You found the same product for sale loose in a bin, priced as five for $3.89. Is one price better than the other?

Record students’ strategies as you consolidate the problem and draw out the usefulness of finding the unit rate (i.e., the price for one pen) and multiplying or scaling it by five to find the cost of five pens.

Pose a follow-up question to draw out a different comparison strategy:

• A six-pack of paper towels costs $4.59, but the store also has a two-pack of the same paper towels on sale for $1.39. Are these prices proportional? How do you know?

Students can determine the price for one roll, two rolls, or three rolls to show proportionality. Ask them to make conclusions about which option is the better deal.

Other possible contexts include:

• If six bananas cost $2.40, what should the cost of five bananas be?
• Determine how much four ride tickets cost at a fair if six ride tickets cost $8.80.
• If three apples cost $1.98, how much do six apples cost? How much do seven apples cost?
• If two shirts cost $30.50, how much does one shirt cost? How much would you need to pay for four shirts? Five shirts?
• If five graduation tickets cost $35.00, how many could you buy with $14.00?
• You get six pieces of sugar-free gum in a package that costs $0.87. If you could buy a package with three pieces of gum, how much would it cost? How much would you need to pay for a package of eight?