B2. Powers
Specific Expectations
Powers
B2.1
analyse, through the use of patterning, the relationship between the sign and size of an exponent and the value of a power, and use this relationship to express numbers in scientific notation and evaluate powers
- relationship between exponent of a power and the value of the power:
- patterns can be used to represent relationships among numbers expressed as powers
- as illustrated in the chart below, when the size of an exponent decreases by 1, the value of the power is divided by the base, in this case, 2:
- patterns can be used to represent relationships among numbers expressed as powers
- as illustrated in the chart below, when the size of an exponent decreases by 1, the value of the power is divided by the base, in this case, a:
- numbers expressed in scientific notation:
- a DNA strand has a width of about 2.5 × 10−9 m:
0.000 000 002 5 = 2.5 × $$\frac{1}{10}$$ × $$\frac{1}{10}$$ × $$\frac{1}{10}$$ × $$\frac{1}{10}$$ × $$\frac{1}{10}$$ × $$\frac{1}{10}$$ × $$\frac{1}{10}$$ × $$\frac{1}{10}$$ × $$\frac{1}{10}$$
= 2.5 × 10−9 m - −4 000 000 000 = −4.0 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10
= −4.0 × 109
- a DNA strand has a width of about 2.5 × 10−9 m:
- evaluating powers:
- powers expressed with rational number bases and integer exponents:
- $$-\left(2^{4}\right)=-(2 \times 2 \times 2 \times 2)$$
$$=-16$$ - $$(-4)^{4}=(-4) \times(-4) \times(-4) \times(-4)$$
$$= 256$$ - $$\left(\frac{1}{3}\right)^{2}=\frac{1}{3} \times \frac{1}{3}$$
$$=\frac{1}{9}$$ - $$2^{-4}=\left(\frac{1}{2}\right)^{4}$$
$$=\frac{1}{16}$$
- $$-\left(2^{4}\right)=-(2 \times 2 \times 2 \times 2)$$
- powers expressed in formulas:
- when the length of one side of the base of a cube = 12.5 cm, then
volume of a cube = (side length)3
= (12.5 cm)3
= 12.5 cm × 12.5 cm × 12.5 cm
= 1953.1 cm3
- when the length of one side of the base of a cube = 12.5 cm, then
- powers expressed with rational number bases and integer exponents:
Teachers can:
- use a variety of strategies, including representing using visual models, patterning, and coding, to build students’ understanding of the relationship between the sign and size of an exponent and the value of a power;
- begin the learning with patterns involving bases that are natural numbers, then extend the learning to bases that are integers, unit fractions, and proper and improper fractions;
- extend a pattern from numeric bases to variable bases to build students’ understanding of how to generalize the relationship between the sign and size of an exponent and the value of a power;
- create opportunities to illustrate why scientific notation can be useful for representing extremely large and extremely small numbers, including negative numbers; for example, instead of writing 0.000 000 002 5 m, it is simpler and more efficient to write 2.5 × 10−9 m.
- What happens to the value of a power when the size of the exponent decreases by 1?
- What happens to the exponent of the power when the value of the power is multiplied by the base of the power? (For example, what happens to the exponent of 74 when you multiply this power by 7?)
- What is the value of a power with an exponent of 0? How do you know?
- How would you show in a different way that 3−2 = $$\frac{1}{9}$$?
- What are some situations where you have noticed numbers written in scientific notation?
- What do you notice when you divide or multiply a number repeatedly by 10? What is the connection to showing this in scientific notation?
- How would you use patterning to write 23.7 × 105 in scientific notation?
Give students different base numbers. Have each student make their own patterning chart like those in the example above.
Have students use powers to form as many expressions equal to 1000 as they can.
Have students determine, using a strategy of their choice, which number is smaller, −4 × 103 or 4 × 10−3.
Have students explain how they would write 0.005 689 and 479 000 in scientific notation.
B2.2
analyse, through the use of patterning, the relationships between the exponents of powers and the operations with powers, and use these relationships to simplify numeric and algebraic expressions
- simplifying operations with powers by examining patterns in the expanded forms of powers:
- product of powers (addition of exponents):
- 35 × 32 = (3 × 3 × 3 × 3 × 3) × (3 × 3) = 3(5 + 2) = 37
- 35 × 3−2 = (3 × 3 × 3 × 3 × 3) × $$\frac{1}{3 \space × \space 3}$$ = 35 + (−2) = 33
- ax × ay = ax + y
- quotient of powers (subtraction of exponents):
- $$\frac{4^{5}}{4^{3}}=\frac{4 \space \times \space 4 \space \times \space 4 \space \times \space 4 \space \times 4}{4 \space \times \space 4 \space \times \space 4}$$
= 4 × 4
= 42
- $$\frac{4^{5}}{4^{3}}=\frac{4 \space \times \space 4 \space \times \space 4 \space \times \space 4 \space \times 4}{4 \space \times \space 4 \space \times \space 4}$$
- power of a power (multiplication of exponents):
- (52)3 = 52 × 52 × 52 = (5 × 5) × (5 × 5) × (5 × 5) = 56
- (ax)y = a(x × y)
- product of powers (addition of exponents):
- numeric expressions:
- expressions composed of rational bases and integer exponents; for example, $$2^{-4} \times 2^{5}$$, $$10^{7} \div 10^{5}$$, (($$\frac{1}{3}$$)2)4, $$\frac{(-2.3)^{5} \space \times \space (-2.3)^{-2}}{(-2.3)^{3}}$$
- algebraic expressions:
- expressions involving integer exponents; for example, x5 × x−2, (y−2)2, (xy2)2, $$\frac{x^{−3} \space \times \space x^{4}}{x^{2}}$$
Teachers can:
- emphasize the reasoning behind the relationships between multiplication, division, and power of a power involving exponents;
- use numeric expressions based on real-life examples to highlight bases that are commonly used (e.g., base 10 in scientific notation);
- organize the learning in stages to develop students’ understanding of the relationships (e.g., begin the learning with exponents that are positive integers and then move to negative integers; begin with simplifying numeric expressions and then move to algebraic expressions);
- consider connecting the learning in this expectation to future learning about domain; for example,
m5 × m−2 does not allow m = 0, since division by 0 is undefined, but simplifying to m3 seems to make m = 0 allowable.
- How does using the expanded forms of powers help you to understand the rules for operations with powers?
- How are 4500 and 21000 the same? How are they different? Can you identify other pairs of powers that exhibit the same characteristics?
- What two powers could you multiply together to get 9−4? Is there another set of two powers that you could use?
- How would you use a patterning approach to simplify 59 divided by 53?
- What strategies would you use to determine a possible way to complete the mathematical statement $$\frac{\left(x^{\square}\right)^{\square}}{x^{\square}}=x $$?
Have students use a patterning approach to show that 32 × 34 = 36.
Have students use a patterning approach to justify that (−0.5)0 = 1.
Give students the answer 49, and have them come up with potential questions.
Have students fill in the blanks to make the statements true:
- a5 × □□ × □□ = □2
- □□ ÷ □3 = b□
- (□□)2 = c□