002i - Powers of 10 and Expanded Notation

Lesson 2 — Powers of 10 and Expanded Notation

Strand: Number | Descriptor: AC9M7N03 | Duration: 45 minutes

Learning Intentions

• To understand exponent notation as repeated multiplication.
• To represent whole numbers in expanded notation using powers of .

Success Criteria

I can:

1. Write a power of in exponent form and evaluate it, e.g. .
2. Explain the relationship between the exponent and the number of zeros.
3. Write a numeral in expanded notation using powers of , e.g. .
4. Convert between expanded power-of-10 form and standard numeral form.

Warmup

(5 minutes — chorus response, then mini whiteboards)

Complete the pattern on the board, one row at a time:

Product	Value	Number of tens multiplied
		1
		2
		3
	?	?
?		?

1. What do you notice about the number of zeros?
2. Predict: how many zeros will (seven tens) have?
3. Is there a shorter way to write “seven tens multiplied together”?

Activities

Activity 1 — Explicit Instruction: Exponent Notation (10 min)

Define the notation precisely, naming each part:

• is the base.
• is the exponent (or index/power).
• is read “ten to the power of four”.

Build the reference table with the class:

Exponent form	Expanded product	Value	Place value name
	—		ones
			tens
			hundreds
			thousands
			ten thousands
	five factors		hundred thousands
	six factors		millions

Justifying . Do not assert it — derive it by descending the pattern. Each step down divides by :

Warning to state aloud: . The exponent counts factors, not a multiplier.

Activity 2 — Expanded Notation with Powers of 10 (12 min)

I do: Model .

Emphasise the three-stage progression: place value sum → multiplicative form → exponent form. Students should show all three lines initially.

We do: and .

You do: Express in expanded form using powers of :

Then reverse: write as a single numeral.

Activity 3 — Inquiry: how Big is a Million? (10 min)

Small groups, calculator permitted.

A student claims: “A million seconds and a billion seconds are basically the same — they’re both just a lot.”

Investigate. About how long is seconds? About how long is seconds? Express each in a unit that makes the size feel real.

Socratic scaffolding:

Prompt	Purpose
What is the unknown?	A duration expressed in a familiar unit — days or years, not seconds.
What are you given?	seconds and seconds.
What connects seconds to days?	seconds per minute, minutes per hour, hours per day.
Can you do it in stages?	Seconds → minutes → hours → days → years. Do one conversion at a time.
Devise a plan	Divide by , then , then , then about .
Carry it out	See below.
Looking back	Is the answer reasonable? How many times bigger is than ? Does the ratio match your two answers?

Key discussion point: is not “a bit more” than — it is times larger. Roughly 12 days versus 32 years.

Checks for Understanding

(5 minutes — exit ticket)

1. Evaluate .
2. Write in expanded notation using powers of .
3. Write as a numeral: .
4. A student writes . Identify the error and correct it.
5. How many times larger is than ? Explain without a calculator.

Answers: 1. ; 2. ; 3. ; 4. The student multiplied the base by the exponent; the exponent counts factors, so ; 5. times, since — three extra factors of ten.

Common Misconceptions

Misconception	How to pre-empt it
(multiplying base by exponent).	Always read exponent form aloud as the full product before evaluating. Insist on the middle line of working.
”The exponent equals the number of zeros” applied to any base, e.g. has 4 zeros.	State clearly the zero rule holds only for base 10, and demonstrate .
.	Derive it via the descending division pattern rather than asserting it. Revisit whenever it appears.
Omitting the : writing for .	Insist on the multiplication sign in every line of written work.
Losing a column when a digit is zero, e.g. miscounted as .	Require students to write the full place-value line first, including the zero terms, before condensing.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). What is the value of ?

Answer

Alternatively .

E2 (AMC Junior style). How many digits does the number have when written out in full?

Answer

is a followed by zeros — digits. Multiplying by gives followed by zeros, still 13 digits.

E3 (Challenge). The number satisfies . What is the sum of the digits of ?

Answer

, so the digit sum is .

E4 (Reasoning challenge). Without evaluating either number, explain which is larger: or .

Answer

, and , so is larger. A single extra power of ten outweighs any single-digit multiplier.

Homework

1. Evaluate: (a) (b) (c) (d) .
2. Write in expanded notation using powers of : (a) (b) (c) (d) .
3. Write as a single numeral: (a) (b) (c) .
4. Complete: (a) (b) (c) .
5. Reasoning. Marcus says . Explain why he is wrong and give the correct value.
6. Challenge. The distance from the Earth to the Sun is roughly km. Written out in full, how many digits would this number have? How many zeros?

Answers: Q5 — adding does not add exponents; . Q6 — has 9 digits and 7 zeros.