Mr. Ward's Teaching Materials

001i - Place Value and Expanded Notation

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Lesson 1 — Place Value and Expanded Notation

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

Learning Intentions

  • To understand that the value of a digit depends on its position within a numeral.
  • To represent whole numbers in expanded notation using place value.

Success Criteria

I can:

  1. State the place value of any digit in a numeral up to the millions.
  2. Write a whole number in expanded notation as a sum of place value parts, e.g. .
  3. Reconstruct a numeral from its expanded form.
  4. Explain why and are different values despite using the same digit.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

Display the numeral .

  1. How many digits does this number have?
  2. Read it aloud. Where do the spaces (or commas) fall, and why?
  3. Point to the digit . What is it worth?
  4. Point to the digit . Is it doing any work? Convince me.

Teacher note: Question 4 is the hook. Ask two students with opposing answers to justify. Do not resolve it yet — return to it in the Checks for Understanding.

Activities

Activity 1 — Explicit Instruction: the Place Value Chart (10 min)

Model on the board using a labelled place value chart:

MillionsHundred ThousandsTen ThousandsThousandsHundredsTensOnes
6357204

I do: Write into the chart. Say aloud: “The 3 sits in the hundred thousands column, so its value is .”

Model the full expansion:

Draw attention explicitly: the zero term contributes nothing to the sum, but the zero digit is essential to hold the column.

We do: Together expand and .

You do: Students expand , , , .

Activity 2 — Reversing the Process (10 min)

Students reconstruct numerals from expanded form. Include deliberately scrambled and gap-containing examples:

  2. (out of order)
  3. (gaps in several columns)
  4. (introduces multiplicative form, previewing Lesson 2)

Activity 3 — Inquiry Task: the Digit Swap (12 min)

Pairs, then whole-class share.

Start with the number . You may swap any two of its digits.

  • Which swap makes the number as large as possible?
  • Which swap makes it as small as possible?
  • By how much does each swap change the number?
  • Can you predict the change before you calculate it?

Extend to a four-digit number, e.g. .

Socratic scaffolding for pairs who stall on “predict before calculating”:

PromptPurpose
What are you trying to find?Restate the goal: the difference caused by a swap, not the new number.
What do you know?Only two digits move; the others keep their place value.
Have you seen a related problem?Compare with the warmup: how much is a digit worth in each column?
Can you try a simpler case?Swap the digits of a two-digit number, e.g. . What changed?
What stays the same?The two digits themselves. Only the columns they occupy change.
Can you write that as an expression?If digits and swap between the hundreds and tens columns, the change is becoming .
Looking back — does it generalise?The difference is . Test it against your earlier results.

For a swap between the hundreds and the tens in (, ):

Check: . ✓

Checks for Understanding

(6 minutes — exit ticket, collected)

  1. Write in expanded notation.
  2. In the number , what is the value of the digit ?
  3. True or false, with a reason: “In , the zero can be left out because it is worth nothing.”
  4. Two students expand . Ali writes . Bree writes . Who is correct? Explain.

Answers: 1. ; 2. ; 3. False — the zero holds the tens column; removing it gives ; 4. Both are mathematically correct, since changes nothing. Bree’s form makes the empty column visible; Ali’s is conventional.

Common Misconceptions

MisconceptionHow to pre-empt it
”The digit is three.”Insist on the phrase “the digit 3 has a value of 300” throughout modelling. Never say “the 3” alone.
Zero is optional because it is worth nothing.Contrast , and side by side in a place value chart. Zero is a placeholder, not a quantity here.
Longer numeral means larger number.Compare style leading zeros and ; also compare and .
Reading as “six three five seven two zero four”.Model reading in groups of three from the right; practise chorally.
Expanded form must be in descending order.Show and ask whether the sum has changed. Reinforce commutativity of addition.

Enrichment — Competition-Style Problems

Suitable for early finishers or a challenge board. Styles drawn from the Australian Mathematics Competition (AMC) Junior division and the Mathematical Kangaroo.

E1 (Kangaroo style). A four-digit number uses each of the digits , , and exactly once. What is the difference between the largest and smallest such numbers?

Answer

(Students may enjoy knowing is Kaprekar’s constant.)

E2 (AMC Junior style). In a three-digit number, the hundreds digit is twice the units digit, and the tens digit is . How many such numbers are there?

Answer

The units digit must satisfy and , so , giving , , , — 4 numbers.

E3 (Extension). The digits of a two-digit number are reversed, and the new number is larger than the original. List all possible original numbers.

Answer

Let the number be . Then:

With and : , so the numbers are , , , .

Homework

  1. Write each number in expanded notation: (a) (b) (c) (d) .
  2. Write the numeral for each: (a) (b) (c) .
  3. State the value of the underlined digit: (a) 45,921 (b) 7,308,000 (c) 620,145.
  4. Arrange in ascending order: , , , , .
  5. Reasoning. Explain, using place value, why is one thousand times larger than .
  6. Challenge. Using the digits , , and exactly once each, write the largest possible four-digit number and the smallest possible four-digit number. (Careful: a number cannot begin with .)

Answers to Q6: largest ; smallest .

Graph View

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