003i - Problem Solving with Place Value

Lesson 3 — Problem Solving and Consolidation: Place Value and Powers of 10

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

Learning Intentions

• To apply place value and powers of to solve unfamiliar problems.
• To communicate reasoning about the structure of numerals.

Success Criteria

I can:

1. Use place value reasoning to compare and order large numbers.
2. Solve digit puzzles by reasoning about column values.
3. Justify an answer in writing using place value language.
4. Recognise and correct place value errors in another person’s work.

Warmup

(6 minutes — “Find the error”, pairs)

Four students have each made one mistake. Find and correct each.

Student	Work
Ana
Ben
Cleo
Dai	because

Answers: Ana — the is in the hundreds column, so . Ben — the exponent counts factors: . Cleo — the is in the ones column: . Dai — compare the number of digits first; has 7 digits and has 6, so is larger.

Activities

Activity 1 — Ordering and Comparing (8 min)

Explicit protocol, then independent practice.

Model the comparison protocol on the board:

1. Count the digits. More digits (with no leading zeros) means larger.
2. If the digit count is equal, compare from the left, column by column, and stop at the first difference.

We do: Order from smallest to largest.

You do: Three more sets, including one with a decimal-free trap such as .

Activity 2 — Digit Puzzles (12 min)

Pairs. Each puzzle is solved by reasoning about columns, not by trial and error.

Puzzle A. I am a four-digit number. My thousands digit is more than my ones digit. My hundreds digit is . My tens digit is twice my ones digit. My ones digit is . What number am I?

Puzzle B. A three-digit number has digit sum . Its hundreds digit is . Its tens digit is one less than its ones digit. What is the number?

Puzzle C. When a certain two-digit number is subtracted from the number formed by reversing its digits, the result is . List all possibilities.

Socratic scaffolding for Puzzle C:

Prompt	Purpose
Understand: what is the unknown?	A two-digit number — really, two unknown digits.
What is the condition?	Reversed number minus original equals .
Can you name the digits?	Let the tens digit be and the ones digit be .
How do you write the number using place value?	Original ; reversed .
Devise a plan	Write the condition as an equation and simplify.
Carry it out	See below.
Looking back — is the answer complete?	Have you checked every valid digit pair? Must ?
Can you generalise?	Why is the difference always a multiple of ?

With and , the pairs are , giving the numbers , , , , , .

Answers: A — . B — , i.e. (since with gives , , ). C — as above.

Activity 3 — Investigation: the Reversal Difference (10 min)

Extends Puzzle C into a generalisation task.

Take any two-digit number. Reverse its digits. Subtract the smaller from the larger.

• Repeat for at least six different starting numbers.
• What do all your answers have in common?
• Can you prove your conjecture using place value?

Expected conjecture: Every result is a multiple of .

Proof, developed with the class:

Since is a whole number, the difference is always a multiple of .

Extension for fast groups: Try the same with three-digit numbers, e.g. . What is the pattern now? (The difference is always a multiple of : .)

Checks for Understanding

(6 minutes — exit ticket)

1. Order from largest to smallest: , , , .
2. Write in expanded notation using powers of .
3. I am a three-digit number. My hundreds digit is times my tens digit, my tens digit is , and my ones digit is . What number am I?
4. Reasoning. Explain, using place value, why reversing the digits of and subtracting gives a multiple of .

Answers: 1. ; 2. ; 3. ; 4. ; . Reversing swaps a digit worth with one worth , so the change is .

Common Misconceptions

Misconception	How to pre-empt it
Comparing large numbers from the right instead of the left.	Enforce the two-step protocol: count digits first, then compare left to right.
Ignoring leading-zero constraints in digit puzzles (e.g. offering as a three-digit number).	State the constraint explicitly at the start of every puzzle: the leading digit cannot be zero.
Stopping at the first solution when a puzzle has several.	Ask routinely: “Is that the only answer? How do you know?”
Treating a single verified example as a proof.	After the investigation, ask “Have you shown it works for all numbers, or just the ones you tried?” before introducing the algebraic argument.
Sign slips when expanding .	Model the subtraction line-by-line; require the intermediate line before collecting like terms.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A three-digit number is times the sum of its digits. What is the number?

Answer

Let the number be with digit sum . Then . Since , ; and since , . Checking multiples of from to where holds: gives and . Systematically, with three-digit requires ; testing each: (digit sum , no); (digit sum , no); (digit sum , no); (digit sum , no); (digit sum , no). Testing all from 15 to 27 yields no solution — there is no such three-digit number. Use this as a discussion of when a problem has no solution and how to prove it.

E2 (Kangaroo style). How many three-digit numbers have all three digits different and are made only from the digits , , and ?

Answer

choices for the hundreds digit, for the tens, for the ones:

E3 (Challenge). The number (a two-digit number with digits and ) satisfies . Find all possible values of .

Answer

So always, giving the pairs and their reverses.

Homework

1. Order from smallest to largest: , , , , .
2. Write in expanded notation using powers of : (a) (b) .
3. I am a four-digit number. My thousands digit is , my hundreds digit is half my thousands digit, my tens digit is , and my ones digit is more than my hundreds digit. What am I?
4. Find all two-digit numbers where reversing the digits increases the number by .
5. Reasoning. Priya says ” is five hundred times bigger than .” Is she right? Show your reasoning using powers of .
6. Challenge. Using each of the digits , , , , exactly once, form a five-digit number that is as close as possible to . Explain how you know it is closest.

Answers: Q3 — . Q4 — . Q5 — Yes: . Q6 — (difference ); compare against (difference ) — in fact is closer, so the answer is . Encourage students to check both sides of .