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🦘 Kangaroo Grade 3 · session 1 of 24 · 60 minutes · free

Place value and digit puzzles

Kangaroo questions love digits: make the biggest number, count how many digits a book needs for its page numbers, find the missing digit. Learn what each place is worth and the tricks follow.

🎯 Practice mode🖨 Print a worksheet

A complete lesson · 60 minutes

Teach it, see it, practise it, all on this page.

Open this page with your learner and work down it, or press Teach this lesson to go through it one step at a time on a big screen. Every section has something to do.

See the teacher's plan
  1. Warm-up4 min
  2. Place value8 min
  3. Biggest and smallest9 min
  4. Digit sums7 min
  5. Missing digits7 min
  6. Activity 17 min
  7. Activity 27 min
  8. Activity 36 min
  9. Quiz4 min
  10. Set the project1 min

By the end you can

  • Say what each digit is worth in a 3- and 4-digit number
  • Make the biggest and smallest numbers from digit cards
  • Count digits when numbers are written in a row
  • Solve missing-digit sums

Before you start

  • Reading and writing numbers to 1,000
  • Adding 2-digit numbers

Have ready

  • Paper and pencil
  • Digit cards 0 to 9 (paper squares are fine)

Grade 3 · ages 8 to 9 · contest level 3–4

Warm-up · think first

Using the digits 4, 0 and 7 once each, how many different 3-digit numbers can you make?

Show the answer

4 numbers: 407, 470, 704 and 740. A 3-digit number cannot start with 0, so only 4 or 7 can go first.

Teacher tip: Most learners say 6. Ask them to write every number down and check which ones are really 3-digit numbers.

More warm-ups for every class →

Part 1 of 4 · learn, see, practise

Hundreds, tens and ones

Every digit has a value that depends on its place. In 374 the 3 is worth 300, the 7 is worth 70 and the 4 is worth 4.

A 0 keeps a place empty: 609 has 6 hundreds, 0 tens and 9 ones. Ten ones make a ten and ten tens make a hundred, so 15 tens is the same as 1 hundred and 5 tens.

Example 1

What number has 6 hundreds, 0 tens and 9 ones?

  1. 6 hundreds is 600, 0 tens is 0 and 9 ones is 9.
  2. 600 + 0 + 9 = 609. The 0 holds the tens place.

Answer: 609

Example 2

What number is 3 hundreds, 15 tens and 4 ones?

  1. 15 tens is 150, which is 1 hundred and 5 tens.
  2. 300 + 150 + 4 = 454.

Answer: 454

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up

Warm-up

Core

Core

Stretch 🔥

Stretch 🔥

Answers (for the teacher)
  1. 528
  2. 70
  3. 523
  4. 405
  5. 447
  6. 100

Part 2 of 4 · learn, see, practise

The biggest and the smallest number

To make the biggest number from some digits, put the biggest digit in the biggest place, then the next biggest, and so on.

To make the smallest, put the smallest digit first, but never 0: a number cannot start with 0. Put the 0 straight after the first digit.

Example 1

Make the biggest and the smallest 3-digit numbers from 5, 2 and 8.

  1. Biggest: 8 first, then 5, then 2: 852.
  2. Smallest: 2 first, then 5, then 8: 258.

Answer: 852 and 258

Example 2

Make the smallest 3-digit number from 0, 6 and 3.

  1. 0 cannot go first, so the smallest digit that can is 3.
  2. Then 0, then 6: 306.

Answer: 306

Digit cards

Tap the cards to place them in order. Tap a placed card to take it back.

  1. Make the biggest 4-digit number.
  2. Make the smallest 4-digit number.
  3. Make the biggest odd 4-digit number.

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up

Warm-up

Core

Core

Stretch 🔥

Stretch 🔥

Answers (for the teacher)
  1. 941
  2. 268
  3. 1059
  4. 427
  5. 73
  6. 522

Part 3 of 4 · learn, see, practise

Digit sums and counting digits

The digit sum of a number is what its digits add up to: the digit sum of 507 is 5 + 0 + 7 = 12.

To count the digits needed to write many numbers, split them into groups: 1 to 9 use one digit each, 10 to 99 use two digits each.

Example 1

What is the digit sum of 507?

  1. Add the digits: 5 + 0 + 7.
  2. That is 12.

Answer: 12

Example 2

How many digits do you write to number the pages of a book from 1 to 25?

  1. Pages 1 to 9: 9 pages with one digit each, 9 digits.
  2. Pages 10 to 25: 16 pages with two digits each, 32 digits.
  3. 9 + 32 = 41.

Answer: 41

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up

Warm-up

Core

Core

Stretch 🔥

Stretch 🔥

Answers (for the teacher)
  1. 15
  2. 90
  3. 51
  4. 39
  5. 15
  6. 910

Part 4 of 4 · learn, see, practise

Missing digits

In a sum with missing digits, work one column at a time, starting with the ones. Remember a carry when a column adds to 10 or more.

When the same digit hides in two places, try the digits 0 to 9 and check. Kangaroo answers are only five choices, so testing each choice is a smart move too.

Example 1

2☐ + ☐5 = 69. The same digit goes in both boxes. What is it?

  1. Ones: ☐ + 5 ends in 9, so ☐ is 4.
  2. Check: 24 + 45 = 69.

Answer: 4

Example 2

A 2-digit number has a tens digit 3 more than its ones digit, and its digit sum is 11. What is it?

  1. Try ones digits: 4 gives a tens digit of 7, and 7 + 4 = 11.
  2. Check: 74 has tens digit 7, which is 3 more than 4.

Answer: 74

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up

Warm-up

Core

Core

Stretch 🔥

Stretch 🔥

Answers (for the teacher)
  1. 6
  2. 6
  3. 63
  4. 4
  5. 8
  6. 92

Watch out

Common mistakes

Starting a number with 0

0 can never be the first digit. 0378 is really the 3-digit number 378.

Mixing up a digit and its value

In 374 the digit is 7 but its value is 70. Questions ask for one or the other: read carefully.

Counting pages, not digits

Page 25 needs two digits. Count the 1-digit and 2-digit pages separately.

Class activity 1 of 3

Digit card challenge

Use the digit cards 2, 5, 8 and 1, each once, to make 4-digit numbers.

  1. For the biggest, put the biggest digit first.
  2. For even or odd, choose the last digit first, then make the rest as big or as small as possible.
  3. For the closest number, try numbers just above and just below the target.

Answers (for the teacher)
  1. 8521
  2. 1258
  3. 8512
  4. 1285
  5. 5128

Class activity 2 of 3

Page number detective

A book's pages are numbered 1, 2, 3, … with no gaps. Count the digits the printer needs.

  1. Count 1-digit pages, 2-digit pages and 3-digit pages separately.
  2. Multiply each group by the number of digits it uses.
  3. For the last two, work backwards from the digits.

Answers (for the teacher)
  1. 9
  2. 189
  3. 252
  4. 24
  5. 20

Class activity 3 of 3

Secret number clues

Each set of clues fits exactly one number. Find it.

  1. Write what each clue tells you.
  2. Use the strongest clue first to make a short list.
  3. Check your number against every clue.

Answers (for the teacher)
  1. 436
  2. 52
  3. 666
  4. 126

After class · project

My digit puzzle cards

Make six puzzle cards about digits for someone at home, with the answer on the back.

What to make: Six cards with a puzzle on the front and the answer and working on the back.

Stretch: Make a clue puzzle whose answer is a 4-digit number and check that only one number fits.

How it is marked

Look forDone well when
VarietySix different kinds of puzzle
CorrectEvery answer is right
WorkingThe back of each card shows how to get the answer

Project template

Copy it onto paper or print the practice sheet.

My digits: ___ ___ ___ ___
Card 1 (biggest): ____________________   Answer: ______
Card 2 (smallest): ___________________   Answer: ______
Card 3 (digit sum): __________________   Answer: ______
Card 4 (pages): _____________________   Answer: ______
Card 5 (secret number clues): ______________________
Card 6 (secret number clues): ______________________

Homework

Practise at home

Try each on paper, then tap to see the answer and working. Or print the practice sheet.

  1. What is the biggest 4-digit number with four different digits?

    9876

  2. How many digits are needed to number the pages 1 to 50?

    91 (9 one-digit pages and 41 two-digit pages: 9 + 82)

  3. What is the smallest 3-digit number with digit sum 20?

    299 (a hundreds digit of 1 would need 19 from two digits, which is too much)

  4. How many 3-digit numbers can be made from 1, 2 and 3 using each once?

    6: 123, 132, 213, 231, 312, 321

  5. Swapping the digits of a 2-digit number makes it 27 bigger, and its digit sum is 9. What is the number?

    36 (63 − 36 = 27)

🔥 Challenge round · finished early?

The hardest questions of the lesson, no notes

Cover the notes and try these from memory. They are meant to make you think hard: draw, try small cases, and check your answer.

Stretch 🔥

Stretch 🔥

Stretch 🔥

Stretch 🔥

Stretch 🔥

Stretch 🔥

Stretch 🔥

Stretch 🔥

Answers (for the teacher)
  1. 447
  2. 100
  3. 73
  4. 522
  5. 15
  6. 910
  7. 8
  8. 92

🧠 Think-hard question

How many digits are needed to number the pages 1 to 1,000?

Talk it through or use the drawing board. There is more than one way to get there.

🎯 Practice mode: one question at a time🧩 Brain warm-ups🎮 Maths and logic games

Quiz · 8 questions

Final quiz: check what you learned

Pick an answer to see if it is right and why. Your best score is saved on this device.

1Which number is 7 hundreds, 3 tens and 12 ones?
Show the answer

742. 12 ones is 1 ten and 2 ones, so 700 + 30 + 12 = 742.

2Using 3, 0, 6 and 9 once each, what is the smallest 4-digit number?
Show the answer

3069. 0 cannot go first, so start with 3, then 0, 6, 9.

3How many digits are needed to write all the numbers from 1 to 20?
Show the answer

31. 1 to 9 use 9 digits, 10 to 20 are 11 numbers with 2 digits each: 9 + 22 = 31.

4What is the biggest 3-digit number whose digits add up to 5?
Show the answer

500. Put as much as possible in the hundreds: 5 + 0 + 0 = 5.

52☐ + ☐2 = 55, with the same digit in both boxes. What is the digit?
Show the answer

3. 23 + 32 = 55.

6How many 2-digit numbers have both digits the same?
Show the answer

9. 11, 22, 33, …, 99: nine numbers.

7Kanga writes the numbers from 1 to 30. How many times does she write the digit 2?
Show the answer

13. In the ones place: 2, 12, 22 (3 times). In the tens place: 20 to 29 (10 times). 3 + 10 = 13.

8The digits of a 3-digit number add up to 3. How many such numbers are there?
Show the answer

6. 300, 210, 201, 120, 111 and 102: six numbers.

🖨 Print what we learned

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For parents and teachers

The 60-minute lesson plan

Use it as it is, or stretch any part over two sessions. Each row links to the part of the page you need.

TimePartThe teacherThe learner
0–4 minWarm-upAsk the warm-up question and wait for a guess before any hint.Guess, then explain.
4–12 minPlace valueExplain, reveal the examples one step at a time, then the practice.Practise up to the stretch question.
12–21 minBiggest and smallestHand out digit cards, then use the on-screen cards for the tasks.Practise; try the stretch.
21–28 minDigit sumsExplain and show the examples.Practise.
28–35 minMissing digitsExplain and show the examples.Practise; the stretch can be homework.
35–42 minActivity 1Let the learner work alone; hint only after a wrong check.Solve and check.
42–49 minActivity 2Do the first item together, then let them finish.Finish the activity.
49–55 minActivity 3Set as homework if time is short.Solve the puzzles.
55–59 minQuizRun the quiz like the contest: no calculator, pick one of five answers.Answer and read why.
59–60 minSet the projectShow the template and agree a due date.Start the project.

Questions to ask

  • Why can't a number start with 0?
  • How do you know which place a digit should go in for the biggest number?
  • Why do we count 1-digit and 2-digit pages separately?

If it feels hard

Use real digit cards and a place-value chart (hundreds, tens, ones) for every question.

If it feels easy

How many digits are needed to number the pages 1 to 1,000?

FAQ

Place value and digit puzzles: questions

What place value questions come in Math Kangaroo for Grade 3?

Making the biggest or smallest number from digits, digit sums, counting digits in page numbers and missing digits in sums. This lesson practises each kind.

How do you find the smallest number from digits that include 0?

Put the smallest digit that is not 0 first, then the 0, then the rest from smallest to biggest: from 0, 6 and 3 you get 306.

Is this lesson free?

Yes, the first two lessons of the course are free to teach, with the practice sheet.

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