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Contents
Introduction
1. Odd and even numbers
Assessment task 1
2. Counting techniques: Consecutive numbers
3. Counting techniques: How many digits?
Assessment task 2
4. Number chains
5. The Pigeonhole Principle – 1
6. The Pigeonhole Principle – 2
Assessment task 3
7. Divisibility
8. Revision questions
9. Challenge questions
10. Problem solving without a calculator – 1
11. Problem solving without a calculator – 2
12. Prime and composite numbers
13. Problem solving without a calculator – 3
14. Problem solving: Working backwards
15. Logical questions: Balancing scales
16. Super operations
Assessment task 4
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Answer keys
1. Odd and even numbers
2. Counting techniques: Consecutive numbers
3. Counting techniques: How many digits?
4. Number chains
5. The Pigeonhole Principle – 1
6. The Pigeonhole Principle – 2
7. Divisibility
8. Revision questions
10. Problem solving without a calculator – 1
11. Problem solving without a calculator – 2
12. Prime and composite numbers
13. Problem solving without a calculator – 3
14. Problem solving: Working backwards
15. Logical questions: Balancing scales
16. Super operations
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52
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Answer keys to the assessment tasks
Answer keys to the challenge questions
Achievement Profile
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Introduction
The content of this book builds on the mathematics topics covered in Teaching and
Assessing Working Mathematically Book 1.
Teaching and Assessing Working Mathematically Book 2 consists of 16
interrelated topics presented in 31 teaching sessions.
Complete solutions and answers are provided on all questions.
The book presents challenging, but achievable curriculum that links workingmathematically processes and familiar Level 3 mathematics content. Most of the
questions are suitable for mathematically able and talented mid-primary or upperprimary students.
The questions aim at prompting students to use a range of problem-solving
strategies, such as exploring possibilities, using diagrams, working backwards,
constructing counter examples, and systematic counting, in familiar contexts.
Most of the sessions include representative examples with the solution method
shown. It is recommended that all representative examples are discussed and
explained in detail at the beginning of the session.
The book includes four assessment tasks and seven challenge questions.
Complete solutions and appropriate marking keys are provided.
The Achievement Profile gives an overview of the individual student achievement
of the skills assessed cross the 16 mathematics topics.
The following general structure for the teaching sessions is recommended:
• direct instruction: up to 15 - 20 minutes. This part of the lesson should be
focused on: revision of appropriate skills and understandings related to the
given topic, followed by an introduction of the new mathematics concepts or
problem-solving techniques by using additional examples;
•
individual work: up to 15 - 20 minutes. Students work on solving
questions similar to the one already discussed. The teacher helps individual
students on a one-to-one basis;
•
whole-class discussion: up to 15 - 20 minutes. At the centre of the
discussion are the types of solution strategies used and verbal justifications
of the way the answer was obtained. The teacher should prompt students to
compare and comment on different solution strategies and pose questions
similar to the ones already solved.
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1. Odd and even numbers
Mathematics language:
counting numbers, even numbers, odd numbers, two-digit numbers, three-digit numbers,
consecutive odd numbers, consecutive even numbers.
Example:
Write the next three numbers in the patterns below:
Counting numbers:
1, 2, 3, 4, 5, ....., ....., ....., . . .
Three-digit numbers:
100, 101, 102, 103, ....., ....., ....., . . .
Four-digit numbers:
1 000, 1 001, 1 002, ....., ....., ....., . . .
Even numbers:
0, 2, 4, 6, ....., ....., ....., . . .
Odd numbers:
1, 3, 5, 7, 9, ....., ....., ....., . . .
Consecutive odd numbers:
11, 13, 15, 17, ....., ....., ....., . . .
Consecutive even numbers:
22, 24, 26, 28, ....., ....., ....., . . .
Exercise 1a
1. Write the first 10 even numbers. Find the difference between the largest and the smallest.
Answer: ......................................
2. Write a number twice as large as the tenth odd number.
Answer: ......................................
3. In your own words explain what is an even number. What is the smallest even number?
………………………………………………………………………………………...……..
4. Write the largest one-digit odd number as a sum of different odd numbers.
Answer: ......................................
5. Write next to each other all one-digit numbers greater than 3.
Delete all even digits. The sum of the remaining odd digits is: ................
6. Write the smallest two-digit number with the sum of its digits equal to 2: ................
7. Write the smallest three-digit number with the sum of its digits equal to 2: ..............
8. Write the greatest three-digit number that has different digits: ................
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9. By using all of the digits 4, 3, 2 and 1 write:
a) the greatest possible four-digit even number:
....................
b) the smallest possible four-digit odd number:
....................
c) two numbers with the greatest possible sum:
……………..
d) two two-digit numbers with the largest possible difference:
……………..
e) two two-digit numbers with the smallest possible difference:
……………..
10. By using all of the digits 0, 1, 2 and 3 write:
a) the smallest possible four-digit number:
....................
b) the greatest possible four-digit odd number:
....................
c) the smallest possible four-digit even number:
....................
d) two numbers with the smallest possible sum:
………...…...
e) two two-digit numbers with the smallest possible difference:
……………..
11. By using all the digits of 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 write:
a) the smallest possible ten-digit even number that ends on 8:
………...…...
b) the greatest possible ten-digit odd number that ends on 9:
………...…...
c) three odd numbers with the greatest possible sum:
………...…...
d) two five-digit numbers with the smallest possible difference:
……………..
e) two five-digit odd numbers with the greatest possible sum:
……………..
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Exercise 1b
Example:
The sum of two consecutive even numbers is 22. What is the smaller of the two numbers?
Solution: If we halve 22, then 22 = 11 + 11. The two even numbers closest to 11 are 10 and 12,
and 10 + 12 = 22. The smaller of these two numbers is 10.
12. The sum of two consecutive even numbers is 222. The smaller number is:
A) 108
B) 110
C) 111
D) 112
E) 200
13. The sum of two consecutive odd numbers is 220. The smaller number is:
A) 109
B) 110
C) 111
D) 113
E) 115
14. The sum of three consecutive odd numbers is 33. The smallest of the three numbers is:
A) 7
B) 9
C) 10
D) 11
E) 13
15. The sum of three consecutive even numbers is 48. The smallest of the three numbers is:
A) 12
B) 14
C) 16
D) 18
E) 20
16. Maria said: ‘The result of 0 × 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10 × 11 is an even number.’
Maria is correct. Explain why.
……………………………………………………………………………………………………………
17. How many different even numbers can be obtained from the number 51
replaced with a digit?
A) 2
B) 4
C) 5
D) 6
E) 10
* 2 , when the * is
18. How many different even numbers can be obtained from the number 52
replaced with a suitable digit?
A) 2
B) 4
C) 5
D) 9
E) 10
* , when the * is
19. Find three consecutive numbers such that the sum of the first and the third is 376.
Answer: ......................................
20. Helen deleted three of the digits in the number 12 376. She obtained an even number between
20 and 30. The sum of the deleted digits is:
A) 6
B) 10
C) 11
D) 14
E) None of these
21. How many two-digit numbers have different digits?
A) 180
B) 99
C) 90
D) 81
E) None of these
22. Find the sum of 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 +11 + 12 in as many ways as you can.
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Assessment task 1
Odd and even numbers, consecutive numbers
1. The sum of two odd consecutive numbers is 36. The sum of the digits of the smaller number
is:
A) 7
B) 8
C) 9
D) 17
E) 18
Show your working:
……………………………………………………………………………………………………………………
…………………………………………………………………………………………………………………..
2. How many different even numbers can be obtained from the number 7 31
replaced with a suitable digit?
A) 0
B) 4
C) 5
D) 9
E) 10
*,
when the
* is
3. Find three consecutive numbers such that the sum of the first and the third is 56.
Answer: ......................................
Show your working:
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
4. By using all of the digits 0, 3, 5 and 9 write:
a) the smallest possible four-digit number:
....................
b) the greatest possible four-digit odd number:
....................
c) two two-digit numbers with the smallest possible difference:
....................
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