Contents page Introduction 4 1. Odd and even numbers 5 Assessment task 1 8 2. Counting techniques: Consecutive numbers 9 3. Counting techniques: How many digits? 11 Assessment task 2 13 4. Number chains 14 5. The Pigeonhole Principle 1 17 6. The Pigeonhole Principle 2 19 Assessment task 3 21 7. Divisibility 22 8. Revision questions 24 9. Challenge questions 26 10. Problem solving without a calculator 1 27 11. Problem solving without a calculator 2 29 12. Prime and composite numbers 31 13. Problem solving without a calculator 3 33 14. Problem solving: Working backwards 36 15. Logical questions: Balancing scales 38 16. Super operations 41 Assessment task 4 43 Answer keys 44 1. Odd and even numbers 44 2. Counting techniques: Consecutive numbers 45 3. Counting techniques: How many digits? 46 4. Number chains 47 5. The Pigeonhole Principle 1 49 6. The Pigeonhole Principle 2 50 7. Divisibility 52 8. Revision questions 53 10. Problem solving without a calculator 1 54 11. Problem solving without a calculator 2 55 12. Prime and composite numbers 56 13. Problem solving without a calculator 3 57 14. Problem solving: Working backwards 58 15. Logical questions: Balancing scales 59 16. Super operations 60 Answer keys to the assessment tasks 61 Answer keys to the challenge questions 63 Achievement Profile 64 3
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. 4
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. 2. Write a number twice as large as the tenth odd number. 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. 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:... 5
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:.. 6
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 * 2, when the * is replaced with a digit? A) 2 B) 4 C) 5 D) 6 E) 10 18. How many different even numbers can be obtained from the number 52 *, when the * is replaced with a suitable digit? A) 2 B) 4 C) 5 D) 9 E) 10 19. Find three consecutive numbers such that the sum of the first and the third is 376. 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. 7
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 *, when the * is replaced with a suitable digit? A) 0 B) 4 C) 5 D) 9 E) 10 3. Find three consecutive numbers such that the sum of the first and the third is 56. 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:... 8