Problems To solve in Primary school maThemaTics b henry
AM T P u b l i s h i n g
ii
Problems to Solve in Primary School Mathematics
How to Use This Book It is generally considered that there are two aspects to the teaching of problem solving: 1. Teach techniques (such as working backwards or making a table) and provide problems which practise these techniques. 2. Provide a miscellany of problems, have students try to solve these and discuss techniques as they arise. This book is useful in both situations. When practice with a particular strategy is required, by consulting the tables on the next pages, the teacher can select appropriate problems. On the other hand, when a miscellany of problems is required, the teacher can select problems from a suitable area of study. Within each area of study, the problems are arranged in approximate order of difficulty. A class can all be given the same problem, or different groups given different problems, perhaps with the better students working on the harder parts of a problem. The problems could be part of class work or be set as homework, or could be part of a ‘Problem of the Week’ program.
iii
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69
• • • • • •
• •
• • • • • •
• •
•
• •
• • • • •
• • • •
Look for a Pattern
Use Logic
Draw a Diagram
List All Possibilities
Make a Table
Work Backwards • •
Find a Rule
A-mazing Buckets and Beanbags Lucy Likes Darts Numbers in a Line Doubles Adding Even and Odd Numbers Multiplying Even and Odd Numbers Digits Consecutive Counting Numbers Plus and Minus 5 and 7 4×4 Ten Cards Decode 1 Decode 2 Number Tower Threedub Square Numbers Sums of Square Numbers Patterns of Consecutive Numbers Plus, Minus, Times Brian’s Numbers Numbers in Words Sum and Product Square Sums Counters Divisibility Stamps Steps Triangular Numbers Tables Crossnumber The Funpark Multiplying Digits Digits in a Square Mystery Times Table
Solve a Simpler Problem
Number
Guess and Check
Page
Preface
•
•
• • • • • • • • • • • • • •
• • • • • • • • • • • • •
• • •
•
•
•
iv
Space
Measurement
Logic
Animal Photos Maths Word Trail Mystic Rose Rectangle Jigsaws Tetrominoes Coloured Cube Tiling Chessboard Pieces Triangular Rectangles Shapes on a Triangle Grid Tetracubes Four Shapes Units Paper Clips and Counters The School Sports Biggest Domino Boxes Two Clocks Maths Words Counters and Squares Number Crossword Pets and Cards Three Boys Four Girls Weighings Seven Birthdays
71 73 75 78 80 83 85 89 92 95 97 99 103 105 107 109 111 113 115 117 120 122 124 126 128 130
•
•
•
•
• •
• •
•
• • • •
•
• • • • • • • • • • • • • • •
• • • •
• • • • • • • •
• • • • • • • • • • •
• •
Find a Rule
Solve a Simpler Problem
Look for a Pattern
Use Logic
Draw a Diagram
List All Possibilities
Make a Table
Work Backwards
Guess and Check
Page
Problems to Solve in Primary School Mathematics
Number
1
Number
A-mazing Lee drew this maze. You enter at the top on the left, move either to the right or down and exit bottom right. You add the numbers in the boxes you pass through.
John went this way: His total was 8 + 3 + 5 + 7 + 2 = 25.
.................................................................................................................................. ... ... ... .... ... ... .... ... ... ... ... ... . . . . ... ... .. .. . . .. ... ... ... . . .................. ................................ ................................ ................... .. . . .. . . ... . . ..... .. ... .... . . .... ... ... ... . ... .. ... . ... ... .. ... . .. ... ... . . . ................. ................................ ................................ .................. .... ..... .... .... ... ... ... ... ... ... ... . .. ... . . .... .... .. .... ..................................................................................................................................
ENTER → 8
1
6
3
5
7
4
9
2 → EXIT
........................................................................................................................... ... ..... ... .. .. .. ... ... ... ... ... ... ... ... . ... . ... . . . . ... .... .. .. . . .. ... .. .. . . ................. ................................. ................................. .................. ... ... . . . . . . . . .... ... .... ... . .... ... ... ... .. ... .. .... . ... ... ... .... . . .. . . . ................. ................................. ................................ ................... .. . ... ..... .... .... ... .. ... .. ... .. ... .. .. .... . . .... ... ... .. ..................................................................................................................................
1 6 ENTER → 8 ↓ 3→5→7 ↓ 4 9 2 → EXIT
a Find a path which gives you a total of 24.
........................................................................................................................... .. ..... ... ... ... .. .. .... .. ... ... ... ... ... ... . . ... . . . . ... .... .. .. . . . . . .. . . . . .................. ................................ ................................ ................... ... . . ... . . . . ... . . ... ..... .... ... .... .... ... ... ... ... .... .... ... .. ... .... .................. ................................. ................................ ................... ... ... . . . . . .... .... ... .... . .. .. .. ... ... ... .. .... .... . . ... .... ... .... ............................................................................................................................
ENTER → 8
1
6
3
5
7
4
9
2 → EXIT
b Find the path which gives the largest total. ENTER → 8
1
6
3
5
7
4
9
2 → EXIT
................................................................................................................................ ... ... ... ... ... ... ... ... . . ... . .. ... . .. . . .... .... .... ..... ... .. .. ... .................. ................................ ................................ .................. ... ... ... .... . ... . .... .. . . . .. .... ... ... ... ... . ... .. . . ... ... .. ... . .. ... ... .... ................. ................................ ................................ .................... .... .... ..... .... ... ... .. .... . ... ... ... . . . . .... .... .. .... .... ... . . .. ..............................................................................................................................
c Find the path which gives the smallest total.
.................................................................................................................................. ... ... ... ... ... ... ... ... . . . ... ... ... . ... . . . ... ... .. ... . . ... ... . .. . . ................... ................................. ................................. ................... . . ... .. . . . . .... .... ..... .... ... . ... ... ... .. .... ... ... . ... ... ... ... .. ... ... ... ................. ................................ ................................ .................. .... .... ..... ..... .. ... ... .... ... ... ... . . . ... . .... ... .... .... . . . ... .............................................................................................................................
ENTER → 8
1
6
3
5
7
4
9
2 → EXIT
2
Problems to Solve in Primary School Mathematics
Teacher’s Notes – A-mazing The problem provides practice in addition and in checking all possibilities. The maze is in fact a magic square of order 3 - all the rows, columns and diagonals add to 15.
Solutions There are 6 possible paths: 8 + 1 + 6 + 7 + 2 = 24 8 + 1 + 5 + 7 + 2 = 23 8 + 1 + 5 + 9 + 2 = 25 8 + 3 + 5 + 7 + 2 = 25 8 + 3 + 5 + 9 + 2 = 27 8 + 3 + 4 + 9 + 2 = 26 Hence the solutions are: a 8 + 1 + 6 + 7 + 2 = 24 b 8 + 3 + 5 + 9 + 2 = 27 c 8 + 1 + 5 + 7 + 2 = 23
Extensions 1. Suppose you can go up, down, left or right in the maze, but cannot visit the same square more than once. What is the largest total you can get now? 2. Enter the maze in the original problem as before, travel either right or down. This time, multiply the numbers in the boxes you pass through until you go out the exit. What is the largest product you can get?
Solutions to Extensions 1. 8 + 3 + 4 + 9 + 5 + 1 + 6 + 7 + 2 = 45 OR 8 + 1 + 6 + 7 + 5 + 3 + 4 + 9 + 2 = 45. You can visit each square once. 2. For the 6 paths given above, the largest product is 8 × 3 × 5 × 9 × 2 = 2160.
3
Number
Buckets and Beanbags Rose and Holly each had four small beanbags. They were trying to throw them into a bucket. Rose threw her four bags. Two went in. Her score was 2. Holly threw her four bags. Some went in the bucket. Rose threw her four bags again. Three went in. Her total score was now 2 + 3 = 5. Holly threw her four bags again. Some went in. Her total score was now 6. a How many bags might Holly have got in on her first and second throws? Is there another way she could have scored on each throw? Is there a third way she could have scored on each throw? b Rose and Holly each had another turn. Their total scores were now the same. Complete this table to show what Rose must have scored for each of Holly’s possible scores. If Holly scored ... then Rose scored ... 0 1 2 3 Explain why Holly could not have scored 4. c At the start of another game, Holly took two turns and scored a total 4. List all the ways she could have scored.
4
Problems to Solve in Primary School Mathematics
Teacher’s Notes – Buckets and Beanbags The problem gives practice in partitioning and checking all cases. The children could be given beanbags or tennis balls and buckets and act it out.
Solutions a All the possibilities are 4 and 2, 2 and 4, 3 and 3. b
If Holly scored ... 0 1 2 3
then Rose scored ... 1 2 3 4
If Holly scored 4, Rose could not score enough to make the total scores equal. c (0, 4), (1, 3), (2, 2), (3, 1), (4, 0).
Extension Holly now took all 8 beanbags and had two turns at throwing them into the bucket. Her total score was 10. Make a list of all the ways this could have happened (for example, 3 on the first turn, 7 on the second turn).
Solution to Extension (2, 8), (3, 7), (4, 6), (5, 5), (6, 4), (7, 3), (8, 2).
5
Number
Lucy Likes Darts Lucy throws darts at this target. She never misses the target. Her darts always score.
..................................... .................. ........... ......... ........ ........ ....... ....... ..... ..... ..... . . . ..... ... . . . .... ............................................. .. . . . . . . . . . .... . . . . ........ ...... ... ... . . . . . . . . . . ..... ... .. .... . . . . ... . . . ..... ... ... . ... . . . . .... ... ... .. . . . . ... ... ...................................... .. . . .. . . . . . . . . . . . ...... .. ... ...... .. . ... . . . . . . ... . .... ... ... .. . . ... . . .. . .... .. .. .. .. . ... . . . . ... .. .. .. ... . . . . . . . ... . . . . . . .. . ......... .. .. ........ .. . . . . ... . .. . ... . .. .... .. .. ... . . . ... . . .. ... ... .. .. .. .. ... .. .. . . ... . .. ... ... ... ... ..... .... . . . . .. ... . . ... .. .. . . .... . .. . .. . . ... ... . ... . . .... . .. . . . .. . ... . . . . ... . . . . ... ... . . . . ... . . . . . ... . . . . . .. .... . ... ... .... .. ...... .. ... ... .. ........................ ... .. .. ... .. ... ... ... .. .. .. ... ... .. ... .. ... . . . . . .. . . . . .... ... .. .. .. .... ..... ... .. .. .. ..... ........ ... ... .. ... ........................................... .... ... ... ... .... ... . ... . . . . . . .... ... .. .... ..... ... ... ..... ...... ... ... ...... ......... .... ......... ............. .... .... ...................................... .... . .... . . ..... ... ..... .... ....... ...... ........ ........ .......... ......... . . . . . ................. . . . . . . ..................................
7 5 3 1
She throws 2 darts. She adds the 2 numbers together. Her score is 8. a What could her numbers have been? Is there another way she could have scored? b After her third throw, her total score was 13. Where did her third dart hit? c Lucy throws a fourth dart. Explain why her total score is now even. d Lucy threw 17 darts for a total of 80. Explain why she must have made a mistake in her adding up.
6
Problems to Solve in Primary School Mathematics
Teacher’s Notes – Lucy Likes Darts The problem provides practice in addition and subtraction in an unusual setting. It also gives an opportunity to discuss generalisation. A discussion about odd and even numbers after the problem has been completed is worthwhile.
Solutions a (3 and 5), (5 and 3), (1 and 7) or (7 and 1). b 5. c Alternative i If she hits 1, her score is 13 + 1 = 14 (even). If she hits 3, her score is 13 + 3 = 16 (even). If she hits 5, her score is 13 + 5 = 18 (even). If she hits 7, her score is 13 + 7 = 20 (even). So it is always even. Alternative ii Each of 1, 3, 5 and 7 is odd, and 13 is odd. But two odd numbers add to an even number, so the total is always even. d Lucy is adding 17 odd numbers together. The total must be odd.
Extensions 1. What is the smallest total Lucy could have after she has thrown 4 darts and all the darts hit the board? What is the largest total she could have after throwing 4 darts and all the darts hit the board? 2. Lucy throws some darts and gets a total of 15. What is the smallest number of darts she could have thrown?
Solutions to Extensions 1. Smallest total = 1 + 1 + 1 + 1 = 4. Largest total = 7 + 7 + 7 + 7 = 28. 2. Largest score from 2 darts is 7 + 7 = 14, so she must throw at least 3 darts. 3 darts is enough – she might throw 7 + 7 + 1 = 15, or 7 + 5 + 3, etc.
