E-book Code: REAU0009
Active Maths Problem Solving Maths for 10 - 12 year old students.
Written by Ken Smith. Illustrated by Rod Jefferson. Published by Ready-Ed Publications (1997) PO Box 276 Greenwood Perth Australia 6024 E-mail:
[email protected]
Web Site: www.readyed.com.au
COPYRIGHT NOTICE Permission is granted for the purchaser to photocopy sufficient copies for non-commercial educational purposes. However this permission is not transferable and applies only to the purchasing individual or institution.
ISBN 1 86397 139 4
Active Maths
Page 2
Ready-Ed Publications
Active Maths
Contents Teachers’ Notes ...................................................................................................4 Pool Table Maths - 1 ............................................................................................ 5 Pool Table Maths - 2 ............................................................................................ 6 Pool Table Maths - 3 ............................................................................................ 7 Pool Table Maths - 4 ............................................................................................ 8 Pool Table Maths - 5 ............................................................................................ 9 Pool Table Maths - 6 ..........................................................................................10 Number Tracks - 1 ............................................................................................. 11 Number Tracks - 2 .............................................................................................12 Number Tracks - 3 .............................................................................................13 Map Colouring - 1 ..............................................................................................14 Map Colouring - 2 ..............................................................................................15 Map Colouring - 3 ..............................................................................................16 Murder at the Lodge - 1 .....................................................................................17 Murder at the Lodge - 2 .....................................................................................18 Murder at the Lodge - 3 .....................................................................................19 Stamp Study - 1 .................................................................................................20 Stamp Study - 2 .................................................................................................21 Unmagic Squares - 1 .........................................................................................22 Unmagic Squares - 2 .........................................................................................23 Maths With Caps - 1 ..........................................................................................24 Maths With Caps - 2 ..........................................................................................25 Pentominoes - 1 ................................................................................................26 Pentominoes - 2 ................................................................................................27 Pentominoes - 3 ................................................................................................28 Pentominoes - 4 ................................................................................................29 Pentominoes - 5 ................................................................................................30 Pentominoes - 6 ................................................................................................31 Pentominoes - 7 ................................................................................................32 Stepping Stones - 1 ...........................................................................................33 Stepping Stones - 2 ...........................................................................................34 Stepping Stones - Template ...............................................................................35 Shapes Within Shapes - 1 .................................................................................36 Shapes Within Shapes - 2 .................................................................................37 Delivering the Post - 1 .......................................................................................38 Delivering the Post - 2 .......................................................................................39 Delivering the Post - 3 .......................................................................................40 Dartboard Maths - 1 ...........................................................................................41 Dartboard Maths - 2 ...........................................................................................42 Seven Lines ......................................................................................................43 Farmland Maths - 1 ............................................................................................44 Farmland Maths - 2 ............................................................................................45 Answers ...................................................................................................... 46-48
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Active Maths
Teachers’ Notes The Active Maths booklet represents a response to current trends in mathematics for the development of problem solving skills in primary school students. The activities contained within the booklet are designed to interest and stimulate children in the 10 to 12 years age range. They are presented as blackline masters which are able to be photocopied for use in the classroom. Wherever possible the activities are stand alone worksheets although occasionally other materials such as grid paper, glue or card may be required. Problem Solving Strategies The activities in the Active Maths booklet are thematically grouped and are so structured as to provide an increasing level of difficulty with each successive sheet in the theme. Obviously gifted children in lower years, or less able children in higher years, will both find the structured problems equally challenging. Knowledge of the problem solving abilities of the students is essential, in order that each child can be presented with an activity which he or she feels comfortable solving and not become frustrated with, because of inappropriate matching. Initially problem solving activities could be tackled in class groups. This establishes a framework from which the children can branch out to work in smaller problem solving groups and then ultimately, independently. This step by step approach uses a structured framework for tackling problems: 1. 2. 3. 4.
Understand the nature of the problem. Develop a strategy for solving the problem. Carry out the chosen strategy. Look back and check.
1. Understand the nature of the problem. Ask questions about the problem. Edit out irrelevant details. Re-word the problem in simpler terms. Highlight key words or phrases. Find similar problems to model from. 2. Develop a strategy for solving the problem. Discuss alternative strategies. Use concrete aids. Use pictures or scenarios. Use tables or patterns. Use logic. Guess, check and alter strategy accordingly. Use trial and error techniques. Eliminate inappropriate solutions. 3. Carry out the chosen strategy. Use aids/materials to assist in the calculation. 4. Look back and check. Check that the problem has been fully answered. Discuss the solution and its feasibility. Be aware of alternative methods of solving the problem. Be able to present the steps leading to the solution. Page 4
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Active Maths Shape and space: Position and direction.
Name.......................
Pool Table Maths - 1
This is a special mathematical pool table. It only has pockets at the four corners and the surface is marked out as a 7 by 4 grid. Top Left
Top Right
Bottom Left
Bottom Right
To play on this table you use only ONE ball. This is hit from the bottom left-hand corner to strike and rebound off each cushion at an angle of 45o. Complete the path of the ball until it meets a pocket. Which corner pocket did the ball fall into? ........................................................................ How many times did the ball hit a side cushion? ............................................................. How many squares has the ball travelled across? (Don’t forget to count the starting square.) ............................................................................................................................ Here is a second pool table marked as a 4 by 9 grid. Once again, the ball is hit from the bottom left-hand corner. Predict which pocket the ball will fall into. ........................... Test your prediction by tracing the path of the ball on the grid. Were you right? ...................................................................
Follow-up Find three pool table sizes where the ball will fall into the top left-hand pocket each time. (Always start from the bottom left-hand corner!) Draw your pool tables on grid paper. Trace the path of each ball across the table.
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Active Maths
Name.......................
Shape and space: Position and direction.
Pool Table Maths - 2 Here is another special mathematical pool table. It only has pockets on the four corners and the surface is marked out as a 3 by 4 grid. A ball was hit from the bottom left-hand corner of the table. Its path was intersected 3 times (circled). What will happen to the number of intersections if the length of the pool table is increased to 5 units? ...................................................... ...................................................... ...................................................... Investigate using grid paper. Increase the length by another unit. Find the number of intersections and write them into this table of results. Repeat by increasing the length a further unit. Length
Width
1
3
2
3
3
3
4
3
5
3
6
3
7
3
8
3
9
3
10
3
Intersections
3
Explain any pattern you found in the number of intersections: ......................................................................................................................................... .........................................................................................................................................
Follow-up Change the width of the pool table and repeat the investigation. Note any new pattern found.
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Active Maths
Name.......................
Shape and space: Position and direction.
Pool Table Maths - 3 This special mathematical pool table has pockets at the four corners and the surface is marked out as a 7 by 4 grid.
A white ball is hit from the bottom left-hand corner, grid position (0,0) to strike and o rebound off each cushion at an angle of 45 . A black ball is at grid position (3,1). Trace the path of the white ball on the grid. Did the white ball strike the black ball? ............................................................................ How many squares did the white ball travel across before it struck the black ball? ......... Upon contact, the motion of the white ball is transferred to the black ball. Continue to trace the path of the black ball. Into which pocket does the black ball fall? ....................................................................... How many squares has the black ball travelled across? .................................................
Follow-up Place the white ball in the bottom left-hand corner (0,0) and the black ball at the grid position (3,1). Investigate which of these pool table sizes will: 1. Pocket the black ball. 2. Pocket the white ball. a) 7 by 5 ..................................................................... b) 6 by 6 ..................................................................... c) 5 by 3 ..................................................................... d) 4 by 6 ..................................................................... e) 9 by 5 ..................................................................... f) 4 by 8 .....................................................................
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Active Maths
Name.......................
Shape and space: Position and direction.
Pool Table Maths - 4 This mathematical pool table has four corner pockets and two centre pockets. The surface is marked out as an 8 by 5 grid.
A ball is hit from the bottom left-hand corner, grid position (0,0), to strike and rebound off each cushion at an angle of 45o. Predict which pocket the ball will fall into. ........................................................................ Test your prediction by tracing the path of the ball. Were you right? ................................................................................................................ Predict the pocket the ball will fall in for these table sizes: a. 6 by 5 .......................
b. 12 by 7 .......................
c. 10 by 7 .......................
Across how many squares has the ball travelled on each of the above tables? Calculate, remembering to count the starting square! Write your results in this chart. Length
Width
Distance Travelled
8
5
squares
6
5
squares
12
7
squares
10
7
squares
8
7
squares
10
9
squares
Predict the distances travelled for the other table sizes in the chart. Is there a pattern? ............................................................................................................ Will the pattern work for all table sizes? ........................................................................... If not, give reasons. .......................................................................................................... Page 8
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Active Maths Shape and space: Position and direction.
Name.......................
Pool Table Maths - 5 This mathematical pool table has four corner pockets and two centre pockets. The surface is marked out as a 10 by 7 grid.
The ball is now hit with enough energy to allow it to travel across 25 squares before it falls in a pocket or stops on the table. Trace the path of the ball for a distance of 25 squares. 1. Does the ball end up in a pocket? 2. If so, which pocket? 3. How far had the ball travelled? 4. If the ball didn’t reach a pocket, what is its grid position now? Determine the position of the ball on each of these tables if the ball only travels 25 squares. a) 12 by 7
b) 14 by 5
d) 14 by 8 c) 10 by 8
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Active Maths
Name.......................
Shape and space: Position and direction.
Pool Table Maths - 6 These mathematical pool tables each have only four corner pockets and no centre pockets. Their surfaces are marked out in a grid. Table A - 5 by 8
Table B - 10 by 4
The surface area of each table is 40 square units. A ball is hit from the bottom left-hand corner of each table, to strike and rebound off each cushion at an angle of 45o. Into which pocket will the ball travel on: Table A ............................................................................................................................. Table B ............................................................................................................................. Investigate (Use grid paper or the back of this sheet for these problems.) Find a pair of pool tables with an equal surface area such that a ball struck from the bottom left-hand corner of each will finish in an identical pocket. The pair ........................ by ....................... and ....................... by ....................... . both finished with the ball in the .................................................................... pocket.
Follow-up Find three more pairs of tables which perform the same feat. Draw them.
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Active Maths
Name.......................
Calculations: Mental calculation strategies.
Number Tracks - 1
The Rules Draw a line through any 20 numbers then add up the total. Your line may start and finish anywhere but not cross itself. The line must not go through any number more than once. Only lines going across, up or down are allowed. Here is an example: 7
2
0
6
5
1
9
4
5
9
0
2
8
3
5
0
7
1
3
8
4
7
2
6
5
9
1
8
1
7
2
9
4
0
5
120 My line is worth ..................................... Now try this one for yourself. Can you beat a score of 100? 9
1
7
4
8
2
6
0
5
3
4
8
2
6
0
8
3
5
1
7
7
0
9
2
5
4
8
3
6
1
2
6
1
7
4
3
5
9
2
8
1
7
4
8
2
6
0
5
3
9
7
4
8
2
6
0
9
3
5
1
0
9
2
5
4
8
3
6
1
7
2
6
1
7
4
3
5
8
2
8
2
3
5
9
2
8
6
1
7
4
7
4
8
9
1
2
6
0
5
3
I scored a total of ..............................
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Active Maths
Name.......................
Calculations: Mental calculation strategies.
Number Tracks - 2 The Rules Draw a line through any 20 numbers then add the total. Your line may start and finish anywhere but not cross itself. The line must not go through any number more than once. Only lines going across, up or down are allowed. What is the highest total you can make? ................................................................... 2
6
1
7
4
3
5
9
0
8
2
6
0
5
3
9
1
7
4
8
7
4
8
2
6
0
9
3
5
1
4
8
3
6
1
7
0
9
2
5
6
1
7
4
3
5
9
2
8
0
0
8
6
2
3
5
9
1
7
4
8
7
4
9
1
2
6
0
5
3
7
4
8
2
9
1
6
0
5
3
4
8
2
6
0
9
3
5
1
7
3
6
1
7
0
9
2
5
4
8
What is the lowest total you can make? 2
6
1
7
4
3
5
9
0
8
2
6
0
5
3
9
1
7
4
8
7
4
8
2
6
0
9
3
5
1
4
8
3
6
1
7
0
9
2
5
6
1
7
4
3
5
9
2
8
0
0
8
6
2
3
5
9
1
7
4
8
7
4
9
1
2
6
0
5
3
7
4
8
2
9
1
6
0
5
3
4
8
2
6
0
9
3
5
1
7
3
6
1
7
0
9
2
5
4
8
(Highest) ...................................................................................................... (Lowest) ....................................................................................................... (Difference) .................................................................................................. The difference between my two lines is .......................................................
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Active Maths Calculations: Mental calculation strategies.
Name.......................
Number Tracks - 3 The Rules Draw a line through any 20 numbers then add up the total. Your line may not cross itself at any point. The line must not go through any number more than once. Only lines going across, up or down are allowed. In this number track you add all double digit numbers. ... and subtract all single digit numbers. Begin at number 20. 12
6
10
7
14
3
15
9
10
8
2
16
0
15
3
19
1
17
4
18
17
4
18
2
16
0
19
3
15
1
7
14
8
12
9
11
6
10
5
13
14
8
12
6
20
9
13
5
11
7
3
16
1
17
0
19
2
15
4
18
16
1
17
4
13
5
19
2
18
0
0
18
6
12
3
15
9
11
7
14
18
7
14
9
11
2
16
0
15
3
4
18
3
16
1
17
0
19
2
15
Use a coloured pencil to show your number track. Can you find a line that will total more than 100? My line was worth ............................................................................................................. Use a different coloured pencil. Can you beat your previous total by beginning anywhere on the grid? My second line was worth ................................................................................................ I beat my previous total by ...............................................................................................
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Active Maths
Name.......................
Shape and space: Properties of 2D shapes.
Map Colouring - 1 When a cartographer (map-maker) is colouring a map it is usual to give different colours to any two countries, states or regions which have a common border. Here is a simple example...
Where areas meet at a single point they are not considered to have a common border and so may be coloured the same. For example:
Here is a simple map which requires colouring. Using as few colours as possible, colour in the map.
I used a total of ...................... colours to colour the map. Page 14
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Active Maths
Name.......................
Shape and space: Properties of 2D shapes.
Map Colouring - 2 It is firmly believed that any map can be coloured in using only four colours. Investigate this claim. Below are four outlines of the same island country. Use Map A. to divide the country up into smaller states or districts. Colour in each state or district remembering not to give the same colour to any which have a common border. Repeat the activity with the other outlines, dividing the island into more sections each time. (Tip: Plan out your colouring before you begin!)
A.
B.
I used a total of ................. colours to colour the map.
I used a total of ................. colours to colour the map.
C.
D.
I used a total of ................. colours to colour the map.
I used a total of ................. colours to colour the map.
Your conclusion: I discovered the claim to be TRUE / FALSE (circle).
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Active Maths
Name.......................
Shape and space: Properties of 2D shapes.
Map Colouring - 3 Welcome to the Isle of Hues. The island has been divided among the sheep farmers who live on Hues. A map maker has drawn the divisions on a map. Your task is to colour in the map using as few colours as possible. Remember, you can’t use the same colour for any divisions that have a common border. Good luck!
I used a total of .................. colours to colour in the map.
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Active Maths
Name.......................
Understanding and appying mathematics.
Murder at the Lodge - 1 A murder has been committed at the Lodge. You’ve been sent to solve the murder and bring the criminal to justice. The Lodge contains four rooms, four suspects and four weapons. At the time of the murder each suspect and each weapon was in a different room. Here is a plan of the lodge.
Kitchen Dining R oom Room
Hall L ounge
Ten pieces of information are known to be true. 1. Miss Bailey was in the lounge. 2. Mr Allen was in the same room as the walking stick. 3. The knitting needle was in the dining room. 4. Mr Dale was not in the dining room. 5. The murder took place in the room next door to where the duelling pistol was. 6. The victim was not killed by a blow to the head. 7. Mr Allen was in a room next door to Mrs Carrow. 8. The rat poison was not in the hall. 9. The murderer wasn’t married. 10. Mr Dale was not in a room next door to Mrs Carrow. Using these logic tables, find out who committed the murder, where and with which weapon. Duelling Knitting Pistol Needle
Rat Walking Poison Stick
Hall
Kitchen
Dining Lounge Room
Mr Allen Miss Bailey Mrs Carrow Mr Dale The murderer was: Hall Kitchen Dining Room Lounge Ready-Ed Publications
................................................. The weapon used was: ................................................. The murder room was: ................................................. Page 17
Active Maths Understanding and appying mathematics.
Name.......................
Murder at the Lodge - 2 Another murder has been committed at the Lodge! You’ve been sent to solve the murder and bring the criminal to justice. The Lodge contains four rooms, four suspects and four weapons. At the time of the murder each suspect and each weapon was in a different room. Here is a plan of the lodge.
Kitchen
Hall L ounge
Dining R oom Room
Ten pieces of information are known to be true. 1. Dr Graham was in the kitchen. 2. Mr Ellis was in the same room as the carving knife. 3. The rifle was in the dining room. 4. The murder wasn’t committed in the lounge. 5. Neither male suspect was in a room next to the kitchen. 6. The victim was not shot. 7. Ms Fisher was not in the same room as the rifle. 8. The silk scarf was not in the kitchen. 9. Mrs Hardy was not in the room next door to where the murder was committed. 10. The doctor is innocent. Using these logic tables, find out who committed the murder, where and with which weapon. Baseball Carving Bat Knife
Rifle
Silk Scarf
Hall
Kitchen
Dining Lounge Room
Mr Ellis Ms Fisher Dr Graham Mrs Hardy The murderer was: Hall Kitchen Dining Room Lounge Page 18
................................................. The weapon used was: ................................................. The murder room was: ................................................. Ready-Ed Publications
Active Maths
Name.......................
Understanding and appying mathematics.
Murder at the Lodge - 3 Yet another murder has been committed at the Lodge! You’ve been sent to solve the murder and bring the criminal to justice. The Lodge contains four rooms, four suspects and four weapons. At the time of the murder each suspect and each weapon was in a different room. Here is a plan of the lodge.
Kitchen
Hall
Dining R oom Room
L ounge
Ten pieces of information are known to be true. 1. Ms Keaton was in the lounge. 2. Dr Jones was in the same room as the paperweight. 3. The scissors were in the kitchen. 4. Mr Lawson was not in a room next door to the doctor. 5. The murder took place in the room next door to where the tow rope was. 6. The victim was not killed by a woman. 7. The scissors were in a room next door to Mr Lawson. 8. The shotgun was not with Mr Lawson. 9. The paperweight was not in the dining room. 10. The tow rope was in a room next door to Ms Keaton. Using these logic tables, find out who committed the murder, where and with which weapon. Paper Weight
Scissors Shotgun
Tow Rope
Hall
Kitchen
Dining Lounge Room
Mr Isaacs Dr Jones Ms Keaton Mr Lawson The murderer was: Hall Kitchen Dining Room Lounge Ready-Ed Publications
................................................. The weapon used was: ................................................. The murder room was: ................................................. Page 19
Active Maths
Name.......................
Shape and space: Position and direction.
Stamp Study - 1 In the grid below, draw all the different ways you could buy three attached stamps.
In the grid below, draw all the different ways you could buy four attached stamps.
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Active Maths
Name.......................
Making sense of no. problems; using no. operations to solve word problems.
Stamp Study - 2 The local post office has run short of stamps. It now finds itself with only sheets of 5 cent and 7 cent stamps.
It is possible to place up to 8 stamps on an envelope.
Investigate What is the minimum value of stamps that can be used on any one envelope? ................... What is the maximum value? ............................................................................................. Is it possible to make 45 cents in value? ............................................................................ How many of each stamp did you use? ............................................................................. Investigate the values that can be made using just 5 cent and 7 cent stamps, placing from 1 to 8 stamps on any single envelope. The values I found that can be made are: ..................
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Active Maths
Name.......................
Making sense of no. problems; reasoning about numbers through number operations.
Unmagic Squares - 1 Here is a special kind of number square called a magic square.
4
9
2
3
5
7
8
1
6
All the rows, columns and both diagonals add up to 15. Here is another grid. Using the same digits from 1 to 9, design an unmagic square. (All the rows, columns and diagonals must add up to a different number in each case but not 15!)
Here is another unmagic square. With the help of a pair of scissors, turn it into a proper magic square once again!
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4
15
10
5
14
11
8
1
7
12
13
2
9
6
3
16
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Active Maths
Name.......................
Making sense of no. problems; reasoning about numbers through number operations.
Unmagic Squares - 2 Here is our magic square once again.
4
9
2
3
5
7
8
1
6
We will use the nine squares and the digits from 1 to 9 in our puzzles.
If we rearrange the nine squares into the pattern of a wheel, can you write in the digits from 1 to 9 so that each spoke adds up to the same total?
Now let’s rearrange the nine squares again into a cross pattern. Write in the digits from 1 to 9 so that each line adds up to the same total.
This time we’ve rearranged the squares to form a triangle shape. Write the digits from 1 to 9 so that each side adds up to the same total.
Here the squares have been arranged into a pyramid. Write in the digits from 1 to 9 so that no two consecutive numbers are in squares that touch in any way! (Consecutive numbers are 1 & 2, 2 & 3, etc.)
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Active Maths
Name.......................
Understanding and appying mathematics.
Maths With Caps - 1 The top of this cap is made from six pieces of material, each the same shape and size. How many different caps can you make if: a) only 3 pieces of yellow and 3 pieces of red material are used; b) each piece can either be yellow or red to form a different pattern? Draw your different caps on these blanks. Be careful not to duplicate or rotate a pattern.
a)
a)
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Active Maths
Name.......................
Understanding and appying mathematics.
Maths With Caps - 2 The top of this cap is made from six pieces of material, each the same shape and size. Using red and yellow colour combinations design and show 36 patterns on the caps below. Two have been done for you.
Y Y
R R
R Y
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R Y
Y R
R Y
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Active Maths Shape and space: Properties of 2D shapes.
Name.......................
Pentominoes - 1 Pentominoes are shapes formed by arranging five squares. Complete the chart below using up to five 1 cm squares. Remember - join along full sides only.
=
Rotations are not new shapes.
=
Reversals are not new shapes. Shape
Number of squares
Draw the combination/s
How many examples?
Domino
2
1
Triominoes
3
2
Quadominoes
4
3
Pentonimoes
5
Estimate ......... Found.............
Draw all your pentominoes here.
How many did you create? .................................................
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Active Maths
Name.......................
Shape and space: Properties of 2D shapes.
Pentominoes - 2
Master Board of Pentomino Shapes Here are the TWELVE pentomino shapes. Glue this sheet to a piece of card and then cut out the pentomino shapes.
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Active Maths
Name.......................
Shape and space: Properties of 2D shapes.
Pentominoes - 3 Arrange the 12 pentomino shapes into this 6 by 10 grid. A few pieces have been placed to give you a start. Some shapes look different when they are turned over.
Arrange 5 of the 12 pentominoes into the lattice below. One piece has been placed to give you a start. Some shapes look different when they are turned over.
a) b) c) d) e)
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Try placing all 12 pentominoes in these other grid sizes. 5 by 12 3 by 20 (only 2 solutions) 4 by 15 two 5 by 6 rectangles one 5 by 7 rectangles with one 5 by 5 square
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Active Maths
Name.......................
Shape and space: Properties of 2D shapes.
Pentominoes - 4 GOLOMB ’S G AME GOLOMB’S GAME
This is a very simple game, yet skill is needed to play it. Each player in turn places a pentomino piece anywhere on the board. The last player able to do so wins. No pieces may overlap. One set of pentominoes is used.
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Active Maths
Name.......................
Shape and space: Properties of 2D shapes.
Pentominoes - 5
GOLOMB ’S G AME - Investigation (Using previous sheet) GOLOMB’S GAME In this game each player in turn places a piece anywhere on the board. The last player able to do so wins. No pieces may overlap. One set of pentominoes is used. What is the minimum number of pieces which can be used to complete “Golomb’s Game”? Draw your answer.
If the maximum number of pieces which can be used is 12, draw the completed game board leaving empty the four squares marked in black.
On grid paper, draw winning end-of-game boards where 7, 8, 9 and 10 pentominoes have been used in turn.
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Active Maths
Name.......................
Making sense of number problems; calculating perimeter & area using standard units.
Pentominoes - 6 PERIMETER AND AREA
Pick out the following eight pentomino pieces from a set.
These eight pieces can be used to construct a perimeter pathway around an enclosed area. They can be linked like this:-
But not like this:-
Here is an example of a complete perimeter pathway:-
Note the continuous “pathway” (dotted line) through each of the perimeter pentominoes. The area enclosed by this pathway is 36 squares.
Use your eight pentomino pieces to create your own perimeter pathways. What is the largest area that can be enclosed? ............................................. What is the smallest area that can be enclosed? ........................................... Draw each of your solutions on grid paper. Ready-Ed Publications
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Active Maths
Name.......................
Shape and space: Properties of 2D shapes.
Pentominoes - 7 TES SELL ATIONS TESSELL SELLA
One of the twelve pentominoes pieces has been selected below. By using it to draw around, a tessellation pattern has been produced.
Choose another of the pentomino pieces and use it to draw around. Try and produce a tessellation pattern as in the example above. Is it possible to make a tessellation pattern with each of the twelve pentominoes? ....................... If not, draw the piece(s) you don’t think will tessellate in this space:-
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Ready-Ed Publications
Active Maths Making sense of no. problems; reasoning about numbers.
Name.......................
Stepping Stones - 1
Here is a set of numbered stepping stones. To travel over the stones you can only move ACROSS, UP or DOWN. No diagonal moves! Also, you cannot retrace your steps at any time or visit a stone more than once.
S
6
7
8
5
4
9
3
7
2
6
6
3
F
Can you move from the start (S) to the finish (F) in a total of 32? Write the number sentence of your pathway. Draw it with a coloured pencil. ................................................................................................................................... Can you find two more ways of making 32? Draw each in a different colour. Write the number sentence of each pathway here:................................................................................................................................... ................................................................................................................................... Look at this set of stepping stones.
Find pathways that will total: a) 29 (draw it in red)
S
7
1
8
2
4
3
6
7
6
7
5
9
F
...............................................................................................
b) 39 (draw it in black) ............................................................................................... c) 49 (draw it in green) ............................................................................................... d) 59 (draw it in blue) ............................................................................................... What is the lowest total pathway you can make over these stepping stones? .......... What is the highest total ? ........................................................................................ Ready-Ed Publications
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Active Maths
Name.......................
Shape and space: Position and direction.
Stepping Stones - 2
Here is a group of stepping stones. The start is marked by (S) while the finish is marked by (F). One pathway has been marked from S to F.
S
F Investigate How many different pathways can be taken to get from S to F? Each pathway may travel ACROSS, UP or DOWN but not diagonally. Also, you may not retrace your steps at any time or visit a stone more than once. Either use grid paper or dotted paper to draw your pathways as shown below.
S
S
F
F
In all I found ................ pathways from (S) to (F).
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Active Maths
Name.......................
(Teachers’ Blank Template)
Stepping Stones Here is a set of numbered stepping stones. To travel over the stones you can only move ACROSS, UP or DOWN. No diagonal moves! Also, you cannot retrace your steps at any time or visit a stone more than once.
S
F Find pathways that will total: a) ..................... (draw it in red) b) ..................... (draw it in black) c) ..................... (draw it in green) d) ..................... (draw it in blue) What is the lowest total pathway you can make over these stepping stones? ............................................................................ What is the highest total ? ........................................................................................
Ready-Ed Publications
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Active Maths Shape and space: Properties of 2D shapes.
Name.......................
Shapes Within Shapes - 1
Look very carefully at each drawing. Try and determine how many of each shape can be found within the drawings. Use colour to help outline your solution. How many triangles? ................................
How many rectangles? ................................
How many triangles? ................................
How many triangles? ................................
How many rectangles? ................................
How many triangles? .................................
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Active Maths
Name.......................
Shape and space: Position and direction.
Shapes Within Shapes - 2
Look very carefully at the drawings below. Try and determine how many of each shape can be found within each drawing. How many equilateral triangles?
....................
....................
How many isosceles triangles?
....................
....................
How many right-angled triangles?
....................
....................
How many regular hexagons?
....................
....................
How many rectangles?
....................
....................
Ready-Ed Publications
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Active Maths
Name.......................
Shape and space: Position and direction.
Delivering the Post - 1
Bill is a postie in Anytown. Below is a map of the streets where he delivers letters.
Find a route to enable Bill to visit each street once only. Use arrows to show the path you select. Jenny is also a postie in Anytown. Below is a map of the streets where Jenny delivers letters.
Find a route to enable Jenny to visit each street once only. Use arrows to show the path you select. Page 38
Ready-Ed Publications
Active Maths Shape and space: Position and direction.
Name.......................
Delivering the Post - 2
Pete is a postie in Anytown. Below is a map of the streets where Pete delivers letters. A B D
C
E H
G
F
J
I
Investigation Starting at each letter in turn (A to J) find out whether Pete can complete his round without visiting any street more than once. Fill in the table below. Starting position
Finishing position
Possible? (Yes/No)
.................. A ................................................................................................................... .................. B ................................................................................................................... .................. C .................................................................................................................. .................. D .................................................................................................................. .................. E ................................................................................................................... .................. F ................................................................................................................... .................. G .................................................................................................................. .................. H................................................................................................................... ................... I ................................................................................................................... ................... J ................................................................................................................... Ready-Ed Publications
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Active Maths
Name.......................
Making sense of no. problems; reasoning about numbers.
Delivering the Post - 3
Wendy is a postie on Two Rocks Island. Below is a map of the island where she delivers letters. There are eleven cottages on the island. Wendy’s cottage, the island post office, is at (C). Each road on the island is one kilometre in length.
Work out a route for Wendy to take so that she visits each cottage at least once. She must finish back at home! (Show your route like this: - D - A - B - C - A - E etc.) Wendy’s route is
........................................................................................................
By the time she reaches home she has travelled ...................... kms. How many roads did Wendy have to travel along more than once? ................................. Page 40
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Active Maths
Name.......................
Making sense of no. problems; reasoning about numbers.
Dartboard Maths - 1
Using two darts: find one way to score 19. ................ + ................. find one way to score 37. ................ + ................. See if you can work out how many different ways there are to score 4. Remember, some darts might miss! Try to find the ways to score 5, 6 and 7, and fill in the table. Score 0 1 2 3 4 5 6
Different ways 0+0 1+0 2+0, 1+1 3+0, 2+1
Total ways 1 1 2 2
Note: This is a special mathematical dartboard.
7 Did you discover a pattern? .................. Describe this pattern. Use it to continue the table on the back of this sheet This time using three darts: find one way to score 19. ................ + ................. +................. find one way to score 37. ................ + ................. +................. See if you can work out how many different ways there are to score 4, 5, and 6. Complete the table below. 1, 2 and 3 have been done for you. Score 0 1 2 3 4
Different ways 0+0+0 1+0+0 2+0+0, 1+1+0 3+0+0, 2+1+0,1+1+1
Total ways 1 1 2 3
5 6 Predict how many ways you could make 12. ............... (Use the back of the sheet for working.) Ready-Ed Publications
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Active Maths
Name.......................
Making sense of no. problems; reasoning about numbers.
Dartboard Maths - 2 Here is the special ‘mathematical dartboard’ again. Using three darts: find one way to score 60. ................ + ................. + .................. How many other ways are there to score 60? ................ See if you can work out how many different ways there are to score 59. Remember, some darts might miss! Try to find the ways to score 58, 57 and 56, and fill in the table. Score 60 59 58
Different Ways 20+20+20
Total Ways 1
57 56 Do you see a pattern in the Total Ways column? ................. Where have you seen this pattern before? ............................................................ ............................................................................................................................
Challenge If you were using three darts which score could be made in the greatest number of different ways? ............................ Page 42
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Active Maths
Name.......................
Shape and space: Properties of 2D shapes.
Seven Lines
Using seven lines, what is the largest number of non-overlapping triangles that can be produced. This diagram shows how six triangles (shaded black) have been produced.
On the back of this page draw your own seven lines and shade the number of triangles you produced. Write the number here below. I produced ............... triangles with my seven lines. Here are another seven lines. They are drawn to intersect each other. The intersections have been circled. The total number of intersections is 14.
On the back of this page draw seven lines of your own. Try and make as many intersections as you can, and circle the intersections. I made ................. intersections with my seven lines. Using seven lines, what is the largest number of non-overlapping quadrilaterals that can be produced. This diagram shows how four quadrilaterals (shaded black) have been produced.
On another piece of paper draw your own seven lines and shade the number of quadrilaterals you produced. I produced ............... quadrilaterals with my seven lines. Ready-Ed Publications
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Active Maths
Name.......................
Shape and space: Properties of 2D shapes.
Farmland Maths - 1
Here is a plot of land. It has been divided into 16 squares. The land is to be divided between the four sons of the farmer so that each son gets a piece of land the same shape and area as each of his brothers. Using colour, show how the land can be divided correctly.
Another farmer is dividing his land between his four daughters so that each daughter gets a piece of land, with a house (H) and a pond (P), the same shape and area as each of her sisters. Using colour, show how the land can be divided correctly.
H
P P
P H H H P Below is a plot of land. It has been divided into 36 squares. The land is to be divided between the farmer’s four children so that each gets a piece of land, with a house (H) and a pond (P), the same shape and area as each of the others. Using colour, show how the land can be divided correctly.
P P P H P H H H Page 44
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Active Maths
Name.......................
Shape and space: Properties of 2D shapes.
Farmland Maths - 2
Another farmer is dividing his land between his four children so that each gets a piece of land, with a house (H) and a pond (P), the same shape and area as each other. Using colour, show how the land can be divided correctly.
P H H H H P P P
Here is a plot of land. It has been divided into 64 squares. The land is to be divided between the farmer’s four children so that each gets a piece of land, with a house (H) and a pond (P), the same shape and area as the others. Using colour, show how the land can be divided correctly.
P
H
P
H H H
P Ready-Ed Publications
P Page 45
Active Maths
ANSWERS POOL TABLE MATHS P5.
Act. 1
Top left; 9; 28
P6.
Act. 2
Table: 0, 1, 0, 3, 4, 0, 6, 7, 0, 9
P7.
Act. 3
Yes; 17 Top left; 11 Black ball White ball
P8. P9.
Act. 4 Act. 5
P10. Act. 6
P21. Act. 2 These values can be made (numbers expressed in cents): 5, 7, 10, 12, 14, 15, 17, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 54, 56. UNMAGIC SQUARES
(a), (c), (d), (e) (b), (f)
P22. Act. 1
20, 15, 42, 35, 28, 45 It will not work on all tables.
3 6 7
No, none, 25 squares, (5,3) (a) (1,3) (b) (3,5) (c) (5,7) (d) (3,7) Table A - top left; Table B - top left e.g. 12 by 2 and 8 by 3; bottom right.
NUMBER TRACKS P11-12
9 4 8
Using scissors - bottom activity: 4 14 7 9
15 1 12 6
P23. Act. 2
P14. Act. 1
A minimum of 6 colours is needed. Acts. 2 & 3 - Answers will vary. MURDER AT THE LODGE
10 8 13 3
5 11 2 16
Turn these three numbers upside down.
examples of answers: 2
6
Miss Bailey, Lounge, Rat Poison. Ms Fisher, Hall, Silk Scarf. Mr Isaacs, Kitchen, Scissors.
1
3
1
2 3 4 9 5 6 8 7
4
5 7
9 8 4
STAMP STUDY P20. Act. 1
2 1 5
Answers will vary.
MAP COLOURING
P17. Act. 1 P18. Act. 2 P19. Act. 3
There are numerous answers here is an example:
8
(top)
3 5
5 8 1 7 2 6 4 9 3
1 9
7
2
6
MATHS WITH CAPS Act. 1
(bottom)
P24. Act. 1
(a) 4 solutions (b) 16 solutions
P25. Act. 2 Answers will vary. PENTOMINOES P28 Act. 3 6 by 10 grid - numerous solutions. 5 by 12 and 4 by 15 3 by 20 - only 2 solutions possible Solutions for 5 by 12 grid will give 5 by 6 (twice) and 5 by 5 with 5 by 7. PERIMETER PATHWAYS
P31. Act. 6 - Perimeter Pathways Largest - 58 squares
Smallest - 14 squares Page 46
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Active Maths STEPPING STONES P33. Act. 1a S, 6, 7, 4, 9, 6, F. S, 8, 3, 7, 5, 4, 2, 3, F. S, 8, 5, 4, 9, 6, F. Act. 1b (a) S, 8, 2, 4, 6, 9, F. (b) S, 8, 2, 4, 3, 7, 6, 9, F. (c) S, 7, 1, 4, 3, 7, 6, 7, 5, 9, F. (d) S, 8, 2, 7, 1, 4, 3, 7, 6, 7, 5, 9, F. P34. Act. 3 There are 48 pathways. SHAPES WITHIN SHAPES P36. Act. 1 top to bottom 5 triangles; 15 rectangles; 8 triangles; 20 triangles; 51 rectangles; 27 triangles. P37. Act. 2 Left shape: Equilateral triangles - 14 Isosceles triangles - 24 Rt angled triangles - 72 Regular hexagons - 2 Rectangles - 9 Right shape: Equilateral triangles - 16 Isosceles triangles - 24 Rt angled triangles - 36 Regular hexagons - 3 Rectangles - 18 DELIVERING THE POST P38. Act. 1 Answers will vary. P39. Act. 2 Only B and H are possible. P40. Act. 3 Wendy’s route: CBABFAEFEDACDKIDGHIGIJHJKDC She travelled 26kms. 4 roads more than once. DARTBOARD MATHS P41. Act. 3 Two darts 4 4+0; 3+1; 2+2 = 3 ways 5 5+0; 4+1; 3+2 = 3 ways 6 6+0; 5+1; 4+2; 3+3 = 4 ways 7 7+0; 6+1; 5+2; 4+3 = 4 ways Three darts 4 4+0+0; 3+1+0; 2+2+0; 2+1+1 = 4 ways 5 5+0+0; 4+1+0; 3+2+0; 3+1+1; 2+2+1 = 5 ways 6 6+0+0; 5+1+0; 4+2+0; 4+1+1; 3+3+0; 3+2+1; 2+2+2 = 7 ways. 12 can be made 16 ways. Ready-Ed Publications
DARTBOARD MATHS (cont.) P42. Act. 2
59: 20+20+19 = 1 way 58: 20+20+18; 20+19+19 = 2 ways 57: 20+20+17; 20+19+18; 19+19+19 = 3 ways 56: 20+20+16; 20+19+17; 20+18+18; 19+19+18 = 4 ways Challenge: 30 can be made in the most different ways.
SEVEN LINES P43. Top 11 triangles maximum Centre 21 intersections Bottom 6 quadrilaterals FARMLAND MATHS P44. Act. 1 Numerous solutions - these are examples Middle problem e.g. Bottom problem e.g. H
P P P P P H P
P H H H P
H H H
P45. Act. 2 Top problem e.g.
Bottom problem e.g. P
P
H
P
H H H
H H H H P P P
P
P
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