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### Number and algebra

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# Working Systematically

*Working Systematically is part of our Working Mathematically collection.*

### Noticing Patterns

### Finding All Solutions

### Consecutive Numbers

### Two and Two

### Gabriel's Problem

### Sociable Cards

### Special Numbers

### M, M and M

### Can They Be Equal?

### Multiples Sudoku

### Cinema Problem

### Add to 200

### Sticky Numbers

### Number Daisy

### Fence It

### Shifting Times Tables

### Shady Symmetry

### American Billions

### Isosceles Triangles

### Summing Consecutive Numbers

### 1 Step 2 Step

### Squares in Rectangles

### Where Can We Visit?

### Peaches Today, Peaches Tomorrow...

### Weights

### Triangles to Tetrahedra

### Cuboids

### Consecutive Negative Numbers

### How Many Miles to Go?

### Working Systematically - Short Problems

### Charlie's Delightful Machine

### Wipeout

### Product SudokuLive

### Shopping Basket

### Nine Colours

### Ben's Game

### River Crossing

### Warmsnug Double Glazing

### What's Possible?

### Pick's Theorem

### LCM Sudoku

### Difference Sudoku

### Latin Numbers

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Mathematicians often talk about the importance of working systematically. This means that rather than working in a haphazard and random way, there is a methodical, organised and logical approach. The problems below will challenge students to work systematically, and will help them appreciate the benefits of working in this way.

Scroll down to see our complete collection of problems, or explore the two sub-collections.

The key to solving these problems is to notice patterns or properties. Encouraging students to organise their work systematically allows them to notice what might not otherwise be obvious.

These problems challenge students to find all possible solutions. One of the best answers to "How do you know you have found them all" is to be able to say "I worked systematically!"

Age 7 to 14

Challenge Level

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Age 7 to 16

Challenge Level

How many solutions can you find to this sum? Each of the different letters stands for a different number.

Age 11 to 14

Challenge Level

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

Age 11 to 14

Challenge Level

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

Age 11 to 14

Challenge Level

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

Age 11 to 14

Challenge Level

This problem offers the students an opportunity to consolidate what they are expected to know about mean, mode and median whilst also challenging them to work systematically, and justify their reasoning

Age 11 to 14

Challenge Level

Sometimes area and perimeter of rectangles are taught separately, and are often confused. In this problem students consider the relationship between them.

Age 11 to 14

Challenge Level

Each clue in this Sudoku is the product of the two numbers in adjacent cells.

Age 11 to 14

Challenge Level

A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.

Age 11 to 14

Challenge Level

By selecting digits for an addition grid, what targets can you make?

Age 11 to 14

Challenge Level

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

Age 11 to 14

Challenge Level

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

Age 11 to 14

Challenge Level

This problem challenges students to work systematically while applying their knowledge of areas of rectangles.

Age 11 to 14

Challenge Level

This problem encourages students to think about the properties of numbers. It could be used as an introduction to work on linear sequences and straight line graphs.

Age 11 to 14

Challenge Level

This problem is a good activity for the visualisation of symmetry, and for encouraging students to work systematically.

Age 11 to 14

Challenge Level

This is an engaging context in which to reinforce rules of divisibility and challenge students to reason mathematically and work systematically.

Age 11 to 14

Challenge Level

This problem encourages students to use coordinates, area and isosceles triangles to solve a non-standard problem. To find all possible solutions they will need to work systematically.

Age 11 to 14

Challenge Level

This problem offers a simple context for students to explore, make generalisations and prove conjectures, working numerically and algebraically.

Age 11 to 14

Challenge Level

This problem is inaccessible without looking at simpler cases, and thus helps students to see the value of specialising in order to generalise.

Age 11 to 14

Challenge Level

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Age 11 to 14

Challenge Level

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

Age 11 to 14

Challenge Level

This problem could replace repetitive textbook work on calculating fractions of integers. It offers plenty of practice of these calculations while requiring students to come up with problem-solving strategies.

Age 11 to 14

Challenge Level

Different combinations of the weights available allow you to make different totals. Which totals can you make?

Age 11 to 14

Challenge Level

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Age 11 to 14

Challenge Level

Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

Age 11 to 14

Challenge Level

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Age 11 to 14

Challenge Level

How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

Age 11 to 16

A collection of short Stage 3 and 4 problems on Working Systematically.

Age 11 to 16

Challenge Level

This problem could be used to reinforce work on recording and describing linear sequences.

Age 11 to 16

Challenge Level

Can you do a little mathematical detective work to figure out which number has been wiped out?

Age 11 to 16

Challenge Level

This problem offers an engaging context in which to apply knowledge of factors, multiples and prime factor decomposition.

Age 11 to 16

Challenge Level

The items in the shopping basket add and multiply to give the same amount. What could their prices be?

Age 11 to 16

Challenge Level

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

Age 11 to 16

Challenge Level

Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?

Age 11 to 18

Challenge Level

There is nothing half so much worth doing as simply messing about in boats...

Age 14 to 16

Challenge Level

This problem offers students a chance to develop strategies for organising and understanding mixed up information within the context of calculating areas and perimeters of rectangles.

Age 14 to 16

Challenge Level

As well as introducing the difference of two squares, this problem allows students to explore, conjecture and use algebra to justify their results.

Age 14 to 16

Challenge Level

This problem allows students to consolidate their understanding of how to calculate the area of irregular shapes, while offering an opportunity to explore and discover an interesting result.

Age 14 to 16

Challenge Level

Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.

Age 14 to 16

Challenge Level

Use the differences to find the solution to this Sudoku.

Age 14 to 16

Challenge Level

Can you create a Latin Square from multiples of a six digit number?