There are 366 different Starters of The Day, many to choose from. You will find in the left column below some starters on the topic of Problem Solving. In the right column below are links to related online activities, videos and teacher resources.

A lesson starter does not have to be on the same topic as the main part of the lesson or the topic of the previous lesson. It is often very useful to revise or explore other concepts by using a starter based on a totally different area of Mathematics.

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### Problem Solving Starters:

Add 'em: Add up a sequence of consecutive numbers. Can you find a quick way to do it?

All for 100: Can you write an expression for 100 which uses all the digits 1 to 9?

Area Two: How many different shapes with an area of 2 square units can you make by joining dots on this grid with straight lines?

Bizarre Triangle: By how much would the area of this triangle increase if its base was enlarged to 8cm?

Boxing Day: Work out the contents and the cost of the Christmas boxes from the given clues

Broken Calculator: Use only the 1, 5 and 0 keys on a calculator to make given totals.

Calculator Nim: A game in which players take turns to add a single-digit number to what is already in the calculator. The winner is the player who makes the display show 30.

Cars: Calculate the total cost of four cars from the information given.

Chin-Ups: Work out the number of chin ups the characters do on the last day of the week give information about averages.

Christmas Presents: Work out the total cost of five Christmas presents from the information given.

ClockEquate: Can you use the digits on the left of this clock along with any mathematical operations to equal the digits on the right?

Coins On The Table: A puzzle about the number of coins on a table given information about fractions of them.

Consecutive: Three consecutive numbers multiplied together give a given product. Pupils are asked to figure out what the numbers are.

Consecy Brothers: Which three consecutive numbers multiplied together give the given answer.

Cracked Clock: How did the clock break if the numbers on each of the pieces added up to the same total?

Cube Calendar: What numbers should be on each face of the two cubes to make this perpetual calendar?

Dice Nets: Determine whether the given nets would fold to produce a dice.

Digivide: Arrange the digits 1 to 6 to make a three digit number divided by a two digit number giving a one digit answer.

Family Buses: Fit families onto eleven seater buses without splitting up the families.

Faulty Button: Find out which of the calculator keys is faulty from the given information. A mathematical puzzle requiring good problem solving strategies.

For Thought: Add up the answers to the four real life questions.

Four Gone: An activity involving a broken calculator which is missing the four button. Can you evaluate the given expressions without using the four?

Four to Seven: Which of the numbers from one to twenty can you make with the digits 4, 5, 6 and 7?

Four's Independence: A clock face containing only the number 4. Can you make a clock face containing any other single number?

Freemason's Cipher: Find symmetric words in this ancient cipher.

Giraffe: The height of this giraffe is three and a half metres plus half of its height. How tall is the giraffe?

Handshakes: If all the students in this room shook hands with each other, how many handshakes would there be altogether?

Hole in One: If six girls can plant 90 trees in a day. How many trees can ten girls plant in a day? The unitary method.

How Do You Do?: A little lateral thinking will help you solve this number puzzle.

How Many Triangles? 1: How many Triangles can you find in the diagram?

How Many Triangles? 2: How many triangles are hidden in the pattern? What strategy might you use to count them all to ensure you don't miss any out?

How Many Triangles? 3: Find a systematic way of counting the number of triangles in the given diagram.

Justundera Quid: Find a word whose letters would cost exactly ninety nine pence.

Know Weigh: Find the weight of one cuboid (by division) of each colour then add your answers together.

Letter Clue Calculations: Work out what the calculations might be from the letter clues.

Light Shopping: A lamp and a bulb together cost 32 pounds. The lamp costs 30 pounds more than the bulb. How much does the bulb cost?

Match Fish: A classic matchstick puzzle designed to challenge your spacial awareness.

Missing digits: Work out what the missing digits in the calculations are.

Missing Pound: A puzzle about a restaurant bill. Exactly where did the missing pound go?

Nine Digit Sum: Arrange the digits one to nine to make a correct addition calculation.

Nine Nine Nine: Arrange the numbers 1-9 to make three 3 digit numbers that add up to 999.

Not Too Close: This activity requires eight students to sit non consecutively on a grid of chairs.

One Digit 100: How many ways can you write an expression for 100 which only uses the same digit repeated and any operations?

One Torch Tunnel: Work out the least amount of time for four people to walk through a tunnel?

Peanuts and Buttons: Two questions involving estimating a quantity.

Pentadd: Five numbers are added together in pairs and the sums shown. What might the five numbers be?

Plane Numbers: Arrange numbers on the plane shaped grid to produce the given totals

Product Placement: Arrange the numbers to produce the largest product.

Pyramid Puzzle: Arrange numbers at the bottom of the pyramid which will give the largest total at the top.

Rabbits and Chickens: There are some rabbits and chickens in a field. Calculate how many of each given the number of heads and feet.

Rail Weigh: Record the weights of the trains to work out the weight of a locomotive and a coach. A real situation which produces simultaneous equations.

Render Digitful: Find a calculation for the current year which uses all of the digits 1 to 9.

Roundabout: Go around the roundabout performing each of the operations. Which starting point gives the largest answer?

Rows of Roses: Can you draw 4 straight lines, without taking your pencil off the paper, which pass through all 9 roses?

Santa's Sleigh: Work out the number of clowns and horses given the number of heads and feet.

Sea Shells: A question which can be best answered by using algebra.

Shadow Sums: Make sums from the three digit numbers given.

Shunting Problem: Allow two trains to pass by using the limited amount of siding space.

Siam Symbols: Can you work out what each of the strange symbols represents in these calculations?

Simple Nim: The classic game of Nim played with a group of pens and pencils. The game can be extended to the multi-pile version.

Simultaneous Occasions: A problem which can best be solved as a pair of simultaneous equations.

Six Discrimination: An activity involving a calculator which is missing the six button. Can you evaluate the given expressions without using the six?

Small Satisfaction: Arrange the digits one to nine in the grid so that they obey the row and column headings.

Snowman and Woman: How many different ways can you spell out the word "Snowman" by moving from snowflake to snowflake.

Spinsum: Arrange the numbers on the grid of squares so that the totals along each line of three squares are equal.

Square and Even: Arrange the numbers on the cards so that each of the three digit numbers formed horizontally are square numbers and each of the three digit numbers formed vertically are even.

Square Pairs: Arrange the numbered trees so that adjacent sums are square numbers.

Squared Animals: Separate three rows of three animals using three squares.

Stair Perimeter: Use the information implied in the diagram to calculate the perimeter of this shape.

Stencils: Which of the coloured stencils will fit over the numbered card to produce correct calculations?

Step Perimeter: Is it possible to work out the perimeter of this shape if not all the side lengths are given?

Suko: Interactive number-based logic puzzle similar to those featuring in The Times and Telegraph newspapers.

Sum of the Signs: Each traffic sign stands for a number. Some of the sums of rows and columns are shown. What numbers might the signs stand for?

Sum Square: Arrange the numbers one to eight into the calculations to make the totals correct..

Team Age: Work out who is in which team from the information given.

Tindice: How can you put the dice into the tins so that there is an odd number of dice in each tin?

Tran's Hats: In how many different ways might Tran decide to wear his hats in one week?

Two Twins and Tim: A tricky problem set on a coordinate grid.

Unmagic Square: Arrange the numbers 1 to 9 in a 3 by 3 grid so that none of the line totals are the same.

Word Sum: Each letter stands for a different digit. Can you make sense of word sum?

Biased Coin: Use a biased coin to obtain a fair result

Cheryl's Birthday: Use a process of elimination to work out the correct date from the clues given.

Cuboid: Find the dimensions of a cuboid matching the description given

Difference Cipher: Find the mathematical word from the cipher

Fence Optimisation: Find the length of a rectangle enclosing the largest possible area.

Fermi Problem: The classic Fermi problem using standard estimation techniques

Hands Together: The hands of a clock are together at midnight. At what time are they next together?

Multiple of sum of digits: What number is six times the sum of its digits?

Road Connections: Design roads to connect four houses that are on the corners of a square, side of length one mile, to minimise the total length of the roads.

Same Series Sum: Find an arithmetic series and a geometric series that have the same sum of the first five terms.

Same Three Digits: Find expressions which contain indices and evaluate to numbers containing the same digits.

#### Xmas Presents

Work out the cost of the Christmas presents from the information given.

Transum.org/go/?to=presents

### Curriculum for Problem Solving:

#### Year 5

Pupils should be taught to solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why more...

Pupils should be taught to use all four operations to solve problems involving measure [for example, length, mass, volume, money] using decimal notation, including scaling more...

Pupils should be taught to solve problems involving multiplication and division including using their knowledge of factors and multiples, squares and cubes more...

Pupils should be taught to solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the equals sign more...

Pupils should be taught to solve problems involving number up to three decimal places more...

Pupils should be taught to solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates more...

#### Year 6

Pupils should be taught to solve problems involving the relative sizes of two quantities where missing values can be found by using integer multiplication and division facts more...

Pupils should be taught to solve problems involving the calculation of percentages [for example, of measures, and such as 15% of 360] and the use of percentages for comparison more...

Pupils should be taught to solve problems involving unequal sharing and grouping using knowledge of fractions and multiples. more...

Pupils should be taught to find pairs of numbers that satisfy an equation with two unknowns more...

Pupils should be taught to solve problems involving addition, subtraction, multiplication and division more...

Pupils should be taught to use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy. more...

Pupils should be taught to solve problems which require answers to be rounded to specified degrees of accuracy more...

### Feedback:

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"I think this is a brilliant website as all the students enjoy doing the puzzles and it is a brilliant way to start a lesson."

Comment recorded on the 1 May 'Starter of the Day' page by Phil Anthony, Head of Maths, Stourport High School:

"What a brilliant website. We have just started to use the 'starter-of-the-day' in our yr9 lessons to try them out before we change from a high school to a secondary school in September. This is one of the best resources on-line we have found. The kids and staff love it. Well done an thank you very much for making my maths lessons more interesting and fun."

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Best wishes from Inger Kisby"

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### Notes:

What good is being a master of calculation if you cannot apply your skills to problem solving? This topic provides lots of examples, activities and situations in which pupils can practise their problem solving skills.

### Problem Solving Teacher Resources:

Countdown: How close can you get to the target by making a calculation out of the five numbers given?

Ice-cream Combinations: How many two-scoop ice creams can you make from the given flavours?

Satisfaction: This is quite a challenging number grouping puzzle requiring a knowledge of prime, square and triangular numbers.

Word Sums Guide: A step by step guide showing how to solve a Word Sum where each letter stands for a different digit.

### Problem Solving Activities:

Area and Perimeter of a Rectangle: Questions on the areas and perimeters of rectangles which will test your problem solving abilities.

Area Maze: Use your knowledge of rectangle areas to calculate the missing measurement of these composite diagrams.

Area Wall Puzzles: Divide the grid into rectangular pieces so that the area of each piece is the same as the number it contains.

Arithmagons: An extensive collection of Arithmagons arranged in levels according to their features. Find the missing numbers in these online, self-checking exercises and discover the wonders of Arithmagons.

Bidmaze: Find your way through the maze encountering mathematical operations in the correct order to achieve the given total.

Brainbox: A puzzle requiring the arrangement of numbers on the function machines to link the given input numbers to the correct output.

Broken Calculator: Some of the buttons are missing from this calculator. Can you make the totals from 1 to 20?

Car Park Puzzle: Can you get your car out of the very crowded car park by moving other cars forwards or backwards?

Centexpression: Arrange the numbers from 1 to 9 to make an expression with a value of 100.

Clue Sudoku: A different way to complete a Sudoku puzzle with clues available at every stage.

Code Cracker: Crack the code by replacing the encrypted letters in the given text. There are lots of hints provided about code breaking techniques.

Consecutive Numbers: Find the consective numbers that are added or multiplied to give the given totals

Cracked Clock Quiz: A self marking set of ten mathematical questions about a clock which cracked!

Digivide: Arrange the numbers from 1 to 6 in the spaces to make the division calculation correct.

Divisive: Arrange the digits one to nine on the spaces provided to make two division calculations containing multiples of three.

Dominoes Puzzle: Arrange the dominoes in seven squares. The number of dots along each side of the square must be equal to the number in the middle

Double Treble: Arrange the digits to make three 3 digit numbers such that the second is double the first and the third is three times the first.

Fleur-De-Lis: Click on six fleur-de-lis to leave an even number in each row and column.

Four Sum: Arrange the given number tiles to make two 2 digit numbers that add up to the given total.

Go Figure: Arrange the digits one to nine to make the four calculations correct.

Green Fingers: Choose the amount of liquid from each bottle needed to make the watermelon grow as big as possible.

How Many of Each?: Work out how many items were bought from the information given.

How Many Triangles?: A self marking step by step approach to calculating the number of triangles in a design.

Ice-cream Combinations: How many two-scoop ice creams can you make from the given flavours?

Jugs: Can you make 4 litres if you only have 7 and 5 litre jugs?

Largest Product: A drag and drop activity challenging you to arrange the digits to produce the largest possible product.

Ludicross: Arrange the given numbers on the cross so that the sum of the numbers in both diagonals is the same.

Magic Square: Each row, column and diagonal should produce the same sum.

Mine Find: Find where the mines are hidden without stepping on one.

Mixpressions: Arrange the cards to create a valid mathematical statement.

Multi-step Problems: Solve multi-step problems in contexts, deciding which operations and methods to use and why.

Multitude: Arrange the given digits to make three numbers such that the third is the product of the first and the second.

Nine Digits: Arrange the given digits to make three numbers such that two of them add up to the third.

Nine Nine Nine: Use the digits 1 to 9 to make three 3 digit numbers which add up to 999.

Not Too Close: The students numbered 1 to 8 should sit on the chairs so that no two consecutively numbered students sit next to each other.

Number Jigsaws: Online, interactive jigsaw puzzles of grids of numbers.

Numskull: Interactive, randomly-generated, number-based logic puzzle designed to develop numeracy skills.

One Digit Only: Find expressions using only one digit which equal the given targets.

One Torch Tunnel: Solve the problem of getting four people through a tunnel with one torch in the minimum amount of time.

Pancake Day: Toss the pancakes until they are neatly stacked in order of size. Find how to do this using the smallest number of moves.

Pentadd Quiz: Find the five numbers which when added or multiplied together in pairs to produce the given sums or products.

Pentominoes: Arrange the twelve pentominoes in the outline of a rectangle.

Pick Up Sticks: If you were to pick up the sticks from this pile so that you were always removing the top stick what calculation would you create?

Plane Numbers: Arrange numbers on the plane shaped grid to produce the given totals

Plus: A number arranging puzzle with seven levels of challenge.

Polygon Pieces: Arrange the nine pieces of the puzzle on the grid to make different polygons.

Prime Square: Drag the numbers into the red cells so that the sum of the three numbers in each row and each column is a prime number.

Puzzle Cube Net: A jumbled moving-block puzzle cube is shown as a net. Can you solve it?

Pyramid Puzzle: Numbers in the bricks are found by adding the two bricks immediately below together. Can you achieve the given target?

River Crossing: The traditional River Crossing challenge. Can you do it in the smallest number of moves?

Roman Numerals Jigsaw: An online interactive jigsaw puzzle of a grid of Roman numerals.

Satisfaction: This is quite a challenging number grouping puzzle requiring a knowledge of prime, square and triangular numbers.

Satisfy: Place the nine numbers in the table so they obey the row and column headings about the properties of the numbers.

Scheduling Puzzle: Make a schedule for the 24-hour Darts Marathon which will take into account everyone's requests and keep everyone happy.

Scouts in Boats: Arrange a rota for the Scouts to travel in boats so that they are with different people each day.

Sheep Herding: Arrange the sheep in the field according to the instructions. An introduction to loci.

Shunting Puzzles: Move the trams to their indicated parking places in the shunting yard as quickly as possible.

Six Discrimination: The six button has dropped off! How could these calculations be done using this calculator?

Spinsum: Arrange the numbers on the squares so that the totals along each line of three squares are equal.

Square and Even: Arrange the numbers on the cards so that each of the three digit numbers formed horizontally are square numbers and each of the three digit numbers formed vertically are even.

Squorder: The Transum version of the traditional sliding tile puzzle.

Stable Scales: Ten balance puzzles to prepare you for solving equations.

Stamp Sticking: Drag stamps onto the envelopes to make the exact postage as shown at the top left of each envelope.

Suko Sujiko: Interactive number-based logic puzzles similar to those featuring in daily newspapers.

Superfluous: Find a strategy to figure out the values of the letters used in these calculations.

T Puzzle: Use the pieces of the T puzzle to fit into the outlines provided. A drag, rotate and drop interactive challenge.

Tangram Table: Use the pieces of the tangram puzzle to make the basic shapes then complete the table showing which shapes are possible.

Tantrum: A game, a puzzle and a challenge involving counters being placed at the corners of a square on a grid.

The Miller's Puzzle: This is an interactive version of the puzzle described by Henry Ernest Dudeney in The Canterbury Puzzles

Thrice: Can you arrange all of the counters on the grid to form 10 lines of three counters?

Tools: In how many different ways can the numbers be arranged to give the same totals?

Tower of Hanoi: Move the pieces of the tower from one place to another in the minimum number of moves.

Trafalgar Square: Solve the number puzzles drawn on the pavement of Trafalgar Square in London.

Tran Towers: An adventure game requiring students to solve puzzles as they move through the old mansion.

Tran Tunnels: An adventure game requiring you to answer questions and solve puzzles as you move through the tunnels.

Triside Totals: Arrange the digits 1 to 9 on the triangle so that the sum of the numbers along each side is equal to the given total.

Unitary Method: Ten questions which can be solved using the unitary method.

Unmagic Square: Like the magic square but all of the totals should be different.

Vector Cops: Help the cops catch the robbers by finding the vectors that will end the chase.

Where's Wallaby?: Find the hidden wallaby using the clues revealed at the chosen coordinates.

Window View: Drag the 20 flowers into the gardens so that 9 flowers are visible from each window of the house.

Xmas Presents: Work out the cost of the Christmas presents from the information given.

Finally there is Topic Test, a set of 10 randomly chosen, multiple choice questions suggested by people from around the world.

### Problem Solving Investigations:

Aunt Sophie's Post Office: Investigate the ways of making up various postage amounts using 3p and 8p stamps. An online stamp calculator is provided for you to check your working.

Braille: Investigate the possibility of redesigning the Braille alphabet to make it easier to learn.

Crossing the River: Two men and two boys want to cross a river and they only have one canoe which will only hold one man or two boys.

Green Fingers: Choose the amount of liquid from each bottle needed to make the watermelon grow as big as possible.

How Many Rectangles?: Investigate the number of rectangles on a grid of squares. What strategies will be useful in coming up with the answer?

Jugs: Can you make 4 litres if you only have 7 and 5 litre jugs?

Search for Infinity: Manipulate the Lissajou curve to produce a perfectly symmetrical (vertically and horizontally) infinity symbol.

Tantrum: A game, a puzzle and a challenge involving counters being placed at the corners of a square on a grid.

Tower of Hanoi: Move the pieces of the tower from one place to another in the minimum number of moves.

Window View: Drag the 20 flowers into the gardens so that 9 flowers are visible from each window of the house.

Featured Investigations

Anti-Calculator: A puzzle about a two digit number divided by the sum of its digits.

### Problem Solving Videos:

Problems with Zero: Dividing by zero, zero divided by zero and zero to the power of zero - all pose problems!

Puzzles In The Simpsons: The Simpson's 26th season finale. 'Mathlete's Feat' is full of mathematical problems that Presh solves.

The Train Fly Problem: This video from Mind Your Decisions teaches a trick so you can solve the puzzle quickly.

### Problem Solving Worksheets/Printables:

How Many Rectangles?: A worksheet containing many copies of the How Many Rectangles Starter diagram allowing students to record their findings.

How Many Squares? 1: A printable grid containing many copies of the design used in the shape counting Starter.

How Many Squares? 2: A printable grid containing many copies of the design used in the second shape counting Starter.

Lemon Law Worksheet: Can you find the solution to these Lemon Law challenges using a spreadsheet?

Line Drawings: Six line drawings that may or may not be able to be traced without lifting the pencil or going over any line twice.

Missing Operations Worksheet: Each red box represents a missing operation (plus, minus, times or divide). Can you work out what they are?

Pyramid Worksheet: Put the numbers 1 to 5 in the bottom row of the pyramid then each other brick is the sum of the two below.

Siam Symbols Worksheet: Work out which digit each of the ten symbols represents from the six given calculations.

Tools Worksheet: A worksheet containing the grids to fill in for the Tools puzzles.

Two Twins Printable: This worksheet extends the puzzle in the July 21st Starter of the Day.

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#### Area and Perimeter of a Rectangle

Questions on the areas and perimeters of rectangles which will test your problem solving abilities.

Transum.org/go/?to=oblongs

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