maths problem solving activities year 6

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Year 6 Maths Worksheets UK Hub Page

Welcome to our Year 6 Maths Worksheets area.

Here you will find a wide range of free printable Year 6 Maths Worksheets for your child to enjoy.

Come and take a look at our rounding decimal pages, or maybe some of our adding and subtracting fractions worksheets. Perhaps you are looking for some worksheets about finding angles in a triangle, or need some ratio problem worksheets to help your child learn about ratio?

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• This page contains links to other Math webpages where you will find a range of activities and resources.
• If you can't find what you are looking for, try searching the site using the Google search box at the top of each page.

Year 6 Maths Learning

Here are some of the key learning objectives for the end of Year 6:

• know and use Place value up to 10 million
• Counting on and back in steps of powers of 10 from any number up to 10 million
• Round numbers to any given degree of accuracy.
• Count forwards and backwards through zero with positive and negative numbers.
• Read Roman numerals to 1000 and recognise years written in Roman numerals
• solve multi-step problems using addition and subtraction in a range of contexts
• identify multiples and factors including common factors
• multiply and divide up to 4-digit numbers by up to 2 digits
• Use their knowledge of the order of operations to carry out calculations involving the four operations.
• Identify common factors, common multiples and prime numbers.
• solve problems involving addition, subtraction, multiplication and division
• simplify fractions
• compare and order fractions including mixed numbers
• add and subtract fractions with different denominators including mixed numbers
• multiply simple fractions together and simplify the answer
• divide proper fractions by whole numbers
• recall and use equivalence between simple fractions, decimals and percentages.
• Multiply and divide whole numbers and decimals up to 3dp by 10, 100 or 1000
• read, write, order and compare numbers up to 3dp
• round decimals with up to 3dp to the nearest whole
• solve problems with numbers up to 3dp
• work out percentages of different amounts
• solve problems using percentages
• use simple formulae
• express missing number problems using algebra
• find pairs of numbers that satisfy equations with two variables
• solve problems involving simple ratios
• solve problems involving similar shapes where the scale factor is known
• use, read, write and convert between standard units of measure
• measure, compare and calculate using different measures
• know that shapes with the same area can have different perimeters
• find the area of parallelograms and right triangles
• find the volume of cubes and cuboids
• convert between miles and km
• name and understand the parts of circles - radius, diameter and circumference
• draw 2D shapes accurately using dimensions and angles
• compate and classify 2D shapes by a range of properties
• find missing angles in triangles, quadrilaterals and regular shapes
• use coordinates in all 4 quadrants
• draw and translate simple shapes in all 4 quadrants
• interpret and construct pie charts and line graphs
• calculate the mean as an average

Please note:

Our site is mainly based around the US Elementary school math standards.

Though the links on this page are all designed primarily for students in the US, but they are also at the correct level and standard for UK students.

The main issue is that some of the spelling is different and this site uses US spelling.

Year 6 is generally equivalent to 5th Grade in the US.

On this page you will find link to our range of math worksheets for Year 6.

Quicklinks to Year 6 ...

• Place Value Zone
• Mental Math Zone

Word Problems Zone

Fractions percents ratio zone.

• Percentages Zone
• Measurement Zone

Geometry Zone

Data analysis zone.

• Fun Zone: games and puzzles

Coronavirus Stay At Home Support

For those parents who have found themselves unexpectedly at home with the kids and need some emergency activities for them to do, we have started to develop some Maths Grab Packs for kids in the UK.

Each pack consists of at least 10 mixed math worksheets on a variety of topics to help you keep you child occupied and learning.

The idea behind them is that they can be used out-of-the-box for some quick maths activities for your child.

They are completely FREE - take a look!

• Free Maths Grabs Packs

Place Value & Number Sense Zone

Year 6 number worksheets.

Here you will find a range of Free Printable Year 6 Number Worksheets.

Using these Year 6 maths worksheets will help your child to:

• use place value with numbers up to 10 million;
• use place value with up to 3 decimal places;
• understand how to use exponents (powers) of a number;
• understand and use parentheses (brackets);
• understand and use multiples and factors;
• extend their knowledge of prime and composite (non-prime) numbers up to 100;
• know and be able to use the PEMDAS (or PEDMAS) rule.
• Place Value Worksheets to 10 million
• Place Value to 3dp
• Ordering Decimals Worksheets
• PEMDAS Rule Support Page
• PEMDAS Problems Worksheets
• Balancing Math Equations
• Roman Numerals worksheets

Ordering Large Numbers and Decimals to 3dp

The sheets in this section involve ordering lists of decimals to 3 decimal places and also large numbers up to 100 million.

There are sheets with decimals up to 10, and also sheets with numbers from -10 to 10.

• Ordering Large Numbers up to 100 million
• Ordering Decimals to 3dp

Rounding Decimals

• Rounding to the nearest tenth
• Rounding Decimal Places Sheets to 2dp
• Rounding Decimals Worksheet Challenges

Year 6 Decimal Counting Worksheets

Using these sheets will support you child to:

• count on and back by multiples of 0.1;
• fill in the missing numbers in sequences;
• count on and back into negative numbers.
• Counting By Decimals

Year 6 Mental Maths Zone

Each worksheet tests the children on a range of math topics from number facts and mental arithmetic to geometry, fraction and measures questions.

A great way to revise topics, or use as a weekly math quiz!

• Year 6 Mental Maths Tests

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Year 6 Addition Worksheets

• add decimals including tenths and hundredths mentally;
• add a columns of multi-digit numbers, including decimals.
• Decimal Addition Fact Worksheets
• 5th Grade Addition Worksheets BIG Numbers
• Decimal Column Addition Worksheets
• Money Worksheets (randomly generated)

Year 6 Subtraction Worksheets

Using these sheets will help your child to:

• subtract decimals including tenths and hundredths mentally;
• subtract multi-digit numbers, including decimals using column subtraction.
• Subtracting Decimals Worksheets (mental)
• Subtraction Worksheets up to Billions (columns)
• Column Subtraction with Decimals

Year 6 Multiplication Worksheets

• extend their knowlege of multiplication to decimals;
• use their multiplication tables to answer related facts, including decimals;
• multiply a range of decimals with up to 2 decimal places (2dp) by a whole number;
• multiply different money amounts by a whole number.
• Multiplying Decimals by 10 and 100
• Multiplication Fact Sheet Decimals
• Decimal Multiplication Worksheets to 1dp
• Decimal Multiplication Worksheets to 2dp
• Free Multiplication Worksheets (randomly generated)
• Multiply and Divide by 10 100 (decimals)
• Multiplication & Division Worksheets (randomly generated)
• Multiplication Word Problems

Division Worksheets 5th Grade

Using these Year 6 maths worksheets will help your child learn to:

• divide any whole number up to 10000 by a two digit number;
• express any division with a remainder in the form of a mixed number (a number with a fraction part).
• Long Division Worksheets (whole numbers)
• Long Division of Decimal Numbers
• Decimal Division Facts
• Division Facts Worksheets (randomly generated)

Year 6 Maths Problems

• apply their addition, subtraction, multiplication and division skills;
• apply their knowledge of rounding and place value;
• solve a range of problems including "real life" problems and ratio problems.

These sheets involve solving one or two more challenging longer problems.

• Year 6 Math Problems (5th Grade)

These sheets involve solving many 'real-life' problems involving data.

• Year 6 Math Word Problems (5th Grade)

These sheets involve solving a range of ratio problems.

Year 6 Fraction Worksheets

Year 6 percentage worksheets, year 6 ratio worksheets.

• compare and order fractions;
• add and subtract fractions and mixed numbers;
• understand how to multiply fractions by a whole number;
• understand how to multiply two fractions together, including mixed fractions;
• understand the relationship between fractions and division;
• know how to divide fractions and mixed fractions;
• convert decimals to fractions.
• Comparing Fractions Worksheet page
• Adding Fractions Worksheets
• Adding Improper Fractions
• Subtracting Fractions Worksheets
• Adding Subtracting Fractions Worksheets
• Improper Fraction Worksheets
• Converting Decimals to Fractions Worksheets
• Fractions Decimals Percents Worksheets
• Multiplying Fractions Worksheets
• Dividing Fractions by Whole numbers
• Divide Whole numbers by Fractions
• Simplifying Fractions Worksheets
• Free Printable Fraction Riddles (harder)

Take a look at our percentage worksheets for finding the percentage of a number or money amount.

We have a range of percentage sheets from quite a basic level to much harder.

• Percentage of Numbers Worksheets
• Money Percentage Worksheets
• Percentage Word Problems

These Year 6 Ratio worksheets are a great way to introduce this concept.

We have a range of part to part ratio worksheets and slightly harder problem solving worksheets.

• Ratio Part to Part Worksheets
• Ratio and Proportion Worksheets

Year 6 Geometry Worksheets

• know how to find missing angles in a range of situations;
• learn the number of degrees in a right angle, straight line, around a point and in a triangle;
• know how to calculate the area of a triangle;
• know how to calculate the area of a range of quadrilaterals.
• learn the formulas to calculate the area of triangles and some quadrilaterals;
• write and plot coordinates in all 4 quadrants.
• (5th Grade) Geometry - Angles
• Area of Quadrilaterals
• 5th Grade Volume Worksheets
• Coordinate Worksheets (1st Quadrant)
• Coordinate Plane Worksheets (All 4 Quadrants)
• Parts of a Circle Worksheets

Measurement Zone, including Time & Money

Year 6 measurement worksheets.

Using these sheets will help your child understand how to:

• learn how to read a standard scale going up in different fractions: halves, quarters, eighths and sixteenths;
• learn how to read a metric scale going up in 0.1s, 5s, 10s, 25s, 50s & 100s;
• learn how to estimate a measurement of length, weight or liquid;
• convert temperatures in Celsius and Fahrenheit.
• (5th Grade) Measurement Worksheets

Time Puzzles - harder

Here you will find our selection of harder time puzzles.

• Time Word Problems Worksheets - Riddles (harder)

Using these sheets will help you to:

• find the mean of up to 5 numbers;
• find a missing data point when the mean is given.
• Mean Worksheets

Fun Zone: Puzzles, Games and Riddles

Year 6 maths games.

• Year 6 Math Games (5th Grade)

Year 6 Maths Puzzles

The puzzles will help your child practice and apply their addition, subtraction, multiplication and division facts as well as developing their thinking and reasoning skills in a fun and engaging way.

• Printable Math Puzzles

Math Salamanders Year 6 Maths Games Ebook

Our Year 6 Maths Games Ebook contains all of our fun maths games, complete with instructions and resources.

This ebooklet is available in our store - use the link below to find out more!

• Year 6 Maths Games Ebook

Other UK Maths Worksheet pages

See below for our other maths worksheets hub pages designed for children in the UK.

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National Curriculum In England: Mathematics Programmes Of Study

Students are entering a new section of math that consists of algebra, geometry, decimals, and more complicated division problems. When there is not a parent or teacher around, students have access to on-demand videos for their sixth-grade mathematical lessons. Our video tutorials are taught by math teachers who go through the problem solving process.

• The key to learning math is repetition, and our math teachers go through multiple examples. 
• Math teachers show students step-by-step directions on how to solve these more complex problems.
• Students learn money, ratios, decimals, graphing, and other concepts from the 6th grade curriculum.

Teachers | Pupils | Parents

Collect 1 Mission Credit for each correct Fluency answer or solution…

Collect 3 Mission Credits for each correct iQ answer or solution…

Please use the search bar to filter and find the learning objective required.

Welcome to the Year 6 Mini-Maths homepage. Daily maths questions for Year 6 are available for every week of the year, providing free help for teachers, pupils and parents. 1000s of free tasks and activities are available to support children in their learning both in the classroom and at home. Use the search and filter box below to easily find the outcomes you require.

It is recommended that a topic is selected per week and daily maths tasks accessed using the links provided below. Use each day’s answers to help identify and guide the required support in preparation for the following day’s task. Watch your understanding, application and confidence grow throughout the week. New resources are always being added, so keep checking for updates.

More about Year 6 Maths Questions …

The Year 6 curriculum completes the Key Stage 2 maths programme of learning bringing understanding and confidence together from Year 3 through to the end of Year 6. This journey culminates in the KS2 SATs assessments where learners are able to demonstrate their progress and development in the subject.

Year 6 NUMBER …

The four operations reach a natural conclusion in Year 6 – addition, subtraction, multiplication and division involve more complex examples before introducing decimals values . Learners are looking to extend their understanding of units, tens, hundreds, thousands, etc, with the Place value of digits in decimal places, supporting the wider curriculum such as calculating decimal remainders using short division.

Year 6 SHAPE …

Learners have been developing a sound understanding of 2D shape and 3D shape allowing them to sort and classify by their properties. Being able to identify and use these properties also improves an awareness of a shape’s area and perimeter or volume .

Year 6 DATA …

Year 5 offers further coverage of tables, charts and graphs, where learners are expected to both read and interpret information in all types of graphs and charts including pie charts . Drawing graphs and charts from given data is also introduced in Year 6 maths.

All learning objectives are supported by a Home Learning pack and iQ mastery questions challenge learners’ understanding, application and confidence. The Home Learning packs provide first class opportunities for effective intervention and an ever growing library of help videos for Year 6 maths is available on the MyMiniMaths YouTube channel in support of quality learning.

Year 6 SATs …

MyMiniMaths also offers specialised support for Year 6 SATs with a programme of practice KS2 SATs papers for the arithmetic paper . Easy identification of key skills in need of further support to maximise attainment is available through the Arithmetic Target Questions .

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Year 6 Maths Problems

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

22 February 2018

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A bundle is a package of resources grouped together to teach a particular topic, or a series of lessons, in one place.

Year 6 Maths Scheme of Work (based on White Rose Maths)

A scheme of learning with planning and worksheets included for Year 6 Maths. Included are 90 lessons that cover: Place Value Rounding Negative Numbers Roman Numerals Indices Inverse operations Written addition and subtraction Mental addition and subtraction Written multiplication and division Order of operations Finding fractions of amounts Comparing fractions Converting between improper and mixed number fractions Add and subtract fractions Multiply fractions together Multiply fractions by whole numbers Divide fractions by whole numbers Percentages Fraction, decimal and percentages Ratio Algebra including missing information, expressing problems, satisfying equations, satisfying two variables and sequences Multiplying and dividing by 10, 100 and 1000 Multiplying decimal numbers Finding percentages of amounts x 2 Fraction, decimal and percentage equivalents Converting measurements Miles and kms Time Shape with same area but different perimeter and vice versa Volume Area Area of triangles Addition and subtraction (Same as 1st planning scheme as it is revision) Multiplication (Same as 1st planning scheme as it is revision) Division (Same as 1st planning scheme as it is revision) Worksheets are differentiated three ways with a mastery aim running throughout. Having used a similar format in my own lessons, these sheets often provide enough challenge for two days lessons, with children taking on the easier work then moving on to harder challenges within the other sheets. These have all been made with mastery in mind and so follow this pattern. Worksheet One - Works on practise and fluency (LA) Worksheet Two - Practise, fluency and reasoning (MA) Worksheet Three - Practise, fluency, reasoning and problem solving (HA) Worksheet Four - Same as worksheet three as a spare for anybody who finishes problem solving Answers are included for each sheet.

Maths resources

A range of Maths resources including fully planned and resourced lessons that can all be used in Years 5 and 6.

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Reasoning/Problem Solving Maths Worksheets for Year 6 (age 10-11)

Problem solving.

A variety of problem solving activities including time and money.

A perimeter/area investigation.

A good knowledge of the order of operations is needed for this problem solving task.

Find the cost of different numbers of houseplants.

Tricky addition which may need jottings to complete.

Here are sets of word problems only needing one step to solve, but it is not always obvious what calculation to carry out to reach the answer.

Here are some longer problems written in words. You need to work out more than one thing to get the answers.

Quite tricky word problems involving money, often requiring several steps to complete.

A good knowledge of metric measures is needed to complete these questions.

Questions which need an understanding of a.m., p.m., 24 hour clock, leap years as well as using graphs.

An atlas with time zones is needed for the first activity and there are line graphs on time to interpret.

When answering these problems say which operation is needed and whether it can be done mentally, on paper or with a calculator.

More on choosing the most appropriate method of calculating.

Tricky calculations working out the best and worst meal deals.

Purchasing various items from a vending machine. Tricky calculating with money.

Investigate Numbers and Patterns

Investigate numbers and patterns, including sequences.

Quick and easy way to multiply 2-digit numbers by 11 - why does it work?

Show all your calculating skills with just four fours.

A first look at triangle numbers.

More on triangle numbers.

Further investigations involving triangle numbers.

Some great patterns can be found when dividing by 9 or 11, and using square numbers.

Plenty of ideas to help with investigations and inventing similar problems.

More investigations on adding consecutive numbers, dividing by a half, writing a simple formula as well as shape patterns.

Investigate numbers and shapes; quite tricky.

Complete a variety of number sequences and say what the rule for each sequence is. Some negative number work as well.

Using a multiplication square to look for number patterns: there are certainly plenty of them! Further sequence work, including Fibonacci.

Try the Ancient Egyptian method of multiplying. Great, if you are good at adding up!

Tips and hints on how to use a calculator to ensure correct answers.

Holiday Problem Solving

A tricky selection of resources all linked to a holiday theme.

Very tricky to find the best options.

Very tricky calculations to find the best deal.

Finding the best deal is harder than it might seem!

A holiday which might cost more than it should unless there is some careful calculating.

This is complicated with special offers and percentages off making finding the cheapest deal difficult to work out.

More difficult than you might think to find the cheapest deal, including percentages.

Much trickier than it first looks, with many possible options to choose from. Some time is needed to complete this correctly.

A collection of puzzles to solve - some easy and some quite tricky. Ideal for the end of term.

Place the numbers 1 to 9 in the boxes so that each side of the triangle adds up to 20.

Replace the hexagons in the calculation with numbers: logical thinking needed.

Find the mathematical words in these anagrams.

Turn your calculator upside down to read the words!

A calculator is needed for this fun challenge.

This might not seem possible but with a little imagination it can be done.

How many moves to get all the coins on heads?

Nifty little puzzle with more than one solution.

More to do with square numbers than shapes!

Finding pairs of numbers that add up to square numbers. Tricky!

How many ways of making 15. Have you got all of them?

Can you make prime numbers by adding two square numbers?

Lots of coins and not change for £1.

Neat little code-breaker. Can you come up with others?

Logical thinking needed here.

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Resources tagged with: NC Yr 6

There are 50 NRICH Mathematical resources connected to NC Yr 6 , you may find related items under NC .

4 by 4 Mathdokus

Can you use the clues to complete these 4 by 4 Mathematical Sudokus?

Different Deductions

There are lots of different methods to find out what the shapes are worth - how many can you find?

Name That Triangle!

Can you sketch triangles that fit in the cells in this grid? Which ones are impossible? How do you know?

Number Lines in Disguise

Some of the numbers have fallen off Becky's number line. Can you figure out what they were?

Price Match

Can you find pairs of differently sized windows that cost the same?

Finding 3D Stacks

Can you find a way of counting the spheres in these arrangements?

Extending Fraction Bars

Can you compare these bars with each other and express their lengths as fractions of the black bar?

More Fraction Bars

What fraction of the black bar are the other bars? Have a go at this challenging task!

Always, Sometimes or Never? Shape

Are these statements always true, sometimes true or never true?

Always, Sometimes or Never? Number

Round the Three Dice

What happens when you round these three-digit numbers to the nearest 100?

Round the Four Dice

This activity involves rounding four-digit numbers to the nearest thousand.

Use the information on these cards to draw the shape that is being described.

Dicey Perimeter, Dicey Area

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

Domino Sets

How do you know if your set of dominoes is complete?

Making Spirals

Can you make a spiral for yourself? Explore some different ways to create your own spiral pattern and explore differences between different spirals.

Round a Hexagon

This problem shows that the external angles of an irregular hexagon add to a circle.

Have a look at this data from the RSPB 2011 Birdwatch. What can you say about the data?

So It's Times!

How will you decide which way of flipping over and/or turning the grid will give you the highest total?

After training hard, these two children have improved their results. Can you work out the length or height of their first jumps?

Button-up Some More

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Counting Cogs

Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?

Doughnut Percents

A task involving the equivalence between fractions, percentages and decimals which depends on members of the group noticing the needs of others and responding.

Next Size Up

The challenge for you is to make a string of six (or more!) graded cubes.

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

What can you see? What do you notice? What questions can you ask?

Treasure Hunt

Can you find a reliable strategy for choosing coordinates that will locate the treasure in the minimum number of guesses?

First Connect Three

Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?

This challenge is a game for two players. Choose two of the numbers to multiply or divide, then mark your answer on the number line. Can you get four in a row?

Factor-multiple Chains

Can you see how these factor-multiple chains work? Find the chain which contains the smallest possible numbers. How about the largest possible numbers?

Fraction Fascination

This problem challenges you to work out what fraction of the whole area of these pictures is taken up by various shapes.

Triangles All Around

Can you find all the different triangles on these peg boards, and find their angles?

Diagonal Sums

In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?

Ten Hidden Squares

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

Orange Drink

A 750 ml bottle of concentrated orange squash is enough to make fifteen 250 ml glasses of diluted orange drink. How much water is needed to make 10 litres of this drink?

Each of the nets of nine solid shapes has been cut into two pieces. Can you see which pieces go together?

Sponge Sections

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.

Factor Lines

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

Would You Rather?

Would you rather: Have 10% of £5 or 75% of 80p? Be given 60% of 2 pizzas or 26% of 5 pizzas?

Plenty of Pens

Amy's mum had given her £2.50 to spend. She bought four times as many pens as pencils and was given 40p change. How many of each did she buy?

Mystery Matrix

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

The Moons of Vuvv

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

Where Are They?

Use the isometric grid paper to find the different polygons.

Rectangle Tangle

The large rectangle is divided into a series of smaller quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the shapes?

Pumpkin Pie Problem

Peter wanted to make two pies for a party. His mother had a recipe for him to use. However, she always made 80 pies at a time. Did Peter have enough ingredients to make two pumpkin pies?

Quadrilaterals

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

Two and Two

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

Making Cuboids

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

Round and Round the Circle

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Doplication

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

The home of mathematics education in New Zealand.

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Year 6 tasks

The Ministry is migrating nzmaths content to Tāhurangi.             Relevant and up-to-date teaching resources are being moved to Tāhūrangi (tahurangi.education.govt.nz).  When all identified resources have been successfully moved, this website will close. We expect this to be in June 2024.  e-ako maths, e-ako Pāngarau, and e-ako PLD 360 will continue to be available. 

For more information visit https://tahurangi.education.govt.nz/updates-to-nzmaths

To solve maths problems at year 6 it is very important for your child to be able to instantly recall addition and subtraction facts up to 20, and multiplication and division facts up to 10 x 10.

e-ako maths provides free online basic fact learning tools which will help your child learn their basic facts, as well as a collection of games to improve their speed and accuracy. If you think this would benefit your child, click to read how to access this resource  (show instructions).

• To access e-ako maths, your child will need an account. You can make a new account for free, but check first whether they already have one that they use at school. If they do, they can use the same username and password to use e-ako maths at home.
• You can log in to e-ako maths or register a new account from the e-ako maths home page . If you are registering a new account to use at home you will need to click the "No join code? Click here." button and then "I am learning at home".

• The two large green boxes at the top are 'Learning tools' which will test the basic facts your child knows, and then help them learn the ones they don't already know.
• The blue buttons below are a collection of games designed to practise basic facts to improve speed and accuracy. 
• In Year 3 your child should mostly be concentrating on learning and practising their addition and subtraction facts, the buttons on the left.

Click to read about the purpose of the tasks below  (show purpose).

The tasks in this section are intended:

• for students in year 6 
• to complement similar class tasks
• to be a selection only of enjoyable learning opportunities for your child (and for you) 
• to generate useful explanations and discussions between you and your child.

The tasks in this section are NOT: 

• intended to be comprehensive in their coverage of all maths at year 6
• drill and practice exercises to learn basic facts.

These Year 6 tasks recognise that students:

• are mostly working at level 3 of the NZ Curriculum, growing in their understanding of number, algebra, geometry, measurement and statistics
• are all different, and some find parts of maths easier or harder than others
• are learning a range of approaches to solving problems that often involve several steps
• learn from and appreciate tasks presented in a variety of ways
• respond well to supportive questions from adults, such as, ‘What is the question asking you to do?’, or ‘What do you already know that might help you (solve the problem)?’
• enjoy exploring open tasks for which there can be a range of appropriate approaches and solutions (accordingly, set answers are not included for a number of these tasks).

We suggest that you ask your child what they are learning in maths at school, and select a task from this menu.

Click for  versions with Māori content .

Measurement

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Year 6 Reasoning

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35 SATs Maths Questions And Answers With Worked Examples: Essential Maths Reasoning Practice For Year 6 [FREE]

Anantha Anilkumar

For Year 6, the toughest of their SATs maths questions are the reasoning questions. 

No matter how good pupils’ subject knowledge is or how effective your SATs revision lessons are, the examiners always manage to come up with maths questions that can baffle and flummox even the hardiest year 6 pupil.

To mitigate against this for SATs 2024 your pupils need plenty of exam practice and more specifically exam question practice to be familiar with both the types of SATs reasoning questions that can come up and the skills needed to answer them.

So to make life easy for you we’ve put together here a comprehensive collection of 35 SATs maths questions, organised by the sorts of question that pupils can expect to encounter.

All these practice SATs questions have been based on a mix of questions from past SATs papers , our own free year 6 maths SATs papers , and our collections of year 6 reasoning questions from the Rapid Reasoning resource collection. The answers are all taken from the answer sheets we provide for each test paper.

Other useful SATs revision resources

Before we get into the year 6 maths questions you might find it helpful to know that we have hundreds of other free KS2 SATs revision resources, including free SATs papers , and SATs intervention packs for those wanting to use the Year 6 SATs revision lessons we use in one to one tutoring in their own boosters.

You should also make sure you’re up to date with the latest analysis on the 20 most ‘valuable’ topics to study for SATs this year and also this analysis of the KS2 SATs Maths Papers Question Breakdown in 2023 .

You may also wish to read this guide to SATs first to give you some background information about the tests.

Why the focus on maths reasoning questions for SATs? 

Ever since the new national curriculum Key Stage 2 SATs in 2016, the emphasis in the all SATs papers has been very much on SATs reasoning questions. While there is one arithmetic paper, there are two reasoning papers; and the variety, breadth and level of challenge in the reasoning paper continues to impress us.

Most Year 6 pupils find the reasoning SATs maths questions the hardest part of these maths papers. Unsurprisingly! We teach thousands of pupils every week in the run up to SATs, and teaching them maths reasoning skills at KS2 is a big part of what we do.

We even recently took the decision to restructure our SATs lessons to introduce maths reasoning questions earlier in the learning journey as the level of challenge just at the end of the lesson was too high. So we feel the Year 6 teacher’s pain!

Whatever level pupils go on to perform at in maths, maths reasoning questions and numerical reasoning tests (such as those used by grammar and private schools) are likely to be a part of the practice they require.

If you find you have children in your class with much further to catch-up than the others then we we would be happy to support them with some personalised online one to one maths tuition .

35 SATs maths questions for KS2 year 6 SATs

For the KS2 SATs tests, there are 7 types of maths reasoning question that are likely to come up:

• Single step worded problems
• Multiple step worded problems
• Problems involving measures
• Problems involving drawing
• Explanation questions
• Sequence questions
• Ordering questions

For each of these types we’ll examine an example SATs maths question from a previous SATs paper, looking at the question, the correct answer, and how to go about answering this question.

We’ll also look at further examples of each type of maths reasoning questions and answers from Third Space’s Rapid Reasoning resource, again with worked examples and an explanation of how to answer each.

Finally, at the end of this article we provide links to further Year 6 maths questions, assessments and other SATs papers that you may find useful including plenty of arithmetic practice too.

Our aim is to provide you as part of your SATs practice with a sample of the types of KS2 SATs questions pupils can expect in the reasoning papers and how to teach the reasoning and problem solving  skills they’ll need to answer them.

For more word problems like this, check out our collection of 2-step and multi-step word problems for you as well as tips on how to use the bar model to answer Year 6 word problems . For advice on how to teach children to solve problems like this, check out these maths problem solving strategies.

SATs Maths Question Type 1: Single step worded problems

The simplest type of reasoning question pupils are likely to encounter in the reasoning papers, single step problems are exactly that: pupils are asked to interpret a written question and carry out a single mathematical step to solve it.

Have a look at the question below:

Reasoning Question 1

Answer: 65p

A relatively easy question to interpret and solve – the first step is to recognise £2 and £1.35 as equivalent to 200 and 135. From here the simple mathematical step is subtraction i.e. 200-135=65.

The most crucial skill for primary school pupils in this question is a solid understanding of money as relating to place value. If this understanding is present, the mathematical step itself is quite easy.

Below are several more examples, taken from Third Space Learning’s Rapid Reasoning resources:

Reasoning Question 2

Answer : 7 hours 24 minutes

Pupils need to understand that one hour is equal to 60 minutes. From here the single mathematical step is short division: 444/60, with a remainder.

Reasoning Question 3

Answer : 48 cm 3

Pupils must calculate length by breadth by height, using the figures provided by the question.

Reasoning Question 4

Answer : 124 cm

A simple enough calculation (doubling) if pupils are aware that the diameter is twice the radius.

Reasoning Question 5

Answer : 7,590

A single, relatively simple rounding problem – pupils should recognise that ’94’ is the operative part of this figure.

SATs Maths Question Type 2: Multiple step worded problems

A more complex version of the single step worded problem, multi-step problems require pupils to interpret a written problem, but solving it then requires the use of two or three maths skills,

For example, consider this question from the 2019 KS2 maths SATs:

Reasoning Question 6

Answer: £1.85

This question encompasses three different maths skills: multiplying (and dividing) mixed numbers, addition and subtraction. Pupils can choose to work out the multiplication or division first, but must complete both before moving on.

Once these values have been worked out the next steps are relatively simple – adding the two values together, and subtracting the total from £5.

Multi-step problems are particularly valuable to include in practice tests because they require children to apply their knowledge of maths language and their reasoning skills several times across the course of a single question, usually in slightly different contexts.

More examples:

Reasoning Question 7

Answer : £5,520

There are two steps to this problem, but both are multiplications. The first is to work out how much money is made per day – 92 x £15. This sum is then multiplied by 4 – the number of days – to get to the solution.

Reasoning Question 8

Answer : 2,160 km

Another two step problem. The first step is to work out 10% of 5400 km. Then multiply this by 4 to solve 40%.

Reasoning Question 9

Answer : £43.50

There are three steps involved in solving this problem: multiplication (doubling £51 and £36 to find the cost of two adult and two child tickets), addition (putting the two costs together) and division (dividing the total by four to obtain the mean cost).

Given the number of steps involved it can be easy for pupils to make arithmetic mistakes, and the mark scheme accounts for this by allowing for one mistake – but no more.

Read more: Mean median mode

Reasoning question 10.

Answer : 11.45 kg

A two-step problem again: multiplying 3.45 kg by 4, then subtracting 2.35 kg from the total. As with the previous problem, the mark scheme again allows for at most one arithmetic error, assuming the method is correct.

Year 6 Rapid Reasoning (Weeks 1-6)

Download 6 weeks of Rapid Reasoning slides for your Year 6 pupils and help them get a head start on preparing for the SATs reasoning papers.

SATs Maths Question Type 3: Problems involving measures

As their name suggests, these questions ask pupils to solve a problem that includes one or more units of measurement.

Take a look at this question from 2018’s Reasoning Paper 3:

Reasoning Question 1 1

Answer: 40 washes

This is a two step problem; pupils must first be able to read and convert kilograms to grams (and therefore know the relationship between the two units), then divide 2600 by 65 to work out the number of washes possible.

Questions involving measures tend to be few in number in the KS2 exam papers, but they often provide an excellent way to couch key maths skills such as the four operations.

Further examples:

Reasoning Question 12

Answer : 50g

A relatively simple division problem, relying on pupils having knowledge that 200g is one fifth of a kilogram.

Reasoning Question 13

Answer : 1.1kg

Another three step problem – multiplying 500 by 4 to get the total mass of the four melons, multiplying 300 by 3 to get the total mass of the remaining three melons, and then subtracting 2000 from 900 to obtain the mass of the fourth melon.

It’s worth noting that the mark scheme allows either 1.1kg or 1,100g as acceptable answers – the units of measurement are not as important as obtaining the current figure.

Reasoning Question 14

Answer : 216cm

Interesting to note that in this problem (unlike the previous example), the units for the answer are specified – an answer given in metres will be marked as wrong, since cm is specified in the answer box. This is why we encourage pupils to keep an eye on whether units are provided in the answer box.

Reasoning Question 15

Answer : 170g

As with the melon question there are three steps involved to solve this problem: working out the mass of the four cars (4 x 80), working out the mass of the remaining three cars (3 x 50) and subtracting 150 from 320 to get the mass of the fourth car.

SATs Maths Question Type 4: Problems involving drawing

Problems involving drawing require pupils to construct an  accurate  drawing by following a set of instructions, or through reflection, translation, or scaling.   

This type of question is quite rare, but there are some notable exceptions, such as the infamous Question 21 in Paper 2 of the 2019 Reasoning SATs:

Reasoning Question 1 6

Answer: Any pair of lines that make a square of 4 units, a rectangle of 6 units, and a square of 25 units.

This question is considerably more complex than it appears, and incorporates aspects of multiplication as well as spatial awareness. One potential solution is to work out the area of the card (35), then work out the possible square numbers that will fit in (understanding that square numbers produce a square when drawn out as on a grid), and which then leave a single rectangle behind.

A lot of work for a single mark!

Some further examples:

Reasoning Question 17

Answer : Any quadrilateral made by joining the dots that has 3 acute angles e.g. an arrowhead shape.

Reasoning Question 18

Answer : An accurately drawn angle.

The mark scheme here allows some room for error – “between 34 and 36 degrees” is acceptable .

Reasoning Question 19

As with the question above, a small amount of room for error is given – “between 139 and 141 degrees”.

Reasoning Question 20

Answer : a new triangle drawn with points at (2,1), (5,1) and (2,4).

Translation can be tricky for pupils. Encourage them to look at the triangle as three points, and to translate each point separately rather than trying to move ‘the whole triangle’.

SATs Maths Question Type 5: Explanation questions

An early form of the ‘Prove X’ questions that come up in GCSEs, these problems ask children to explain a mathematical statement or error.

As an example:

Reasoning Question 21

Answer: If the distance from P to R is 800m and the distance from P to Q is (Q -> R x 4), it must be 4/5 of 800 = 640m. Therefore Olivia is wrong.

More than most problems, this type requires pupils to actively demonstrate their  reasoning skills  as well as their mathematical ones. Here pupils must articulate either in words or (where possible) numerically that they understand that Q to R is 1/5 of the total, that therefore P to Q is 4/5 of the total distance, and then calculate what this is via division and multiplication.

Further examples from TSL’s Rapid Reasoning resources:

Reasoning Question 22

Answer : No; 20/100 is the same as 20 divided by 100, which equals 0.2.

Reasoning Question 23

Answer : No; multiplication and division have the same priority, so in a problem like 40 x 6 ÷2, you would carry out the multiplication first as it occurs first.

The mark scheme notes that vague answers or any answers with a mathematical error are unacceptable.

Reasoning Question 24

Answer : No

Any explanation that provides a counter-example is acceptable e.g. “Not if the number is 1”, “Not for 0” etc.

Reasoning Question 25

Answer : Any answer that refers to the fact that there is a 5 in the hundreds place, AND a 9 in the thousands place, so that the number has to be rounded up as far as the ten-thousands place.

SATs Maths Question Type 6: Sequence questions

Another relatively simple kind of reasoning question, sequence problems involve pupils completing mathematical sequences.

Consider this example:

Reasoning Question 26

Answer:   35 , 42, 49,  56 , 63,  70

Number sequence questions, particularly those that involve linear sequences or (as in this case) times tables, come up relatively frequently in the SATs maths tests. The question’s instructions point clearly to the solution: work out what the increase between numbers is, then apply this via addition or subtraction to find the missing numbers.

Higher attaining pupils might quickly pick up that this is in fact the 7 times table and rely on their knowledge of multiplication facts to obtain the answer – this should be encouraged so long as they then check their answer in the normal method to ensure they haven’t made a mistake.

Reasoning Question 2 7

Answer(s) : 5/8 and 2 1/8 (OR 17/8)

Both answers must be correct to receive the mark. Pupils must recognise that 3/4 is the same as 6/8, so the following number must be three eighths higher.

Reasoning Question 28

Answer(s) : -19 and 9

Reasoning Question 29

Answer(s) : 128, 135 and 156.

Reasoning Question 30

Answer(s) : -10 and 22

This question can be a little tricky; pupils need to work out that the marks on the line represent increments of 4, and count backwards and forwards in 4s to obtain the missing numbers.

SATs Maths Question Type 7: Ordering questions

A slightly more complex variation of the sequence question, ordering problems require pupils to put a set of numbers, fractions or measures in the correct order.

A good example is this question from Paper 2 of the 2018 SATs:

Reasoning Question 31

Answer: 3/5, 3/4, 6/5

This question throws a stick in the wheels by including an improper fraction, but this is hardly unusual. These sorts of questions are just the place to find other ‘curveballs’ such as equivalent fractions, mixed numbers and decimals and fractions combined.

A good knowledge of the fundamentals of fractions is essential here: pupils must understand what a larger denominator means, and the significance of a fraction with a numerator greater than its denominator.

Reasoning Question 32

Answer : D,C,A,B

Encourage pupils to convert all the fractions to one denominator value to make ordering easier.

Reasoning Question 33

Answer : (descending down the ‘Place’ column) 3rd, 5th, 2nd, 4th

As with the example above, pupils should be encouraged to convert the fractions to make it easier to order them.

Reasoning Question 34

Answer : C, B, D, A

Reasoning Question 35

Answer : D, A, C, B

7 top tips for answering SATs questions

Now that we’ve covered how to answer some specific types of reasoning questions, here are some more generic tips for success in the reasoning papers. They may not all be applicable to every single question type, but will apply to at least two, usually more.

• Get pupils in the habit for any practice paper of identifying what information they’re given in a question, and what they need to know to solve the problem. This helps them start to form the steps needed to find the solution.
• Ask pupils to ‘spot the maths’ in a question – which calculations or skills do they actually need to use to solve the problem? This is useful even for arithmetic questions – it’s no surprise how often children can misread a question.
• Check the units! Especially in questions involving multiple measures, it can be easy to give the answer in the wrong one. The answer box might give a specific unit of measurement, so pupils should work to give their answer in that unit.
• In a similar vein, remind pupils to convert different units of measurement in a question into the same unit to make calculations easier e.g. kg to g.
• Encourage numerical answers where possible. Even in explanation questions demonstrating the mathematical calculation is a better explanation than trying to write it out.
• The bar model can be a useful way of visualising many different types of questions, and might make it easier to spot the ‘steps’ needed for the solution.
• Check your working out! Even if the working is ultimately irrelevant to the question, you can lose marks if it is wrong.

More free SATs questions (all with answers)

• 75 KS2 SATs maths questions Reasoning and Arithmetic practice questions by topic – includes fractions, decimals, percentages; place value; addition and subtraction.
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Learn about the SATs revision programme or request a personalised quote from your school to speak to us about your school’s needs and how we can help.

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Get ready for KS2 SATs tests with this set of 6 maths SATs practice papers.

Includes 2 Arithmetic and 4 Reasoning Papers, with mark schemes and answer booklets, that follow the National Curriculum Assessments.

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Grade 6 Maths Curriculum

Introduction to Grade 6 Mathematics – This course marks an exciting stage in your mathematical journey, where you’ll delve into more advanced concepts and applications. Grade 6 Mathematics builds upon the fundamental skills you’ve acquired in previous years and introduces new topics in arithmetic, algebra, geometry, and data analysis. Through interactive lessons, hands-on activities, and problem-solving challenges, you’ll develop a deeper understanding of mathematical concepts and sharpen your critical thinking skills.

The Year 6 Mathematics curriculum introduces students to more advanced topics including algebraic expressions, geometric properties, data analysis, and problem-solving strategies. Through engaging lessons and hands-on activities, students deepen their understanding of mathematical concepts and further develop their critical thinking skills, preparing them for future academic challenges.

FAQs on Year 6 Maths Curriculum

How can i improve my problem-solving skills in math.

– Practice solving a variety of math problems regularly. – Break down complex problems into smaller, more manageable steps. – Use different problem-solving strategies such as drawing diagrams, making tables, or working backward.

What resources are available for extra help with math?

– Seek assistance from your teacher during office hours or after class. – Utilize online tutorials, educational websites, or math apps for additional practice. – Form study groups with classmates to collaborate and support each other.

How can I overcome math anxiety?

– Practice relaxation techniques such as deep breathing or visualization before tackling math problems. – Focus on your progress and effort rather than comparing yourself to others. – Adopt a growth mindset and believe in your ability to improve with practice and perseverance.

Why is it important to learn math concepts beyond basic arithmetic?

– Advanced math concepts develop critical thinking, problem-solving, and analytical skills. – Math is essential in various fields such as science, engineering, finance, and technology. – Understanding complex math concepts prepares you for academic success and future career opportunities.

How can I apply math skills in everyday life?

– Use math skills for budgeting, calculating discounts, and managing finances. – Measure ingredients for cooking or baking recipes accurately. – Analyze data from news articles or scientific studies to make informed decisions.

Embrace the challenges and opportunities that Grade 6 Mathematics presents. By approaching math with curiosity, perseverance, and a willingness to learn, you’ll not only excel in the classroom but also develop valuable skills that will benefit you throughout your academic and professional journey. Let’s embark on this mathematical adventure together!

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Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

May 5, 2024 / 7:00 PM EDT / CBS News

As the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years. 

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson:   So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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