problem solving in maths year 4

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Year 4 Maths Worksheets UK Hub Page

Welcome to our Year 4 Maths Worksheets Hub page.

Here you will find our selection of printable maths worksheets for Year 4 children, for your child will enjoy.

Take a look at our times table colouring pages, or maybe some of our fraction of shapes worksheets. Perhaps you would prefer our time worksheets, or learning about line or block symmetry?

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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 4 Maths Learning

Here are some of the key learning objectives for the end of Year 4:

• know and use Place value up to 4 digits
• Compare and order numbers up to 10,000
• Counting on and back in 1s, 10s, 100s and 1,000s from different starting points
• Position numbers on a number line up to 10,000
• Round numbers to the nearest 10, 100 or 1000.
• Count backwards through zero and use negative numbers.
• add and subtract with up to 4 digits in columns
• add or subtract 1s, 10s, 100s or 1000s from a 4-digit number
• solve 2-step problems using addition and subtraction
• recall and use multiplication and division facts up to 12x12
• recognise and use factor pairs
• multiply 2-digit and 3-digit numbers by a 1-digit number
• solve problems using multiplication and division
• count up and down in hundredths
• recognise and use equivalent fractions
• add and subtract fractions with the same denominator
• solve fraction problems including with non-unit fractions
• write common fractions such halves and quarters as decimals;
• understand tenths and hundredths as decimals
• round decimals with 1dp to the nearest whole
• compare numbers with up to 2dp
• solve simple money and measure problems
• measure, compare and calculate using different measures
• find the area and perimeter of squares and rectangles
• convert between 12- and 24-hour clock
• convert between different units of measure
• identify and order acute and obtuse angles
• compare and classify 2D and 3D shapes according to their properties
• identify lines of symmetry in 2D shapes
• use coordinates in the first quadrant
• translate shapes up, down, left and right
• interpret and present data in bar graphs, pictograms and tabels
• solve 1-step and 2-step problems using data in tables, pictograms and graphs

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 4 is generally equivalent to 3rd Grade in the US.

On this page you will find link to our range of math worksheets for Year 4 pupils.

Quicklinks to Year 4 ...

• Online 3rd Grade Practice
• Place Value Zone
• Mental Math Zone

Word Problems Zone

Fractions 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 4 numbers & place value worksheets.

Using these Year 4 maths worksheets will help your child to:

• learn their place value with 4 digit numbers;
• use place value models to understand how to combine thousands, hundreds, tens and ones;
• understand the value of each digit in a 4 digit number;
• learn to use standard and expanded form with 4 digit numbers.
• learn to read and write Roman numerals
• Place Value Models 4 Digits
• Place Value 4 Digit Numbers Worksheets (conversion)
• Ordering 4-Digit Numbers
• Ordering Negative Numbers -10 to 10
• Roman Numerals worksheets

Year 4 Counting & Sequences Worksheets

Each worksheets consists of a sequence which has been partially filled in. The rest of each sequence must be completed.

At this grade, the focus is on counting on and back in constant steps of a digit.

• Counting on and back by digits

Rounding, Inequalities, Multiples and Balancing Equations

Using these Year 4 Maths worksheets will help your child to:

• round a number to the nearest 10, 100 or 1000;
• use the > and < symbols correctly for inequalities;
• use multiples and apply them to solve problems.
• learn to balance math equations
• Rounding to the nearest 10 Worksheets
• Rounding to the nearest 100 worksheets
• Rounding to the nearest 1000 worksheets
• Rounding Inequalities Multiples Worksheets
• Balancing Math Equations

Year 4 Mental Math Zone

Here you will find a range of printable Year 4 mental maths quizzes for your child to enjoy.

Each quiz 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 4 Mental Maths Test sheets

Number Bonds Worksheets

The worksheets on this page will help to develop children's knowledge of numbers bonds to 20, 50 and 100.

There are a range of matching and wordsearches for children to enjoy whilst developing their number bond knowledge.

• Number Bonds to 50 and 100

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Year 4 Addition Worksheets

• learn to add numbers mentally to 100;
• add on 1, 10, 100 and 1000 to different numbers;
• learn to add 4 digit numbers in columns;
• Addition Facts Worksheets to 100+100
• 4-Digit Addition Worksheets
• Money Addition Worksheets (£ )

Year 4 Subtraction Worksheets (3rd Grade)

Using these subtraction worksheets will help your child to:

• learn to subtract numbers mentally to 100;
• learn to do 4 Digit column subtraction.
• Third Grade Subtraction Worksheets to 100
• 4 Digit Subtraction Worksheets
• Money Subtraction Worksheets UK (£ )

Year 4 Multiplication Worksheets

• learn their multiplication tables up to 12 x 12;
• understand and use different models of multiplication;
• solve a range of Year 4 Multiplication problems.

Online Times Table Practice

• Times Tables Practice Zone

Understanding Multiplication

• Understanding Multiplication Facts Worksheets to 10x10

Multiplication Table Worksheets

• Multiplication Table Worksheets - 2 3 4 5 10
• Multiplication Drill Sheets 6 7 8 9
• Fun Multiplication Worksheets to 10x10
• Times Table Worksheets Circles 1 to 12 tables
• Multiplying (integers) by 10 and 100 Worksheets
• Multiplying by Multiples of 10 and 100

2-Digit Multiplication

• 2 Digit Multiplication Worksheets

Multiplication Word Problems

• Year 4 Multiplication Word Problem Worksheets (3rd Grade)

Randomly Generated Multiplication Worksheets

Using our random worksheet generator, you can:

• Choose the tables you want to test;
• Choose how big you want the numbers to go - up to 5 times, 10 times or bigger!
• Choose how many questions per page.
• Times Tables Worksheets (randomly generated)
• Free Multiplication Worksheets (randomly generated)
• Single Digit Multiplication Worksheets Generator
• Multiplication & Division Worksheets (randomly generated)

Year 4 Division Worksheets

Using these Year 4 Maths worksheets will help your child learn to:

• understand how division and multiplication relate to one another;
• know their division facts to 10x10;
• begin to learn 2-digit by 1-digit long division.
• Division Facts to 10x10 Worksheets
• Divding by Multiples of 10 and 100 Worksheets
• Year 4 Long Division Worksheets (3rd grade)
• Division Facts Worksheets (randomly generated)

Using the 3rd Grade Math worksheets will help your child to:

• apply their addition, subtraction, and multiplication skills;
• develop their knowledge of fractions;
• apply their knowledge of rounding and place value;
• solve a range of 'real life' problems.

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

• Year 4 Math Problems (3rd Grade)

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

• Year 4 Math Word Problems for kids (3rd Grade)

These sheets involve solving 3-digit and 4-digit addition word problems.

• Addition Word Problems 3rd Grade (3- and 4-digits)

These sheets involve solving 3-digit and 4-digit subtraction problems.

• Subtraction Word Problems 3rd Grade

These sheets involve solving a range of multiplciation problems.

These sheets involve solving a range of division problems.

• Division Worksheets Grade 3 Word Problems

Year 4 Fraction Worksheets

Using these sheets will help your child to:

• understand what fractions are;
• relate fractions to everyday objects and quantities;
• place different fractions on a number line;
• shade in different fractions of a shapes;
• work out unit fractions of numbers.

• What is a Mixed Number Support page
• Finding Fractions - Fraction Spotting
• Fractions of Shapes Worksheets
• Unit Fraction of Numbers
• Halves and Quarters (up to 100)
• Fraction Number Line Sheets
• Adding Fractions with Like Denominators
• Subtracting Fractions with like denominators
• Fraction Riddles for kids (easier)

Year 4 Geometry Worksheets

The following worksheets will help your child to:

• Identify and name a range of 2d and 3d shapes;
• Draw 2d shapes;
• Use reflective symmetry to reflect shapes in a mirror line.
• recognise and identify right angles and lines of symmetry;
• recognise and identify parallel lines;
• identify the faces, edges, vertices and nets of 3d shapes;
• Year 4 Free Printable Geometry Worksheets
• Block Symmetry Worksheet
• Line Symmetry Worksheets
• Symmetry Activities
• Geometry Nets Information and Worksheets

Measurement Zone, including Time & Money

Year 4 measurement worksheets.

• Year 4 Measurement Worksheets - reading scales
• Metric Conversion Worksheets

Year 4 Money Worksheets

Using challenges is a great way to get kids to use their thinking skills and extend learning by applying the knowledge they have.

• count a range of coins up to £10
• compare money amounts
• apply their existing skills to puzzle out clues;
• understand money terminology;
• develop their thinking skills.
• Year 4 Money Challenges
• Column Addition Money Worksheets (UK)
• Column Subtraction Money Worksheets (UK)

Area and Perimeter Worksheets

• understand area and perimeter;
• learn how to find the area and perimeter of rectangles.
• Area Worksheets
• Perimeter Worksheets

Time Worksheets

Using the sheets in this section will help your child to:

• tell the time to the nearest 5 minutes;
• become familiar with both digital and analogue times;
• use the words 'past' and 'to' to describe the time correctly.
• add and subtract time intervals from times and work out time intervals.
• Add and Subtract Time Worksheets
• Elapsed Time Worksheets
• Printable Time Worksheets - Time Puzzles (easier)
• 24 Hour Clock Conversion Worksheets

On this page there are a selection of bar and picture graphs, including bar graphs with real-life data such as tree heights.

• Year 4 Bar Graph Worksheets (3rd grade)
• Year 4 Venn Diagram Worksheets

Fun Zone: Puzzles, Games and Riddles

Year 4 maths games.

The following games involve different Year 4 Maths activities which you and your child can enjoy together.

• Year 4 / Third Grade Math Games

Year 4 Math Puzzles

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

• Year 4 Math Puzzle Worksheets (3rd Grade)

Math Salamanders Year 4 Maths Games Ebook

Our Year 4 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 4 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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20 Word Problems For Year 4: Develop Their Problem Solving Skills Across Single and Mixed KS2 Topics

Emma Johnson

Word problems for Year 4 play an important role in Year 4 maths. In Year 4, the main focus is to ensure that pupils are becoming more fluent with whole numbers and the four operations. Students work to develop efficient written methods and to be accurate with their calculations. Pupils in Year 4 are exposed to a wider range of problem-solving questions and progress from one to two-step problems.

It is important that all children are given regular opportunities to access reasoning and word problem style questions. Fluency, reasoning and problem solving should be intertwined through every lesson, with all children having the opportunity to tackle each of these question types. 

All Kinds of Word Problems Multiplication

Strengthen your students' problem solving and multiplication skills with this pack of multiplication word problems

Place value 

Addition and subtraction , multiplication and division, fractions, decimals and percentages, measurement, why are word problems important in year 4 maths, how to teach problem solving in year 4, addition word problems for year 4, subtraction word problems for year 4, multiplication word problems for year 4, division word problems for year 4, fraction and decimal word problems in year 4, time word problems in year 4, multi-step word problems in year 4., more primary word problems resources.

There can sometimes be a tendency for reasoning and problem solving questions to be treated as extension activities for only the higher attaining pupils to attempt, but children of all abilities need to be accessing them on a regular basis.

To help you with this, we have put together a collection of 20 word problems aimed at Year 4 pupils. For more Year 4 maths resources, take a look at our collection of Year 4 maths worksheets .

Year 4 Maths Word Problems in the National Curriculum

In Year 4, pupils progress from solving one-step problems, to also being exposed to two-step problems across a range of topics, as set out in the National Curriculum.

Solve word problems involving counting in multiples of 6,7,9, 25 and 100; finding 1000 more or less than a given number; counting backwards through 0 to include negative numbers; ordering and comparing numbers beyond 1000 and rounding numbers to the nearest 10. 100 and 1000

Solve addition and subtraction word problems with up to 4 digits, including two-step word problems, deciding which operations and methods to use and why.

Solve problems involving multiplying and adding, including using the distributive law to multiply two-digit numbers by 1 digit, integer scaling problems and harder correspondence problems.

Solve problems involving increasingly harder fractions word problems to calculate quantities, and fractions to divide quantities, including non-unit fractions where the answer is a whole number. Also Solve simple measure and money problems involving fractions and decimals to 2 decimal places.

Solve problems involving converting from hours to minutes, minutes to seconds, years to months, weeks to days.

Solve comparison, sum and difference problems using information presented in bar charts, pictograms, tables and other graphs.

Word problems are increasingly important as pupils move through Key Stage 2. As they become more confident with some of the core concepts pupils need to be applying this knowledge to a range of situations. By the end of Year 4, pupils should have memorised their multiplication tables up to and including the 12 times table and should be showing precision and fluency in their work.

Word problems in Year 4 should be fun and engaging for students. There are many ways to do this, including:

• acting out the problem;
• using manipulatives and visual images to help children understand the maths within the problem;
• use of talk partners to encourage children to discuss the question and share strategies for reaching a solution;
• using relatable problem solving situations.

Children need to be encouraged to read word problem questions carefully, to ensure they have identified the key information needed to be able to solve the problem. Pupils need to think about what they already know and how that information can help them to answer the question. They should also be encouraged to draw pictures and visual images, where appropriate, to help them to understand what the question is asking.

Here is an example:

A shop has an 8m roll of fabric.

The first customer buys 125cm of fabric and the second customer buys 3m from the same roll.

How much fabric is left on the roll, once the two customers have taken theirs?

How to solve:

What do you already know?

• The amount the first customer buys is given in cm, the amount the second customer buys is given in m. These needed to be converted to the same unit.
• Pupils in Year 4 need to be able to convert cm to m and vice versa. In this question, both the cm can be given as m or the m changed to cm to solve it.
• Once the units are the same, the two amounts need to be added together, to work out the total amount bought by the two customers.
• We can see this is a two-step question. To calculate how much fabric is left on the roll, the total amount bought by the two customers needs to be subtracted from the initial amount of fabric on the roll.

How can this be drawn/represented pictorially?

We can draw a bar model to represent this problem:

• To calculate the total amount of fabric bought, we need the units to be the same. We can either calculate in cm (300cm + 125cm = 425cm) or in m (3m + 1.25m = 4.25m)
• The amount of material bought needs to be subtracted from the original amount. This can again be solved in m or cm. Either in m: 8m – 4.25m = 3.75m or in cm: 800cm – 425cm = 375cm
• The total amount of material left is 3.75m or 375cm

In Year 4, addition word problems involve questions up to 4-digit numbers. They can include one and two-step addition and incorporate a range of concepts, such as measures and money word problems  

Addition question 1

It is 4164 miles to travel from London to Doha and 3266 miles to travel from Doha to Bangkok.

How far is it to travel from London to Bangkok, if the flight stops in Doha first?

Answer (1 mark): 7430 miles

Addition question 2

Fill in the missing numbers in this calculation.

Answer (1 mark): 6840

Addition question 3

On Saturday, 5486 fans attended a football game and 3748 fans attended a rugby game.

How many fans watched the two games in total?

Answer (1 mark): 9234

Subtraction word problems in Year 4, also involve numbers up to 4-digits, including both one and two-step problems, covering a range of concepts. By this stage, children should be confident in estimating and using the inverse, to check calculations.

Subtraction question 1

3241 people visited the zoo on Saturday.

On Sunday 2876 people visited.

How many more people visited the lake on Saturday than on Sunday?

Answer (1 mark): 365

 3241 – 2876 = 365

Subtraction question 2

 A teacher prints out 1242 worksheets in a term.

If 435 were maths worksheets, how many did she print out for the other subjects?

Answer (1 mark): 807

Subtraction question 3

The temperature in Toronto dropped to minus 15 degrees celcius in December. 

In July the temperature was 47 degrees celsius warmer than it was in December. What was the temperature in July? 

Answer (1 mark): 32 degrees warmer

47 – 15 degrees = 32 degrees celsius

Counting on 47 degrees from minus 15 degrees  = 32 degrees celsius

In Year 4, multiplication word problems can include recalling facts for times tables up to 12 x 12 and multiplying two and three-digit numbers by a 1-digit number, using formal written layout.  

Multiplication question 1

All the pupils in Year 4 complete a mental maths test.

27 pupils score 9 marks out of 10.

What is the total number of marks scored by the 27 pupils?

Answer (1 mark): 243

27 x 9 = 243

Multiplication question 2

Year 3 and 4 children from a local primary school go on a school trip. 

Six mini buses are used to transport the children.

 There are 17 children on each minibus.

How many children go on the school trip?

Answer (1 mark): 102

17 x 6 = 102 children

Multiplication question 3

Biscuits come in packs of 18.

Mrs Smith buys 8 packs for the parents at the Y4 maths workshop.

How many biscuits does she buy altogether?

Answer (1 mark): 144 biscuits

Division word problems in year 4 require pupils to be able to recall division facts for multiplication tables up to 12 x 12. Formal written method of division isn’t a requirement until Year 5 however, many schools choose to teach the formal method in Year 4. Pupils need to understand the concept of grouping and sharing and to understand the link between multiplication & division.

Division question 1

Sam has 28 friends coming to his birthday party.

Each child will receive a cupcake, which come in packs of 4.

How many packs of cupcakes will Sam need to buy?

Answer (1 mark): 7 packs

28 ÷ 4 = 7 

Division question 2

4 children raised £96 between them on a sponsored walk.

If they split the money evenly between the four of them, how much did each pupil raise?

Answer (1 mark): £24 each

96 ÷ 4 = 24

Division question 3

Ahmed is thinking of a number

He says, ‘when I divide my number by 12, the answer is 108.

What number was Ahmed thinking of?

Answer (1 mark): 9

108 ÷ 12 = 9

In Year 4, decimal and fraction problems involve increasingly harder fractions to calculate quantities and fractions to divide quantities, including non-unit fractions, where the answer is a whole number. Decimal word problems include measure and money problems involving fractions and decimals up to 2 decimal places.

Fraction and decimal question 1

Jamie has 18 sweets. 

He gives \frac{1}{6} of the sweets to his friend and keeps the rest himself.

How many sweets does Jamie have now?

Answer (1 mark): 15 sweets

\frac{1}{6} of 18 = 3

18 – 3 = 15

Fraction and decimal question 2

Jaxon collected 36 conkers. 

\frac{1}{4} of the conkers fell out of a hole in his bag, when he was walking home.

How many conkers did Jaxon have left, when he got home?

\frac{1}{4} of 36 = 9

36 – 9 = 37      or      \frac{3}{4} of 36 = 29 (3 x 9)

Fraction and decimal question 3

Sara ate \frac{3}{12} of a chocolate bar and gave \frac{2}{12} to her friend.

What fraction of the chocolate bar did she have left? 

 Answer (1 mark): \frac{7}{12}

In Year 4, time word problems include: converting from hours to minutes, minutes to seconds, years to months and weeks to days.

Time question 1 

A cake was put in the oven at 4:35pm and taken out at 4:57pm.

How long was the cake in the oven?

Answer (1 mark): 22 minutes

57-35 = 22 minutes

Using an number line: 

Time question 2

It took Evie 25 minutes to complete a page of number problems. 

If she started at 2:45pm. What time did she finish?

Answer (1 mark): 3:10pm

In Year 4, children are introduced to multi-step word problems requiring up to two steps. These problems cover a range of concepts, including the four operations, fractions, decimals and measures.

Third Space Learning’s online one-to-one tutoring frequently incorporates multi-step questions to test students’ knowledge and problem solving skills. Our personalised tutoring programme works to identify gaps in students’ learning, fill those gaps, reinforce students’ knowledge and build confidence.

Multi-step question 1

There are 6 handwriting pens in each pack.

A class has 30 children and each child needs 2 handwriting pens.

How many packs will the teacher need to buy?

Answer (2 marks): 10 packs of handwriting pens.

30 x 2 = 60

Multi-step question 2

 Sophie has £4.50.

She buy 3 books at a carboot sale, costing 50p, 65p and £1.20.

How much money does she have left?

Answer (2 marks): £2.15 left

Multi-step question 3

Abullah is thinking of a number.

He doubles the number and adds 7.

He gets an answer of 25. 

What was his original number?

Answer (2 marks): 9

Third Space Learning offers word problems for all primary year groups. Take a look at our word problems for year 3 , word problems for year 5 and word problems for year 6 . Our word problems span a range of topics such as ratio word problems and percentage word problems .

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Year 4 Maths Problems

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

22 February 2018

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30 year 4 maths questions

In year 4, students are introduced to important maths concepts that become the building blocks for future mathematical learning. Let’s practise those concepts with this list of year 4 maths questions.

Author Katie Wickliff

Published February 2024

30 year 4 math questions

• Key takeaways
• In year 4 maths, students develop a deeper understanding of multiplication and division, fractions, and measurements.
• Year 4 word problems require more advanced comprehension skills.
• Place Value: Understanding place value helps us read and write numbers correctly and compare their values.
• Look for opportunities for your child to practise their maths at home–worksheets, maths games , or a maths practice app will help strengthen new mathematical knowledge. While consistency is key, maths practice for year 4 should be engaging and fun.

Table of contents

• Subtraction
• Multiplication
• Multi-step word problems

Answer sheet

In addition to reviewing and strengthening the skills learned in previous years, year 4 maths students learn multiplication and division, explore fractions, calculate measurements, and more. Also, word problems at this level are often multi-step, requiring students to understand what operations to perform and how to perform them correctly. 

This list of year 4 maths questions is intended to reinforce concepts students have already been taught. The list is divided into sections: addition, subtraction, multiplication, division, fractions, measurement, and multi-step problems. These sections allow students to specifically focus their practice, but feel free to skip around and choose the maths practice problems that best suit your child’s needs. 

30 maths questions for year 4

Section 1: addition.

At the school carnival, Hattie won 25 tickets at ring toss and 40 tickets at balloon darts. How many tickets did she win altogether?

There are 15 baby chicks at the barn. 12 more chicks are born. How many baby chicks are at the barn now?

How can you represent the number line with an equation?

Using the number line, solve 7+ _ = 22

Amir practises basketball three days a week. On Tuesday, he practises basketball for 60 minutes. On Wednesday, he practises for 37 minutes. On Thursday, he practises for 15 minutes. Which expression shows how to find the total number of minutes Amir practised? Circle the correct answer

Section 2: Subtraction

Using the number line, solve 17-6=

There are 224 Lions fans and 212 Tigers fans at the football game. How many more Lions fans than Tigers fans are at the game?

An animal shelter has 27 kittens available for adoption. On Friday, 8 kittens are adopted. How many kittens are left at the shelter?

Gemma has 130 beads for bracelets. She gives 75 beads to Piper and 6 beads to Claire. Which expression shows how to find the total number of beads Gemma has left? Circle the correct answer.

Using the number line, solve 30- _ = 14

Section 3: Multiplication

Using the picture below, write a multiplication equation to find the number of shapes

Francis and Jane are playing a card game. Francis has 7 cards. Jane has 3 times as many cards as Francis. How many cards does Jane have?

Using the number line to show hops, complete the multiplication sentence

Solve the equation

Mariela has 8 pages of maths homework. There are 4 questions on each page. How many total problems does Mariela have for maths homework?

Section 4: Division

Connor has 80 crayons that he will put into crayon boxes. Each crayon box can hold 8 crayons. How many crayon boxes does he need?

Using the number line to show hops, complete the division sentence

Natalie divided 12 marbles into equal groups of 3. Draw a representation to show equal grouping.

Craig’s mom brought 12 cookies to the soccer game. She shared them equally among Craig and 3 teammates. Which expression will solve the problem correctly? Circle the answer.

Section 5: Fractions

Using the number line, work out what A, B and C represent.

Plot ¼ on this number line:

A cheese pizza has 8 slices. Marco ate 2 of the slices. His dad at the rest.  What fraction of the pizza did Marco eat? What fraction of the pizza did his dad eat?

There are 15 flowers in a garden. 10 flowers are yellow and the rest are red. What is the fraction of red flowers in the garden?

Using the number line, find the value of K and L. Which is bigger?

Section 6: Multi-step word problems

Sarah has two £10 notes. Her sister has one £5 note. They combine their money to buy a game that costs £22 pounds. How much change will they receive?

Dad baked 36 cookies for the bake sale. Jonas baked 24 cookies for the bake sale. His brother ate 5 cookies before the bake sale started. How many cookies do they have left to sell?

The school bus seats 75 students in all. On Friday, the school bus was full. At the first stop, 7 students got off. At the second stop, 15 students got off. How many children were left on the bus?

George has 25 water balloons. He wants to divide the water balloons equally between himself and 2 friends. How many water balloons will each get? Will there be any water balloons left over?

On Monday, Julia walked 5 kilometres. On Tuesday, she walked 3 kilometres. On Wednesday, she walked 7 kilometres. A marathon is 42 kilometres. How many more kilometres would Julia have to walk to have walked a full marathon?

Addition answers

25+40= 65. Hattie won 65 tickets

15+12= 27. 27 baby chicks are at the barn

7+ 15= 22. This should be represented by 15 “hops” on the number line

Subtraction answers

17-6=11. This should be represented by 11 “hops” on the number line

224-212= 12. There are 12 more Lions fans than Tiger fans

27-8= 19. There are 19 kittens left at the shelter.

Multiplication answers

2×3=6; 3×2=6

7×3=21. Jane has 21 cards.

8×4= 32

Division answers

40÷8= 5. It will take Max 5 days to pay Bennett.

Drawing should show 3 groups of 4 marbles each.

Fractions answers

The ¼ should be plotted on the point between 0 and ½

Marco ate 2/8 (or ¼) of the pizza. His dad ate 6/8 (or ¾) of the pizza

5/15 (or ⅓)

K= ⅜, L= 6/8 (or ¾). L is larger.

Multi-step word answer answers

£10+£10= £20. 

£20+£5=£25.

£25-£22= £3

 Sarah and her sister will receive £3 in change

They have 55 cookies left to sell

53 students are left on the bus

George and his friends each get 6 water balloons, with one left over.

5 + 3 + 7 = 15

42 – 15 = 27

Julia must walked 27 more kilometres.

Parents, sign up for a DoodleMaths subscription and see your child become a maths wizard!

Lesson credits

Katie Wickliff

Katie holds a master’s degree in Education from the University of Colorado and a bachelor’s degree in both Journalism and English from The University of Iowa. She has over 15 years of education experience as a K-12 classroom teacher and Orton-Gillingham certified tutor. Most importantly, Katie is the mother of two elementary students, ages 8 and 11. She is passionate about math education and firmly believes that the right tools and support will help every student reach their full potential.

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Problem-Solving Booklet: Year 4

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This Problem-Solving Booklet is a great resource for your students to practise utilising problem-solving skills.

This booklet has 18 pages of questions, having students apply addition, subtraction, division, and multiplication operations to answer the problem. Each page has the students identify the operation needed and the problem-solving strategy they will use, then record the number sentence to solve the problem. Some problems have multiple steps to extend students. It also contains two cover pages as it may be suitable to use with other year grades.

This would work well in small group activities, extension work, or as an independent task.

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Year 4 maths worksheets

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Your child can learn, practice and test their year 4 (age 8 - 9) maths skills with these automatically marked, interactive year 4 maths worksheets. We’ve created these math worksheets for year 4 as close to the National Curriculum as possible to help your child learn and develop their understanding of key topics. These revision materials should help your child start to build on their knowledge of year 3 maths and help them progress towards more difficult topics. This year they'll be working towards consolidating the Lower Key Stage 2 (years 3 and 4) math topics, which you can do with our year 4 counting  and year 4 decimals worksheets amongst many others, helping to prepare for year 4 maths tests and assessments.

Download free year 4 maths PDF worksheets

Brand new for 2024, we've created a range of 100% free year 4 maths PDF worksheets for you to work through with your child. Download the printable worksheet PDF, and your child can both learn and answer questions.

Year 4 maths tests & assessments

Revise popular maths topics, this terms popular year 4 maths worksheets.

We know that finding the perfect resources to help your child isn't easy, which is why we've put together a selection of easy to pick up year 4 maths worksheets to get them started!

Adding Fractions

Identify Lines Of Symmetry

Plotting Simple Shapes

More year 4 maths worksheets

Identify Quadrilaterals TRY WORKSHEET FOR FREE

Follow Coordinate Directions Correctly to Find the Treasure TRY WORKSHEET FOR FREE

Give Coordinates Correctly to Find the Treasure TRY WORKSHEET FOR FREE

Identify Different Quadrilaterals TRY WORKSHEET FOR FREE

Addition and Subtraction

Check Answers to Calculations Using the Inverse Operation TRY WORKSHEET FOR FREE

Find the Answer by Doing the Inverse Operation TRY WORKSHEET FOR FREE

Improve Your Ability to Solve Two-Step Word Problems TRY WORKSHEET FOR FREE

Practise Solving Word Problems TRY WORKSHEET FOR FREE

Order Angles and Identify Regular and Irregular Shapes TRY WORKSHEET FOR FREE

Practise Ordering Angles and Identifying Regular and Irregular Shapes TRY WORKSHEET FOR FREE

Practise Sorting Angles into Acute or Obtuse TRY WORKSHEET FOR FREE

Recognise Right Angles in Shapes TRY WORKSHEET FOR FREE

Back-to-school

Assessment: Back-to-school (Y4) PREVIEW WORKSHEET

Combined Assessment

Assessment: English, Maths and Science Combined (Y4) PREVIEW WORKSHEET

Coordinates

Plot Missing Points on a Grid TRY WORKSHEET FOR FREE

Plot Points on a Graph to Form Letters TRY WORKSHEET FOR FREE

Plot Simple Shapes on a Grid TRY WORKSHEET FOR FREE

Practise Plotting Missing Points on a Grid TRY WORKSHEET FOR FREE

Count Back Through Zero TRY WORKSHEET FOR FREE

Count Backwards and Forwards Through Zero TRY WORKSHEET FOR FREE

Count Below Zero TRY WORKSHEET FOR FREE

Cross Zero When Counting Backwards and Forwards TRY WORKSHEET FOR FREE

Add Numbers with Two Decimal Places TRY WORKSHEET FOR FREE

Add Three Numbers with Two Decimal Places TRY WORKSHEET FOR FREE

Compare Decimal Numbers TRY WORKSHEET FOR FREE

Count in Hundredths TRY WORKSHEET FOR FREE

End of Term 1

Assessment: End of Term 1 PREVIEW WORKSHEET

End of Term 2

Assessment: End of Term 2 PREVIEW WORKSHEET

End of Term 3

Assessment: End of Term 3 PREVIEW WORKSHEET

End of Year

Assessment: End of Year PREVIEW WORKSHEET

Add and Subtract Fractions Including Beyond One Whole TRY WORKSHEET FOR FREE

Add and Subtract Mixed Number Fractions with the Same Denominator TRY WORKSHEET FOR FREE

Add Fractions with the Same Denominator TRY WORKSHEET FOR FREE

Convert Between Decimals and Both Mixed and Improper Fractions TRY WORKSHEET FOR FREE

Graphs, Charts and Tables

Compare Data Presented in a Line Graph TRY WORKSHEET FOR FREE

Read and Compare Data Presented in Line Graphs TRY WORKSHEET FOR FREE

Compare Data in Pie Charts to Answer Questions TRY WORKSHEET FOR FREE

Interpret Information in Time Tables TRY WORKSHEET FOR FREE

Graphs, Tables, Charts and Data

Interpret Line Graphs TRY WORKSHEET FOR FREE

Read Data From a Bar Chart TRY WORKSHEET FOR FREE

Retrieve Data about Cars From a Table of Results TRY WORKSHEET FOR FREE

Understand Data in a Pictogram TRY WORKSHEET FOR FREE

Hundreds, Tens and Units

Arrange Numbers in Ascending Order TRY WORKSHEET FOR FREE

Arrange Numbers in Descending Order TRY WORKSHEET FOR FREE

Complete Sequences by Filling in Gaps TRY WORKSHEET FOR FREE

Create the Largest and Smallest Four-Digit Numbers TRY WORKSHEET FOR FREE

Measurement

Select the Most Appropriate Unit of Measure TRY WORKSHEET FOR FREE

Calculate Areas and Perimeters of Rectangles TRY WORKSHEET FOR FREE

Calculate Perimeter by Finding Missing Measures TRY WORKSHEET FOR FREE

Calculate the Area of a Rectangle TRY WORKSHEET FOR FREE

Mental Maths

Subtract 1,000 from a Four-Digit Number TRY WORKSHEET FOR FREE

Add 1,000 to Four-Digit Numbers TRY WORKSHEET FOR FREE

Add 1,000 to Numbers that are Less than 1,000 TRY WORKSHEET FOR FREE

Add or Subtract 1,000 With Four-Digit Numbers TRY WORKSHEET FOR FREE

Solve Money Problems TRY WORKSHEET FOR FREE

Calculate Fractions of Amounts of Money TRY WORKSHEET FOR FREE

Calculate Problems Involving Fractions of Money TRY WORKSHEET FOR FREE

Convert Pence into Pounds TRY WORKSHEET FOR FREE

National Curriculum Objectives

Assessment: Addition and Subtraction PREVIEW WORKSHEET

Assessment: Fractions and Decimals PREVIEW WORKSHEET

Assessment: Geometry - Position and Direction PREVIEW WORKSHEET

Assessment: Geometry and Properties of Shape PREVIEW WORKSHEET

Place Value

Find the Value of Digits in Numbers TRY WORKSHEET FOR FREE

Identify the Value of Digits in Numbers TRY WORKSHEET FOR FREE

Identify the Value of Digits in Numbers Including Zero as a Place Holder TRY WORKSHEET FOR FREE

Understand the Use of Zero as a Place Holder in 4-Digit Numbers TRY WORKSHEET FOR FREE

Read and Write Numbers

Count Backwards Through Zero TRY WORKSHEET FOR FREE

Count Backwards to Include Negative Numbers TRY WORKSHEET FOR FREE

Count in Multiples of 1,000 TRY WORKSHEET FOR FREE

Count in Multiples of 25 TRY WORKSHEET FOR FREE

Roman Numerals

Convert Numbers up to 100 to Roman Numerals TRY WORKSHEET FOR FREE

Convert Numbers up to 50 to Roman Numerals TRY WORKSHEET FOR FREE

Practise Reading and Writing Roman Numerals up to 100 TRY WORKSHEET FOR FREE

Read, Write, and Match Roman Numerals up to 50 TRY WORKSHEET FOR FREE

Practise Rounding Numbers to the Nearest 10, 100 or 1,000 TRY WORKSHEET FOR FREE

Round Numbers to the Nearest 10, 100 or 1,000 TRY WORKSHEET FOR FREE

Interpret Data from a Bar Graph TRY WORKSHEET FOR FREE

Read and Interpret Information in a Table TRY WORKSHEET FOR FREE

Count Lines of Symmetry TRY WORKSHEET FOR FREE

Find Lines of Symmetry TRY WORKSHEET FOR FREE

Find Lines of Symmetry in Letters TRY WORKSHEET FOR FREE

Identify Lines of Symmetry TRY WORKSHEET FOR FREE

Practise Using Digital and 24-hour Time TRY WORKSHEET FOR FREE

Calculate How Many Weeks in a Number of Days TRY WORKSHEET FOR FREE

Change Times from Analogue to Digital TRY WORKSHEET FOR FREE

Convert From Analogue to Digital Time including Quarter Past and Quarter To the Hour TRY WORKSHEET FOR FREE

Times Tables

Practise Your 11 Times Table TRY WORKSHEET FOR FREE

Practise Your 12 Times Table TRY WORKSHEET FOR FREE

Practise Your 6, 7, 9, 11 and 12 Times tables TRY WORKSHEET FOR FREE

Practise Your 9 Times Table TRY WORKSHEET FOR FREE

Above, you'll find a list of interactive and printable year 4 maths worksheets, assessments and revision materials by topic which have been designed by teachers to help your child improve their understanding and comprehension of the topics taught in schools.

All of our worksheets are available to complete online or in a pdf printable format so you can decide how to plan lessons yourself. Our expert team of teachers make sure our year 4 maths worksheets are always kept up to date so they’re 100% aligned to the National Curriculum and can provide you visibility to your child's progress throughout the year, and can help prepare for year 4 maths tests. If you need support across other subjects, you can also take a look at our year 4 science worksheets and our year 4 English worksheets . Try one of the above sample free year 4 maths worksheets to test your child's attainment levels - we're here to help your child succeed!

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Solving Problems (Year 4)

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In KS2 Maths, children have fun solving problems from real-life situations. For example, in Year Four, they tackle challenges like converting measurements, such as centimetres to metres or minutes to hours.

Real problems are everywhere! Like figuring out how long it takes to walk to school or how much money you'll have left after buying sweets. But, how do we solve these problems? It's all about finding solutions! This quiz asks questions about everyday situations, testing your skills in addition, subtraction, multiplication, division, weights, distances, capacities, and time.

Life is full of exciting problems to solve. Take this quiz to practice and see how well you can tackle real-life challenges. Get ready for the adventures of KS2 Maths!

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Reasoning/Problem Solving Maths Worksheets for Year 4 (age 8-9)

This category is all about children choosing the most appropriate method to achieve a quick and accurate answer and to be able to explain their methods and reasoning.

Problem solving: finding the missing digits in 4-digit numbers. Tricky!

Problem solving: finding the values of missing counters. Logical thinking and careful work needed here!

Reasoning about numbers on a number line and rounding numbers.

More reasoning about number and place value.

Not only do the questions have to be answered, but explanations given. These pages really show how well children understand what they are doing.

More written explanations on which operation to use and whether the question should be done mentally or on paper.

Lots of questions, most of which can be done mentally. The important thing is to explain how they were tackled.

Decide which operation is needed (add, subtract, multiply, divide) and whether a calculator, or pencil and paper are needed, or whether it can be done, 'in your head'.

Solving problems

A selection of problem solving activities and word problems.

Here are some problems written in words. Fairly straightforward as they require just one mathematical step to work out the answer

Some word problems that need at least two steps to find the answer. This makes them much harder, so careful thinking is needed.

5 pages of word problems, all involving money. Some are suitable for calculator work as they are specially written to make the process the most important factor: it is no use having a calculator if you don't know what calculation to carry out!

A range of number and money problems, typical of those found in KS2 test papers.

More number and money problems, typical of those found in KS2 test papers.

Tricky word problems involving time and money.

More word problems; this time on measurement of length, mass and capacity. These really test children's understanding of when to use addition, subtraction, multiplication or division to solve a problem.

Time problems are especially tricky, especially when finding lengths of time that cross the hour boundary. A good knowledge of the number of days in each month is also needed.

Not many problems on a page, but there is an awful lot within each question. Each is a mini investigation, which can be explored in depth.

Number patterns, remainders, number machines and much more in this mixed selection of problems.

Working out the missing value.

More working out missing values.

Using the instructions on the packet what is the maximum number of bulbs that can be grown in the garden?

Using the instructions on the packet what is the maximum number of seeds that can be sown in the garden?

Using the instructions on the packet what is the maximum number of vegetables that can be grown in the garden?

An addition challenge suitable for year 4 on.

The fish are all called numbers. Can you work out their names by solving the clues?

Investigations

More open ended activities where children choose the approach to the task.

Making rules for sequences is an early step towards understanding algebra, and an important part of Year 5 maths. More 'How would you...?' type questions as well.

How many different three course meals can Sam choose?

Investigate patterns made by placing consecutive numbers in a number square.

A great little investigation which has plenty of extension possibilities.

Fun addition investigation using the date: 22.02.2022.

Some thoughts on how important this area of maths is.

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Resources tagged with: NC Yr 4

There are 51 NRICH Mathematical resources connected to NC Yr 4 , you may find related items under NC .

Satisfying Four Statements

Can you find any two-digit numbers that satisfy all of these statements?

Representing Numbers

Find as many different ways of representing this number of dots as you can.

Ordering Journeys

How would you put these journey lengths in order?

What Distance?

Can you use addition and subtraction to answer these questions about real-life distances?

Count Me In

How do you know whether you will reach these numbers when you count in steps of six from zero?

Dicey Operations in Line

Who said that adding, subtracting, multiplying and dividing couldn't be fun?

Seeing Squares

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Reasoned Rounding

Four strategy dice games to consolidate pupils' understanding of rounding.

Round the Dice Decimals 1

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

Let Us Divide!

Look at different ways of dividing things. What do they mean? How might you show them in a picture, with things, with numbers and symbols?

Discuss and Choose

This activity challenges you to decide on the 'best' number to use in each statement. You may need to do some estimating, some calculating and some research.

Bryony's Triangle

Watch the video to see how to fold a square of paper to create a flower. What fraction of the piece of paper is the small triangle?

Sorting Logic Blocks

This activity focuses on similarities and differences between shapes.

Light the Lights Again

Each light in this interactivity turns on according to a rule. What happens when you enter different numbers? Can you find the smallest number that lights up all four lights?

A task which depends on members of the group noticing the needs of others and responding.

What Shape?

This task develops spatial reasoning skills. By framing and asking questions a member of the team has to find out what mathematical object they have chosen.

Counters in the Middle

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

Table Patterns Go Wild!

Nearly all of us have made table patterns on hundred squares, that is 10 by 10 grids. This problem looks at the patterns on differently sized square grids.

Times Tables Shifts

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

Reflector ! Rotcelfer

Can you place the blocks so that you see the reflection in the picture?

Nice or Nasty

There are nasty versions of this dice game but we'll start with the nice ones...

The Remainders Game

Play this game and see if you can figure out the computer's chosen number.

Four-digit Targets

You have two sets of the digits 0-9. Can you arrange these in the five boxes to make four-digit numbers as close to the target numbers as possible?

Venn Diagrams

How will you complete these interactive Venn diagrams?

Eight Hidden Squares

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

Shape Times Shape

These eleven shapes each stand for a different number. Can you use the number sentences to work out what they are?

Multiplication Square Jigsaw

Can you complete this jigsaw of the multiplication square?

Twice as Big?

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

Multiples Grid

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

Coordinate Challenge

Use the clues about the symmetrical properties of these letters to place them on the grid.

Torn Shapes

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

Take Your Dog for a Walk

Use the interactivity to move Pat. Can you reproduce the graphs and tell their story?

Fractional Wall

Using the picture of the fraction wall, can you find equivalent fractions?

Stringy Quads

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

Nine-pin Triangles

How many different triangles can you make on a circular pegboard that has nine pegs?

Carrying Cards

These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers?

Andy's Marbles

Andy had a big bag of marbles but unfortunately the bottom of it split and all the marbles spilled out. Use the information to find out how many there were in the bag originally.

How Big Are Classes 5, 6 and 7?

Use the two sets of data to find out how many children there are in Classes 5, 6 and 7.

Fractional Triangles

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

The Deca Tree

Find out what a Deca Tree is and then work out how many leaves there will be after the woodcutter has cut off a trunk, a branch, a twig and a leaf.

Symmetry Challenge

How many symmetric designs can you make on this grid? Can you find them all?

Let Us Reflect

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?

A Cartesian Puzzle

Find the missing coordinates which will form these eight quadrilaterals. These coordinates themselves will then form a shape with rotational and line symmetry.

Fractions in a Box

The discs for this game are kept in a flat square box with a square hole for each. Use the information to find out how many discs of each colour there are in the box.

Shapes on the Playground

Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?

Zios and Zepts

On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Four Triangles Puzzle

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

There are three tables in a room with blocks of chocolate on each. Where would be the best place for each child in the class to sit if they came in one at a time?

Ninja Maths

Weekly Maths Worksheets

Year 4 Free Maths Worksheets

They’re not very difficult and are aimed at instilling the ethos of repeated practice.

Summary of Year 4 Maths Content

New & updated free sample worksheets coming soon, based on the UK curriculum;

Click to open each PDF – all of which are one page and contain questions and answers:

In the mean time you can subscribe here for free worksheets as they’re written:

International Equivalence

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English Year 6 for school ages 10-11 equates to:

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What Does “Squared” Mean in Math? Explain It to a 10-Year-Old

This beginner-friendly overview of square numbers in math is for everyone, from seven- to seventy-year-olds, looking to learn new mathematical concepts or refresh their memory. Read on to find simple definitions, and fun applications and exercises.

What Is a Square Number?

A square number is the result of multiplying a number by itself.

For instance, if you take the number 3 and multiply it by 3, you get 9.

9 is a square number because it’s the product of multiplying 3 by itself.

In math, we write this as 3² (we read it as “three to the second power” or “three squared”). We call the “ 3 ” in 3² the base and we call “ ² ” the exponent .

Let’s look at some examples to understand square numbers better:

• 2² which can also be expressed as 2 x 2 = 4
• 4² which can also be expressed as 4 x 4 = 16
• 5² which can also be expressed as 5 x 5 = 25

Refresh Your Memory:  What is a Factor in Math?

What’s the Difference Between a Square Number and a Square Root?

In addition to “square numbers,” you might have also heard about “square roots.”

Square numbers and square roots are like two sides of the same coin — they represent opposite actions.

Since square roots are a whole new topic, let’s look at a brief example:

As we said, a square number is a result of multiplying a number by itself. For example, 3 multiplied by 3 equals 9 (3² = 9), so the number 9 is a square number.

A square root is the opposite of a square number. A square root  is the number you multiply by itself to get another number. We show it with this symbol: √ (radical symbol or surd). For example, the square root of 9 is 3 (√9 = 3), and the square root of 25 is 5 (√25 = 5).

Why Are They Called Square Numbers?

If you thought square numbers must have something to do with squares, you would have been correct!

Square numbers are called “squared” because they make the shape of a square.

Squares have sides of equal length. To find the area of a square, you just need to multiply one side by itself, or “square” it.

How Do We Use Square Numbers in Math?

Squares are like puzzle pieces in math. They fit into various math classes and fields.

Pre-Algebra and Algebra : In algebra, we use square numbers in equations. Let’s say that we have the expression . In algebra, this means “x squared” or “x multiplied by itself.” For example, if , then would be or which equals .

Geometry : In geometry we use square numbers to measure the area inside of a square based on its side lengths. Let’s look at a square with sides of length that are 4 centimeters long for this example. To find the area of this square, we square the length of one of its sides. In this case, the length of one side is 4 centimeters. So, the area of the square would be 16 square centimeters.

Number Theory : In number theory, we use square numbers to study patterns and relationships between numbers. For example, the sequence 1, 4, 9, 16, 25, 36 is a list of square numbers. How do we know this? Each number in the sequence is made by multiplying a natural number by itself.

Here’s proof:

Check out   this video   demonstrating a cool trick for squaring numbers ending in 0 or 5.

Examples of Square Numbers

Let’s see how numbers from 1 to 12 are squared.

• 1² = 1 which can also be expressed 1 × 1 = 1
• 2² = 4 which can also be expressed 2 × 2 = 4
• 3² = 9 which can also be expressed 3 × 3 = 9
• 4² = 16 which can also be expressed 4 × 4 = 16
• 5² = 25 which can also be expressed 5 × 5 = 25
• 6² = 36 which can also be expressed 6 × 6 = 36
• 7² = 49 which can also be expressed 7 × 7 = 49
• 8² = 64 which can also be expressed 8 × 8 = 64
• 9² = 81 which can also be expressed 9 x 9 = 81
• 10² = 100 which can also be expressed 10 x 10 = 100
• 11² = 121 which can also be expressed 11 × 11 = 121
• 12² = 144 which can also be expressed 12 × 12 = 144

Check out   this short guide to squaring any number (no matter how big!) using mental math and Number Sense.

How Can We Use Square Numbers Outside of Classroom?

We can easily use (and practice!) square numbers outside the classroom, especially when we want to calculate the surfaces.

Here are a couple of examples of how you can put square numbers to practice, plus how we use them to plan and build the spaces you are using every day:

Practice with Toy Bricks

Take your toy bricks with studs and a studded baseplate to place them on. The rule of the game is: count the number of studs on a piece and put that many bricks one next to another on the baseplate.

• Start with the smallest brick which has 1 stud. Since the brick only has 1 stud, leave it alone on your baseplate.
• Next, take the brick with 2 studs. Since we have 2 studs, we’ll put 2 bricks on the baseplate, one next to another.
• Now let’s take a brick with 3 studs. Since we have 3 studs, we’ll put 3 bricks next to one another on the baseplate.

As you start to count the number of studs in each group of bricks, you will get square numbers of studs on single bricks.

• 1 2  (1 brick only) equals 1 stud
• 2 2  (or 2 studs x 2 studs) equals 4 studs
• 3 2  (or 3 studs x 3 studs) equals 9 studs

Refresh Your Memory of Square Numbers While Building Virtual Houses

Kids over the age of 12 might like this one!

If you are familiar with virtual house-building games, you’ll know that these games often provide you with “plots” overlaid with square patterns

Next time you outline your square living room or bedroom, count the number of squares and you’ll get a square number. Simple as that!

Learn How Engineers and Architects Use Square Numbers to   Plan & Build Spaces

From planning playgrounds to rooms and buildings, you can see square numbers in action in any square-shaped space.

Take the measuring tape and measure one side of your room. Multiply that number by itself and you’ll get the size of your floor if the room is perfectly square. For example, if each side of your room is 10ft wide, the surface of your square floor would be 10x10ft = 100ft 2 (read as “square feet”).

Is 100 a square number?

Yes! Yes, it is.

Ask your parents, engineers, or architects in your family to see how they use square numbers to plan and build the spaces you use every day.

Practice Square Numbers in Math

Your time to shine! Let’s review what we’ve learned with these simple exercises.

Exercise 1 : Square Number Multiplication

Let’s calculate these:

7 2 =___________

9 2 =___________

5 2 =___________

11 2 =___________

Exercise 2 : Missing Square Numbers

Fill in the missing square numbers in the sequence:

1, __, 9, __, 25, __, 49, 64, __, 100

Exercise 3: Square Number Word Problems

Try to solve these 3 word problems:

• If a square garden has an area of 64 square feet, what is the length of each side? (Hint: What number can we multiply by itself to get 64?)
• Sarah wants to build a square picture frame with an area of 100 square inches. What should be the length of each side? (Hint: What number can we multiply by itself to get 100?)
• A square rug has an area of 49 square meters. What is the length of one side of the rug? (Hint: What number can we multiply by itself to get 49?)

Completed the exercises?

Scroll to the end to check your answers.

A Neat Trick:  Learn How to Square Any Number

Frequently Asked Questions About Square Numbers

Find answers to common queries regarding the properties and applications of square numbers.

1.   When do students learn about square numbers at school?

Students usually encounter the basic square numbers when learning addition in early elementary school. They then formally begin to learn about square numbers later in elementary school, typically around grades 4 to 6.

Mathnasium works with elementary school students of all ages and skill levels to help them master math, including square numbers.

2. Is zero a square number?

Yes, zero is a square number because 0 x 0 = 0.

3. Is every positive number a square number?

Not all positive numbers can be square numbers. For example, 7 is a positive number, but it’s not a square number because you can’t make 7 by multiplying a number by itself.

4. Can you square a negative number?

Yes, you can square a negative number. Squaring a negative number also means multiplying the number by itself. When you square a negative number, the result is always positive because multiplying a negative number by a negative number always makes a positive number.

Let’s look at these examples:

-2 squared is (-2) * (-2) = 4

-3 squared is (-3) * (-3) = 9

-4 squared is (-4) * (-4) = 16

Learn & Master Square Numbers with Math Tutors Near You

Mathnasium’s specially trained tutors work with students of all skill levels to help them learn and master any K-12 math topic, including squared numbers.

Our tutors assess each student’s skills to create personalized learning plans that will put them on the best path to math mastery.

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Answers to Square Number Practice Exercises

Completed your square number exercises? Check your answers:

5 2  = 25

7 2  = 49

9 2  = 81

1, _ 4 _, 9, _ 16 _, 25, _ 36 _, 49, 64, _ 81 _, 100

• If a square garden has an area of 64 square feet, what is the length of each side? (Hint: What number can we multiply by itself to get 64?) The length of each side is 8 ft, because 64 = 8 2 .
• Sarah wants to build a square picture frame with an area of 100 square inches. What should be the length of each side? (Hint: What number can we multiply by itself to get 100?) The length of each side should be 10 inches, because 100 = 10 2 .
• A square rug has an area of 49 square meters. What is the length of one side of the rug? (Hint: What number can we multiply by itself to get 49?) The length of one side of the rug is 7 meters, because 49=7 2 .

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The Supreme Court Is Not Done Remaking America

Some of the rulings that came before the justices’ decision on presidential immunity could prove to have just as big an impact..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

From “The New York Times,” I’m Michael Barbaro. This is “The Daily.”

When the Supreme Court wrapped up its term last week, much of the focus was on the ruling that gave Donald Trump sweeping immunity from criminal prosecution. But as my colleague Adam Liptak explains, a set of rulings that generated far less attention could have just as big an impact on American government and society.

It’s Monday, July 8.

Adam, welcome back. It hasn’t been very long, but we want to talk to you about the rest of the Supreme Court’s decisions that happened over the past few weeks, the rest meaning the non-Trump decisions. There were a lot of other cases, many of which we covered on the show over the past year, but we haven’t yet talked about where the justices landed as they issued their rulings on these cases over the past few weeks. So I wonder if you can walk us through some of the bigger decisions and what, taken as a whole, this entire term really means. So where should we start?

Well, this term had so many major cases, Michael, on so many important issues touching all aspects of American politics and society, that it’s a little hard to know where to start. But I think one way to think about the term is to ask, how much is this a 6-3 court? There are six conservatives in the majority, the three liberal justices in dissent. Are we going to get that kind of classic lineup time after time after time?

And one way to start answering that question is to look at two areas which are kind of part of the court’s greatest hits, areas where they’ve done a lot of work in the last few terms — guns and abortion.

OK, let’s start with guns.

The court had two big guns cases. One of them involved the Second Amendment and broke 8 to 1 against Second Amendment rights. Only Justice Clarence Thomas, the most avid supporter of gun rights, was in dissent. So let me tell you just a little bit about this case.

There’s a federal law that says people subject to domestic violence restraining orders, it’s a crime for them to have guns. A guy named Zackey Rahimi was subject to such a domestic violence restraining order, but he goes to court and says, this law violates my Second Amendment rights. The Second Amendment protects me and allows me to have a gun even if I’m in this status.

And that goes to the Supreme Court. And the way the Supreme Court analyzes this question is it looks to a test that it established only a couple of years ago, in 2022, which said you judge the constitutionality of gun control laws using history. You kind of go back in time and you see whether the community and the founding era disarmed people in the same way that the current law disarms people.

And you might think that actually, back in the 1700s, there were no such things as domestic violence restraining orders. So you might think that the answer is, this contemporary law is unconstitutional. But Chief Justice John Roberts, writing for an eight-justice majority, says, no, that’s not quite right. We’re going to kind of roll back the specificity of the test and look at very general principles. Can you disarm dangerous people back then? And if you can do that, then you can disarm Rahimi, even under this law that the founding generation could not have contemplated.

That’s really interesting. So the court, its conservative majority especially, seems to be saying that our last big decision made it too hard to regulate guns. We need to fix that. So we’re going to search really hard for a way to make sure that somebody with a restraining order for domestic abuse can’t legally have a gun.

Right. On the other hand, there was a second guns case, not involving the Second Amendment, but posing an important issue. The question in the case was whether the Trump administration was allowed to enact a gun control regulation in 2017 after the Las Vegas shooting in which, at an outdoor music festival, a gunman killed 58 people, wounded 500 more.

And the Trump administration, prompted by this massacre, they issue a regulation that tries to outlaw bump stocks. What are bump stocks? They’re devices that turn semi-automatic weapons into weapons that can fire at rates approaching a machine gun. And drawing on the authority of a 1934 law which bans, for the most part, civilian ownership of machine guns, it said bump stocks are basically the same thing, and we will, by regulation, outlaw them.

And the question for the court was, did the 1934 law authorize that? And here — and this is a typical split on this kind of stuff — the majority, the conservative majority, takes a textualist approach. It bears down on the particular words of the statute. And Justice Thomas looks at the words that Congress said a machine gun is one where a single function of the trigger causes all of these bullets to fly. And a bump stock, he said, is not precisely that. Therefore, we’re going to strike down this regulation.

So how do you reconcile these two divergent gun rulings, one where the court works really hard to allow for gun restrictions in the case of domestic abusers, and another where they seem to have no compunction about allowing for a bump stock that I think most of us, practically speaking, understand as making a semiautomatic weapon automatic in the real world?

I think the court draws a real distinction between two kinds of cases. One is about interpreting the Constitution, interpreting the Second Amendment. And in that area, it is plowing new ground. It has issued maybe four major Second Amendment cases, and it’s trying to figure out how that works and what the limits are. And the Rahimi case shows you that they’re still finding their way. They’re trying to find the right balance in that constitutional realm where they are the last word.

The bump stocks case doesn’t involve the Constitution. It involves an interpretation of a statute enacted by Congress. And the majority, in those kinds of cases, tends to read statutes narrowly. And they would say that that’s acceptable because unlike in a constitutional case, if it’s about a congressional statute, Congress can go back and fix it. Congress can say whatever it likes.

Justice Samuel Alito said, in the bump stocks case, this massacre was terrible, and it’s a pity Congress didn’t act. But if Congress doesn’t act, a regulator can’t step in and do what Congress didn’t do.

That’s interesting, because it suggests a surprising level of open-mindedness among even the court’s most conservative justices to an interpretation of the Constitution that may allow for a greater level of gun regulation than perhaps we think of them as being interested in.

Yeah. When we’re talking about the Constitution, they do seem more open to regulating guns than you might have thought.

OK. You also mentioned, Adam, abortion. Let’s talk about those decisions from this court.

So the Court, in 2022, as everyone knows, overturned Roe v. Wade, eliminated the constitutional right to abortion. But in two cases this term, they effectively enhanced the availability of abortion.

One of them involves emergency rooms. There’s a federal statute that says that emergency rooms that receive federal money have to treat patients and give them stabilizing care if they arrive in the emergency room. That seems to conflict with a strict Idaho law that prohibits abortions except to save the life of the mother.

The court agrees to hear the case, it hears arguments, and then it dismisses the case. It dismisses it as improvidently granted, which is judicial speak for “never mind.” But it’s very tentative. The court merely dismissed the case. It said it was too early to hear it. They’re going to look at it later. So it’s a very tentative sliver of a victory for abortion rights.

But nonetheless, the effect of this is to suspend the Idaho law, at least to the extent it conflicts with the federal law. And it lets emergency abortions continue. Women in Idaho have more access to emergency abortions as a consequence of this decision than if the court had gone the other way.

And of course, the other abortion case centered on the abortion pill, mifepristone.

Right. And that pill is used in a majority of abortions. And the availability of that pill is crucial to what remains of abortion rights in the United States. Lower courts had said that the Food and Drug Administration exceeded its authority in approving these abortion pills. And the case comes to the Supreme Court.

And here, again, they rule in favor of abortion rights. They maintain the availability of these pills, but they do so, again, in a kind of technical way that does not assure that the pills will forever remain available. What the court says, merely — and unanimously — is that the particular plaintiffs who challenged the law, doctors and medical groups who oppose abortion, didn’t have standing, hadn’t suffered the sort of direct injury, that gave them the right to sue.

And it got rid of the case on standing grounds. But that’s not a permanent decision. Other people, other groups can sue, have sued. And the court didn’t decide whether the FDA approval was proper or not, only that the lawsuit couldn’t go forward. And here, too, this case is a victory for abortion rights, but maybe an ephemeral one, and may well return to the court, which has not given an indication of how it will turn out if they actually address the merits.

Got it. So this is a court, the one you’re describing in these rulings, acting with some nuance and some restraint?

Yeah, this picture is complicated.

This is not the court that we’re used to thinking about. There are a lot of crosscurrents. There are a lot of surprises. And that was true, in those cases, on big issues, on guns and abortion. But in another set of cases, the court moved aggressively to the right and really took on the very power and structure of the federal government.

[MUSIC PLAYING]

We’ll be right back.

So Adam, tell us about these cases where the court was less nuanced, less, perhaps, judicious, and really tried to move aggressively to the right and take on the power of government?

So it’s been a long-term goal of the conservative legal movement to weaken the power of regulators, of taking on what they call the administrative state. And this term, the court really vindicated that decades-old project primarily by overruling the foundational precedent in this area, a precedent that gave expert agencies the power to interpret federal laws and enact regulations to protect consumers, investors, all manner of people.

And the court overruled that decision called Chevron. It was as important as the court, two years ago, overruling the right to abortion, one year ago, overruling affirmative action in higher education. This decision will reshape the way the federal government does its work.

Right. And Adam, as I recall, because we did a whole episode about this with you, Chevron created a framework whereby if a law has any ambiguity about how it’s supposed to play out, that the experts within the federal government, within the EPA or the FDA, you name the agency, that we collectively defer to them and their wisdom, and that that becomes the basis for how these laws get interpreted and carried out.

That’s right. And if you think about it, Michael, Congress can’t anticipate every circumstance. Congress will, on purpose sometimes, and inevitably at other times, leave gaps in the law. And those gaps need to be filled by someone. And the choice that the Chevron decision made was to say, we’re going to let the expert regulator fill in those gaps. If there are ambiguities in statutes, the reasonable interpretation of the regulator will get deference from courts.

Experts, not judges, will decide this matter, is what Chevron said 40 years ago. And it’s really hard to overstate the consequences of overruling Chevron. It will open countless, countless regulations to judicial challenge. It may actually kind of swamp the courts. The courts have relied very heavily on Chevron to make difficult decisions about complicated stuff, questions about the environment, and food safety, and drugs, and securities, questions that really often require quite technical expertise.

So what was the court’s rationale for changing that Chevron framework that’s been in place for so long?

What the six-justice majority opinion written by Chief Justice Roberts says is that Chevron was a wrong turn from the outset, that unelected bureaucrats should not be empowered to say what the law means, that that’s the job of judges. So it moves from the expert agency to federal judges the determination of all sorts of important issues. And it probably has the effect of deregulating much of American society.

I mean, in the old world, the regulator had a thumb on the scale. The regulator’s interpretation of an ambiguous statute was the one that counted. And now, the judge will have a fresh look at it. That doesn’t mean that, in every case, the challenger wins, and in every case, the regulator loses. But it shifts the balance and it makes challenges more likely to succeed.

Where else did we see this instinct by the court to challenge the government’s authority in this term?

So right after the court overturns Chevron, it issues a second decision that really amplifies the power of that decision, because it says that challenges can be brought not only in the usual six-year statute of limitations from when a regulation is issued, but six years from when it first affects a company.

And bear with me, because that’s a big difference. If I start a company tomorrow, I have six years to sue over a regulation that affects it, even though that regulation may have been in place for 30 years. So it restarts the clock on challenges, and that one-two punch, both of them decided by 6-3 majorities, go even further in reshaping the ability of the federal government to regulate.

I just want to be sure I understand something. So in the past, let’s say the Clean Water Act was passed in the 1970s. Under the old statute of limitations, a company could sue and say that regulation is a problem for six years. But you’re saying a new company formed right now could go back and sue over something in a 30 - or 40-year-old law and how it’s being interpreted. In other words, this ruling means there really isn’t a statute of limitations on challenging these regulations any longer.

That’s right. And it’s not as though you can’t form a company just for the purpose of litigation. I mean, it completely opens up the ability of industries, trade groups just to set up a trivial nothing company that will then be said to be affected by the regulation and then can sue from now until the end of time.

And the liberal justices sure understood what was happening here, that this one-two punch, as Justice Ketanji Brown Jackson wrote in dissent, was a catastrophe for regulators. She wrote, “At the end of a momentous term, this much is clear — that tsunami of lawsuits against agencies that the court’s holdings in this case have authorized has the potential to devastate the functioning of the federal government.”

It’s a pretty searing warning.

Yeah. I mean, talking about regulations and administrative law might put some people to sleep, but this is a really big deal, Michael. And as if those two cases were not a substantial enough attack on the federal government’s regulatory authority, the court also issues a third 6-3 decision undoing one of the main ways that regulators file enforcement actions against people who they say have violated the law.

They don’t always go to court. Sometimes, they go to administrative tribunals within the agency. The court says, no, that’s no good. Only courts can adjudicate these matters. So it’s just another instance of the court being consistently hostile to the administrative state.

Adam, all three of these decisions might sound pretty dangerous if you have a lot of confidence in the federal government and in the judgments of regulators and bureaucrats to interpret things. But if you’re one of the many Americans who doesn’t have a whole lot of faith in the federal government, I have to imagine all of these rulings might seem pretty constructive.

That’s an excellent point. Lots of people are skeptical of regulators, are skeptical of what they would call the deep state, of unelected bureaucrats, of even the idea of expertise. And so for those people, this is a step in the right direction. It’s taking power away from bureaucrats and handing it to what we would hope are independent, fair-minded judges.

What does seem clear, Adam, is that even though this episode was supposed to be about the rest of the Supreme Court’s rulings this year, the less sexy-sounding decisions than Trump and immunity and how much power and protection all future presidents have, the rulings that you’re describing around the government’s administrative power, they seem like they’re going to have the greatest long-term impact on how our government functions, and in a sense, what our society looks like.

Well, the biggest case of the term is obviously the Trump immunity case. That’s a decision for the ages. But close behind these decisions, reshaping the administrative state and vindicating a long-held goal of the conservative legal movement going back to the Reagan administration, that the Federalist Society, the conservative legal group, has been pushing for decades, and really unraveling a conception of what the federal government does that’s been in place since the administration of Franklin Delano Roosevelt and his New Deal.

So as much as we’ve been talking about other cases where the court was tentative, surprising, nuanced in the biggest cases of the term, all delivered by six to three votes, all controlled by the conservative supermajority, the court was not nuanced. It was straightforward, and it reshaped American government.

In the end, a hard right court is going to, no matter how much it might deviate, operate like a hard right court.

Yes, Michael. It’s possible to look at the balance of the decisions and draw all kinds of complicated conclusions about the court. But when you look at the biggest cases, the picture you see is a conservative court moving the law to the right.

Well, Adam, thank you very much. We appreciate it.

Thank you, Michael.

Here’s what else you need to know today. “The Times” reports that four senior Democratic house members have told colleagues that President Biden must step aside as the party’s nominee over fears that he is no longer capable of winning. They include the top Democrats on the House Judiciary Committee, the Armed Services Committee, and the Veterans Affairs Committee.

Those top Democrats joined five rank and file House Democrats who have publicly called for Biden to step down. The latest of those was representative Angie Craig of Minnesota, who represents a swing district in the state. In a statement, Craig said that after watching Biden in the first debate, quote, “I do not believe that the president can effectively campaign and win against Donald Trump.” Senate Democrats remain largely quiet on the question of Biden’s future.

Now, you probably heard, I had a little debate last week. I can’t say it was my best performance.

In several appearances over the weekend, Biden acknowledged the growing skepticism of his candidacy —

Well, ever since then, there’s been a lot of speculation. What’s Joe going to do?

— but emphatically rejected the calls to step aside.

Well, let me say this clearly as I can. I’m staying in the race.

And in a surprise electoral upset, France’s political left was projected to win the largest number of seats in the National Assembly after the latest round of voting. The anti-immigrant far right had been expected to make history by winning the most seats, but a last-minute scramble by left wing parties averted that result.

Today’s episode was produced by Rikki Novetsky, Shannon Lin, and Rob Szypko. It was edited by Devon Taylor and Lisa Chow. Contains original music by Dan Powell and Sophia Lanman, and was engineered by Chris Wood. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Michael Barbaro. See you tomorrow.

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Hosted by Michael Barbaro

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Produced by Rikki Novetsky ,  Shannon M. Lin and Rob Szypko

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Engineered by Chris Wood

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When the Supreme Court wrapped up its term last week, much of the focus was on the ruling that gave former President Donald J. Trump sweeping immunity from criminal prosecution. But another set of rulings that generated less attention could have just as big an impact on American government and society.

Adam Liptak, who covers the Supreme Court for The Times, explains.

On today’s episode

Adam Liptak , who covers the Supreme Court for The New York Times and writes Sidebar, a column on legal developments.

Background reading

In a volatile term, a fractured Supreme Court remade America .

Here’s a guide to the major Supreme Court decisions in 2024 .

In video: How a fractured Supreme Court ruled this term .

There are a lot of ways to listen to The Daily. Here’s how.

We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page.

The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Mike Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, John Ketchum, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Summer Thomad, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Maddy Masiello, Isabella Anderson, Nina Lassam and Nick Pitman.

Adam Liptak covers the Supreme Court and writes Sidebar, a column on legal developments. A graduate of Yale Law School, he practiced law for 14 years before joining The Times in 2002. More about Adam Liptak

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