reasoning and problem solving year 2 addition

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Problem-solving maths investigations for year 2.

Hamilton provide an extensive suite of problem-solving maths investigations for Year 2 to facilitate mathematical confidence, investigative inquiry and the development of maths meta skills in 'low floor – high ceiling' activities for all.

Explore all our in-depth problem solving investigations for Year 2 .

Use problem-solving investigations within every unit to encourage children to develop and exercise their ability to reason mathematically and think creatively.

Investigations provide challenges that offer opportunities for the development of the key mathematical skills while deepening conceptual understanding. They are designed to be accessible in different ways to all children. An added bonus is the substantial amount of extra calculation practice they often incorporate! The problems are designed to help children identify patterns, to explore lines of thinking and to reason and communicate about properties of numbers, shapes and measures.

Hamilton provide a mix of our own specially commissioned investigations, that include guidance for teachers together with a child-friendly sheet to guide your pupils through the investigation, as well as links to investigations on other highly regarded websites.

I am very grateful for Hamilton Trust resources, particularly the maths investigations. Julia, teacher in Wiltshire

You can find Hamilton's investigations for Year 2:

• Individually, they are incorporated into every unit in our Year 2 flexible maths blocks .
• Collectively, they appear on our resources page where you can explore all our in-depth problem solving investigations for Year 2 .

Do read our extensive range of advice for more information about the investigations and for tips on how to use them effectively.

Hamilton’s problem-solving investigations are 'low floor, high ceiling' activities that give all children opportunities to develop mastery and mathematical meta-skills. Explore a set for a whole year group.

Hamilton’s Problem-solving Investigations provide school-wide solutions to the challenges of building investigative skills from Early Years to Year 6.

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Reasoning/Problem Solving

Year 2 Reasoning

Many schools are using some of the ideas of the ‘mastery programme’ as used in Singapore and other parts of the Far East. The aim of this programme is to give children a deep understanding and fluency in the fundamentals of maths. Because this understanding is deep it will not have to be re-taught at a later time; something which happens all too often at the moment.

This new approach needs new resources, especially geared towards reasoning and problem solving. Thankfully we have always considered reasoning to be a key part of the maths curriculum and many of our resources in all categories are designed for this. But we also have a separate category in each year group called, ‘Reasoning and Problem Solving’ where you will find a rich source of material.  In Year 2 we have a large selection of  ‘How many ways ….’ type of practical exercises (e.g. How many different ways can you score 12 with 3 dice?) These encourage children to think logically and order their results so that they can be sure that they have all possible answers. A favourite of mine is the ‘Two Coin Challenge’. Progressing from these practical tasks are the more abstract challenges such as making 2-digit numbers from a set of number cards.

Finally, take some time out to play the maths games such as Nim which originated in China. They are great at developing logical thinking and developing a winning strategy. You should never lose if you go first! These pages really are not to be missed and we thoroughly recommend them.  

Money and shopping

Solving problems involving money and shopping.

What coins could you receive when given 10p change?

What totals can be made from just two coins?

I've got three coins. They are all different. What possible totals could I have?

Can I use just 7 coins to pay all amounts up to 20p?

Solving problems about shopping: quite tricky.

More solving problems about shopping: quite tricky.

Buy shapes which are priced according to the number of sides.

'How many ways' investigations

Challenges which ask how many possible ways there are to solve a problem.

Three boxes and 20 buttons. How many different ways can the buttons be put in the boxes so that each box has an even number?

19 buttons to sort into odd numbers.

How many presents could each of the three girls have got from a total of 20. They all got an even number of presents!

How many ways can the apples be put in the boxes so that there is an even number of apples in each box?

How many ways can the balloons be coloured? A logical thinking exercise.

How many different addition, subtraction and multiplication questions can be made with the cards shown?

How many different addition sentences can be made using the numbers shown?

Create a number plate for the gorillas new car: but you can only use the digits 1, 2 or 3!

How many different ways can various jars be filled with jam?

How many different ways can various buckets be filled with water?

How many different ways can various containers be filled with shampoo?

How many different ways can various containers be filled with bubble bath?

Investigating combinations of fruit.

Investigating combinations of packs of toy farm animals.

Investigating combinations of groups of children.

Find all the ways of jumping forward.

Three very athletic runners sharing a 19 mile relay. They each run an odd number of miles. How could they do this?

Mini investigations into how many different ways totals can be made from three numbers.

A mini investigation for superbrains. How many ways can you make 100, just using the numbers and signs shown?

Can you find 10 different ways of making 12 with 3 cards? (One card is zero, which makes it trickier!)

Number challenges

A variety of number problems and challenges

Making 2-digit numbers.

Using only the numbers 3 and 2 and the + and - signs, can all the numbers up to 20 be made?

A colourful set of number problems involving beads.

Colourful pages on solving number problems involving buttons.

Roll two dice and add the totals. Record and sort the results. What do you notice?

How many different ways can you score a total with two dice?

Using grids to reason about place value of numbers.

Use three dice to see how many different ways a score of 12 can be made.

Solve number problems with dice (1)

More number problems with dice.

Find as many ways as possible to complete the number sentence.

Make and order 2-digit numbers from a selection of number cards.

Number sentences with missing digits using addition, subtraction, multiplication and division.

Recognising that subtraction is the inverse of addition and the relationship between multiplication and division.

All the signs are missing. Find the correct signs to complete the number sentences.

Put the addition or subtraction signs in to make the number sentences correct. Hard, as there is more than one sign!

Problem solving: finding the right sign.

How many patterned tiles are there altogether?

Find how many tiles there are in the pattern.

Further challenges and investigations

More great investigations and challenges for children in Year 2.

Place the toys on the correct shelves.

Use the balance scales to work out which shelves the items go on.

These pages are all to do with sequencing.

Finding the lengths of pencils in a line.

Finding more than the given numbers.

Monsters and legs problems.

Writing addition statements as multiplication.

Put the numbers in the boxes so that the total of the row across adds up to the same as the total of the column down. Several ways to do this, leading to further investigations.

How long will it take to complete a page of addition, subtraction, multiplication and division questions?

Find a route through the maze from IN to OUT adding the numbers as you go.

Dominoes are a great resource for sorting. Try these ideas to begin with.

Roll two dice and add the totals. Record and sort results.

Sam's got a great aim but how many can he score with four beanbags?

What possible scores could I get with four darts?

Harder four dart totals.

Word problems

Use knowledge of maths to solve problems written in words.

Some tricky word problems to solve.

A great selection of one and two-step word problems.

More tricky one-step and two-step word problems.

Addition and subtraction related to the six digits shown.

A selection of mathematical games for one or more players.

An easy version of the game of Nim. Great for logical thinking and you can always win (if you know what you are doing!)

The very simplest forms of magic squares and some ideas on how to turn them into colour patterns.

How many counters can you place on the grid without getting 3 in a row?

Fill in the puzzle so that every row, column and diagonal contains the numbers 1 to 4.

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Maths Resources & Worksheets >

Year 2 Resources and Worksheets

About these resources.

These Year 2 maths lessons for KS1 include teacher led IWB and PowerPoints, worksheets and activities for Year 2 primary teachers. There are lessons, worksheets and activities for each area of the curriculum. Our lessons are organised into blocks and steps to ensure full coverage for the National Curriculum in an order that builds and progresses across the year. Place Value, Addition and Subtraction, Money, Multiplication and Division, Statistics, Properties of Shapes Fractions, Length and Height and Time resources are covered. In addition, we have different question types including varied fluency, reasoning and problems solving, investigations and consolidation .

Maths Stem Sentences

Additional Supporting Step →

Other Maths Resources

Spring maths year 2 fractions resources & worksheets, spring maths year 2 length & perimeter resources & worksheets, spring maths year 2 multiplication & division resources & worksheets, spring maths year 2 properties of shape resources & worksheets, spring maths year 2 statistics resources & worksheets, summer block 1 (position and direction), summer block 2 (problem solving), summer block 3 (time), summer block 4 (mass, capacity and temperature), summer block 5 (investigations), times tables quizzes, year 2 autumn maths - addition and subtraction, year 2 autumn maths - money, year 2 autumn maths - multiplication and division, year 2 autumn maths - place value 1, year 2 consolidation packs.

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Language of Position – Extension

This worksheet includes a range of varied fluency and reasoning and problem solving questions for pupils to further extend and practise the main skill of using the language of position.

Language of Position – Discussion Problem

This worksheet includes one discussion problem that can be used in pairs or in small groups to enable pupils to further their understanding of the concepts linked to using language of position.

Language of Position – Teaching PowerPoint

These lesson slides guide pupils through the prior learning of using everyday language to describe position, before moving on to the main skill of using language of position. There are a number of questions to check pupils' understanding throughout.

Tables – Teaching PowerPoint

These lesson slides guide pupils through the prior learning of making tally charts, before moving on to the main skill of understanding tables.

Tables – Discussion Problem

This worksheet includes one discussion problem that can be used in pairs or in small groups to enable pupils to further their understanding of the concepts linked to using tables.

Tables – Extension

This worksheet includes a range of varied fluency and reasoning and problem solving questions for pupils to further extend and practise the main skill of using tables.

Hours in a Day – Teaching PowerPoint

These lesson slides guide pupils through the prior learning of telling the time to o'clock and half past, before moving on to the main skill of understanding hours in a day.

Hours in a Day – Extension

This worksheet includes a range of varied fluency and reasoning and problem solving questions for pupils to further extend and practise the main skill of understanding hours in a day.

Hours in a Day – Discussion Problem

This worksheet includes one discussion problem that can be used in pairs or in small groups to enable pupils to further their understanding of the concepts linked to understanding hours in a day.

Year 2 Minutes in an Hour Homework

Download this Y2 Minutes in an Hour Homework today to help consolidate the work you have done in class on converting times.

Year 2 Minutes in an Hour Discussion Problem

Download this Y2 Minutes in an Hour Discussion Problem to provide your pupils with a fun challenge. How quickly can they find the matching pairs?

Year 2 Minutes in an Hour Teaching PPT

Download this Y2 Minutes in an Hour Teaching PowerPoint to introduce the number of minutes in an hour and how to use this fact to convert times.

Year 2 Tell the Time to the Hour Homework

Extend children's learning from your class to their home by using our Y2 Tell the Time to the Hour Homework mixed question resource with answers.

Year 2 Tell the Time to the Hour Discussion Problem

Let children apply all the skills they've learnt in this lesson by using this ready-to-complete Year 2 Tell the Time to the Hour Discussion Problem.

Year 2 Tell the Time to the Hour Teaching PPT

Use our Y2 Tell the Time to the Hour Teaching PowerPoint for a step-by-step guide on how to complete this tricky concept with knowledge check questions!

Year 2 Tell the Time Past the Hour Teaching PowerPoint

Download this Y2 Tell the Time Past the Hour Discussion Problem to find out just what your pupils know and understand about telling the time.

Year 2 Tell the Time Past the Hour Homework

Make homework both meaningful and fun with this Year 2 Tell the Time Past the Hour Homework resource. Download it today!

Year 2 Tell the Time Past the Hour Discussion Problem

Year 2 Recognise Three Quarters Teaching PPT

Use this Y2 Recognise Three Quarters Teaching PPT to teach children how to recognise three quarters in a range of different representations.

Year 2 Recognise Three Quarters Discussion Problem

Download this challenging matching activity to help your Y2 pupils to recognise three quarters. This activity is a fun practical way to embed learning.

Year 2 Recognise Three Quarters Homework

Download this Year 2 Recognise Three Quarters Homework to embed children's learning at home. See how much your children really know!

Equal and Unequal Parts – Extension

This worksheet includes a discussion problem that can be used in pairs or in small groups to enable pupils to further their understanding of the concepts linked to exploring equal and unequal parts.

Equal and Unequal Parts – Discussion Problem

Equal and Unequal Parts Teaching PowerPoint

These lesson slides guide pupils through the prior learning of recognising parts and wholes, before moving on to the main skill of exploring equal and unequal parts.

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Watch CBS News

How two high school students solved a 2,000-year-old math puzzle

By Bill Whitaker , Aliza Chasan , Sara Kuzmarov, Mariah Campbell

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

A high school math teacher at St. Mary's Academy in New Orleans, Michelle Blouin Williams, was looking for ingenuity when she and her colleagues set a school-wide math contest with a challenging bonus question. That bonus question asked students to create a new proof for the Pythagorean Theorem, a fundamental principle of geometry, using trigonometry. The teachers weren't necessarily expecting anyone to solve it, as proofs of the Pythagorean Theorem using trigonometry were believed to be impossible for nearly 2,000 years.

But then, in December 2022, Calcea Johnson and Ne'Kiya Jackson, seniors at St. Mary's Academy, stepped up to the challenge. The $500 prize money was a motivating factor.

After months of work, they submitted their innovative proofs to their teachers. With the contest behind them, their teachers encouraged the students to present at a mathematics conference, and then to seek to publish their work. And even today, they're not done. Now in college, they've been working on further proofs of the Pythagorean Theorem and believe they have found five more proofs. Amazingly, despite their impressive achievements, they insist they're not math geniuses.

"I think that's a stretch," Calcea said.

The St. Mary's math contest

When the pair started working on the math contest they were familiar with the Pythagorean Theorem's equation: A² + B² = C², which explains that by knowing the length of two sides of a right triangle, it's possible to figure out the length of the third side.

When Calcea and Ne'Kiya set out to create a new Pythagorean Theorem proof, they didn't know that for thousands of years, one using trigonometry was thought to be impossible.  In 2009, mathematician Jason Zimba submitted one, and now Calcea and Ne'Kiya are adding to the canon.

Calcea and Ne'Kiya had studied geometry and some trigonometry when they started working on their proofs, but said they didn't feel math was easy. As the contest went on, they spent almost all their free time developing their ideas.

"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," Cal Johnson, Calcea's dad, said.

Neliska Jackson, Ne'Kiya's mother, says lightheartedly, that most of the time, her daughter's work was beyond her. 

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.

"Well, our teacher approached us and was like, 'Hey, you might be able to actually present this,'" Ne'Kiya said. "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."

Why Calcea' and Ne'kiya's work "blew up"

The reaction was insane and unexpected, Calcea said. News of their accomplishment spread around the world. The pair got a write-up in South Korea and a shoutout from former first lady Michelle Obama. They got a commendation from the governor and keys to the city of New Orleans. 

Calcea and Ne'Kiya said they think there's several reasons why people found their work so impressive. 

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

Ne'Kiya said she'd like their accomplishment to be celebrated for what it is: "a great mathematical achievement."

In spite of the community's celebration of the students' work, St. Mary's Academy president and interim principal Pamela Rogers said that with recognition came racist calls and comments. 

"[People said] 'they could not have done it. African Americans don't have the brains to do it.' Of course, we sheltered our girls from that," Rogers said. "But we absolutely did not expect it to come in the volume that it came."

Rogers said too often society has a vision of who can be successful.

"To some people, it is not always an African American female," Rogers said. "And to us, it's always an African American female."

Success at St. Marys 

St. Mary's, a private Catholic elementary and high school, was started for young Black women just after the Civil War. Ne'Kiya and Calcea follow a long line of barrier-breaking graduates. Leah Chase , the late queen of Creole cuisine, was an alum. So was Michelle Woodfork, the first African American female New Orleans police chief, and Dana Douglas, a judge for the Fifth Circuit Court of Appeals. 

Math teacher Michelle Blouin Williams, who initiated the math contest, said Calcea and Ne'Kiya are typical St. Mary's students. She said if they're "unicorns," then every student who's matriculated through the school is a "beautiful, Black unicorn."

Students hear that message from the moment they walk in the door, Rogers said. 

"We believe all students can succeed, all students can learn," the principal said. "It does not matter the environment that you live in."

About half the students at St. Mary's get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. There's no test to get in, but expectations are high and rules are strict: cellphones are not allowed and modest skirts and hair in its natural color are required. 

Students said they appreciate the rules and rigor.

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

What's next for Ne'Kiya and Calcea

Last year when Ne'Kiya and Calcea graduated, all their classmates were accepted into college and received scholarship offers. The school has had a 100% graduation rate and a 100% college acceptance rate for 17 years, according to Rogers.

Ne'Kiya got a full ride in the pharmacy department at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University. Neither one is pursuing a career in math, though Calcea said she may minor in math.

"People might expect too much out of me if I become a mathematician," Ne'Kiya said wryly. 

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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Outstanding Maths Lesson for Interview/Lesson Observation Problem Solving - Years 5 and 6

Subject: Mathematics

Age range: 7-11

Resource type: Lesson (complete)

Last updated

10 May 2024

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An exceptional maths problem solving lesson, complete with written lesson plan and interactive slides. Crafted to engage and challenge high-ability Year 5/6 students, this lesson offers a rich tapestry of activities and tasks designed to ignite mathematical thinking.

Perfect for a lesson observation or job interview, these comprehensive materials will impress all observers and demonstrate your expertise in delivering high-quality maths education.

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