Vocabulary used in word problems

Solving word problems in elementary school is an essential part of the math curriculum. Here are over 30 math word problems to practice with children, plus expert guidance on how to solve them.

This blog is part of our series of blogs designed for teachers, schools and parents supporting home learning.

What is a word problem?

A word problem in math is a math question written as one sentence or more that requires children to apply their math knowledge to a ‘real-life’ scenario. 

This means that children must be familiar with the vocabulary associated with the mathematical symbols they are used to, in order to make sense of the word problem. 

For example:

Isn’t brilliant arithmetic enough?

In short, no. Students need to build good reading comprehension, even in math. Overtime math problems become increasingly complex and require students to possess deep conceptual understanding and the ability to recall and apply knowledge rapidly and accurately. 

As students progress through their mathematical education, they will need to be able to apply mathematical reasoning and develop mathematical arguments and proofs using math language. They will also need to be dynamic, applying their math knowledge to a variety of increasingly sophisticated problems. 

To support this schools are adopting a ‘mastery’ approach to math

“Teaching for mastery”, is defined with theses component: 

• Math teaching for mastery rejects the idea that a large proportion of people ‘just can’t do math’.  
• All students are encouraged by the belief that by working hard at math they can succeed. 
• Procedural fluency and conceptual understanding are developed in tandem because each supports the development of the other. 
• Significant time is spent developing deep knowledge of the key ideas that are needed to support future learning. The structure and connections within the mathematics are emphasized, so that students develop deep learning that can be sustained.

(The Essence of Maths Teaching for Mastery, 2016)

Mastery helps children to explore math in greater depth

Fluency in arithmetic is important; however, with this often lies the common misconception that once a child has learned the number skills appropriate to their grade level/age, they should be progressed to the next grade level/age of number skills. 

The mastery approach encourages exploring the breadth and depth of these math concepts (once fluency is secure) through reasoning and problem solving. 

How to teach children to solve word problems? 

Here are two simple strategies that can be applied to many word problems before solving them.

1. What do you already know?
2. How can this problem be drawn/represented pictorially?

Let’s see how this can be applied to word problems to help achieve the answer.

Solving a simple word problem

There are 28 students in a class.

The teacher has 8 liters of orange juice.

She pours 225 milliliters of orange juice for every student.

How much orange juice is left over?

1. What do you already know?

• There are 1,000ml in 1 liter
• Pours = liquid leaving the bottle = subtraction
• For every = multiply
• Left over = requires subtraction at some point

2. How can this problem be drawn/represented pictorially?

The bar model, also known as strip diagram, is always a great way of representing problems. However, if you are not familiar with this, there are always other ways of drawing it out. 

Read more: What is a bar model

For example, for this question, you could draw 28 students (or stick man x 28) with ‘225 ml’ above each one and then a half-empty bottle with ‘8 liters’ marked at the top.

Now to put the math to work. This is a 5th grade multi-step problem, so we need to use what we already know and what we’ve drawn to break down the steps.

Solving a more complex, mixed word problem

Mara is in a bookshop.

She buys one book for $6.99 and another that costs $3.40 more than the first book.

She pays using a $20 bill.

What change does Mara get? (What is the remainder?)

1. What do you already know?

• More than = add
• Using decimals means I will have to line up the decimal points correctly in calculations
• Change from money = subtract

2. How can this problem be drawn/represented pictorially?

See this example of bar modelling for this question:

Now to put the math to work using what we already know and what we’ve drawn to break down the steps.

Mara is in a bookshop. 

She buys one book for $6.99 and another that costs $3.40 more than the first book. 1) $6.99 + ($6.99 + $3.40) = $17.38

She pays using a $20 bill.

What change does Mara get? 2) $20 – $17.38 = $2.62

Math Word Problems For Kindergarten to Grade 5

The more children learn about math as they go through elementary school, the trickier the word problems they face will become.

Below you will find some information about the types of word problems your child will be coming up against on a year by year basis, and how word problems apply to each elementary grade. 

Word problems in kindergarten

Throughout kindergarten a child is likely to be introduced to word problems with the help of concrete resources (manipulatives, such as pieces of physical apparatus like coins, cards, counters or number lines) to help them understand the problem.

An example of a word problem for kindergarten would be

Chris has 3 red bounce balls and 2 green bounce balls. How many bounce balls does Chris have in all?

Word problems 1st grade

First grade is a continuation of kindergarten when it comes to word problems, with children still using concrete resources to help them understand and visualize the problems they are working on

An example of a word problem for first grade would be:

A class of 10 children each have 5 pencils in their pencil cases. How many pencils are there in total?

Word problems in 2nd grade

In second grade, children will move away from using concrete resources when solving word problems, and move towards using written methods. Teachers will begin to demonstrate the adding and subtracting within 100, adding up to 4- two-digit numbers at a time.

This is also the year in which 2-step word problems will be introduced. This is a problem which requires two individual calculations to be completed.

Second grade word problem: Geometry properties of shape

Shaun is making shapes out of plastic straws.

At the vertices where the straws meet, he uses blobs of modeling clay to fix them together

Here are some of the shapes he makes:

Shape	Number of straws	Number of blobs of modeling clay
A	4	4
B	3	3
C	6	6

One of Sean’s shapes is a triangle. Which is it? Explain your answer.

Answer: shape B as a triangle has 3 sides (straws) and 3 vertices, or angles (clay)

Second grade word problem: Statistics

2nd grade is collecting pebbles. This pictogram shows the different numbers of pebbles each group finds.

Word problems in 3rd grade

At this stage of their elementary school career, children should feel confident using the written method for addition and subtraction. They will begin multiplying and dividing within 100. 

This year children will be presented with a variety of problems, including 2-step problems and be expected to work out the appropriate method required to solve each one. 

3rd grade word problem: Number and place value

My number has four digits and has a 7 in the hundreds place.

The digit which has the highest value in my number is 2.

The digit which has the lowest value in my number is 6.

My number has 3 fewer tens than hundreds.

What is my number?

Answer: 2,746

Word problems in 4th grade

One and two-step word problems continue in fourth grade, but this is also the year that children will be introduced to word problems containing decimals.

Fourth grade word problem: Fractions and decimals

Stan, Frank and John are washing their cars outside their houses.

Stan has washed 0.5 of his car.

Frank has washed 1/5 of his car.

Norm has washed 2/5 of his car.

Who has washed the most?

Explain your answer.

Answer: Stan (he has washed 0.5 whereas Frank has only washed 0.2 and Norm 0.4)

Word problems in 5th grade

In fifth grade children move on from 2-step word problems to multi-step word problems. These will include fractions and decimals.

Here are some examples of the types of math word problems in fifth grade will have to solve.

5th grade word problem – Ratio and proportion

The Angel of the North is a large statue in England. It is 20 meters tall and 54 meters wide. 

Ally makes a scale model of the Angel of the North. Her model is 40 centimeters tall. How wide is her model?

Answer: 108cm

Fifth grade word problem – Algebra

Amina is making designs with two different shapes.

She gives each shape a value.

Calculate the value of each shape.

Answer: 36 (hexagon) and 25. 

Fifth grade word problem: Measurement

There are 28 students in a class.

The teacher has 8 liters of orange juice.

She pours 225 milliliters of orange juice for every student.

How much orange juice is left over?

Answer: 1.7 liters or 1,700ml

Topic based word problems

The following examples give you an idea of the kinds of math word problems your child will encounter in elementary school

Place value word problem fourth grade

This machine subtracts one hundredth each time the button is pressed. The starting number is 8.43. What number will the machine show if the button is pressed six times? Answer: 8.37

Download free number and place value word problems for grades 2, 3, 4 and 5

Addition and subtraction word problem grade 2

Sam has 64 sweets. He gets given 12 more. He then gives 22 away. How many sweets is he left with? Answer: 54

Download free addition and subtraction word problems for for grades 2, 3, 4 and 5

Addition word problem grade 2

Sammy thinks of a number. He subtracts 70. His new number is 12. What was the number Sammy thought of? Answer: 82

Subtraction word problem fifth grade

The temperature at 7pm was 4oC. By midnight, it had dropped by 9 degrees. What was the temperature at midnight? Answer: -5oC

Multiplication word problem third grade

Eggs are sold in boxes of 12. The egg boxes are delivered to stores in crates. Each crate holds 9 boxes. How many eggs are in a crate? Answer: 108

Download free multiplication word problems for grades 2, 3, 4 and 5. 

Division word problem fifth grade

A factory produces 3,572 paint brushes every day. They are packaged into boxes of 19. How many boxes does the factory produce every day? Answer: 188

Download free division word problems for grades 2, 3, 4 and 5. 

Free resource: Use these four operations word problems to practice addition, subtraction, multiplication and division all together.

Fraction word problem fourth grade

At the end of every day, a chocolate factory has 1 and 2/6 boxes of chocolates left over. How many boxes of chocolates are left over by the end of a week? Answer: 9 and 2/6 or 9 and 1/3

Download free fractions and decimals word problems for grades 2, 3, 4 and 5. 

Money word problem second grade 

Lucy and Noor found some money on the playground at recess. Lucy found 2 dimes and 1 penny, and Noor found 2 quarters and a dime. How many cents did Lucy and Noor find? Answer: Lucy = $0.21, Noor = $0.60; $0.21 + $0.61 = $0.81

Area word problem 3rd grade

A rectangle measures 6cm by 5cm.

What is its area? Answer: 30cm2

Perimeter word problem 3rd grade

The swimming pool at the Sunshine Inn hotel is 20m long and 7m wide. Mary swims around the edge of the pool twice. How many meters has she swum? Answer: 108m

Ratio word problem 5th grade (crossover with measurement)

A local council has spent the day painting double yellow lines. They use 1 pot of yellow paint for every 100m of road they paint. How many pots of paint will they need to paint a 2km stretch of road? Answer: 20 pots

PEMDAS word problem fifth grade

Draw a pair of parentheses in one of these calculations so that they make two different answers. What are the answers?

50 – 10 × 5 =

50 – 10 × 5 =

Volume word problem fifth grade

This large cuboid has been made by stacking shipping containers on a boat. Each individual shipping container has a length of 6m, a width of 4m and a height of 3m. What is the volume of the large cuboid? Answer: 864m3

Remember: The word problems can change but the math won’t 

It can be easy for children to get overwhelmed when they first come across word problems, but it is important that you remind them that while the context of the problem may be presented in a different way, the math behind it remains the same. 

Word problems are a good way to bring math into the real world and make math more relevant for your child. So help them practice, or even ask them to turn the tables and make up some word problems for you to solve. 

We all know what the problem is. It is a more or less difficult situation that is causing us inconvenience and stopping us from doing what we want. Here are different ways for talking about problems in English, including collocations, idioms, phrasal verbs, synonyms, and conversation questions to help you remember the new vocabulary. 

Face/encounter/confront a problem

If you face or encounter or confront a problem, then you are aware of it and you are trying to deal with it.

For example:

• Today, we are facing global warming issues that must be dealt with as soon as possible.
• When they tried to deal with the problem of discipline in their school, they encountered the problem of drugs abuse.
• The Johnsons have been confronted with many problems since they moved to another country.

Address/tackle the problem

If you address or tackle a problem, you are trying to solve it.

For example:

• The member of the parliament addressed the problem of human rights in the last night’s TV programme.
• The United Nations and governments of countries all around the world are trying to tackle environmental problems.

Deal with a problem

If you are dealing with a problem, you are taking concrete actions in order to solve it.

For example:

• Right now, our company is dealing with a problem of safety at work.
• The police have a strategy for dealing with the problem of street riots.

Discuss/debate a problem

To discuss or debate a problem means to exchange your opinions and ideas with another person or a group of people.

For example:

• The residents of the building discussed littering in the neighbourhood.
• One of the most debated problems nowadays is protection against the Coronavirus.

Overcome a problem

If you overcome a problem, you succeed in dealing with it.

For example:

• He’ll get seriously ill unless he overcomes his problem of overeating.
• A way to overcome problems associated with online bullying is to avoid contact with strangers.

The root of the problem

This expression refers to the basic cause of a problem.

For example:

• I believe that the root of the problem lies in his shyness.
• The government is supposed to fight against the root of the problem, not the consequences.

Problem-solving

A  compound noun related to the process of finding a solution to a problem.

For example:

• This course trains people in problem-solving in different situations.
• For this position, they expect their new employee to possess good problem-solving skills.

Other ways to talk about a problem

Hitch 

a small difficulty.

• The job in the advert looked perfect except for the hitch of a low salary. 

Snag

a similar word to hitch. It refers to a small problem or disadvantage.

• The only snag to marriage is that you lose your freedom.

Difficulty

another way to say problem.

• Maria works with children with learning difficulties.

Issue

a problem that people are discussing.

• The scientists met to discuss environmental issues.

Setback

an obstacle that stops something from developing.

• Dorothy has been recovering from her illness successfully except for some small setbacks. 

Complication

when the situation becomes more difficult than it was.

• The risk of complications occurring due to side effects is reduced.

Obstacle

when something stands in the way of achieving something.

• There are so many administrative obstacles that prevent workers from doing their job effectively.

Trouble

a problematic or distressful situation.

• They had a lot of trouble with that Ikea cupboard. The parts just didn’t fit in.

Challenge

a difficult task that may be interesting because it tests your skills and abilities.

• The Olympic basketball team is ready to face the challenges of the competition.

Deal with

to take necessary action in order to solve a problem.

For example:

• We must deal with this problem now before it becomes bigger.
• You have to learn how to deal with all kinds of problems if you want to progress in life.

Run into

to experience a problem unexpectedly.

For example:

• The company ran into some serious financial problems and bankrupted in a year.
• if you run into difficulties while operating this machine, call the mechanic immediately.

Run up against

when you have to deal with an unexpected problem.

For example:

• After they launched the new product, they ran up against some unexpected problems.
• When Lorna tried to talk to her husband about the problem she ran up against silence.

Knuckle down

to begin to work on something in order to solve a problem.

For example:

• The police knuckled down to the task of investigating the details of the robbery.
• If we knuckle down really hard, we’ll have this presentation ready by the end of next week.

Wrestle with ( a problem)

to try hard to deal with a problem or reach a difficult decision.

For example:

• Sam had a hard time wrestling with the problems in her marriage –  should she divorce or not?
• Sara spent most of the afternoon wrestling with her Math homework.

Talk something over

to discuss a problem with someone before making a final decision.

For example:

• First I need to talk it over with my husband and if we decide to buy a house, we’ll contact you.
• If you have digestion problems, you should talk it over with your doctor.

Idioms about problems

Be in dire straits

a very difficult financial situation.

For example:

• They met after the war when the country was in serious financial dire straits.
• The company is facing dire straits so they’re going to borrow some money from a bank.

Vicious circle

when one problem causes another, which creates even more difficulties.

For example:

• People on reduction diets often fall into a vicious circle of losing and then gaining weight again.
• I wish the vicious circle of poverty and stagnation ends in the world.

Put your foot in it

to say something accidentally so that it causes embarrassment and distress.

• I don’t like her at all. Oh sorry, did I put my foot in it? She’s your friend, isn’t she?
• Paul was quick to end the conversation before he put his foot in it even more.

Come up against a (brick) wall

to get to a point when you can not go on because something is preventing you from progressing.

• Many students come up against a brick wall when they apply for scholarships.
• Fiona was working long hours all last week. She’d stopped only when she felt that she came up against a brick wall and couldn’t think clearly.

Fall into place

when things happen without much difficulty.

For example:

• Once I organized my work well, everything fell into place so I had plenty of time for my leisure activities.
• After an extremely complicated plot, everything fell into place at the end of the film.

The light at the end of the tunnel

a ray of hope after a period of worry and difficulties.

• When the vaccine against the Coronavirus was invented, humankind could finally see the light at the end of the tunnel.
• We managed to solve some really serious problems within the company and now we can finally see the light at the end of the tunnel.

Conversation questions for talking about problems in English

The best way to remember new vocabulary is to use it immediately. Therefore, try to discuss these questions with your English speaking partner:

1. What is the greatest challenge/problem you have faced at school or work so far? How did you overcome it?
2. Have you ever come up against a brick wall? When and what happened?
3. Do you deal with your problems by yourself or do you discuss them with someone? Who?
4. Have you ever got embarrassed by putting your foot in it?
5. Who do you usually discuss your problems with? Asking for advice?
6. Have you ever helped anyone solve their problem? Did they take your advice?
7. Do you think the problem of global warming will be solved in this century?
8. Do you think the problem of discrimination will stop existing in this century? Why/Why not?
9. Do you think the problem of poverty and financial difficulties will be solved in this century? Why? Why not?
10. How can social issues affect society?

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