number sentence in math problem solving

Number sentence – Definition, Application, FAQs, Examples

What is a number sentence, application of number sentences, solved examples on number sentence, practice problems on number sentence, frequently asked questions on number sentence.

A number sentence is a mathematical sentence made up of numbers and symbols, as shown below. 

The term “number sentence” is introduced at the elementary school level. However, the application of these sentences extends beyond elementary school because it includes equations and inequalities . These sentences can also be described as the language of mathematics. As shown below, a sentence combines two expressions with a relational symbol $(=, \gt, \lt, \text{etc.})$.

These sentences show the equality or inequality relations using different mathematical operations like addition , subtraction , multiplication , and division . 

The sign of equality and inequality is significant as the sentence is incomplete and makes no sense without them. 

$10 + 8 \gt 15$, is an example of a number sentence. However, if we write $10 + 8$  $15$, it does not make any sense.

A math sentence can be true or false depending on the information provided. 

A mathematical sentence that gives all the information and is known to be either true or false, as shown in the example below. 

Mathematical sentence problems can appear in the form of word problems, asking students how to write a number sentence.

For example: Mary has 10 strawberries. If Dan gives her 15 strawberries, how many strawberries does Mary have in total?

So, Mary has $10 + 15  = 25$ strawberries. 

Related Worksheets

Why Do Students Need to Be Fluent in Math Sentences?

• Mathematical sentences help students understand algebra. This involves weaving algebraic thinking into elementary and middle-school math.
• Math sentences provide flexibility to solve a problem as compared to basic algorithms. Using sentences, students can break the numbers out to see the value of each digit. They can compose and decompose numbers by place value or use other strategies, building their reasoning and mental math skills as shown in the example below.

Number sentences are simply the numerical expression of a word problem.

Example 1: Determine whether the following sentence is true or false.

$12 + 12 + 12 \lt 4 \times 12$

The expression on the right side of the inequality (less than) sign is $12 + 12 + 12$, which is equal to 36.

Solving expressions on the right side of the inequality (less than) sign, we get $4 \times 12$ or 48.

Since $36 \lt 48$, we can say the given sentence $12 + 12 + 12 \lt 4 \times 12$ is true.

Example 2: Complete the math sentence so that it is true.

$6 + 7 = 9$ $+$ $\underline{}$

$6 + 7 = 13$

So, to make the sentence true, $9$ $+$ $\underline{}$ must be equal to 13. Therefore, the missing number must be $13$ $–$ $9$ or 4.

Example 3: Substitute the value into the variable (x) and state whether the resulting sentence is true or false.

$12 –$ x $= 9$ , substitute 4 for x

If we substitute x as 4 in the given sentence, we $12$ $–$ $4 = 9$, which is false, as $12$ $–$ $4 = 8 ≠ 9$.

Example 4: Find the value of the x so that the following sentence is true.

$\text{x}$ $–$ $24 = 10$

Adding the same number to both sides of the equal sign will keep the sentence true.

To find the value of x, we can add 24 to both sides of the equal sign.

$\text{x}$ $–$ $24 + 24 = 10 + 24$

Therefore, $\text{x}$ $= 34$

Number sentence - Definition With Examples

Attend this quiz & Test your knowledge.

Which of the following is not a number sentence?

Select the correct statement for the sentences given below. $40 + 30 = 70$ $90 + 1000 = 1900$, identify the symbol that can fill the blank to make the sentence true 90 ◯ 20 = 70.

Is it important for a number sentence to be true?

A math sentence does not necessarily have to be true. However, every sentence gives us information, and based on the information provided, it is possible to change the statement from false to true.

What is the difference between equations and inequalities?

An equation is a mathematical sentence that shows the equal value of two expressions while an inequality is a sentence that shows an expression is lesser than or more than the other.

Can fractional numbers be written in the form of a number sentence?

Yes, fractional numbers can be written in the form of a sentence. For instance,

$\frac{3}{4}+\frac{5}{4} = \frac{8}{4}$

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Number sentences explained: Definitions and examples

What is a number sentence, what are the different types of number sentences, what is the difference between a true and false number sentence, how do you show that a number sign is not equal, what is a math sentence, are number sentences and ‘sums’ the same thing, why is understanding number sentences so important.

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Number sentences are one of the first things that primary school children are introduced to because it is one of the leading frameworks upon which most mathematics is taught.

Whether you’re solving simple addition problems or complex algebraic problems, you will utilise number sentences.

They help make sense of what calculations need to be done, and without them, you’ll just have numbers without a way to interpret them. But what exactly are number sentences?

That’s what we’ll explain in this article, along with the different types of number sentences, why understanding them is so important, and much more.

A number sentence can be defined as a mix of numbers and signs – also known as simple mathematical symbols – that presents a mathematical problem or equation which needs to be solved. As such, a number sentence is used synonymously with the phrases ‘math problem’ and ‘math equation’. Signs can be anything from operation signs such as multiplication (×) and division (÷) to an equal (=) or inequality (<>) sign.

Number sentences can include numbers and a mathematical operations sign on both sides and are often separated by an equals sign or inequality sign. 1 + 2 = 3 is considered a number sentence because it has all the necessary parts. It has numbers (1 and 2) and an addition sign (+), which is separated from another number (3) by an equal sign.

Due to the wide range of number and sign combinations to choose from, there are essentially an infinite amount of number sentences out there. However, they will typically fall into the following categories:

Addition number sentence

This is when the number sentence has an expression on one side, an equals sign, and then a number after it. To use the same example as above, 1 + 2 = 3 is an addition number sentence.

Subtraction number sentence

A subtraction number sentence usually follows the same format as an addition number sentence. An example of this would be 10 - 7 = 3.

Multiplication number sentence

Again, these follow the same format but with a multiplication sign instead. For instance, 4 × 4 = 16.

Division number sentence

The number sentence 6 ÷ 3 = 2 is a prime example of a division number sentence.

Less than number sentence

This is where the format changes. Instead of an equals sign, there will be a less than (<) sign. The less than sign shows that there is an imbalance in the number sentence, where the left expression is smaller than the right expression. An example of this would be 9 + 3 < 15

We know that 9 + 3 = 12. If we substitute 12 into the number sentence, we will get 12 < 15. This holds true since 12 is indeed less than 15.

Greater than number sentence

The same thing applies here, except we use a greater than sign. The greater than (>) sign shows that the left expression is bigger than the right expression. An example of this would be 20 + 3 > 21.

We know that 20 + 3 = 23. If we substitute 23 into the number sentence, we will get 23 > 21. This holds true since 23 is greater than 21.

Fraction number sentence

Fraction number sentences can include an equals sign or inequality sign but will have fractions instead of whole numbers. For instance, ⅕ + ⅗ = ⅘.

Algebraic number sentence

Lastly, we have algebraic number sentences. These substitute the whole numbers with letters such as a + b = c. As with fraction number sentences, these can vary, with some having equal signs and some having inequality signs.

A valid number sentence can be both true and false and is dependent on the expressions in the number sentence. Let’s explore what each one is and how they differ from each other.

True number sentence

A true number sentence is one where the written sentence is correct and is balanced on both sides. This is usually shown by using an equal sign to show that one side of the equation is equal to the other side of the equation. Examples of number sentences that are true include the following:

• 10 × 5 = 50
• 22 - 7 = 15

The above examples are relatively straightforward since they use the four main basic mathematical operators in an expression on one side, with the answer to the problem on the other. However, you can also have a true number sentence which consists of an expression on both sides. Calculating both expressions separately will result in equal values. For instance, suppose we have the number sentence 8 × 4 = 2 × 16.

If we solve this example problem, we will find that it is a true number sentence. 8 × 4 = 32 and 12 ×16 = 32, which we can write as 32 = 32. Since these are equal, it is a true number sentence. More examples of this are as follows:

• 9 - 8 = 1 ÷ 1
• 3 + 12 = 5 × 3
• 25 × 10 = 1,000 - 750

False number sentence

Whereas true number sentences are balanced on both sides of the equals sign, a false number sentence is one where it is unbalanced. This is often described as an ‘untrue’ problem. For instance, suppose we have the number sentence 5 × 4 = 15. If we calculate this problem, we will find that 5 × 4 is actually 20. 20 does not equal 15, and therefore this number sentence is defined as false.

False number sentences are typically used to test whether a student or person has a sound understanding of mathematical operations and expressions, as this deeper understanding will be required to distinguish between a false and true number sentence. More examples of false number sentences include the following:

• 25 × 9 = 750 ÷ 11

Sometimes, you will find that you don’t want to write a false number sentence, but you still need to show that the number sentence is not equal. This is where using an inequality sign makes sense.

There are four inequality signs to be aware of:

• Greater than (>) – The expression or values on the open side of the sign are said to be bigger than what’s on the closed side
• Less than (<) – The same holds true for this. The expression or values on the open side of the sign is said to be bigger than what’s on the closed side
• Greater than or equal to (≥) – The expression or value on the open side of the sign is said to be bigger or equal to what’s on the closed side
• Less than or equal to (≤) – The same holds true for this. The expression or value on the open side of the sign is said to be bigger or equal to what’s on the closed side

These signs provide much more flexibility since they show the relationship between both sides of an expression without having to create a false number sentence – particularly, the ‘... equal to’ signs as they can make all the difference in statistics and computer science problems where you can have unknown values. Here are a few examples of inequality signs being used in number sentences:

• 9 × 9 > 55
• 55 < 9 × 9
• 23 + 4 < 29
• n + 15 ≥ 20: If you calculate this problem, it means that n must be equal to or greater than the value 5.

So far, the examples we’ve seen of number sentences have been written as numbers and symbols. But, you can also describe number sentences as math sentences, also known as written word problems.

This is where the problem is described using words instead of explicit operational signs. Many schools use math sentences to train their students on how to use their understanding of language to create a number sentence which they can then solve. An example of this would be the following sentence:

Jack has 4 apples, and Liz has 10. If Liz gives 3 apples to Jack, how many apples does she have left?

To calculate this, students would have to connect the dots and translate the words into a number sentence with numbers and operational signs. This particular problem can be solved by doing 10 - 3 = 7. The answer to this problem would be 7 apples.

In the past, teachers and educators would use the word ‘sum(s)’ when referring to number sentences. The problem with this was that sum has different meanings depending on its context.

For instance, the sum is used synonymously with the word ‘total’; you could say ‘the sum of 1 + 1 = 2.' Therefore, referring to number sentences as a sum can confuse those who are new to learning mathematics.

To correct this, in most English-speaking countries such as the USA, UK, Canada, Australia, and New Zealand, math problems are now described as number sentences – or alternatively, problems and equations. This means that when number sentences are mentioned, students immediately know what is being referenced.

Understanding how to write and use number sentences is crucial because it displays fundamental mathematical structures and forms the basis of algebraic equations, which are a key part of any curriculum.

Number sentences also allow students to learn the correct syntax, which is akin to learning the proper punctuation and grammar when writing in English. Without knowing how to write and interpret number sentences, the why behind the calculations remains unclear and can significantly hinder their math progress. This ability to record and analyze connections between numbers is an invaluable method to further your mathematics knowledge.

The benefits of learning how to use number sentences aren’t limited to the classroom. They also have real-life applications. Whether you’re in a bank looking to withdraw money or simply calculating how many ingredients you need to buy for a two-person meal, number sentences will be the framework that you use to solve the problem.

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What Is A Number Sentence: Explained For Primary Parents And Kids!

Sophie Bartlett

In this post, we will be answering the question “ what is a number sentence?” and running through everything you need to know about this particular part of primary maths. We’ve also got number sentence questions and worksheets that you can use to test out your child’s skills.

What is a number sentence?

A number sentence is a combination of numbers and mathematical operations that children are often required to solve. 

Example of a number sentences include:

32 + 57 = ?  

5 x 6 = 10 x ?

103 + ? = 350

They will usually comprise of addition, subtraction, multiplication or division – or a combination of all four!

Remember – You may consider the above simply as “sums”, but referring to them as this can be confusing for children because the word “sum” should only be used when discussing addition. 

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A FREE resource including 20 home learning maths activities and games for years 1 and 2 children to complete on their own or with a partner.

When will my child learn about number sentences?

In the English National Curriculum, number sentences are referred to as ‘mathematical statements’.

These math sentences or statements are introduced as a maths skill in Year 1 , where pupils read, write and interpret mathematical statements involving mathematical symbols including addition (+), subtraction (–) and equals (=) signs.

Number sentences build on what children will have already learnt about number bonds .

Then, children will expand on this:

• Year 2 pupils calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication (×), division (÷) and equals sign (=).
• Year 3 pupils write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental and progressing to formal written methods. These pupils continue to practise their mental recall of multiplication tables when they are calculating mathematical statements in order to improve fluency.
• Year 4 pupils write statements about the equality of expressions (for example, use the distributive law 39 × 7 = 30 × 7 + 9 × 7 and associative law (2 × 3) × 4 = 2 × (3 × 4).
• Year 5 pupils are expected to understand the terms factor, multiple and prime, square and cube numbers and use them to construct equivalence statements (for example, 4 x 35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 81 x 10). 
• They should also recognise mixed numbers and improper fractions and convert from one form to the other and write mathematical statements > 1 as a mixed number [for example, 2/5 + 4/5 = 6/5 = 1 and 1/5].
• Year 6 pupils continue to use all the multiplication tables to calculate mathematical statements in order to maintain their fluency.

So, for example, in Year 1 your child will begin with addition sentences and subtraction number sentences. A number line may be helpful at this stage. In Year 2 they will use division sentences using whole numbers. By Year 4, they will use decimals in their number sentences.

Wondering about how to explain other key maths vocabulary to your children? Check out our Primary Maths Dictionary , or try these other maths terms:

• What Is The Perimeter?
• What Is BODMAS (and BIDMAS)?
• Properties of shapes
• What are 2D shapes ?
• What are 3D shapes ?

Number sentence practice questions

1) Complete the number sentences.

340 ÷ 7 = ____  remainder ____                          

____÷ 3 = 295 remainder 2

2) Here is a number sentence.

____ + 27 > 85

Circle all  the numbers below that make the number sentence correct.

30           40           50           60           70

3) Write in the missing number.

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Learn how the programmes are aligned to maths mastery teaching or request a personalised quote for your school to speak to us about your school’s needs and how we can help.

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FREE Ultimate Maths Vocabulary List [KS1 & KS2]

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Number sentence

A number sentence is a "mathematical sentence" used to express various mathematical relationships, namely equality and inequality. Number sentences are made up of:

• Operations (addition, subtraction, multiplication, division, etc.)
• Equality / inequality symbols

Below are some examples of number sentences.

Number sentences can also be written with fractions, decimals, negative numbers, with powers, and more. We can identify all the examples above as equations based on the use of the "=" sign. It is worth nothing that number sentences do not necessarily have to be true. For example, 2 - 3 = 5 is still a number sentence, albeit a false one. False number sentences can be used to test our understanding of basic arithmetic and all the symbols involved. For example, one thing we could change about the false number sentence is the minus sign. If we changed the minus sign to a plus sign, the number sentence would be true:

Number sentences can also take the form of inequalities . The key inequality symbols that we should recognize are:

• less than: <
• greater than: >
• less than or equal to: ≤
• greater than or equal to: ≥
• is not equal to: ≠

In the false number sentence above, 2 - 3 = 5, instead of changing the minus sign, we could also instead use various inequality signs. 2 - 3 = -1, so we could also write 2 - 3 < 5, and the number sentence would be true. Alternatively, we could write 2 - 3 ≠ 5, and this would also be true.

Change the following false number sentences such that they become true.

1 . 2 + 8 < 10:

2 + 8 ≤ 10

2 + 8 ≥ 10

2 . 5 ≥ 12:

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What is a Number Sentence?

• Teaching Strategies, Tactics, and Methods

A number sentence is an array of numbers and symbols. Also referred to as a “sum” or “problem,” number sentences are a familiar way of arranging questions in K-5 math.

Kids must learn this early, as it is how most of the work in their math lessons will look.

“Number sentence” is the term used in K-5-level math teaching in countries such as the U.S., Canada, the UK, New Zealand, South Africa, and Australia.

Here are some examples of number sentences:

Addition number sentence: 7 + 5 = 12

Subtraction number sentence: 44 – 10 = 34

Multiplication number sentence: 5 x 4 = 20

Division number sentence: 35 ÷ 7 = 5

When are number sentences introduced?

Typically, children will start learning to write and solve number sentences in first grade. They will likely begin by using objects such as counters and small toys to help them understand the value of numbers.

Children also need to be able to turn word problems into number sentences to understand the question. For example:

Steve has $10 and spends $4.50 on his lunch. How much does he have left?

The number sentence is 10 – 4.50 = 5.50

So a child can work out that Steve has $5.50 left using a number sentence.

They are essential for a child to learn early on to develop their math skills, as this is what most of the math problems they’ll be solving will look like.

Number sentences for kids

Here are ten differentiated number sentences for your children to try out. They start friendly and accessible but get harder and harder as you go on. How many can your class solve?

• 3 – 2 = ?
• 6 – ? = 2
• 10 – ? = 3
• 25 + 12 = ?

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Strategy: Write a Number Sentence

Math Problem Solving Strategy: Write a Number Sentence to Solve a Problem

This is another free resource for teachers from The Curriculum Corner.

Looking to help your students learn to write a number sentence to solve a problem?

This math problem solving strategy can be practiced with this set of resources.

Math Problem Solving Strategies

This is one in a series of resources to help you focus on specific problem solving strategies in the classroom.

Within this download, we are offering you a range of word problems for practice.

Each page provided contains a single problem solving word problem.

Below each story problem you will find a set of four steps for students to follow when finding the answer.

This set will focus on the write a number sentence strategy for math problem solving.

What are the 4 problem solving steps?

After carefully reading the problem, students will:

• Step 1:  Circle the math words.
• Step 2:  Ask yourself: Do I understand the problem?
• Step 3:  Solve the problem using words and pictures below.
• Step 4:  Share the answer along with explaining why the answer makes sense.

Write a Number Sentence to Solve a Problem Word Work Questions

The problems within this post are meant to help students solve problems by writing a number sentence.  These problems are designed to be used with first, second or third grade math students.

Within this collection you will find two variations of each problem.

You will easily be able to create additional problems using the wording below as a base.

The problems include the following selections:

• Cookies – easy addition
• Coin Collection – addition with regrouping
• Jewelry – addition with regrouping
• Making Cards – easy subtraction
• Beads for Bracelets – subtraction without regrouping
• Toy Cards – subtraction with regrouping
• Hot Chocolate – easy multiplication
• Pencils – one-digit times two-digit multiplication
• Legos – two-digit times two-digit multiplication

You can download this set of Write a Number Sentence to Solve a Problem here:

Problem Solving

You might also be interested in the following problem solving resources:

• Drawing Pictures to Solve Problems
• Addition & Subtraction Word Problem Strategies
• Fall Problem Solving
• Winter Problem Solving
• Spring Problem Solving
• Summer Problem Solving

As with all of our resources, The Curriculum Corner creates these for free classroom use. Our products may not be sold. You may print and copy for your personal classroom use. These are also great for home school families!

You may not modify and resell in any form. Please let us know if you have any questions.

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Wednesday 2nd of January 2013

Great activities for understanding how to solve word problems.

1.2.2 Number Sentences

Standard 1.2.2.

Represent real-world situations involving addition and subtraction basic facts, using objects and number sentences.

For example: One way to represent the number of toys that a child has left after giving away 4 of 6 toys is to begin with a stack of 6 connecting cubes and then break off 4 cubes.

Determine if equations involving addition and subtraction are true.

For example : Determine if the following number sentences are true or false

5 + 2 = 2 + 5

4 + 1 = 5 + 2.

Use number sense and models of addition and subtraction, such as objects and number lines, to identify the missing number in an equation such as:

2 + 4 = $\square$

3 + $\square$ = 7

5 = $\square$ - 3.

Use addition or subtraction basic facts to represent a given problem situation using a number sentence.

For example : 5 + 3 = 8 could be used to represent a situation in which 5 red balloons are combined with 3 blue balloons to make 8 total balloons.

Standard 1.2.2 Essential Understandings

First graders build on their previous work with the composition and decomposition of numbers by writing number sentences to represent a real world or mathematical situation involving addition and subtraction. In addition, they will write a real world problem to represent a given number sentence.

Work with number sentences continues as first graders determine if a number sentence is true or false.  For example, is 5 + 3 = 8 true or false?  They are also able to write their own true number sentences and false number sentences.

First graders begin their work with variables by determining an unknown in a number sentence.  These unknowns are found in varying positions in the number sentences.

For example, 5 + $\square$ = 8,    5  = $\square$ -  3,   3 + 5 = ∆ .

All Standard Benchmarks - with codes  

1.2.2.1 Represent real-world situations involving addition and subtraction basic facts, using objects and number sentences.

1.2.2.2 Determine if equations involving addition and subtraction are true.

1.2.2.3 Use number sense and models of addition and subtraction, such as objects and number lines, to identify the missing number in an equation such as:                                     2 + 4 = ∆                                     3 + ∆ = 7                                     5 = ∆ - 3. 1.2.2.4 Use addition or subtraction basic facts to represent a given problem situation using a number sentence.

1.2.2.1 Represent real-world situations involving addition and subtraction basic facts, using objects and number sentences. 1.2.2.2 Determine if equations involving addition and subtraction are true. 1.2.2.3 Use number sense and models of addition and subtraction, such as objects and number lines, to identify the missing number in an equation such as:                                     2 + 4 = ∆                                     3 + ∆ = 7                                     5 = ∆ - 3. 1.2.2.4 Use addition or subtraction basic facts to represent a given problem situation using a number sentence.

What students should know and be able to do [at a mastery level] related to these benchmarks

• know the equal sign means "the same as."
• find the number that makes a number sentence true with the unknown in any position:

3 + 6 = ∆  and  ∆ = 3 + 6 3 + ∆ = 9  and  9 = 3 + ∆ ∆ + 6 = 9  and  9 = ∆ + 6 9 - 3 = ∆  and  ∆ = 9 - 3 9 - ∆ = 6  and  6 = 9 - ∆ ∆ - 3 = 6  and  6 = ∆ - 3

• model the above types of equations using manipulatives and number lines.
• create equations to match story problems.
• create story problems to match equations.
• determine the truth value of equations involving addition and subtraction.  

Work from previous grades that supports this new learning includes: 

• compose and decompose numbers to twelve.
• solve addition and subtraction problems to 10 using counters and other visible materials such as fingers and 10-frames.

NCTM Standards

Use mathematical models to represent and understand quantitative relationships.

Pre-K-2 Expectations

• Model situations that involve the addition and subtraction of whole numbers, using objects, pictures, and symbols.

Common Core State Standards

Represent and solve problems involving addition and subtraction.

1.OA.1. Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions; e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Work with addition and subtraction equations.

1.OA.7. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

1.OA.8. Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 +? = 11, 5 = _ - 3, 6 + 6 = _.

Misconceptions

Student Misconceptions and Common Errors

Students may think...

• the equal sign means the answer.
• when they see two numbers they should combine them.
• the only correct format for a problem is a + b = c or a - b = c, not recognizing that it can also be c = a + b or c = a - b or a + b = m + n.
• ∆ = 10 - 3 is read: 3 minus 10 equals ∆.
• they can completely ignore an unknown in an equation.

In the Classroom  

First graders in Mr Xiong's class are determining if number sentences are true.

Mr. Xiong writes 3 + 5 = 8 on the board.

Mr. Xiong:  I want you to think about this number sentence. Is this number sentence true or false?  Skyler, what do you think?

Skyler:  I think it's true. 

Mr. Xiong:      Why do you think it's true?

Skyler:  Well, look. I have five here (flashes five fingers) and three here (flashes three fingers). That's eight!  So it's true.

Mr. Xiong:  Brie, I noticed that you were agreeing with Skyler. Why do you think it's true?

Brie:  I started at the five and counted three more,  5 - 6..7..8..so I know it's true.

Mr. Xiong writes 8 = 3 + 5 on the board.

Mr. Xiong:.  What about this number sentence?  True or false?  He looks around the room and sees some puzzled faces. Alan, what do you think?

Alan:  It's false. But you wrote it wrong, Mr. Xiong. Are you trying to trick us? You can't write it that way. The equal sign goes at the end...and so it's false because 8 + 3 isn't 5, it's 11. Look at the other one, that's how you write it.

Mr. Xiong:  Let's look at the first equation. Alan, can you read it for us? 

Mr. Xiong points to 3 + 5 = 8.

Alan:  Three plus five equals eight.

Mr. Xiong:  Alan, can you read the problem another way?

Alan:               Pauses and thinks for a moment. Three and five make eight.

Mr. Xiong:  Okay...good. How about another way? Could anybody read it another way?

Beth:  Three plus five is the same as eight.

Mr. Xiong:  Good. Now, let's take a look at this problem. Tia, will you read this number sentence. 

Mr Xiong points to 8 = 3 + 5.

Tia:  Eight equals three plus five.

Mr. Xiong asks the class to read the number sentence with Tia as he points to each part of the number sentence.

Brie:  I can read it another way. Eight is the same as three plus five.

Mr. Xiong:   Points to 3 + 5  = 8 . Three plus five is the same as eight.

Points to 8  = 3 + 5. Eight is the same as three plus five.

Does it matter where we put the equal sign, at the beginning or at the end? 

Skyler:  It doesn't matter. It's just like turning things around, but it means the same thing.

Mr. Xiong used Alan's response as a springboard for the conversation about the meaning of the equal sign.

Teacher Notes  

• Students may need support in further development of previously studied concepts and skills .
• Provide opportunities to find an unknown in number sentences with the unknown in various locations.  For example:  2 + 4 = ∆,  3 + ∆ = 7,  5 = ∆ - 3,  ∆ = 6 + 4,
• 6 + 3 = ∆ + 2.
• The number sentence 10 - ∆ = 6 can be read ten minus some number equals six.  When first asked to find an unknown students need help understanding the symbolic language. 
• First graders will apply what they know about composing and decomposing number and the operations of addition and subtraction to help them find the unknown. First graders should initially work with combinations of six or less in order to make the part-whole relationship more visible when finding an unknown.
• A foundational algebraic idea is the equality relationship, which is represented by the equal sign. Students must learn that the equal sign means a balanced relationship. Rather than saying "the answer is," use "is the same as" to help children develop a sense of equality.
• Use concrete materials to explore concepts of equality and inequality.
• Use concrete materials to explore and describe number relationships expressed in open-ended number sentences (e.g. $\square+\square=7$)
• The equal sign can be confusing when students only see it as "what comes before the answer." Number sentences should be presented with the = sign at the beginning, middle, or end of a problem.

and the relationships that exist between and among those operations. The bar model is an effective tool that can be used to represent the relationships in a variety of problem solving situations. For more information on

Questioning

Good questions , and good listening , will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."

Getting Started What do you need to find out? What do you know now? How can you get the information? Where can you begin? What terms do you understand/not understand? What similar problems have you solved that would help?

While Working How can you organize the information? Can you make a drawing (model) to explain your thinking? What are other possibilities? What would happen if...? Can you describe an approach (strategy) you can use to solve this? What do you need to do next? Do you see any patterns or relationships that will help you solve this? How does this relate to...? Why did you...? What assumptions are you making?

Reflecting about the Solution How do you know your solution (conclusion) is reasonable? How did you arrive at your answer? How can you convince me your answer makes sense? What did you try that did not work? Has the question been answered? Can the explanation be made clearer?

Responding (helps clarify and extend their thinking) Tell me more. Can you explain it in a different way? Is there another possibility or strategy that would work? Is there a more efficient strategy? Help me understand this part ...

(Adapted from They're Counting on Us, California Mathematics Council, 1995)

Instructional Resources

• True or False

Asking first graders to determine if a number sentence is true or false is important as first graders develop a sense of equality. It is important to use number sentences with "easy numbers."  Number sentences such as 5 = 5 and number sentences which are false (3 + 5 = 10) challenge student thinking. It is also important for first graders to consider the following types of number sentences:  6 = 4 + 2, 3 = 10 - 7, 4 + 2 = 2 + 4, 5 + 3 = 5 + 3.

The following number sentences might be used the first time first graders are asked to determine if a number sentence is true or false.

     True or False?

            3 + 2 = 5

            2 + 3 = 5

            5 = 3 + 2

            2 + 3 = 4

            5 = 5

            5 = 3 + 3

            2 + 3 = 2 + 3

            2 + 3 = 2 - 3

Additional Instructional Resources    

Greenes, C., Cavanagh, M., Dacey, L., Findell, C., & Small, M. (2001). Navigating through algebra in prekindergarten-grade 2 . Reston, VA: National Council of Teachers of Mathematics.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.   New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA:  Allyn & Bacon.

Van de Walle, J. & Lovin, L.  (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

Von Rotz, L., & Burns, M. (2002). Grades k-2 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications.

number sentence : mathematical sentence written in numerals and mathematical symbols

addition: to join two or more numbers to get one number (called the sum or total)

subtraction: an operation that gives the difference between two numbers. Subtraction can be used to compare two numbers, or to find out how much is left after some is taken away

equal: having the same amount or value, the symbol is =

" Vocabulary literally is the key tool for thinking."      Ruby Payne

Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions.  Students need to have experiences communicating ideas using these words to explain, support, and justify their thinking.

Learning vocabulary in the mathematics classroom is contingent upon the following:

Integration :   Connecting new vocabulary to prior knowledge and previously learned vocabulary.  The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.

Repetition :    Using the word or concept many times during the learning process and connecting the word or concept with its meaning.  The role of the teacher is to provide experiences that will guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.

Meaningful Use:    Multiple and varied opportunities to use the words in context.  These opportunities occur when students explain their thinking, ask clarifying questions, write about mathematics, and think aloud when solving problems.  Teachers should be constantly probing student thinking in order to determine if students are connecting mathematics concepts and relationships with appropriate mathematics vocabulary.

Strategies for vocabulary development

Students do not learn vocabulary words by memorizing and practicing definitions . The following strategies keep vocabulary visible and accessible during instruction.

Mathematics Word Bank :  Each unit of study should have word banks visible during instruction.  Words and corresponding definitions are added to the word bank as the need arises.  Students refer to word banks when communicating mathematical ideas which leads to greater understanding and application of words in context.

Labeled pictures and charts :  Diagrams that are labeled provide opportunities for students to anchor their thinking as they develop conceptual understanding and increase opportunities for student learning.

Frayer Model : The Frayer Model connects words, definitions, examples and non-examples.

Example/Non-example Charts : This graphic organizer allows students to reason about mathematical relationships as they develop conceptual understanding of mathematics vocabulary words.  Teachers should use these during the instructional process to engage student in thinking about the meaning of words.

Vocabulary Strips :  Vocabulary strips give students a way to organize critical information about mathematics vocabulary words.

Encouraging students to verbalize thinking by drawing, talking, and writing increases opportunities to use the mathematics vocabulary words in context.

Additional Resources for Vocabulary Development

Murray, M. (2004). Teaching mathematics vocabulary in context . Portsmouth, NH: Heinemann.

Sammons, L. (2011).  Building mathematical comprehension: Using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

Reflection - Critical Questions regarding the teaching and learning of these benchmarks:

What are the key ideas related to an understanding of equality at the first grade level?  How do student misconceptions interfere with mastery of these ideas?

What kind of number sentences should first graders see related to equality in an instructional setting?

Write a set of number sentences you could use with first graders in exploring their understanding of equality.  Which are the most challenging for first graders?

When checking for student understanding, what should teachers

• listen for in student conversations?
• look for in student work?
• ask during classroom discussions?

Examine student work related to a task involving equality. What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.

What are the key ideas related to identifying an unknown in an equation at the first grade level? How do student misconceptions interfere with mastery of these ideas?

What kind of equations should first graders experience when identifying an unknown in an equation?

Write a set of equations you could use with first graders in exploring their understanding of identifying unknowns in equation.  Which are the most challenging for first graders?

Examine student work related to a task involving identifying unknowns. What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.

How can teachers assess student learning related to these benchmarks?

How are these benchmarks related to other benchmarks at the first grade level?

Professional Learning Community Resources

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.

Carpenter, T., Franke, M., & Levi, L. ( 2003). Thinking mathematically integrating arithmetic & algebra in elementary school , Portsmouth, NH: Heinemann.

Chapin, S., & Johnson, A. (2006). Math matters, understanding the math you teach, grades k-8 . (2 nd ed.). Sausalito, CA: Math Solutions Press.

Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6) . Sausalito, CA: Math Solutions.

Fosnot, C., & Dolk, M. (2010). Young mathematicians at work constructing algebra. Portsmouth, NH: Heinemann.

Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.

Sammons, L. (2011). Building mathematical comprehension-using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions pre-k grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.

Bender, W. (2009). Differentiating math instruction-Strategies that work for k-8 classrooms! Thousand Oaks, CA.: Corwin Press.

Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting english language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.

Burns, M. (Edt).(1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA:  Math Solutions.

Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.

Chapin, S., & Johnson, A. (2006). Math matters, understanding the math you teach, grades k-8 , (2nd ed.).. Sausalito, CA: Math Solutions Press.

Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction k-2. Sausalito, CA: Math Solutions.

Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC.: National Academies Press.

Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course . Sausalito, CA: Math Solutions.

Hyde, Arthur. (2006) Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.

Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.

Leinwand, S., (2000). Sensible mathematics: A guide for school leaders . Portsmouth, NH:  Heinemann.

Murray, M. (2004). Teaching Mathematics Vocabulary in Context . Portsmouth, NH: Heinemann.

Murray, M, & Jorgensen, J. (2007). The differentiated math classroom-A guide for teachers k-8. Portsmouth, NH: Heinemann .

Sammons, L. (2011). Building mathematical comprehension-Using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics.   Reston, VA:  NCTM.

Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., ... & Zbiek, R. M. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: Aquest for coherence. Reston, VA: NCTM.

Seeley, C. (2009). Faster isn't smarter-Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.

West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann .

• Write an equation to represent the following number story.

There are seven turtles in the pond. Three more turtles go into the pond. How many turtles are in the pond?

Solution:          7 + 3 = 10 Benchmark:     1.2.2.4

• Write a number story to match this equation.

7 - 2 = 5 Solution: Contexts will vary but will represent seven minus two equals five. Benchmark:     1.2.2.4

• Look at the number sentences. Tell if they are true or false.

5 - 1 = 6 9 = 4 + 5 6 + 4 = 6 + 4 6 = 6 4 + 1 = 5 + 2 5 + 4 = 4 + 5 Solution:          F, T, T, T, F, T  Benchmark:     1.2.2.2

• What is the missing number in each equation?

3 + 4 = ∆ 10 - ∆ = 6 8 = ∆ + 5 Solution:  7, 4, 3 Benchmark:     1.2.2.3

Differentiation

Students may need to use materials to find the unknown in open number sentences involving addition and subtraction. For example, when presented with 3 + ∆ = 7, a teacher may lay out three counters and ask how many more are needed to make seven. 

Students may need to use materials in order to determine if a number sentence is true or false.

Concrete - Representational - Abstract Instructional Approach

The Concrete-Representational-Abstract Instructional Approach (CRA) is a research-based instructional strategy that has proven effective in enhancing the mathematics performance of students who struggle with mathematics.

The CRA approach is based on three stages during the learning process:

              Concrete     -     Representational     -     Abstract

The Concrete Stage is the doing stage. The concrete stage is the most critical in terms of developing conceptual understanding of mathematical skills and concepts.  At this stage, teachers use manipulatives to model mathematical concepts. The physical act of touching and moving manipulatives enables students to experience the mathematical concept at a concrete level.  Research shows that students who use concrete materials develop more precise and comprehensive mental representations, understand and apply mathematical concepts, and are more motivated and on-task.   Manipulatives must be selected based upon connections to the mathematical concept and the students' developmental level.

The Representational Stage is the drawing stage.  Mathematical concepts are represented using pictures or drawings of the manipulatives previously used at the Concrete Stage. Students move to this level after they have successfully used concrete materials to demonstrate conceptual understanding and solve problems.  They are moving from a concrete level of understanding toward an abstract level of understanding when drawing or using pictures to represent their thinking.  Students continue exploring the mathematical concept at this level while teachers are asking questions to elicit student thinking and understanding.

The Abstract Stage is the symbolic stage.  Teachers model mathematical concepts using numbers and mathematical symbols.  Operation symbols are used to represent addition, subtraction, multiplication and division.  Some students may not make a clean transfer to this level. They will work with some symbols and some pictures as they build abstract understanding.  Moving to the abstract level too quickly causes many student errors.   Practice at the abstract level will not lead to increased understanding unless students have a foundation based upon concrete and pictorial representations. 

Additional Resources:

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k-  2. Sausalito, CA: Math Solutions.

Murray, M. & Jorgensen, J. (2007). The differentiated math classroom-A guide for teachers k-8. Portsmouth, NH: Heinemann .

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.).  Boston, MA: Allyn & Bacon.

Van de Walle, J. & Lovin, L. (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

Using materials to model the action in a story problem will help students write an equation which matches the problem.

• Word banks need to be part of the student learning environment in every mathematics unit of study.
• Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words count, first, second, third, etc.

• Sentence Frames

Math sentence frames provide support that English Language Learners need in order to fully participate in math discussions.  Sentence frames provide appropriate sentence structure models, increase the likelihood of responses using content vocabulary, help students to conceptualize words and build confidence in English Language Learners.

Sample sentence frames related to these benchmarks :

• When assessing the math skills of an ELL student it is important to determine if the student has difficulty with the math concept or with the language used to describe the concept and conceptual understanding. 

Additional ELL Resources:

Bresser, R., Melanese, K., & Sphar, C. (2008).  Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications .

Students expand their skills by writing true or false number sentences on slips of paper and placing them in a container.  After drawing a slip of paper from the container, students determine if the number sentence is true or false.

Students write number stories for equations involving three addends having a sum less than or equal to twelve.

Students write their own number sentences involving unknowns and find the unknown in each other's number sentences.

What Number Goes Where?

Students are given a set of digits 0-9 on small cards. Each playing board has 10 missing numbers involving addition, subtraction or counting. Using each digit only once on the playing card, first graders fill in the missing numbers to make the number sentences true or to complete the counting sequence.

Additional Resources :

Bender, W. (2009). Differentiating math instruction-Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k-2. Sausalito, CA: Math Solutions.

Murray, M. & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann .

Parents/Admin

Administrative/Peer Classroom Observation

What should I look for in the mathematics classroom? ( Adapted from SciMathMN,1997)

What are students doing?

• Working in groups to make conjectures and solve problems.
• Solving real-world problems, not just practicing a collection of isolated skills.
• Representing mathematical ideas using concrete materials, pictures and symbols. Students know how and when to use tools such as blocks, scales, calculators, and computers.
• Communicating mathematical ideas to one another through examples, demonstrations, models, drawing, and logical arguments.
• Recognizing and connecting mathematical ideas.
• Justifying their thinking and explaining different ways to solve a problem.

What are teachers doing?

• Making student thinking the cornerstone of the learning process. This involves helping students organize, record, represent, and communicate their thinking.
• Challenging students to think deeply about problems and encouraging a variety of approaches to a solution.
• Connecting new mathematical concepts to previously learned ideas.
• Providing a safe classroom environment where ideas are freely shared, discussed and analyzed.
• Selecting appropriate activities and materials to support the learning of every student.
• Working with other teachers to make connections between disciplines to show how math is related to other subjects.
• Using assessments to uncover student thinking in order to guide instruction and assess understanding.

Additional Resources

For Mathematics Coaches

Chapin, S. and Johnson, A. (2006).  Math matters: Understanding the math you teach: Grades k-8 , 2nd edition .  Sausalito, CA: Math Solutions.

Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.

Sammons, L., (2011).  Building mathematical comprehension: Using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

For Administrators

Burns, M. (Ed). (1998).  Leading the way: Principals and superintendents look at math instruction. Sausalito, CA:  Math Solutions.

Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC : National Academies Press.

Leinwand, S. (2000). Sensible mathematics: A guide for school leaders . Portsmouth, NH:  Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA:  National Council of Teachers of Mathematics.

Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.

Parent Resources

Cavanagh, M. (2004). Math to Know: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc.

Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc

Helping your child learn mathematics

Provides activities for children in preschool through grade 5

What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN

Help Your Children Make Sense of Math

Ask the right questions

In helping children learn, one goal is to assist children in becoming critical and independent thinkers. You can help by asking questions that guide, without telling them what to do.

While Working How can you organize the information? Can you make a drawing (model) to explain your thinking? What are other possibilities? What would happen if . . . ? Can you describe an approach (strategy) you can use to solve this? What do you need to do next? Do you see any patterns or relationships that will help you solve this? How does this relate to ...? Can you make a prediction? Why did you...? What assumptions are you making?

Responding (helps clarify and extend their thinking) Tell me more. Can you explain it in a different way? Is there another possibility or strategy that would work? Is there a more efficient strategy? Help me understand this part...

Adapted from They're counting on us, California Mathematics Council, 1995

Read Aloud Books:

Domino Addition   by Lynette Long, Ph.D.

The Hershey's Kisses Addition Book  by Jerry Pallotta

One More Bunny: Adding From One to Ten   by Rick Walton

Related Frameworks

• 1.2.2.1 Real-World to Number Sentence
• 1.2.2.2 Equality
• 1.2.2.3 Missing Numbers
• 1.2.2.4 Number Sentence to Real-World

Problem Solving with Number Sentences

Educational game.

Problem Solving with Number Sentences is a game that allows children to demonstrate their proficiency in problem-solving using number sentences. This engaging game is designed to enhance critical thinking skills and mathematical problem-solving abilities. In Problem Solving with Number Sentences, children are presented with a series of mathematical problems that involve number sentences. They need to carefully analyze the problem, manipulate numbers, and devise the correct solution out of the three possible answers given. By exercising critical thinking and logical reasoning, children can strengthen their problem-solving abilities. Through playing this game, children can develop a foundation in mathematical concepts and problem-solving strategies. They will learn to understand the structure of number sentences, learn different operations, and practice applying mathematical principles to various scenarios. Overall, Problem Solving with Number Sentences is an excellent educational game that helps children develop key mathematical skills while honing their problem-solving abilities. ABCmouse.com’s platform provides an extensive and interactive learning environment, helping children have fun while building a strong foundation in mathematics.

Problem Solving with Number Sentences and the rest of the thousands of games on ABCmouse are not just for fun: they also serve a greater purpose. Each game is strategically placed within a well-thought-out learning path that includes a variety of engaging materials such as videos, printable activities, and interactive tasks. By integrating Problem Solving with Number Sentences and other games into a larger educational journey, children have the opportunity to reinforce what they have learned and further develop their skills. ABCmouse educational games not only provide entertainment, but they also serve as a tool for accomplishing specific learning objectives.

This game is a part of the ABCmouse platform.

Explore More Games for 1st Grade

Crazy Race: Telling Time

Listen to a time given to the hour, half hour, and quarter hour, and match the correct words, analog, or digital representation.

Marie’s Pattern Matching

Choose the correct stickers to help Marie extend patterns.

Prairie Mammal Mystery Game

Decide whether or not an animal is a mammal based on three clues.

Storyteller: H is for Help

Identify the characters, setting, problem, and solution in the ABCmouse story, H Is for Help .

Ant Math: Park Picnic

Complete addition and subtraction problems with multiples of 10.

Flying High: Addition and Subtraction

Complete addition and subtraction number sentences for two numbers from 1 to 10.

Historically Speaking: Founding Fathers

Participate in a game show hosted by Happy Harrison about American founding fathers George Washington, John Adams, and Thomas Jefferson.

Pizza Pete’s Pizza Party

Help Pizza Pete seat his guests by forming correct addition and subtraction equations within number families.

Explore More Games for 2nd Grade

Mailbox Measurement

Help Luis and Blinks estimate the lengths of wooden pieces in centimeters to build a mailbox.

Make a Map with Carla

Help Carla build a physical map of the United States by identifying geographic features and symbols.

King Grammar’s Birthday News

Help Rusty Ritter check his newspaper articles on King Grammar’s Birthday for run-on sentences and fix the sentences if needed.

Word-Bot’s Sound Sculptures

Play a game with Word-Bot by spelling words that contain the letters ph or wh .

Build It Matching: Adding Three Numbers

Practice adding three numbers in this short learning challenge.

Sports Day Addition

Help Punctuation Mark fix the Sports Day scores by solving two-digit addition problems that require regrouping.

Pavati Explores the Rainforest

Help Pavati collect pictures of the animals and plants of the rainforest and make note of their specific adaptations to the environment.

Nutly’s Applesauce Emporium

Help Nutly solve two-step addition and subtraction word problems at his Applesauce Emporium.

Unlock endless learning possibilities with ABCmouse!

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Number Sentences

Grab your binoculars to take an expedition in this Number Sentences song, where Edwin learns that Number Sentences are equations - or mathematical sentences - that show statements using numbers and math symbols.

His Number Sentences journey takes him from the jungles of Borneo to the ice shelves off the coast of Greenland on his ship, the SS Mathematica. The only thing Edwin loves as much as the wildlife he encounters along the way is his newfound ability to communicate what he has seen to his friends using Number Sentences.

Number Sentences Song Lyrics

Verse One I’m exploring under the canopy. I see monkeys and count twenty-three. Some climb a tree, now I count twenty; this number sentence describes what I see. (23 - _____ = 20)

How many monkeys climbed? I’ll find out, because that’s the unknown I want to know about: I try a three, and that’s when I see, how many monkeys climbed up a tree. (THREE!)

Chorus Read what the question is asking about; write down what we need to find out, with a statement that makes sense of the problem as a number sentence.

Verse Two I went exploring in Antarctica and saw through my binoculars…. thirty-two penguins on an iceberg. Eighteen more appeared from the ice water.

To find out how many there were, I wrote a number sentence, so I’d be sure. Now I can see the solution is: fifty penguins resting on top of the deep blue ocean.

If you'd like to expand upon this video with other online resources that nicely compliment this video, we recommend following the link to hcpss.instructure.com , which lays out the basic rules and methods that will aid in understanding the concept through more traditional means.

This song targets an important Common Core learning standard for 2nd Grade. Look into the relevant standard to understand the requirements unique to this standard. To look at the specific standard follow this link to learn more about common core standard 2.OA.1.

Or continue browsing Numberock's math video content library to discover more engaging math songs. To gain access to Numberock's growing library of premium content, click here.

We're sorry, but we don't support Internet Explorer anymore. Please use a different browser .

Make number sentences

Online practice for grades 1-4.

You're given numbers (in flowers), and an answer to a math sentence. Drag two flowers to the empty slots so that the math sentence is true.

Choose any of the four operations — addition, subtraction, multiplication, or division.

You can also decide the number range of the numbers used, with the exception of division. This means your addition, subtraction, and multiplication sentences can be with one, two or three-digit numbers, as you wish. Or with negative numbers! For division, the numbers used come from the basic multiplication tables, up to 12 × 12 = 144. (For example, the multiplication fact 6 × 7 = 42 would give rise to the division fact 42 ÷ 7 = 6.)

Note: If you are using a phone, the game works best if you turn your phone to landscape mode.

Number of problems: 5 10 15 20

Screenshots from the game:

🎉 Congratulations! 🎉

How do you solve word problems in math?

Master word problems with eight simple steps from a math tutor!

Author Amber Watkins

Published April 2024

• Key takeaways
• Students who struggle with reading, tend to struggle with understanding and solving word problems. So the best way to solve word problems in math is to become a better reader!
• Mastery of word problems relies on your child’s knowledge of keywords for word problems in math and knowing what to do with them.
• There are 8 simple steps each child can use to solve word problems- let’s go over these together.

Table of contents

• How to solve word problems

Lesson credits

As a tutor who has seen countless math worksheets in almost every grade – I’ll tell you this: every child is going to encounter word problems in math. The key to mastery lies in how you solve them! So then, how do you solve word problems in math?

In this guide, I’ll share eight steps to solving word problems in math.

How to solve word problems in math in 8 steps

Step 1: read the word problem aloud.

For a child to understand a word problem, it needs to be read with accuracy and fluency! That is why, when I tutor children with word problems, I always emphasize the importance of reading properly.

Mastering step 1 looks like this:

• Allow your child to read the word problem aloud to you. 
• Don’t let your child skip over or mispronounce any words. 
• If necessary, model how to read the word problem, then allow your child to read it again. Only after the word problem is read accurately, should you move on to step 2.

Step 2: Highlight the keywords in the word problem

The keywords for word problems in math indicate what math action should be taken. Teach your child to highlight or underline the keywords in every word problem. 

Here are some of the most common keywords in math word problems: 

• Subtraction words – less than, minus, take away
• Addition words – more than, altogether, plus, perimeter
• Multiplication words – Each, per person, per item, times, area 
• Division words – divided by, into
• Total words – in all, total, altogether

Let’s practice. Read the following word problem with your child and help them highlight or underline the main keyword, then decide which math action should be taken.

Michael has ten baseball cards. James has four baseball cards less than Michael. How many total baseball cards does James have? 

The words “less than” are the keywords and they tell us to use subtraction .

Step 3: Make math symbols above keywords to decode the word problem

As I help students with word problems, I write math symbols and numbers above the keywords. This helps them to understand what the word problem is asking.

Let’s practice. Observe what I write over the keywords in the following word problem and think about how you would create a math sentence using them:

Step 4: Create a math sentence to represent the word problem

Using the previous example, let’s write a math sentence. Looking at the math symbols and numbers written above the word problem, our math sentence should be: 10 – 5 = 5 ! 

Each time you practice a word problem with your child, highlight keywords and write the math symbols above them. Then have your child create a math sentence to solve. 

Step 5: Draw a picture to help illustrate the word problem

Pictures can be very helpful for problems that are more difficult to understand. They also are extremely helpful when the word problem involves calculating time , comparing fractions , or measurements . 

Step 6: Always show your work

Help your child get into the habit of always showing their work. As a tutor, I’ve found many reasons why having students show their work is helpful:

• By showing their work, they are writing the math steps repeatedly, which aids in memory
• If they make any mistakes they can track where they happened
• Their teacher can assess how much they understand by reviewing their work
• They can participate in class discussions about their work

Step 7: When solving word problems, make sure there is always a word in your answer!

If the word problem asks: How many peaches did Lisa buy? Your child’s answer should be: Lisa bought 10 peaches .

If the word problem asks: How far did Kyle run? Your child’s answer should be: Kyle ran 20 miles .

So how do you solve a word problem in math?

Together we reviewed the eight simple steps to solve word problems. These steps included identifying keywords for word problems in math, drawing pictures, and learning to explain our answers. 

Is your child ready to put these new skills to the test? Check out the best math app for some fun math word problem practice.

Parents, sign up for a DoodleMath subscription and see your child become a math wizard!

Amber Watkins

Amber is an education specialist with a degree in Early Childhood Education. She has over 12 years of experience teaching and tutoring elementary through college level math. "Knowing that my work in math education makes such an impact leaves me with an indescribable feeling of pride and joy!"

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The math problem that took nearly a century to solve: Secret to Ramsey numbers

Mathematicians unlock the secret to ramsey numbers.

We've all been there: staring at a math test with a problem that seems impossible to solve. What if finding the solution to a problem took almost a century? For mathematicians who dabble in Ramsey theory, this is very much the case. In fact, little progress had been made in solving Ramsey problems since the 1930s.

Now, University of California San Diego researchers Jacques Verstraete and Sam Mattheus have found the answer to r(4,t), a longstanding Ramsey problem that has perplexed the math world for decades.

What was Ramsey's problem, anyway?

In mathematical parlance, a graph is a series of points and the lines in between those points. Ramsey theory suggests that if the graph is large enough, you're guaranteed to find some kind of order within it -- either a set of points with no lines between them or a set of points with all possible lines between them (these sets are called "cliques"). This is written as r(s,t) where s are the points with lines and t are the points without lines.

To those of us who don't deal in graph theory, the most well-known Ramsey problem, r(3,3), is sometimes called "the theorem on friends and strangers" and is explained by way of a party: in a group of six people, you will find at least three people who all know each other or three people who all don't know each other. The answer to r(3,3) is six.

"It's a fact of nature, an absolute truth," Verstraete states. "It doesn't matter what the situation is or which six people you pick -- you will find three people who all know each other or three people who all don't know each other. You may be able to find more, but you are guaranteed that there will be at least three in one clique or the other."

What happened after mathematicians found that r(3,3) = 6? Naturally, they wanted to know r(4,4), r(5,5), and r(4,t) where the number of points that are not connected is variable. The solution to r(4,4) is 18 and is proved using a theorem created by Paul Erdös and George Szekeres in the 1930s.

Currently r(5,5) is still unknown.

A good problem fights back

Why is something so simple to state so hard to solve? It turns out to be more complicated than it appears. Let's say you knew the solution to r(5,5) was somewhere between 40-50. If you started with 45 points, there would be more than 10 234 graphs to consider!

"Because these numbers are so notoriously difficult to find, mathematicians look for estimations," Verstraete explained. "This is what Sam and I have achieved in our recent work. How do we find not the exact answer, but the best estimates for what these Ramsey numbers might be?"

Math students learn about Ramsey problems early on, so r(4,t) has been on Verstraete's radar for most of his professional career. In fact, he first saw the problem in print in Erdös on Graphs: His Legacy of Unsolved Problems, written by two UC San Diego professors, Fan Chung and the late Ron Graham. The problem is a conjecture from Erdös, who offered $250 to the first person who could solve it.

"Many people have thought about r(4,t) -- it's been an open problem for over 90 years," Verstraete said. "But it wasn't something that was at the forefront of my research. Everybody knows it's hard and everyone's tried to figure it out, so unless you have a new idea, you're not likely to get anywhere."

Then about four years ago, Verstraete was working on a different Ramsey problem with a mathematician at the University of Illinois-Chicago, Dhruv Mubayi. Together they discovered that pseudorandom graphs could advance the current knowledge on these old problems.

In 1937, Erdös discovered that using random graphs could give good lower bounds on Ramsey problems. What Verstraete and Mubayi discovered was that sampling from pseudo random graphs frequently gives better bounds on Ramsey numbers than random graphs. These bounds -- upper and lower limits on the possible answer -- tightened the range of estimations they could make. In other words, they were getting closer to the truth.

In 2019, to the delight of the math world, Verstraete and Mubayi used pseudorandom graphs to solve r(3,t). However, Verstraete struggled to build a pseudorandom graph that could help solve r(4,t).

He began pulling in different areas of math outside of combinatorics, including finite geometry, algebra and probability. Eventually he joined forces with Mattheus, a postdoctoral scholar in his group whose background was in finite geometry.

"It turned out that the pseudorandom graph we needed could be found in finite geometry," Verstraete stated. "Sam was the perfect person to come along and help build what we needed."

Once they had the pseudorandom graph in place, they still had to puzzle out several pieces of math. It took almost a year, but eventually they realized they had a solution: r(4,t) is close to a cubic function of t . If you want a party where there will always be four people who all know each other or t people who all don't know each other, you will need roughly t 3 people present. There is a small asterisk (actually an o) because, remember, this is an estimate, not an exact answer. But t 3 is very close to the exact answer.

The findings are currently under review with the Annals of Mathematics .

"It really did take us years to solve," Verstraete stated. "And there were many times where we were stuck and wondered if we'd be able to solve it at all. But one should never give up, no matter how long it takes."

Verstraete emphasizes the importance of perseverance -- something he reminds his students of often. "If you find that the problem is hard and you're stuck, that means it's a good problem. Fan Chung said a good problem fights back. You can't expect it just to reveal itself."

Verstraete knows such dogged determination is well-rewarded: "I got a call from Fan saying she owes me $250."

• Mathematics
• Math Puzzles
• Mathematical Modeling
• Educational Technology
• Computer Modeling
• Distributed Computing
• Quantum computer
• Knot theory
• Massively multiplayer online game
• Supercomputer
• Algebraic geometry
• John von Neumann
• Scientific method

Story Source:

Materials provided by University of California - San Diego . Original written by Michelle Franklin. Note: Content may be edited for style and length.

Journal Reference :

• Sam Mattheus, Jacques Verstraete. The asymptotics of r(4,t) . Annals of Mathematics , 2024; 199 (2) DOI: 10.4007/annals.2024.199.2.8

Cite This Page :

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