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Related Facts – Definition With Examples

Created: December 28, 2023

Last updated: January 5, 2024

Welcome to Brighterly , your trusted source for making complex mathematical concepts simpler and more fun for children. Today, we’re diving into the intriguing world of related facts. From classroom arithmetic to real-life applications, related facts permeate many aspects of our daily lives. But what exactly are related facts? How can we identify and apply them, especially within the realm of mathematics?

In this comprehensive guide, we’re setting out to answer these questions and more. We’ll explore definitions, delve into examples, uncover their unique properties, and highlight their importance in mathematical fluency. Alongside, we’ll also tackle the differences between related and unrelated facts, the art of writing equations with related facts, and wrap it up with some practice problems. Whether you’re a math enthusiast, a curious student, or a dedicated parent, this exploration promises a wealth of understanding to brighten your mathematical journey!

What Are Related Facts?

When we talk about related facts, we are referring to a group of facts that share a relationship through their numbers. For instance, if we take the numbers 2, 3, and 5, we could form the related facts 2+3=5 and 5-2=3. These statements are related because they involve the same numbers but present them in different ways. The same numbers, organized differently, result in a different relationship, thus creating a different fact.

Definition of Related Facts

Defining related facts is straightforward: related facts are math statements that share the same numbers but use different operations, such as addition/subtraction or multiplication/division. They reveal the commutative and reversal properties of these operations. The numbers stay the same, but how they interact changes. This is a valuable concept in early math education, as it helps children understand the inverse relationships that exist in arithmetic.

Examples of Related Facts

Let’s consider a few examples to illustrate the concept of related facts. If we take the numbers 4, 5, and 20, we can form the multiplication fact 4 5=20. If we switch the numbers around, we have 5 4=20. These are related facts because they involve the same numbers and the same operation but in a different order, demonstrating the commutative property. We can also form the related division facts 20/4=5 and 20/5=4, illustrating the reversal property.

Properties of Related Facts

There are a couple of important properties at work in related facts. The commutative property says that you can swap numbers around in addition or multiplication and still get the same answer (3+2 = 2+3 and 4 5 = 5 4, for example). The reversal property shows that you can reverse the operation from multiplication to division (or addition to subtraction) and still find a valid relationship between the numbers (2+3=5 becomes 5-2=3, and 4*5=20 becomes 20/4=5).

Understanding the Nature of Related Facts

Related facts are more than just arithmetic relationships; they reflect a deeper understanding of the nature of numbers and operations. In a sense, they can be thought of as the “grammar” of math – the way numbers and operations combine to form meaningful statements. This understanding is crucial for developing mathematical fluency and comfort with number manipulation.

Properties of Addition and Subtraction in Related Facts

In related facts, addition and subtraction are tightly linked. The addition facts (such as 2+3=5) and their related subtraction facts (5-2=3 and 5-3=2) illustrate the inverse relationship between addition and subtraction. Understanding these properties aids in solving more complex math problems and contributes to numerical fluency.

Properties of Multiplication and Division in Related Facts

Multiplication and division share a similar relationship. If you know the multiplication facts (like 4*5=20), you can use them to derive the related division facts (20/4=5 and 20/5=4). Understanding this connection can dramatically speed up mental calculations and aid in the comprehension of more advanced math concepts.

Difference Between Related and Unrelated Facts

While related facts share common numbers, unrelated facts do not. For example, 2+3=5 and 5-2=3 are related because they involve the same numbers. However, 2+3=5 and 6-4=2 are unrelated because they do not share the same numbers. Understanding the difference between related and unrelated facts can further deepen children’s mathematical intuition.

Equations Involving Related Facts

The related facts concept extends beyond simple number facts to encompass equations. For instance, if we know that a+b=c, we can deduce that c-a=b and c-b=a. Understanding this equivalence is crucial in algebra, where we often need to rearrange equations to solve for a particular variable.

Writing Equations with Related Facts: Addition and Subtraction

In writing equations with related facts, the numbers involved stay the same, but the order and operation can change. If we start with the addition equation a+b=c, we can write the related subtraction equations c-a=b and c-b=a.

Writing Equations with Related Facts: Multiplication and Division

Similarly, if we start with the multiplication equation a*b=c, we can write the related division equations c/a=b and c/b=a. These related equations underscore the inverse relationship between multiplication and division, as well as the commutative property of multiplication.

Practice Problems on Related Facts

To truly grasp related facts, practice is key. Try solving problems that involve identifying and creating related facts. For example, given the numbers 3, 4, and 12, can you write all the related multiplication and division facts? What about the related addition and subtraction facts for the numbers 6, 7, and 13?

As we conclude our journey with related facts, it’s clear that these concepts are more than just an essential mathematical tool. They form an integral part of our cognitive abilities, helping us decipher patterns, make predictions, and solve problems efficiently. At Brighterly, we believe that empowering children with this knowledge and understanding will light up their paths, making them confident, creative thinkers in this numerically driven world. By fostering a deep understanding of related facts, we’re nurturing their ability to see the intricate beauty of numbers and their interrelationships. After all, isn’t learning all about making connections and finding patterns that make sense of the world around us?

Frequently Asked Questions on Related Facts

Why are related facts important in mathematics.

Related facts are important in mathematics because they help establish a fundamental understanding of the relationships between numbers and operations. By grasping this, children can more easily recognize patterns, make predictions, and solve problems more efficiently.

What is the main difference between related and unrelated facts?

Related facts share a common set of numbers but use them in different operations, while unrelated facts do not share the same set of numbers. For example, 2+3=5 and 5-2=3 are related facts because they involve the same numbers in different operations. In contrast, 2+3=5 and 6-4=2 are unrelated because they do not share the same numbers.

How can practicing related facts help improve problem-solving skills?

Practicing related facts can greatly improve problem-solving skills by enhancing a child’s number sense and numerical fluency. They begin to see patterns and connections between different operations and understand the inverse relationships that exist in arithmetic, all of which are critical in solving complex problems.

Are related facts relevant only to addition and subtraction or multiplication and division?

While related facts are most commonly associated with addition/subtraction and multiplication/division, the concept extends to other operations and even algebra. For example, if we know that a+b=c, we can derive that c-a=b and c-b=a. This understanding of the equivalence between different forms of an equation is crucial in algebra.

• Related Facts
• Reversal Property
• Math Education

I am a seasoned math tutor with over seven years of experience in the field. Holding a Master’s Degree in Education, I take great joy in nurturing young math enthusiasts, regardless of their age, grade, and skill level. Beyond teaching, I am passionate about spending time with my family, reading, and watching movies. My background also includes knowledge in child psychology, which aids in delivering personalized and effective teaching strategies.

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Fact Families: Building Fact Fluency with Related Facts

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Building basic fact fluency is a huge focus in 2nd-grade math. Strong fact fluency makes adding and subtracting larger numbers so much easier for students. This is why I usually dedicate tons of time during the first few months of school to building basic fact fluency. One of my favorite and most powerful ways to build fact fluency? Teaching related math facts or fact families. This is also referred to as number bonds. Pick your name… they all refer to the same idea; that numbers work in relation to each other, and that we can use this relationship to help us solve other similar math problems. This is why, when I teach this concept to 2nd-grade students, I really lean into the “fact family” name.

Teaching Related Facts

To start the lesson, I first review what a basic fact is. As we discuss this idea, I write any samples students may give on the whiteboard. Then, I move the discussion towards how we can use a basic fact we know, let’s say, 2 + 3= 5, to help solve other basic facts. For example, if we know that 2 + 3 = 5, then we also know that 3 + 2 = 5. This works because the numbers work in relation to each other.

Teacher note: If your math curriculum uses the terms number bonds or part-part-whole, this is EXACTLY where that key vocabulary comes to life!

So, because these numbers work in relation to each other, they are like a family!

I like to showcase this anchor chart as we learn about fact families and discuss how a fact family can also be referred to as a number bond or the part-part-whole method.

This helps students understand that really, the idea is the same, but the names are just different.

Say Hello to the Fact Family

From there, I draw a fact family house with the numbers in the fact family at the top and four addition and subtraction sentences that are empty. Then, as a class, we work to fill in the sentences, using only numbers from the fact family.

I show students how the 5 is the largest and acts as the “dad” of the family. The “3” is in the middle and is like the “mom,” and the 2 is the smallest and is like the “baby.” Please note, this family vocabulary is modified as we work through more examples in guided practice. To make this concept inclusive, we also use terms like: aunt, uncle, grandpa, grandma, brother, sister, and cousin. But the idea is always the same: they are a family!

We work as a class to build addition and subtraction sentences using only these three numbers. As we build the number sentence, I show students how we only use the three numbers provided and how they work in relation to each other. The two small numbers can be added to make the bigger number. The bigger number is the starting point for subtraction.

We also discuss making sure our number sentences make mathematical sense.

For example, in the beginning, many students want to create the number sentence 3-2=5. Use this as a discussion on how to check the math using different strategies (ie: counting on, fingers, number line).

Don’t Let a Stranger in the House!

A common misconception associated with teaching fact families or related facts is that students often include an extra number. This happens through miscalculation when creating and solving their addition and subtraction facts. To help circumvent this issue, I teach this important line when discussing fact families and creating fact family houses:

Don’t let a stranger in the house!

As we work through guided practice, I’ll have a few students come up to the board and work on filling in their fact family house. As we review their answers, if I notice an additional number, I’ll say, “Don’t let a stranger in the house!” and circle the rogue number. I relate it back to the students: Just like we don’t like strangers in our house, neither do fact family numbers, so we need to be careful not to let them in! If you see a number stranger, be sure to check your math and kick them out!

Teacher friends – this really sticks! Students love building their fact family houses and making sure there aren’t any strangers!

Fact Family with Twins

Now, doubles facts can throw a second grader for a loop when it comes to building related facts.

To help with this, as we’re working together, I’ll build a fact family house with the numbers 10, 5, and 5. Students always get a bit confused about where they should place these numbers because you use the same number twice. To help it stick, we call this a “twins family!”

We discuss how these families have a mom (or dad, auntie, or another inclusive label) and a set of twins. Just like other fact families, we need to keep the relationship between the numbers safe and not let strangers into their house!

Remember: Don’t let a stranger into the house!

And just like twins, there are two of them. They are not the same “person” even if they look alike, so make sure you use the number twice.

This helps students visualize using both 5s from our example to build the related math facts!

Fact Family Activities

Now that students understand how to use related numbers to fill a fact family house with related number sentences, it is time to practice the skill!

Fact Family Group Activity

First up, we have an engaging, creative, and collaborative group fact family activity!

To complete, split students up into groups of three. Then, assign each student in the group a number in a fact family. Then, give each group a sheet of white construction paper.

Have students name their fact family, then draw a fact family house on their paper. The group can then act out creating number sentences using only the numbers in their group. Students record their answers on their fact family house chart. Give students about 10 minutes to complete their number sentences.

After the time is up, have the fact family groups present their posters to the class.

Showcase the house they designed as well as the related addition and subtraction facts they worked to create. Have students fix any mathematical issues on their posters and hang it up in the classroom.

Fact Family Scavenger Hunt

Another fun way to practice fact families?

Send them on a Fact Family Scavenger Hunt!

To complete this scavenger hunt or “read the room” activity, use the fact family cards from the Just the Basic Facts unit here.

Print out and trim the cards. Then, hang them around the room. Print out enough of the recording sheets for each student in your class. (Or save paper and have students work in pairs, changing the “writer” for each card.)

Have your students go on a Fact Family scavenger hunt. As they find the fact family, have students fill in the fact family house with the related facts.

Don’t forget to spot-check the answers as a class, especially those that include twins or possible strangers in their house!

Make it work in Math Stations

After you’ve completed the scavenger hunt activity as a class, don’t forget to place it in your math stations for fun basic facts practice.

Slip the recording sheets into a wipe-off sleeve . Students can use dry-erase markers to fill in the fact family recording sheet. A win for saving paper!

Fun with Related Basic Facts

Building basic fact fluency takes time and practice. These fact family activities are engaging ways to teach one aspect of basic facts so that your students will remember them! Don’t forget to revisit the topic throughout the year to make the idea stick.

That’s one of the main reasons why I love using the activities in this basic facts unit to help build basic fact fluency at any point in the year. Use the included activities to introduce the different basic fact strategies, then, keep the hands-on games in rotation for your math station routine!

You can check out this unit here on TpT or here in my website store !

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Related facts

Related facts are basic mathematical expressions made up of three numbers. Related facts are often taught as part of early math alongside fact families and addition, subtraction, multiplication, and division facts.

Related facts can be used to teach or reinforce a student's understanding of the relationships between the operations of addition and subtraction, and multiplication and divison.

Addition and subtraction

Addition and subtraction are inverse operations. If you add one number, then subtract the same number, they cancel each other out and there is no effect on the expression.

Below are the related addition and subtraction facts using the numbers 2 and 3:

The same related facts can be written for any addition or subtraction problem. If we are given one of the facts, we can write three other facts using the same three numbers and addition or subtraction. Each of the sets of expressions (ex. 2 + 3 = 5 ; 3 + 2 = 5) is referred to as a fact family .

Multiplication and division

Multiplication and division, like addition and subtraction, are inverse operations. They "undo" each other. Multiplying by a number and then dividing by that same number does not change an expression.

Below are the related multiplication and division facts using 2 and 4 as factors:

Like addition and subtraction, each of the sets of three numbers in the multiplication and division facts above makes up a fact family, which can be used to write a total of four related facts.

Remembering related facts, particularly for multiplication and division, can help reduce the number of basic facts a student truly has to memorize. As long as the student knows the solution to a multiplication problem involving two numbers, they can also know the relationship between those numbers and division.

Related Facts – Definition With Examples

What are related facts, fact families, importance of related facts, solved examples, practice problems, frequently asked questions.

Related facts in Math are facts that are related to each other in some way. We can easily find related facts using the given facts. 

Since addition and subtraction are inverse operations, we can form their related facts. Let’s go through a related fact example:

So, A, B, and C are related facts of the given fact, $3 + 4 = 7$. 

Notice that the numbers involved in all three related facts are the same as the numbers involved in the given fact. Only the position and operation changes. 

Since multiplication and division are also inverse operations, we can form related facts for them as well.

Fact families are sets of addition and subtraction facts or multiplication and division facts that involve the same numbers. In other words, it is a collection of related facts. Fact families are of two types:

• Addition and Subtraction
• Multiplication and Division

Related Worksheets

Addition and Subtraction Fact Family

Addition and subtraction are inverse operations. If we first add a number to a given number and then subtract the same number from the result, we get the original number back. For example, consider 10 as the given number, we add and subtract 2.

$10 + 2 = 12$

$12 – 2 = 10$ (original number)

This means that these two operations cancel each other’s effect on another number or expression. 

Let’s have a look at some fact families of addition and subtraction. 

We can assume that fact families are like parts of a whole, where the largest number is the whole, and the other two numbers are the two parts that make up the whole. In the first example above, 10 is the whole and the numbers 4 and 6 are the parts.

Multiplication and Division Fact Families

We know that multiplication and division are also inverse operations, so we can form fact families for these two operations using the related facts. Let’s have a look at some fact families of multiplication and division. 

Related facts are used so that a student can understand the relationships between the operations of addition and subtraction, and multiplication and division. It helps the students to remember that addition and subtraction are inverse operations and multiplication and division are inverse operations.

For example, if we know any one fact from a fact family, we can easily find the other related facts. 

Example 1: Identify which family the given numbers belong to. Also write all four related facts. 

Solution: 

Let’s try adding the two smaller numbers, $9 + 8 = 17$ (third number)

So, these numbers belong to the addition and subtraction fact family. 

The other related facts are: 

$8 + 9 = 17$

$17 － 8 = 9$

$17 － 9 = 8$

Example 2: Identify which fact family the given numbers belong to. Also write all four related facts. 

Let’s try adding the two smaller numbers, $5 + 7 = 12 ≠ 35$. 

Now let’s multiply these numbers. 

$5 \times 7 = 35$ (third number)

So, these numbers belong to the multiplication and division fact family. 

$7 \times 5 = 35$

$35 \div 5 = 7$

$35 \div 7 = 5$

Example 3: Use the given fact to complete the fact family. 

Solution : It is given that $10 – 3 = 7$. 

Using this fact, we can easily determine the other related facts. 

Example 4: Make a fact family of 2, 8 and 16.

Solution: 2, 8 and 16 belong to the multiplication and division fact family. 

Example 5: One of the facts is given: $5 ☐ 3 = 15$ . Identify the operation and write all other related facts. 

Solution : $5 \times 3 = 15$. 

$3 \times 5 = 15$

$15 \div 3 = 5$

#$15 \div 5 = 3$

Related Facts - Definition With Examples

Attend this quiz & Test your knowledge.

Use $6 \times 8 = 48$ to complete the fact: $48 \div$ __ $= 6$

Which of the following does not belong to the fact family of $5 \times 2 = 10$, identify the third number of the addition and subtraction fact family given that the whole is 24 and one part is 8..

What is a fact family?

Fact families are sets of addition and subtraction facts or multiplication and division facts that involve the same numbers. In other words, it is a collection of related facts.

How does multiplication facts help you divide numbers?

Multiplication and division are inverse operations. In multiplication, we combine equal groups to find the total number, whereas in division, we distribute the given number into equal groups. So, knowing a multiplication fact, we can easily find its related division fact.

Can we have a related fact of a given division fact from subtraction?

No, the related facts involve the same numbers and either the same operation or an inverse operation. So, related facts of a division fact will be either of multiplication or division involving the same set of 3 numbers.

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2. Connect with a line the problems that are from the same fact family. You don't need to     write the answers.

3. Fill in. In each fact family, color the marbles so they match the numbers in it.

4. Subtract.

5. Find the missing numbers.

6. Solve the word problems.

7. Figure out the patterns and continue them!

Math Mammoth Add & Subtract 2-A

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Related Subtraction Facts

In this lesson, we are going to learn about  related subtraction facts . There are  three numbers in a family and they are  related in some way.

The best way to understand this is by taking a look at a simple subtraction problem.

5 – 3 = 2

Notice that we have three numbers 5, 3, and 2 and the problem reads 5 – 3 = 2. What the related subtraction fact tells us is that we can switch the numbers 3 and 2 to also make the equation true.

If we switch the numbers 3 and 2, we get  5 – 2 = 3 which is TRUE. We can apply this to any subtraction problem.

Take a look at the example below.

The subtraction problem shows two circles and one circle is being subtracted. We can SUBTRACT to find the difference. We can use the subtraction sentence 3 – 1 = ___ for help. When we SUBTRACT 3 minus 1 , we have 2 left. We say: THREE minus ONE equals TWO. We write it like this: 3 – 1 = 2

We can also use these same three numbers in our math fact: 3, 1, and 2 to make a RELATED  FACT .

This time our math fact will read:  3 – 2 = 1  because we subtracted 2 this time to get a difference of 1.

3 – 1 = 2   is related to  3 – 2 = 1

They include the same three numbers.

Let’s take a look at another example with numbers. This time, we are subtracting the number 7 from the number 9 . The first number in the subtraction sentence is called the MINUEND . The second number in the subtraction sentence is called the SUBTRAHEND . This problem reads as  9 – 7 = 2 . When the SUBTRAHEND 7 is subtracted from the MINUEND  9 , we get a DIFFERENCE of 2 . DIFFERENCE is used to describe the amount that is left.

These facts are related:

9 – 7 = 2

9 – 2 = 7

Related subtraction facts use the same three numbers.

Let’s take a look at these facts.

5 – 1 =

5 – 4 =

They are related and have the same minuend. We can find their differences. The differences are also subtrahends too.

5 – 1 = 4   and  5 – 4 = 1

Here is another example:

8 – 6 =

8 – 2 =

8 – 6 = 2   and  8 – 2 = 6

Sometimes, you might come across a subtraction fact with a missing minuend. You can use related facts to help you solve. Let’s take a look at this pair of facts:

10 – 4 = __

10 – __ = 4

In the first math fact, our difference is missing. We can subtract 4 from 10 to get an answer of 6 . Our second fact is related. It will have the same three numbers as our first fact. Therefore, 10 – 6 = 4 .

Here is another example.

11 – 3 = __

11 – __ = 3

In the first math fact, our difference is missing. We can subtract 3 from 11 to get an answer of 8 . Our second fact is related. It will have the same three numbers as our first fact. Therefore, 11 – 8 = 3 .

We can also write related subtraction facts when we are given three numbers. Let’s look at this example:

5     8     3

We know that the sum in each subtraction fact, the first number must be the largest number. Which number is the largest: 5 , 8 , or 3 ?

The number 8 is the largest. Therefore, the other two numbers will be our subtrahends and difference.

8 – __ = __

Let’s create our related facts using the numbers 5 , 8 , and 3 !

8 – 5 = 3

8 – 3 = 5

Let’s look at one more example:

13      7      6

We know that the sum in each subtraction fact, the minuend must be the largest number. Which number is the largest: 13, 7, or 6?

The number 13 is the largest.

13 – __ = __

Let’s create our related facts using the numbers 13 , 7 , and 6 !

13 – 6 = 7

13 – 7 = 6

Practice Question 1

Let’s practice! Write the subtraction sentences and solve the problems below.

REVEAL THE ANSWER 9 - 6 = 3 9 - 3 = 6

• Subtraction
• Subtraction By 1
• Subtraction By 10
• Adding Doubles
• Adding by 10
• Adding Doubles Minus 1
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Practice Question 2

Let’s try another! Write the subtraction sentences and solve the problems below.

REVEAL THE ANSWER 7 - 2 = 5 7 - 5 = 2

Practice Question 3

Take a look at the math sentence below. Write the related subtraction fact. Then solve.

17 – 8 =

REVEAL THE ANSWER 17 - 8 = 9 17 - 9 = 8

Practice Question 4

Let’s try another one. Take a look at the math sentence below. Write the related subtraction fact. Then solve.

13 – 4 =

REVEAL THE ANSWER 13 - 4 = 9 13 - 9 = 4

Practice Question 5

Let’s try one more problem! Look at the groups below. How many are there in all? Write the related subtraction sentences and solve..

REVEAL THE ANSWER 15 - 9 = 6 15 - 6 = 9

Now you’re ready to take a quick quiz on subtraction related facts!

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Year 2 - Addition and Subtraction – Related Facts - White Rose Maths

Subject: Mathematics

Age range: 5-7

Resource type: Lesson (complete)

Last updated

14 November 2020

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Carefully crafted teaching presentation with everything you need to teach this small step of White Rose Maths Year 2 Autumn Block 2 Addition and Subtraction. This presentation follows the mastery approach of concrete, pictorial and abstract, whilst implementing all aspects of Rosenshine’s principles of instruction.

This presentation contains: *A warm-up to start the children’s thinking with AfL and metacognition style questions. *Guided practice through each new concept, providing models of strategies and worked examples. *Independent practice through differentiated varied fluency, reasoning, and problem-solving questions. *Plus, a greater depth extension activity to fully show students’ understanding. *Answer sheets for teacher or students’ self/peer assessment *Student refection page to discuss the lesson’s objectives and their individual next steps.

The complete Year 2 Addition and Subtraction Unit can be found here.

-This product is not affiliated with nor endorsed by White Rose Maths-

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Good resource over all, very well presented however errors in the presentation needed to be updated and caused some confusion. Was hoping for more materials for the children to work on. Cheers Jack

mrkeechsclassroom

Hi, Jack. Thank you for your comment and sorry for any inconvenience regarding errors. I have updated the resource to fix those. Regarding more materials, what in particular were you looking for? Often, I get comments that there is a lot for Y2 to work on so I would guess you have your class working at a high level.

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Related Facts Lesson

This Year 2 Related Facts lesson covers the prior learning of identifying related facts up to 20 before moving onto the main skill of identifying related addition and subtraction facts up to 100.

The lesson starts with a prior learning worksheet to check pupils’ understanding. The interactive lesson slides recap the prior learning before moving on to the main skill. Children can then practise further by completing the activities and can extend their learning by completing an engaging extension task.

National Curriculum Objective Mathematics Year 2: (2C1)  Recall and use addition and subtraction facts to 20 fluently, and derive and use related facts up to 100

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Resources for teachers

Interactive activities for children

2 Teaching Support

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Modelling Powerpoint

This powerpoint can be used to model the questions that the children will complete on the Varied Fluency and Reasoning & Problem Solving worksheets as part of this lesson.

Lesson Slides

These lesson slides guide pupils through the prior learning of place value of 1s, 10s and 100s, before moving onto the main skill of related facts. There are a number of questions to check pupils' understanding throughout.

These are the same as the lesson slides on Classroom Secrets. You can assign this as an activity for pupils to access individually in school or remotely from home.

Video Tutorial

Katie shows children how to derive the related addition and subtraction calculations in this Related Facts Video Tutorial.

1 Prior Learning

Interactive Activity

This Year 1 Addition and Subtraction Game aims to support pupils’ knowledge of related facts.

2 Varied Fluency

This differentiated worksheet includes varied fluency questions for pupils to practise the main skill of this lesson.

2 Reasoning & Problem Solving

Discussion Problem

This worksheet includes two discussion problems which can be used in pairs or small groups to further pupils' understanding of the concepts taught in this lesson.

This differentiated worksheet includes reasoning and problem solving questions to support the teaching of this step.

This related facts extension task includes a challenge activity which can be used to further pupils' understanding of the concepts taught in the related facts lesson.

Mixed Practice

This worksheet includes varied fluency, reasoning and problem solving questions for pupils to practise the main skill of related facts.

This Related Facts Reasoning activity checks pupils’ understanding of related calculations using similar digits.

This Year 2 Related Facts to 100 Maths Challenge aims to challenge pupils’ understanding of using place value to link additions and subtractions.

This differentiated worksheet includes varied fluency and reasoning and problem solving questions to support the teaching of this step.

2 Additional Activities

Consolidation Pack

This resource is aimed at Year 2 Expected and has been designed to give children the opportunity to consolidate the skills they have learned in Autumn Block 2 – Addition and Subtraction.

Learning Video Clip

Cleo visits London in 1665. She soon discovers that people are quite poorly and the number of sick are recorded using related facts.

Home Learning Pack

This Autumn week 5 Maths pack contains varied fluency, reasoning and problem solving worksheets

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As business leaders we are consequently required to solve problems. But what are the very fundamentals of problem solving?

Read on to find out.

1. Define the Real Problem

This is the biggie. Ensure that you are solving the right problem. Toyota is justifiably famous for its problem-solving savvy in perfecting its production methods. According to Toyota, the key to their method is to spend relatively more time defining the problem and relatively less time on figuring out the solution.

2. Solve the Root Cause of the Problem

Don’t treat the symptoms; solve the root cause of the problem. There are numerous methods to determine root causes—cause mapping, fishbone diagrams, etc. For me, the easiest and most effective is to use the “Five Whys.” Ask a question and to each answer ask. “Why?” again. Doing this for five times should get you to the root cause of the problem.

3. Use a Hypothesis

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5. Keep the solution simple

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6. Do not re-invent the wheel

Plagiarism can be good. If someone has a clever idea or way to solve your problem, by all means legally use it. The “not invented here” syndrome is just sheer arrogance.

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8. Consider time

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Home / Related facts | Worksheets

Related facts | Worksheets

Differentiated worksheets requiring children to complete related facts up to 100 using part-whole models to help them visualise. Children will progress to solving word problems using their knowledge of related facts.

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31 examples of problem solving performance review phrases

Jump to section

You're doing great

You should think of improving

Tips to improve

Use these practical examples of phrases, sample comments, and templates for your performance review , 360-degree feedback survey, or manager appraisal.

The following examples not only relate to problem-solving but also conflict management , effective solutions, selecting the best alternatives, decision making , problem identification, analyzing effectively, and generally becoming an effective problem-solving strategist. Start using effective performance review questions to help better guide your workforce's development. 

Problem solving appraisal comments: you're doing great

• You always maintain an effective dialogue with clients when they have technical problems. Being clear and articulate makes sure our customers' faults are attended to promptly.
• You constantly make sure to look beyond the obvious you never stop at the first answer. You’re really good at exploring alternatives. Well done!
• Keeping the supervisors and managers informed of status changes and requests is important. You’re really good at communicating the changes to the projects at all times. Keep it up!
• You stay cool and collected even when things aren’t going according to plan or up in the air. This is a great trait to possess. Well done!
• You’re excellent at giving an honest and logical analysis. Keep it up! Effectively diagnosing complex problems and reaching sustainable solutions is one of your strong points.
• Your ability to ability to make complex systems into simple ones is truly a unique skill to possess. Well done!
• You often identify practical solutions to every roadblock. You’re a real asset to the team! Great job.
• You always listen actively and attentively to make sure you understand what the exact problem is and you come up with solutions in an effective manner.
• You have an amazing ability to clearly explain options and solutions effectively and efficiently. Well done!
• When driving projects, you can shift to other areas comfortably and easily. making sure the project runs smoothly. Great job!

Problem solving performance review phrases: you should think of improving

• You always seem too overwhelmed when faced with multiple problems. Try to think of ways to make problems more manageable so that they can be solved in a timely and effective manner.
• Avoiding conflicts constantly with people is not a good idea as you will only build up personal frustration and nothing will be done to remedy the situation. Try to face people when there are problems and rectify problems when they occur.
• Don’t allow demanding customers to rattle your cage too much. If they become too demanding, take a step back, regulate your emotions , and try to make use of online support tools to help you rectify problems these tools can help a lot!
• It’s necessary that you learn from your past mistakes . You cannot keep making the same mistakes , as this is not beneficial to the company.
• You tend to ask the same questions over and over again. Try to listen more attentively or take notes when colleagues are answering!
• Providing multiple solutions in an indirect and creative approach will allow you to be more effective at problem-solving . if you struggle with this typically through viewing the problem in a new and unusual light.
• You fail to provide staff with the appropriate amount of structure and direction. They must know the direction you wish them to go in to achieve their goals .
• You need to be able to recognize repetitive trends to solve problems promptly.
• You tend to have problems troubleshooting even the most basic of questions. As a problem solver and customer support person, it’s imperative that you can answer these questions easily.
• Read through your training manual and make sure you fully understand it before attempting questions again.

Performance review tips to improve problem solving

• Try to complain less about problems and come up with solutions to the problems more often. Complaining is not beneficial to progression and innovation.
• As a problem solver, it’s important to be able to handle multiple priorities under short deadlines.
• You need to be able to effectively distinguish between the cause and the symptoms of problems to solve them in an efficient and timely manner.
• Try to anticipate problems in advance before they become major roadblocks down the road.
• Try to view obstacles as opportunities to learn and thrive at the challenge of solving the problem.
• Remember to prioritize problems according to their degree of urgency. It's important that you spend the majority of your time on urgent tasks over menial ones.
• When putting plans into place, stick to them and make sure they are completed.
• When solving problems, try to allocate appropriate levels of resources when undertaking new projects. It is important to become as efficient and as effective as possible.
• Try to learn to pace yourself when solving problems to avoid burnout . You’re a great asset to the team and we cannot afford to lose at this point.
• Meeting regularly with your staff to review results is vital to the problem-solving process.
• Staff that has regular check-ins understand what it is that is required of them, what they are currently achieving, and areas they may need to improve. Try to hold one-on-one meetings every week.

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Madeline Miles

Madeline is a writer, communicator, and storyteller who is passionate about using words to help drive positive change. She holds a bachelor's in English Creative Writing and Communication Studies and lives in Denver, Colorado. In her spare time, she's usually somewhere outside (preferably in the mountains) — and enjoys poetry and fiction.

How a performance review template improves the feedback process

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Investing in Artificial Intelligence (AI) Stocks Can Be Risky, But 2 Spectacular AI ETFs Can Help Solve That Problem

April 24, 2024 — 05:27 am EDT

Written by Anthony Di Pizio for The Motley Fool  ->

Investors watched Nvidia grow from a $360 billion company to a $2 trillion powerhouse since the beginning of 2023, almost entirely on the back of its data center chips, which are designed for processing artificial intelligence (AI) workloads.

Many other stocks also caught the AI wave recently, including Advanced Micro Devices , Microsoft , and Amazon . However, AI alone hasn't been enough to lift stocks like C3.ai , Lemonade , or Upstart from slumps of more than 80% from their all-time highs.

As in every technological revolution in the past, many AI companies won't survive, let alone rise to the heights Nvidia reached. The long-term winners and losers in the AI race aren't obvious right now, which means investors have to be comfortable with a high degree of risk when buying individual stocks.

However, they could take a different approach.

Image source: Getty Images.

Exchange-traded funds can deliver strong AI returns with limited risk

Exchange-traded funds (ETFs) are designed to hold dozens, or even hundreds, of individual stocks from a specific market sector, neatly packaged in a single security. Their portfolios are managed by experts who adjust the holdings as necessary, which means investors can take a passive approach.

By having a large number of holdings, ETFs won't suffer catastrophic losses if a single company fails, which is a great feature in an emerging industry like AI.

Several AI-focused ETFs have come to market in the last few years, but here's why the Global X Artificial Intelligence and Technology ETF (NASDAQ: AIQ) and the Global X Robotics and Artificial Intelligence ETF (NASDAQ: BOTZ) might be two of the best bets.

1. Global X Artificial Intelligence and Technology ETF (AIQ)

The AIQ ETF was established in 2018, and it has deployed $1.6 billion into 84 different stocks on behalf of investors. The objective of this ETF is to invest exclusively in companies that will benefit from the continued development of AI on both the hardware and software side, so it's a great option for investors of all experience levels.

Despite its expansive list of holdings, AIQ is heavily weighted toward its top 10 positions, which account for 33.4% of the total value of its portfolio. That creates concentration risk, because those stocks will have an outsized influence on the overall performance of the ETF; however, they include some of the most popular AI names in the industry:

Data source: Global X. Portfolio weightings are accurate as of April 19, 2024, and are subject to change.

Nvidia is an obvious top holding for an ETF focused on AI. Its H100 data center GPU has consistently been the best performer in the industry for processing AI workloads, and the company is now shipping its new H200, which is twice as powerful. Nvidia will raise the bar again in 2025 when it starts selling chips built on its new Blackwell architecture. In other words, this stock probably still has room to run.

Meta Platforms is the parent of social networks Facebook and Instagram, and it's using AI across both of them to feed more relevant content to users. Plus, Meta continues to develop its Llama large language models (LLMs), which already underpin exciting AI features like its new chatbot, Meta AI.

Netflix is the world's largest streaming service, and it uses AI in its content recommendation engine. Amazon, Alphabet, and Microsoft are home to dominant cloud computing platforms that offer an expanding portfolio of AI products and services to businesses. Oracle, on the other hand, has emerged as one of the go-to data center infrastructure providers for AI developers .

The AIQ ETF has delivered a 38.3% return over the past year, which crushes the 20.1% return of the S&P 500 . The high concentration toward the AI stocks mentioned above is a huge reason for the ETF's outperformance, but it also has a solid long-term track record. AIQ has delivered a compound annual gain of 15.3% since its inception in 2018, which comfortably beats the 10.7% annual return in the S&P 500 over the same period.

If AI lives up to the hype over the coming years, investors might be glad they bought the AIQ ETF today. It has an expense ratio of 0.68%, or $68 annually for each $10,000 invested.

2. Global X Robotics and Artificial Intelligence ETF (BOTZ)

The BOTZ ETF was established in 2016, and it manages $2.5 billion on behalf of investors. However, it's far more concentrated than the AIQ ETF, because it only holds 43 stocks. Its objective is also slightly different, because it focuses on niche segments of the AI space including robotics, automation, and autonomous vehicles.

Plus, the top 10 holdings in the BOTZ ETF represent 62.1% of the total value of its portfolio, making it highly sensitive to the performance of just a few stocks:

Nvidia is the second-largest holding in this ETF, but it has a much higher representation as a percentage of its portfolio compared to AIQ. ABB Ltd, which is a Swedish-Swiss industrial giant with an expansive presence in electrification, automation, and robotics, has top billing in BOTZ. It even has a portfolio of data center solutions designed to make its offerings more energy efficient, cost effective, and automated.

ABB is just one of many foreign holdings in the BOTZ ETF. Keyence manufactures industrial automation equipment, and Yaskawa Electric makes industrial robots -- both companies are based in Japan.

Back in the U.S., Intuitive Surgical is a hybrid healthcare and technology company that designs robotic products to improve the clinical outcomes of patients in surgery, and Dynatrace is a software platform that uses AI to help businesses optimize the performance of its digital applications and infrastructure.

The BOTZ ETF delivered a gain of 25% over the past year, so it underperformed the AIQ ETF, but it's still doing better than the S&P 500. Its compound annual return since its inception in 2016 also stands at 11.1%, but that could improve going forward thanks to the AI tailwind . It also has the same expense ratio as AIQ, at 0.68%.

This ETF carries more risk than most of its peers because of its high concentration. However, that could also be the reason for its potential market-beating performance in the future as the AI revolution unfolds.

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Randi Zuckerberg, a former director of market development and spokeswoman for Facebook and sister to Meta Platforms CEO Mark Zuckerberg, is a member of The Motley Fool's board of directors. Suzanne Frey, an executive at Alphabet, is a member of The Motley Fool’s board of directors. John Mackey, former CEO of Whole Foods Market, an Amazon subsidiary, is a member of The Motley Fool’s board of directors. Anthony Di Pizio has no position in any of the stocks mentioned. The Motley Fool has positions in and recommends Advanced Micro Devices, Alphabet, Amazon, Intuitive Surgical, Lemonade, Meta Platforms, Microsoft, Netflix, Nvidia, Oracle, Qualcomm, Tencent, UiPath, and Upstart. The Motley Fool recommends C3.ai and International Business Machines and recommends the following options: long January 2026 $395 calls on Microsoft and short January 2026 $405 calls on Microsoft. The Motley Fool has a disclosure policy .

The views and opinions expressed herein are the views and opinions of the author and do not necessarily reflect those of Nasdaq, Inc.

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