Number Modelling

Number Modelling

Modelling Numbers


There is an interesting way of teaching numbers to children in our schools. If we really think about it, numbers are just characters. They have no values by themselves, but instead are attached value. It’s similar to a $100 bill and a $1 bill. We know that the $100 bill is worth more because it is a larger value, but we also understand that $100 can buy more things than $1.


A young learner just being introduced to numbers has to develop their understanding of numbers just as we may have forgotten that we did in the past. They have to learn the values of numbers, and usually this is done by representations of numbers using models. One common example is math problems involving cookies. You give a child two sets of cookies: one set has 4 cookies, the other set has 1 cookie. You don’t directly ask which set has more cookies but instead you attach some relatable knowledge they have by asking questions such as “Which set of cookies do you want to take?”


It seems like a silly question to ask, but for a child, the concept of “more” and “less” does not have a strong foundation, but eating 4 cookies is usually better than eating 1 cookie.


So how does this all relate to older learners?


Well, once we get past the initial “learning which numbers are bigger” within 1-10, we find ourselves with a more difficult problem: how do we give a child 1,000 or 10,000 or 100,000 cookies? It’s just not realistic to do. That’s where modelling numbers come in.


Some children find it difficult to decide which number has more when given larger numbers such as “7654” and “7564”. Number modelling breaks down larger numbers into smaller parts that can be represented with 1-10 but have an attached value to them such as ones, tens, hundreds, thousands, and so on. These number models allow us to give children physical representations of larger numbers without having to count out 100 pieces or more each time. Just seeing two abstract values is much more difficult than necessary so number modelling is one way to help your child understand larger number values.


A number model problem, in the third grade, looks something like this:

A number model problem is different across each textbook, school, and teacher. They may use different shapes in the problem, they may use real objects like apples, they may use animals, or they may use different colors. In the case of LI2EI, we opted for different shapes and colors. Each shape represents a specific number. In this case, the yellow squares are “Hundreds”. The green circles are “Tens”. The pink rectangles are “Ones”.


If your child counts each section out they will quickly see that there are:


Two hundreds (200)

Six tens (60)

Seven ones (7)


Putting these numbers together, the representation of the number shown in the number model diagram is 267.


Let’s take a look below at a few different ways your child could solve these number model problems.


Solution 1: Put the numbers together in a line

You can have your child set up the problems like how you see below: They draw a box under each number place and count the values in each column, giving you something like this:


With the numbers falling into place on their own, your child can then see the number this diagram represents is 271 by just putting all the numbers together.


Solution 2: Expanded form numbers

Looking at the same problem, another way your child can solve the problem is by using the expanded form of numbers. They can set up the problem almost the same way but adding a few extras


In this version, they are adding together the true value of each represented column. If they fill out the blanks, they’d end up with something like this:


You may be wondering why we should complicate the number model problem. After all, the first solution type is extremely easy and can be solved in seconds. This second solution type is a way for your child to understand 0 values on numbers, adding numbers together with 0 values, and also to help them practice expanded forms of numbers (which was covered in another blog post). Both solutions help reinforce important concepts in math, so it’s good to practice both!




Comparing Numbers Using Expanded Form Method

Comparing Numbers Using Expanded Form Method

Sometimes, kids will face a problem in math in which big numbers look almost identical. They may see 52323 and 53232 and need to point out which is larger or smaller. Comparing left and right just by looking at numbers can be difficult for some kids, so how should we help them with strategies to find larger or smaller amounts?


The easiest way we’ve found is by adding a few extra steps using the expanded form method.

The expanded form takes everything at a flat 0 place and shows the number as an addition problem. However, our strategy involves not showing an addition problem but putting the values side by side. 


In its simplest and non-applied form, expanded form of numbers looks like this:


Expanded form by itself offers only a few benefits, but using the expanded form in number comparisons with our structure can help your kids greatly.


Expanded form starts by taking each digit and adding 0’s at the end of it based on its place value. If the number is in the thousands place, you add three 0’s to the end. If it’s in the hundreds place, you add two 0’s to the end, and so on. Looking at the example below you can see how we’ve broken it down



So using this, it’s actually easy to compare numbers that look similar.

By putting the expanded form digits side by side, it’s easier to see where the differences are for your child. Once they are side by side, have your child compare each expanded form value. The FIRST symbol that is not an = sign is going to be the answer for the number comparison.

Check out our worksheet below for more practice!

Expanded Form

Expanded Form Answer Key

Comparing Numbers Using Expanded Form Method

Teaching Your Child Addition

Addition is the foundation of arithmetic. Without addition, it becomes very difficult to do many mathematical processes, thereby making it extremely difficult if not impossible to make any progress in learning math.


Addition is a very simple concept – we are taking two groups or more groups, putting them together, and seeing how many there are after. The easiest way to demonstrate addition to your child when they are just learning is to count. It’s a very slow process at first, but it helps to develop your child’s ability to do mental math in the future.


We are easily able to answer 2+2=4 not because we have some mystical ability to add numbers, but as adults we have learned to count by intervals, which is how we are doing addition in our heads when we have that instant reflex of knowing the answer. To develop this skill for your child, there’s a few processes that go into it but to begin with just understanding addition is important.


Teaching your child addition involves using real objects as opposed to just numbers. Real objects help cement value in numbers while also helping your child to visualize addition. Taking for example the problem:



To solve this problem the phrasing of how you ask your child this problem is extremely important. Since addition involves grouping two sets into a single set, it’s important to make sure the groups are somewhat related. Find some examples of what to do and what not to do below:

Examples That Work

  1. I have 5 apples and 3 oranges. How many fruits do I have in all?

This example works because apples and oranges are fruits, and you are seeking the total value of all fruits in the example.

  1. I have 5 chocolate bars in my left hand and 3 chocolate bars in my right hand. How many chocolate bars do I have in all?

This example also works because the items are the same and you are asking how many of that item you have.


Examples that don’t work

  1. I have 5 cats and 3 bananas. How many animals do I have?

This example doesn’t work. The second part of the question does not ask for the sum of the two things.

  1. I have 5 cats and 2 dogs. How many dogs do I have?

This example also doesn’t work as the second part of the question is not asking for the sum of the two things.


The take away from these examples should be:

  • When you are making a problem into an example, make sure there are either the same object or the same type of object within a set such as fruits, animals, people, etc.
  • Asking questions that are not related to addition will not result in the right answer.

Check out our worksheet below to help you practice with your child!

Addition Practice

Comparing Numbers Using Expanded Form Method

How to Multiply Numbers Within 10

Our last post on multiplication involved using the multiplication table. Today, let’s go over how to work with your kids to build up their skills to multiply.


At a basic level, multiplication is just addition in less steps. In our head we know 2×2=4, but the process behind it is a little bit more complex when working with children. We always like to push the importance of teaching your children the why and the how rather than just memorizing something as it will help them build new mental processes and problem solving abilities.


If you think about it, if someone told you to do a basic task you’ve never done before, it’d still be difficult. One example I like to use is to ask someone to describe a color to a person with their eyes closed and tell them to name one object using the color you’ve described. If you try this with your friends you’ll find it’s a lot harder than you think even though colors are something so basic. Multiplication is the same for children. When they see a problem like 4×2, for us, the concept is simple as we’ve already developed the multiplicative reasoning processes in our mind. A step by step process is needed to teach children how to multiply. So let’s start on looking at the process.


Let’s take a look at a simple multiplication problem:



What does 3×2 really mean? In reality, 3×2 actually means “There are TWO 3’s being added together to make 3+3=” OR “There are THREE 2’s being added together to make 2+2+2=”. 


How did we get these math problems? Let’s look closer at the example.


Each multiplication problem has three parts. You have the VALUES which are the amounts being multiplied. You also have the OPERATOR which tells you what mathematical process you are doing. 


First, let’s look at the OPERATOR:

In arithmetic, there are a few different operators that stand for multiplication. 

It is either an X or a between two numbers.

Thus, the problem above could be shown as:

3×2= or 3⋅2= 

It’s important to note that these are the same problem to your children as operators do sometimes cause confusion.


Second, let’s look at the VALUES:

In the problem of 3×2= we have two values

3 and 2. 

Each value tells you two different things. If for example, we take the “3” as “Value 1”, that tells us the base number. The “2”, being “Value 2” tells us how many times we are adding that number to itself. 

Since we have 3 as our first value, and 2 as our second value, we know that we are adding the number “3” two times to itself, giving us the problem 3+3= (which is the same as 3×2). 


If we decide to reverse the values and call “3” as “Value 2” and “2” as “Value 1”, then we know that we are adding the number “2” three times to itself, giving us the problem 2+2+2= (which is also the same as 3×2).


The important process is to understand that multiplication is just telling someone how many times a number should be adding to itself.


So now, let’s break this down into an easy way to explain to your child.

Using the problem 3×4. You can tell your child to pick a number from the problem. That will be their base number. After they have chosen their first number, let them write down the first number as many times as the second number tells them to.


Take a look at this example video below


The important thing to reinforce with your children is to make sure that they recognize they are NOT bringing the second number down, but instead using it as a counter for how many times the first number is adding it to itself.


Check out our printable PDF worksheet below for more practice with your child!

Multiplication Worksheet

Multiplication Worksheet – Answer Key Multiplication Worksheet


Comparing Numbers Using Expanded Form Method

Changing Rewards

Originally, when we began this project, we started out with children earning different amounts of time (much higher than they are now). Over time, we’ve been looking into user usage data and tweaked these values to 75 minutes for a 100% score, 45 minutes for a 90-99% score, and 30 minutes for an 80-89% score. When tuning rewards for children, it’s important to keep in mind that there is a balance that should be had. Our average time we saw spent on exercises was about 25 minutes and as such, we decided that a 3:1 ratio would be ideal for a perfect score. Too much time given, and it removes the incentive to study. We saw that children would complete an exercise or two and continue on to game the whole day away. On the opposite side of the coin, too little time is disheartening for children. Rewarding too little time saw a huge drop off in user activity as the child no longer had any desire to use their devices.

However, given that users are constantly giving us feedback, a commonly requested feature was the ability to tweak the amount of time that their children were earning. Some parents saw their kindergarteners earning too much time, or their 5th graders earning not enough time. As a result, we’ve decided to put some of the power into your hands.

Within your parent dashboard, you will now be able to find a section called “Reward Time”. Under this section, we’ve added three sliders that are tied to the scores your children get. If you feel like they deserve more or less time for completing their exercises, you can now change them.

These times automatically calculate into the scaling system we have in place – even if your child takes too many math or reading exercises, they’ll still be scaled according to the new times you’ve set down to  a minimum of 1 minute.

To access this new feature, visit the log into your parent dashboard at https://app.li2ei.com, head to “Account” and look for “Reward time”


If you haven’t gotten Learn It 2 Earn It yet, head over to www.li2ei.com/download to get it today!