Early Number Sense and Counting to 10

Rote Counting to 10

Today we will learn how to count from 1 to 10 in order. This is called rote counting.

Rote counting means saying the number words in the right order from memory. We say the numbers one after another without skipping any.

When we count to 10, we say:

$$1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10$$

It is important to say the numbers in the correct order. Each number has a place in the counting line.

Let’s look at the counting order:

• 1 comes first
• 2 comes after 1
• 3 comes after 2
• 4 comes after 3
• 5 comes after 4
• 6 comes after 5
• 7 comes after 6
• 8 comes after 7
• 9 comes after 8
• 10 comes after 9

You can think of counting like walking up stairs. You take one step at a time. You do not jump over a step.

When we rote count, we are practicing the number words:

$$\text{one, two, three, four, five, six, seven, eight, nine, ten}$$

We can also match the number words to numerals:

• one = \(1\)
• two = \(2\)
• three = \(3\)
• four = \(4\)
• five = \(5\)
• six = \(6\)
• seven = \(7\)
• eight = \(8\)
• nine = \(9\)
• ten = \(10\)

Main Idea: To count to 10, say every number in order:

$$1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10$$

If we skip a number, the count is not correct. For example, if someone says \(1, 2, 3, 5\), they skipped 4.

If the numbers are out of order, the count is not correct. For example, \(1, 2, 4, 3\) is not the right order because 3 should come before 4.

Here are some helpful ways to remember the counting sequence:

• Say the numbers slowly.
• Tap a finger for each number word.
• Clap as you count.
• Practice the same order again and again.

Worked Example 1

Say the numbers from \(1\) to \(5\).

Answer:

$$1,\ 2,\ 3,\ 4,\ 5$$

We started at \(1\) and said each next number in order.

Worked Example 2

What number comes after \(6\)?

Count forward: \(6, 7\)

Answer: The number after \(6\) is 7.

Worked Example 3

Is this counting correct?

$$1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 8,\ 9,\ 10$$

Answer: No.

The number 7 is missing. The correct counting is:

$$1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10$$

Worked Example 4

Fill in the missing number:

$$1,\ 2,\ 3,\ \_,\ 5,\ 6$$

Think: What number comes after \(3\) and before \(5\)?

Answer: \(4\)

The full counting sequence is:

$$1,\ 2,\ 3,\ 4,\ 5,\ 6$$

Let’s Practice Counting to 10

1. Start at \(1\) and say the numbers to \(10\).
2. Try again without looking.
3. Say the numbers one more time and clap each time you say a number.

Here is the full count again:

$$1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10$$

Summary

Rote counting means saying numbers from memory in the right order.

When we count to 10, we say:

$$1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10$$

Remember:

• Start at 1.
• Say every number in order.
• Do not skip numbers.
• Stop at 10.

With practice, counting to 10 becomes easy and fun!

One-to-One Correspondence means matching one number word to one object when we count.

When we count, we touch or point to each object one time. We say one number for each object: 1, 2, 3, 4...

This helps us know how many objects are in a group.

Let’s learn how to do it carefully and correctly.

Why is one-to-one correspondence important?

• It helps us count correctly.
• It helps us not skip objects.
• It helps us not count the same object twice.
• It helps us find the total number in a set.

How to count with one-to-one correspondence

1. Look at the group of objects.
2. Start with one object.
3. Touch or point to that object.
4. Say one number word: 1.
5. Move to the next object.
6. Touch or point to it and say the next number: 2.
7. Keep going until every object has been counted once.

Important rule: Every object gets one touch and one number.

If we count 4 stars, it can look like this:

★ → 1
★ → 2
★ → 3
★ → 4

There are $$4$$ stars.

Things to remember

• Do not count too fast.
• Do not skip an object.
• Do not count one object two times.
• Point as you count.
• The last number you say tells how many there are.

Worked Example 1: Count a small group

Look at these apples:

🍎 🍎 🍎

Point to each apple one at a time.

First apple: 1
Second apple: 2
Third apple: 3

The last number said is 3.

So, there are $$3$$ apples.

Worked Example 2: Objects in a line

Look at these blocks:

🟦 🟦 🟦 🟦 🟦

Let’s count carefully:

Point to the 1st block and say 1.
Point to the 2nd block and say 2.
Point to the 3rd block and say 3.
Point to the 4th block and say 4.
Point to the 5th block and say 5.

There are $$5$$ blocks.

Worked Example 3: Objects not in a line

Sometimes objects are spread out. We still count each one once.

Look at these dots:

●    ●
    ●    ●

Choose one dot to start. Then point to another dot each time.

Count like this:

1, 2, 3, 4

There are $$4$$ dots.

Even when objects are not in a straight line, we can still use one-to-one correspondence.

Worked Example 4: Find the mistake

Sam is counting 4 toys:

🧸 🧸 🧸 🧸

Sam says: 1, 2, 2, 3

Is that correct? No.

Sam used the number 2 two times. Each object needs the next number word.

The correct count is:

1, 2, 3, 4

So there are $$4$$ toys.

Tips for good counting

• Move objects if you can after you count them.
• Put counted objects in a new group.
• Touch each object gently.
• Say the numbers in order.

Let’s think about zero

If there are no objects, we count 0.

Like this: no cookies = $$0$$ cookies.

There is nothing to point to, so the number is 0.

How one-to-one correspondence helps with numbers to 10

We can use one-to-one correspondence to count groups from $$0$$ to $$10$$.

For example, if you see 6 pencils, you point to each pencil once and say:

1, 2, 3, 4, 5, 6

Then you know the group has $$6$$ pencils.

Quick check

• If you point to 1 object, you say 1 number.
• If you point to 2 objects, you say 2 numbers.
• If you point to 6 objects, you say 6 numbers.

Summary

One-to-one correspondence means one object, one number word.

When you count, point to each object once and say the numbers in order.

Do not skip objects, and do not count any object twice.

The last number you say tells how many objects are in the set.

With practice, you will become a careful and confident counter!

The Cardinality Principle

When we count objects, we say number words in order: 1, 2, 3, 4, 5...

The cardinality principle means something very important: the last number we say tells how many objects there are in all.

So if we count some blocks and say, 1, 2, 3, 4, then there are 4 blocks. The last number said was 4, so the total is 4.

This helps us know the exact amount in a group.

Let’s learn how to count carefully.

• Touch or point to each object one time.
• Say one number word for each object.
• Count in order: 1, 2, 3, 4, 5...
• Listen to the last number you say.
• That last number tells how many.

If we count the same group again, the total stays the same, even if the objects are moved around.

For example, 5 buttons are still 5 buttons if they are in a line, a circle, or a pile. When we count them correctly, the last number said is still 5.

Why this matters

Sometimes children count objects but do not know that the last number means the whole group. They may count 1, 2, 3, 4, 5 and then, when asked “How many?” start counting all over again.

With the cardinality principle, we learn to say: “There are 5.” We know this because 5 was the last number we said.

Example 1: Count a small group

Look at 3 apples.

Count them: 1, 2, 3

The last number said is 3.

So there are 3 apples.

We can write it like this: \(3\)

Example 2: Count and answer “How many?”

Look at 5 stars.

Count them one by one: 1, 2, 3, 4, 5

The last number said is 5.

So, when someone asks, “How many stars are there?” the answer is 5 stars.

We do not need to count again right away. The last number already told us the total.

Example 3: Objects moved around

There are 4 toy cars on the table.

First they are in a row. We count: 1, 2, 3, 4

So there are 4 cars.

Now the cars are moved into a small group. Count again carefully: 1, 2, 3, 4

The last number is still 4.

The group still has 4 cars. Moving objects does not change how many there are.

Example 4: A bigger group up to 10

Look at 8 cubes.

Count each cube once: 1, 2, 3, 4, 5, 6, 7, 8

The last number said is 8.

So there are 8 cubes.

We can show the total with the numeral \(8\).

Helpful counting tips

1. Start at 1.
2. Point to each object as you count.
3. Do not skip any objects.
4. Do not count an object two times.
5. Say the last number again to tell the total.

For example, if you count bears and say 1, 2, 3, 4, 5, 6, then you can say, “There are 6 bears.”

Try thinking about these

• If you count crayons and say 1, 2, there are 2 crayons.
• If you count balls and say 1, 2, 3, 4, 5, 6, 7, there are 7 balls.
• If you count shells and the last number is 10, there are 10 shells.

What about 0?

If there are no objects to count, then there are 0 objects.

For example, if there are no cookies on the plate, the total is \(0\).

Summary

The cardinality principle means: the last number you say when counting tells how many objects are in the group.

Count each object once, say the numbers in order, and listen to the last number.

If you count 1, 2, 3, 4, 5, then there are 5 objects in all.

That last number is the total.

Hierarchical Inclusion means that bigger numbers include smaller numbers.

When we count, each new number has all the ones before it inside it. For example, if you have 7 blocks, you also have 6 blocks in that group, and 5 blocks, and 4 blocks too.

This helps us understand numbers from 0 to 10 in a strong way. It helps us see that numbers grow one at a time.

Let’s think about counting.

If we count: \(1, 2, 3, 4, 5\), each number is one more than the number before it.

That means:

• \(2\) has all of \(1\), and one more.
• \(3\) has all of \(2\), and one more.
• \(4\) has all of \(3\), and one more.
• \(5\) has all of \(4\), and one more.

So bigger numbers are like little numbers tucked inside.

Another way to say it:

If you have a group of \(6\) apples, you can point to any \(5\) of them and say, “Here are \(5\).” The full group of \(6\) still contains that group of \(5\).

This is why we say numbers are nested. They fit inside each other in order.

Look at the pattern:

• \(1 = 0 + 1\)
• \(2 = 1 + 1\)
• \(3 = 2 + 1\)
• \(4 = 3 + 1\)
• \(5 = 4 + 1\)
• \(6 = 5 + 1\)
• \(7 = 6 + 1\)
• \(8 = 7 + 1\)
• \(9 = 8 + 1\)
• \(10 = 9 + 1\)

Each number keeps the smaller number and adds 1 more.

Why is this important?

• It helps us compare numbers.
• It helps us know which number is bigger.
• It helps with adding and taking away later.
• It helps us understand that counting is not just saying number words. Counting tells how many.

Let’s use objects.

Imagine 4 toy cars:

🚗 🚗 🚗 🚗

In this group of \(4\):

• There is a group of \(1\).
• There is a group of \(2\).
• There is a group of \(3\).
• And there is the whole group of \(4\).

So \(4\) includes \(3\), \(2\), and \(1\).

Worked Example 1

Mia has \(3\) crayons.

Can a group of \(3\) crayons include a group of \(2\) crayons?

Yes.

If Mia has 3 crayons, we can pick 2 of them and make a smaller group.

So \(3\) includes \(2\).

We can show it like this:

🖍️ 🖍️ 🖍️

Inside \(3\), there is:

🖍️ 🖍️

Worked Example 2

Sam has \(5\) bears.

Does \(5\) include \(4\)?

Yes.

Because \(5\) is \(4\) and 1 more.

We can write:

\(5 = 4 + 1\)

So a group of \(5\) contains a group of \(4\).

Worked Example 3

There are \(7\) stars.

Does \(7\) include \(6\)? Does it include \(3\)?

Yes and yes.

A group of \(7\) has all of \(6\) and one more.

It also has all of \(3\), because \(3\) is smaller than \(7\).

So inside \(7\), we can find \(6\), \(5\), \(4\), \(3\), \(2\), and \(1\).

Worked Example 4

Lina has \(2\) shells.

Can \(2\) include \(5\)?

No.

A smaller number cannot include a bigger number.

Since \(2\) is less than \(5\), a group of \(2\) cannot contain a group of \(5\).

Let’s remember these big ideas:

1. Numbers grow by 1 more each time.
2. A bigger number has smaller numbers inside it.
3. If a number is greater, it can include a smaller group.
4. If a number is smaller, it cannot include a bigger group.

Try thinking about these:

• Does \(6\) include \(5\)? Yes.
• Does \(6\) include \(2\)? Yes.
• Does \(4\) include \(4\)? Yes, because the whole group is \(4\).
• Does \(3\) include \(8\)? No.

A helpful picture in your mind:

Think of numbers like steps on a staircase.

When you stand on step \(6\), you have already passed steps \(1\), \(2\), \(3\), \(4\), and \(5\).

So step \(6\) is higher, and it includes all the smaller steps before it.

Summary

Hierarchical inclusion means bigger numbers contain smaller numbers.

If you have \(7\), then you also have \(6\), \(5\), \(4\), \(3\), \(2\), and \(1\) inside that group.

This helps us understand counting and know that each number is one more than the number before it.

When you look at a number, ask yourself: What smaller numbers are inside it?

That is hierarchical inclusion.

Perceptual Subitizing to 5

Today we will learn how to look at a small group and know how many right away, without counting one by one.

This skill is called subitizing. That is a big word, but it means something simple: seeing a small number fast.

When you see 1, 2, 3, 4, or 5 things, your eyes and brain can work together to know the amount quickly.

For example, if you see:

● ●

You can say 2 right away. You do not need to count 1, 2. You just see 2.

This helps you become faster and stronger with numbers.

What does subitizing look like?

We are working with groups up to 5. Here are some small groups you can learn to recognize quickly:

• 1: ●
• 2: ● ●
• 3: ● ● ●
• 4: ● ● ● ●
• 5: ● ● ● ● ●

At first, you may want to count. That is okay. But with practice, you will start to know the number right away.

Look for the whole group

When you subitize, you do not point to each object. Instead, you look at the whole group and think, “I know that amount!”

Your brain can learn the look of each small number.

• 1 looks like one dot all alone.
• 2 looks like a pair.
• 3 looks like a small group of three.
• 4 looks like a bigger group, but still small enough to see quickly.
• 5 looks like a full small group.

You can see objects in different ways

The things do not always have to be in a straight line. They can be spread out, close together, or arranged in a shape.

Even if the group looks different, the number stays the same.

Here are some ways to show the same number.

3 can look like this:

● ● ●

or

●
● ●

Both groups still show 3.

4 can look like this:

● ●
● ●

or

● ● ● ●

Both groups still show 4.

5 can look like this:

● ● ●
● ●

or

● ● ● ● ●

Both groups still show 5.

Use quick looks

A good way to practice subitizing is to take a quick look. Look fast, hide the dots, and then say how many you saw.

You can ask yourself:

• Did I see 1?
• Did I see 2?
• Did I see 3?
• Did I see 4?
• Did I see 5?

Worked Example 1

Look at this group:

●

What number is it?

It is 1.

There is only one dot. We can see 1 right away.

In math, we write that as \(1\).

Worked Example 2

Look at this group:

● ●

What number is it?

It is 2.

These two dots make a pair. We can recognize 2 quickly.

In math, we write that as \(2\).

Worked Example 3

Look at this group:

● ●
●

What number is it?

It is 3.

Even though the dots are not in one line, we can still see the whole group and know it is 3.

In math, we write that as \(3\).

Worked Example 4

Look at this group:

● ●
● ●

What number is it?

It is 4.

This is a small square shape. Many children learn to recognize this shape as 4 very quickly.

In math, we write that as \(4\).

One more look at 5

Here is a group of 5:

● ● ●
● ●

If you look at the whole group, you can learn to say 5 without counting each dot.

In math, we write that as \(5\).

Tips to help you subitize

• Take a quick look.
• Look at the whole group, not one dot at a time.
• Practice with groups from 1 to 5.
• Notice shapes and patterns.
• Say the number you see right away.

Why is this important?

Subitizing helps you understand numbers better. It helps you with counting, adding, and comparing numbers later.

When you can quickly see that a group has \(4\) objects or \(5\) objects, math becomes easier.

Let’s review

• Subitizing means knowing how many objects are in a small group without counting one by one.
• We can subitize groups up to 5.
• The same number can look different in different arrangements.
• We look at the whole group and say the number quickly.

Summary

Perceptual subitizing to 5 means you can look at 1, 2, 3, 4, or 5 objects and know the amount right away.

You do not need to count each object. You use your eyes and brain to recognize the small group.

With practice, you will get faster and more confident. Soon, when you see a small group of dots, you will just know how many there are!

Conceptual Subitizing to 10

Sometimes we can know how many without counting one by one. This is called subitizing.

Conceptual subitizing means we see a bigger number by noticing smaller groups inside it. For example, we can see 6 as 3 and 3, or 8 as 4 and 4.

This helps us count faster and understand numbers better.

Big idea: A number up to 10 can be made from smaller parts.

For example:

• 5 can be seen as 2 and 3.
• 6 can be seen as 3 and 3.
• 7 can be seen as 5 and 2.
• 8 can be seen as 4 and 4.
• 9 can be seen as 5 and 4.
• 10 can be seen as 5 and 5.

When we see the parts, we can quickly know the whole number.

How to conceptual subitize

1. Look at the dots, objects, or pictures.
2. Notice small groups you know quickly, like 2, 3, 4, or 5.
3. Put the groups together in your mind.
4. Say the total.

You do not need to count every object one at a time if you can see the groups.

Think about dot patterns

Dot patterns help our eyes see groups. We might see:

• 2 dots and 2 dots, so that is 4.
• 3 dots and 2 dots, so that is 5.
• 4 dots and 3 dots, so that is 7.

We can write these with numbers:

\(2+2=4\)

\(3+2=5\)

\(4+3=7\)

Worked Example 1

You see 4 dots. They are in 2 groups of 2.

First group: 2

Second group: 2

Put them together:

\(2+2=4\)

So the total is 4.

Worked Example 2

You see 5 dots. One group has 3 dots. One group has 2 dots.

Put the groups together:

\(3+2=5\)

So the total is 5.

Worked Example 3

You see 6 dots. They are in 2 groups of 3.

You can think, “3 and 3 makes 6.”

\(3+3=6\)

So the total is 6.

Worked Example 4

You see 9 dots. You notice a group of 5 and a group of 4.

Put the groups together:

\(5+4=9\)

So the total is 9.

More ways to see numbers to 10

• \(4=1+3\) or \(2+2\)
• \(5=4+1\) or \(2+3\)
• \(6=5+1\) or \(3+3\)
• \(7=5+2\) or \(4+3\)
• \(8=5+3\) or \(4+4\)
• \(9=5+4\) or \(3+3+3\)
• \(10=5+5\) or \(4+4+2\)

There can be more than one way to see the same number. That is okay. Different groups can still make the same total.

Why this is helpful

• It helps you know numbers quickly.
• It helps you add small groups together.
• It helps you understand that numbers are made of parts.

Try this thinking:

• If you see 7, ask: “Do I see 5 and 2?”
• If you see 8, ask: “Do I see 4 and 4?”
• If you see 10, ask: “Do I see 5 and 5?”

Tips for students

• Look for groups you know fast.
• Think about doubles, like \(3+3\) or \(4+4\).
• Think about 5 and some more.
• Say the parts, then say the whole.

For example, you can say, “I see 5 and 2, so I know it is 7.”

Summary

Conceptual subitizing means seeing a number by noticing smaller groups inside it.

Instead of counting one by one, you can think about parts and wholes.

For numbers to 10, you can look for easy groups like 2, 3, 4, and 5.

Then put the groups together. For example:

• \(2+2=4\)
• \(3+2=5\)
• \(3+3=6\)
• \(5+4=9\)

When you see groups, you can know the total quickly and become stronger with numbers to 10.

Number Conservation means a group has the same number even if the objects are moved, spread out, or pushed close together.

Sometimes a row that is longer looks like it has more. But in math, we do not guess by how long it looks. We count to find out how many there are.

In this lesson, we will learn that when nothing is added and nothing is taken away, the number stays the same.

Big idea: If we move objects around, the amount does not change.

For example, if you have 5 counters and spread them out, there are still 5 counters.

$$5 = 5$$

The group may look different, but it still has the same quantity.

How to check if the number stayed the same:

• Look at the group.
• Count each object one time.
• Move the objects if you want.
• Count again.
• If nothing was added or taken away, the number will match.

Important rule: Moving is not adding. Moving is not taking away.

If 1 toy is moved to the end of the row, it is still part of the group.

If 6 blocks are close together, there are 6 blocks.

If the same 6 blocks are spread far apart, there are still 6 blocks.

$$6 = 6$$

What can trick our eyes?

• A longer row can look like it has more.
• A shorter row can look like it has less.
• A circle can look different from a line.

That is why we count, not guess.

Worked Example 1

Here are 4 stars in a row:

0000

Count them: \(1, 2, 3, 4\)

Now move them farther apart:

0    0    0    0

Count again: \(1, 2, 3, 4\)

Answer: The number is still 4. Spreading them out did not change the amount.

$$4 = 4$$

Worked Example 2

There are 5 dots close together:



Count them: \(1, 2, 3, 4, 5\)

Now put the same 5 dots in a circle.

Count again: \(1, 2, 3, 4, 5\)

Answer: The shape changed, but the number did not change. There are still 5 dots.

$$5 = 5$$

Worked Example 3

Sam has 7 cubes in a short row.

Then Sam stretches the row to make it long.

Did Sam make more cubes?

Lets count: \(1, 2, 3, 4, 5, 6, 7\)

Count again after moving them: \(1, 2, 3, 4, 5, 6, 7\)

Answer: No. Sam did not make more cubes. Sam still has 7 cubes.

Nothing was added. Nothing was taken away.

$$7 = 7$$

Worked Example 4

Look at two groups:

• Group A: 6 buttons in a tight row
• Group B: the same 6 buttons in a wide row

Which group has more?

Count Group A: \(1, 2, 3, 4, 5, 6\)

Count Group B: \(1, 2, 3, 4, 5, 6\)

Answer: Neither group has more. They have the same number: 6.

$$6 = 6$$

Lets remember:

• If objects are only moved, the number stays the same.
• If objects are spread out, the number stays the same.
• If objects are pushed together, the number stays the same.
• We count to check.

Try it in real life:

1. Put 3 crayons on a table.
2. Count them.
3. Spread them apart.
4. Count again.
5. You will still have 3 crayons.

Now try with 8 small toys or 10 blocks. Move them into a line, a circle, or a pile. Count each time. The number will stay the same if none are added or removed.

Summary

Number conservation means the amount in a group stays the same when the objects are moved around.

A group can look longer, shorter, wider, or different in shape. But if nothing is added and nothing is taken away, the number does not change.

Always remember: count the objects. Counting helps us know the true number.

Understanding Zero as an Empty Set

Today we will learn about the number 0.

Zero means none. Zero means there are no objects. If a group has nothing in it, that group has 0 things.

This is called an empty set. An empty set is just a group with nothing inside.

For example, if you look in a basket and there are no apples, then the basket has 0 apples.

We write zero like this: 0.

Zero is an important number. It tells us that something is not there.

Main Teaching Points

• Zero means none.
• Zero means an empty group.
• The numeral for zero is 0.
• When we count objects and there are no objects, we say 0.

Let’s think about counting.

If you see 1 toy, you say 1. If you see 2 toys, you say 2. But if you see no toys, you say 0.

Zero is less than 1. That means 0 is smaller than every counting number from 1 to 10.

We can show that with math:

\(0 < 1\)

We can also match numbers to groups:

• 0 → no stars
• 1 → ★
• 2 → ★★

For zero, there are no stars at all. That is why zero means an empty set.

How to know if the answer is 0

1. Look at the group.
2. Ask, “Are there any objects?”
3. If there are none, the number is 0.

Worked Examples

Example 1

There are no cookies on the plate.

How many cookies are there?

There are 0 cookies.

We write:

\(0\)

Example 2

Mia has 1 balloon. The balloon pops. Now Mia has no balloons.

How many balloons does Mia have now?

Mia has 0 balloons.

We can show it like this:

Example 3

Look at two boxes.

• Box A has 3 blocks.
• Box B has no blocks.

Which box shows zero?

Box B shows zero because it has no blocks.

So Box B has:

\(0\) blocks

Example 4

Count the crayons in the cup.

The cup is empty.

An empty cup has 0 crayons.

Empty means nothing is inside.

Let’s Practice Thinking About Zero

• If there are no cats in the yard, there are 0 cats.
• If there are no books on the desk, there are 0 books.
• If there are no shoes by the door, there are 0 shoes.

Each time the group has nothing, we use the numeral 0.

Important Idea

Zero is still a number.

Even though zero means none, it is a real number we can write, read, and use.

We say: zero

We write: 0

Quick Check

• No ducks in the pond = 0
• No pencils in the box = 0
• No flowers in the vase = 0

Summary

Zero means none.

Zero tells us a group is empty.

If there are no objects to count, we write 0.

Remember:

• 0 means no objects
• 0 means an empty set
• We read it as zero

Now you know that when nothing is there, the number is 0!

Numeral Recognition and Formation (0–10)

Numbers help us tell how many. We use number words like “one” and “five,” and we also use numerals, which are the number symbols we write.

In this lesson, we will learn to recognize numerals from 0 to 10 and to write them the right way. We will also match each numeral to a group of objects.

What is a numeral?

A numeral is a symbol for a number. For example, the numeral for “three” is \(3\), and the numeral for “ten” is \(10\).

Here are the numerals from 0 to 10:

$$0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10$$

Let’s learn each numeral.

• 0 means none. There are no objects.
• 1 means one object.
• 2 means two objects.
• 3 means three objects.
• 4 means four objects.
• 5 means five objects.
• 6 means six objects.
• 7 means seven objects.
• 8 means eight objects.
• 9 means nine objects.
• 10 means ten objects.

Reading numerals

When you see a numeral, say its number name:

• \(0\) says “zero”
• \(1\) says “one”
• \(2\) says “two”
• \(3\) says “three”
• \(4\) says “four”
• \(5\) says “five”
• \(6\) says “six”
• \(7\) says “seven”
• \(8\) says “eight”
• \(9\) says “nine”
• \(10\) says “ten”

Matching numerals to groups

A numeral tells how many things are in a group. Count the objects carefully, then choose the numeral that matches.

Example groups:

• 🍎🍎 = \(2\)
• ⭐⭐⭐⭐ = \(4\)
• 🍪🍪🍪🍪🍪 = \(5\)
• No balls = \(0\)

Writing numerals

When we write numerals, we make each one in its own special shape. Start carefully and try to make the numeral easy to read.

• 0: Make a round oval shape.
• 1: Make one straight line down.
• 2: Curve across the top, then slant down, then draw a line across the bottom.
• 3: Make two curves, one on top and one on bottom.
• 4: Make a line down, a line across, and one more line down.
• 5: Make a line across the top, a line down, then a curve at the bottom.
• 6: Make a curve around and close it.
• 7: Make a line across the top, then a slant down.
• 8: Make two connected loops.
• 9: Make a small loop on top and a line down.
• 10: Write \(1\), then write \(0\) next to it.

Tip: The numeral \(10\) has two digits: \(1\) and \(0\). All the other numerals from 0 to 9 have just one digit.

How to remember tricky numerals

• 6 and 9 can look alike. 6 has its loop at the bottom. 9 has its loop at the top.
• 2 and 5 are different. 2 starts with a curve and ends with a bottom line. 5 starts with a top line, then goes down, then curves.
• 3 has two bumps. 8 has two closed loops.

Worked Example 1

Look at this group: ⭐⭐⭐

Step 1: Count the stars. There are 3 stars.

Step 2: Match the count to the numeral. The numeral is \(3\).

Answer: ⭐⭐⭐ = \(3\)

Worked Example 2

What numeral matches no objects?

Step 1: No objects means none.

Step 2: The numeral for none is \(0\).

Answer: No objects = \(0\)

Worked Example 3

You count 7 toy cars. What numeral should you write?

Step 1: Count the toy cars: 1, 2, 3, 4, 5, 6, 7.

Step 2: The numeral for seven is \(7\).

Answer: Write \(7\).

Worked Example 4

How do you write ten?

Step 1: Remember that ten is written with two digits.

Step 2: Write \(1\), then write \(0\).

Answer: Ten is written as \(10\).

Practice ideas

1. Say the numerals from \(0\) to \(10\) out loud.
2. Point to each numeral as you say it.
3. Count small objects like blocks, crayons, or buttons.
4. Write the numeral that matches the group.
5. Practice writing each numeral neatly.

Check your thinking

• If you see 5 dots, do you write \(5\)? Yes.
• If there are no dots, do you write \(0\)? Yes.
• If you count 10 cubes, do you write just \(1\)? No. You write \(10\).

Summary

Numerals are the symbols we use to show numbers. We learned the numerals \(0\) through \(10\), how to read them, and how to write them. We also learned that each numeral matches a quantity, and that \(0\) means none while \(10\) is written with two digits: \(1\) and \(0\).

Comparing Quantities: More, Less, and Same

Sometimes we look at two groups of things and want to know which group has more, which group has less, or if they are the same.

This helps us compare numbers and sets of objects. We can compare toys, apples, blocks, stars, or any other things we can count.

In this lesson, we will learn:

• what more means
• what less means
• what same means
• how to compare two groups by counting
• how to compare two groups by matching

1. What does more mean?

More means a group has a bigger number of things.

If one group has 5 blocks and another group has 3 blocks, the group with 5 blocks has more.

We can write:

\(5\) is more than \(3\).

2. What does less mean?

Less means a group has a smaller number of things.

If one group has 2 apples and another group has 6 apples, the group with 2 apples has less.

We can write:

\(2\) is less than \(6\).

3. What does same mean?

Same means both groups have an equal number of things.

If one group has 4 stars and another group has 4 stars, the groups are the same.

We can write:

\(4 = 4\)

4. How can we compare groups?

There are two easy ways to compare groups:

• Count the objects in each group.
• Match one object in the first group to one object in the second group.

Count to compare

Count each group carefully. Then look at the numbers. The bigger number means more. The smaller number means less.

Match to compare

Match one object from one group with one object from the other group.

• If one group has extra objects left over, that group has more.
• The other group has less.
• If no objects are left over, the groups are the same.

Worked Example 1

Look at these two groups:

• Group A: 2 balls
• Group B: 5 balls

First, count each group.

Group A has \(2\).

Group B has \(5\).

Now compare the numbers. Since \(5\) is bigger than \(2\), Group B has more.

Group A has less.

Answer: \(5\) is more than \(2\), and \(2\) is less than \(5\).

Worked Example 2

Look at these two groups:

• Group A: 3 cats
• Group B: 3 cats

Count both groups.

Group A has \(3\).

Group B has \(3\).

Both groups have the same number.

Answer: The groups are the same.

We can write:

\(3 = 3\)

Worked Example 3

Now let us compare by matching.

Group A has 4 pencils.

Group B has 6 pencils.

Match one pencil in Group A to one pencil in Group B:

• 1 matches 1
• 1 matches 1
• 1 matches 1
• 1 matches 1

After matching, Group B still has 2 pencils left over.

That means Group B has more.

Group A has less.

Answer: \(6\) is more than \(4\).

Worked Example 4

Look at these two groups:

• 7 bears
• 5 bears

Count the bears in each group.

One group has \(7\).

The other group has \(5\).

Since \(7\) is bigger than \(5\), the group with 7 bears has more.

The group with 5 bears has less.

Answer: \(7\) is more than \(5\).

Tips for Comparing Quantities

• Count slowly and carefully.
• Touch each object one time as you count.
• Say the number names in order: \(1, 2, 3, 4, 5, 6, 7, 8, 9, 10\).
• If two groups are hard to compare, try matching the objects.
• Look for leftovers when you match.

Words to Remember

• More = bigger amount
• Less = smaller amount
• Same = equal amount

Let’s Think

If one plate has 8 cookies and another plate has 8 cookies, are they more, less, or same?

They are the same.

If one box has 1 toy and another box has 4 toys, which box has more?

The box with 4 toys has more.

Summary

We can compare two groups to find out which has more, which has less, or if they are the same.

To compare, we can:

• count the objects in each group
• match the objects one by one

Remember:

• The bigger number means more.
• The smaller number means less.
• If the numbers are equal, the groups are the same.

Great job learning how to compare quantities from \(0\) to \(10\)!

Comparing Numerals within 10

Today we will learn how to compare numerals from 0 to 10. Comparing means we look at two numbers and decide which number is greater (more), which number is less (smaller), or if they are equal (the same).

A numeral is a number written down, like \(3\), \(7\), or \(10\). When we compare numerals, we are comparing the amounts they stand for.

Here are the numerals from 0 to 10 in order:

$$0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10$$

If a numeral comes later in this list, it is usually greater. If it comes earlier in this list, it is less.

For example, \(8\) comes after \(5\), so \(8\) is greater than \(5\). Also, \(2\) comes before \(6\), so \(2\) is less than \(6\).

Main Idea 1: Use counting order

You can compare two numerals by saying the counting numbers in order. The number you say later is the greater number.

• \(1\) is less than \(4\)
• \(6\) is greater than \(3\)
• \(7\) and \(7\) are equal

Main Idea 2: Think about more and less

Each numeral tells how many. A bigger numeral means a bigger amount. A smaller numeral means a smaller amount.

• \(9\) means more than \(2\)
• \(5\) means less than \(8\)
• \(4\) means the same amount as \(4\)

Main Idea 3: Learn the comparison words

• Greater than means more
• Less than means smaller
• Equal to means the same

We can write comparisons like this:

$$7 > 4$$

$$2 < 6$$

$$5 = 5$$

The symbol \(>\) means greater than. The symbol \(<\) means less than. The symbol \(=\) means equal to.

How to compare numerals

1. Look at the two numerals.
2. Say the counting numbers in order, if needed.
3. Decide which numeral comes later, which comes earlier, or if they are the same.
4. Say the comparison: greater than, less than, or equal to.

Worked Example 1

Compare \(3\) and \(5\).

When we count, we say \(3, 4, 5\). The numeral \(5\) comes after \(3\).

So, \(5\) is greater than \(3\), and \(3\) is less than \(5\).

$$3 < 5$$

Worked Example 2

Compare \(8\) and \(6\).

When we count, \(8\) comes after \(6\).

So, \(8\) is greater than \(6\).

$$8 > 6$$

Worked Example 3

Compare \(4\) and \(4\).

Both numerals are the same. They stand for the same amount.

So, \(4\) is equal to \(4\).

$$4 = 4$$

Worked Example 4

Compare \(0\) and \(2\).

The numeral \(0\) means none. The numeral \(2\) means two. Two is more than none.

So, \(0\) is less than \(2\).

$$0 < 2$$

Helpful tips

• If you are not sure, count from 0 up to 10.
• The numeral said later when counting is greater.
• The numeral said earlier when counting is less.
• If both numerals are the same, use \(=\).

Let’s practice thinking

Which is greater: \(9\) or \(7\)? Since \(9\) comes after \(7\), \(9\) is greater.

Which is less: \(1\) or \(6\)? Since \(1\) comes before \(6\), \(1\) is less.

Are \(10\) and \(10\) equal? Yes. They are the same numeral, so they are equal.

Summary

To compare numerals within 10, look at two written numbers and decide which one is more, which one is less, or if they are the same. You can use the counting order from 0 to 10 to help. Numbers that come later are greater, numbers that come earlier are less, and matching numbers are equal.

Ordinal numbers tell the place of something in a line or order.

They are different from counting numbers. Counting numbers tell how many. Ordinal numbers tell which one.

For this lesson, we will learn these ordinal numbers:

• first = 1st
• second = 2nd
• third = 3rd
• fourth = 4th
• fifth = 5th

When things are in a line, we can say where each one is.

Look at this order from left to right:

⭐ ⭐ ⭐ ⭐ ⭐

If we name the stars by place, we get:

1. first
2. second
3. third
4. fourth
5. fifth

Important: We must know where to start. Most of the time, we look from left to right unless someone tells us a different side.

Here is a helpful way to think about it:

• first means the one at the beginning
• second means the one after first
• third means the one after second
• fourth means the one after third
• fifth means the one after fourth

Let’s match the words and short forms:

$$1st=\text{first} \qquad 2nd=\text{second} \qquad 3rd=\text{third}$$

$$4th=\text{fourth} \qquad 5th=\text{fifth}$$

Worked Example 1

Five children are standing in a line from left to right.

Ana, Ben, Cam, Dina, Eli

Who is first?

We start at the left. Ana is at the beginning of the line.

Answer: Ana is first.

Who is third?

Let’s count places:

1. Ana
2. Ben
3. Cam

Answer: Cam is third.

Worked Example 2

Look at these toys in a line from left to right:

car, ball, kite, doll, drum

Which toy is second?

Count the places:

1. car
2. ball
3. kite
4. doll
5. drum

Answer: The ball is second.

Which toy is fifth?

The fifth toy is the one in place 5.

Answer: The drum is fifth.

Worked Example 3

Look at these animals in a line from left to right:

cat, dog, pig, hen, fox

Which animal is fourth?

Count carefully:

1. cat
2. dog
3. pig
4. hen
5. fox

Answer: The hen is fourth.

Now let’s ask a different question: What place is the pig in?

Count to the pig:

1. cat
2. dog
3. pig

Answer: The pig is third.

Worked Example 4

Look at these shapes in a line from left to right:

circle, square, triangle, rectangle, heart

If the square is second, what shape is first?

The first shape is the one before second.

Answer: The circle is first.

What shape is fifth?

Count all the way to place 5:

1. circle
2. square
3. triangle
4. rectangle
5. heart

Answer: The heart is fifth.

Tips to help you

• Find where to start.
• Move one by one in order.
• Say the place words as you count: first, second, third, fourth, fifth.
• Check carefully so you do not skip anything.

Let’s remember the difference:

• How many? Use counting numbers: 1, 2, 3, 4, 5
• Which one? Use ordinal numbers: first, second, third, fourth, fifth

Example:

• There are 5 apples. That tells how many.
• The apple at the end is fifth. That tells which one.

Summary

Ordinal numbers tell the place of something in a line or order.

The first five ordinal numbers are first, second, third, fourth, and fifth.

We can also write them as 1st, 2nd, 3rd, 4th, and 5th.

When you solve a problem, start at the correct side and count each place carefully.