Elementary Mathematics: Arithmetic & Fractions

The big picture

Today we learn to count and to tell which number is bigger. This is fun because you use it every day. You count your toys, your snacks, and your friends!

What counting means

To count means to say numbers in order, one at a time. You say 1, 2, 3, 4, 5. Each number is one more than the number before it.

Picture 3 apples on a table. You add 1 more apple. Now you say 4. You counted up by one.

Key idea: Each number is just one more than the one before it.

Counting up and counting back

You can count two ways. Counting up makes numbers bigger: 6, 7, 8, 9.

Counting back makes numbers smaller: 9, 8, 7, 6. A rocket counts back: 3, 2, 1, blast off!

Try it. Count back from 5. You say 5, 4, 3, 2, 1, 0. Good job!

Key idea: Up means bigger. Back means smaller.

Skip counting is faster

Skip counting means counting in jumps. It is faster than counting by ones.

• Count by 2s: 2, 4, 6, 8, 10.
• Count by 5s: 5, 10, 15, 20, 25.
• Count by 10s: 10, 20, 30, 40, 50.

Say you have coins worth 5 cents each. Skip count by 5s to add them fast!

Key idea: Skip counting jumps by the same number each time.

Which is bigger? Comparing numbers

To compare means to tell which number is bigger, or if they are the same. We use three signs.

• The greater than sign is >. It means the first number is bigger. Example: 8 > 3.
• The less than sign is <. It means the first number is smaller. Example: 4 < 9.
• The equal sign is =. It means both numbers are the same. Example: 6 = 6.

Here is a fun trick. The open, wide side is a hungry mouth. It wants to eat the bigger number! In 8 > 3, the open side faces the 8.

Key idea: The open mouth of the sign always faces the bigger number.

Using a number line

A number line is a line with numbers in order. On a number line, bigger numbers are to the right. Smaller numbers are to the left.

Which is bigger, 7 or 5? Start at 5 and count up: 5, 6, 7. You had to go up to reach 7. So 7 is bigger. We write 7 > 5.

See how 7 sits to the right of 5? That is another way to know 7 is bigger.

Key idea: On a number line, farther right means bigger.

Putting numbers in order

To order numbers means to line them up from small to big, or big to small. From least to greatest means smallest first.

Let us order 8, 3, and 6. Find the smallest first. That is 3. Next is 6. Biggest is 8. In order: 3, 6, 8.

Here is one more. Order 10, 2, 7, 5 from least to greatest. Smallest to biggest: 2, 5, 7, 10.

Key idea: Least to greatest means smallest number first.

Watch out for

• Do not let the sharp point trick you. The point touches the smaller number. The open mouth faces the bigger number.
• For whole numbers, more digits means bigger. 100 is bigger than 99.
• When you count back, remember the numbers get smaller, not bigger.

Recap

• Counting means saying numbers in order. Each one is one more.
• Skip counting jumps by 2s, 5s, or 10s to count fast.
• The signs > and < and = tell us which number is bigger, smaller, or the same.
• On a number line, bigger numbers are to the right.
• Least to greatest puts the smallest number first.

Sources

1. Khan Academy Kids, counting and comparing numbers activities (Khan Academy).
2. NCTM Illuminations, number sense lessons (National Council of Teachers of Mathematics).
3. PBS LearningMedia, early number and counting resources.
4. Bedtime Math, playful counting and comparing problems.

The big picture

Today we learn about place value. Place value is a clever trick that lets us write any number using just ten little symbols. It is fun because it makes big numbers easy!

Digits are our building blocks

A digit is one of the ten symbols we use to write numbers. The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Just like a few letters can make many words, these ten digits can make every number in the world.

Key idea: There are only ten digits, from 0 to 9.

Where a digit sits matters

Place value means how much a digit is worth because of where it sits. The same digit can mean different amounts.

Look at the digit 5. In 5, it means five. In 50, it means fifty. In 500, it means five hundred. Same digit, but its place changed its worth!

Key idea: A digit is worth more when it sits farther to the left.

The places

Starting from the right, the places are ones, then tens, then hundreds, then thousands.

• The ones place is on the far right. It counts single things.
• The tens place is next. Each one here is worth 10.
• The hundreds place comes next. Each one here is worth 100.

Each place is 10 times bigger than the place to its right. This is called our base-ten system. It is called base ten because we have ten fingers to count on!

Key idea: Each place to the left is 10 times bigger.

Reading a number by its places

Let us look at 256. We can put each digit in its place.

Hundreds	Tens	Ones
2	5	6

1. The 2 is in the hundreds place. It is worth 2 hundreds, which is 200.
2. The 5 is in the tens place. It is worth 5 tens, which is 50.
3. The 6 is in the ones place. It is worth 6 ones, which is 6.

Key idea: Read each digit by the place it sits in.

The mighty zero

Zero is a special digit. It is a placeholder. It holds a place open so the other digits stay where they belong.

Look at 408. The 0 in the tens place tells us there are no tens. But it keeps the 4 in the hundreds place and the 8 in the ones place. Without the zero, 408 would turn into 48. That is a very different number!

Key idea: Zero holds a place open so the other digits stay put.

Expanded form

Expanded form means writing a number as the sum of its place values. It shows the number stretched out.

For 256, the expanded form is 200 + 50 + 6.

Let us check. 200 + 50 is 250. Then 250 + 6 is 256. It matches! Checking your work is a smart habit.

Key idea: Expanded form breaks a number into its place value parts.

Bigger numbers and the comma

Here is a bigger number: 1,437.

• The 1 is in the thousands place. It is worth 1,000.
• The 4 is in the hundreds place. It is worth 400.
• The 3 is in the tens place. It is worth 30.
• The 7 is in the ones place. It is worth 7.

So 1,437 = 1,000 + 400 + 30 + 7. We use a small comma to help us read big numbers. We put it three digits from the right. It groups the thousands so the number is easier to read.

Key idea: A comma every three digits makes big numbers easy to read.

Watch out for

• Do not forget the zero. It keeps the other digits in the right place.
• The far right place is ones, not tens. Always start counting places from the right.
• In expanded form, use the place value amount, like 50, not just the digit 5.

Recap

• The ten digits are 0 through 9.
• Place value tells how much a digit is worth by where it sits.
• The places from the right are ones, tens, hundreds, thousands.
• Zero is a placeholder that holds a spot open.
• Expanded form shows a number as the sum of its parts, like 200 + 50 + 6.

Sources

1. Khan Academy, place value and expanded form lessons (Khan Academy).
2. NCTM Illuminations, base-ten and place value activities (National Council of Teachers of Mathematics).
3. PBS LearningMedia, place value learning resources.
4. Bedtime Math, number and place value problems.

The big picture

Today we learn to add bigger numbers. Adding puts groups together to find how many in all. It is useful when you count all your toys or all your treats!

What adding means

To add means to put groups together. If you have 2 apples and get 3 more, you now have 5 apples.

The answer to an adding problem is called the sum. The numbers we add are called addends. In 2 + 3 = 5, the addends are 2 and 3, and the sum is 5.

Key idea: Adding puts groups together to find the total.

Line up the places

To add big numbers, we stack them so the ones are under the ones and the tens are under the tens. This lineup is called a column.

Let us add 23 + 14. We line them up like this:

  2 3
+ 1 4
-----

1. Add the ones first: 3 + 4 = 7. Write 7 in the ones place.
2. Add the tens next: 2 + 1 = 3. Write 3 in the tens place.
3. The sum is 37.

Key idea: Always start adding from the ones place on the right.

When a column is 10 or more, we carry

Sometimes a column adds up to 10 or more. Then we carry. To carry means to move a group of ten to the next place on the left. This is also called to regroup.

Let us add 27 + 15.

  2 7
+ 1 5
-----

1. Add the ones: 7 + 5 = 12. That is more than 9! 12 is 1 ten and 2 ones.
2. Write the 2 in the ones place. Carry the 1 ten to the top of the tens place.
3. Add the tens: 1 (carried) + 2 + 1 = 4. Write 4 in the tens place.
4. The sum is 42.

Key idea: When a column reaches 10, carry the ten to the next place.

Adding three-digit numbers

The same steps work for bigger numbers. Let us add 148 + 126.

  1 4 8
+ 1 2 6
-------

1. Ones: 8 + 6 = 14. Write 4, carry 1 ten.
2. Tens: 1 (carried) + 4 + 2 = 7. Write 7.
3. Hundreds: 1 + 1 = 2. Write 2.
4. The sum is 274.

Key idea: Work one column at a time, right to left, and carry when needed.

Estimating to check

To estimate means to make a smart guess that is close to the real answer. It helps you check if your sum makes sense.

For 27 + 15, think 27 is close to 30 and 15 is close to 15. 30 + 15 is about 45. Our real answer 42 is close to 45, so it looks right!

Key idea: Estimate first to check that your answer is reasonable.

Watch out for

• Do not forget to add the carried ten. It is easy to leave it out.
• Keep your columns lined up neat, ones under ones.
• Always start on the right, in the ones place.

Recap

• Adding puts groups together. The answer is the sum.
• Line up ones under ones and tens under tens.
• Start adding from the ones place on the right.
• When a column reaches 10, carry the ten to the next place.
• Estimate to check your answer.

Sources

1. Khan Academy, addition and regrouping lessons (Khan Academy).
2. NCTM Illuminations, addition strategy activities (National Council of Teachers of Mathematics).
3. PBS LearningMedia, addition learning resources.
4. Bedtime Math, adding-in-real-life problems.

The big picture

Today we learn to subtract bigger numbers. Subtracting takes some away to find how many are left. It helps when you share treats or count what is gone!

What subtracting means

To subtract means to take some away. If you have 5 apples and eat 2, you have 3 apples left.

The answer to a subtraction is called the difference. In 5 - 2 = 3, the difference is 3.

Key idea: Subtracting takes away to find what is left.

Line up the places

Just like adding, we stack the numbers so ones are under ones and tens are under tens. The bigger number goes on top.

Let us subtract 48 - 23.

  4 8
- 2 3
-----

1. Subtract the ones first: 8 - 3 = 5. Write 5.
2. Subtract the tens: 4 - 2 = 2. Write 2.
3. The difference is 25.

Key idea: Start subtracting from the ones place on the right.

When the top digit is too small, we borrow

Sometimes the top digit is too small to take away from. Then we borrow. To borrow means to take one ten from the next place and give it to the ones. This is also called to regroup.

Let us subtract 42 - 15.

  4 2
- 1 5
-----

1. Look at the ones: 2 - 5. We cannot take 5 from 2. So we borrow.
2. Take 1 ten from the 4 tens. Now the tens place has 3. Give that ten to the ones, so 2 becomes 12.
3. Now subtract the ones: 12 - 5 = 7. Write 7.
4. Subtract the tens: 3 - 1 = 2. Write 2.
5. The difference is 27.

Key idea: If the top digit is too small, borrow a ten from the next place.

Borrowing with three-digit numbers

The same steps work for bigger numbers. Let us subtract 253 - 128.

  2 5 3
- 1 2 8
-------

1. Ones: 3 - 8. Too small, so borrow. Take 1 ten from the 5 tens (now 4 tens). The 3 becomes 13.
2. Now 13 - 8 = 5. Write 5.
3. Tens: 4 - 2 = 2. Write 2.
4. Hundreds: 2 - 1 = 1. Write 1.
5. The difference is 125.

Key idea: Borrow only from the place right next door, on the left.

Check by adding back

Adding and subtracting are opposites. We call them inverse operations. That means you can check by adding the answer back.

We found 42 - 15 = 27. To check, add 27 + 15. That is 42, our starting number. It matches, so we are right!

Key idea: To check subtraction, add the answer to the number you took away.

Watch out for

• Do not just take the small digit from the big one. If the top is smaller, you must borrow.
• When you borrow, remember the next place goes down by one.
• Keep your columns lined up neat.

Recap

• Subtracting takes away. The answer is the difference.
• Line up ones under ones and start on the right.
• If the top digit is too small, borrow a ten from the next place.
• Borrowing makes the next place go down by one.
• Check your answer by adding it back.

Sources

1. Khan Academy, subtraction and regrouping lessons (Khan Academy).
2. NCTM Illuminations, subtraction strategy activities (National Council of Teachers of Mathematics).
3. PBS LearningMedia, subtraction learning resources.
4. Bedtime Math, taking-away problems for kids.

The big picture

Today we learn about multiplication. Multiplication is a fast way to add equal groups. It is fun because it saves you lots of counting!

What multiplication means

To multiply means to add equal groups quickly. We use the sign x, which means times.

Say you have 3 bags. Each bag has 2 apples. That is 3 groups of 2. You can add: 2 + 2 + 2 = 6. Or you can multiply: 3 x 2 = 6. Both give 6 apples!

Key idea: Multiplication is adding the same number over and over.

The parts of a multiplication

The numbers we multiply are called factors. The answer is called the product.

In 3 x 2 = 6, the factors are 3 and 2, and the product is 6.

Key idea: Factors are the numbers you multiply. The product is the answer.

Multiplication is repeated addition

Repeated addition means adding the same number again and again. This is exactly what multiplication does.

4 x 5 means 5 added 4 times: 5 + 5 + 5 + 5 = 20. So 4 x 5 = 20.

You can also skip count to find it. Count by fives four times: 5, 10, 15, 20. Same answer!

Key idea: You can find a product by skip counting the groups.

Equal groups

Equal groups means every group has the same amount. Multiplication only works with equal groups.

3 groups of 4 crayons is 3 x 4 = 12 crayons. Each group must have 4. If the groups are different sizes, we cannot multiply them together like this.

Key idea: Multiplication needs groups that are all the same size.

Arrays help us see it

An array is a picture of rows and columns, like dots lined up in a neat box. Arrays make multiplication easy to see.

Picture 2 rows of dots with 4 dots in each row:

* * * *
* * * *

Count them: there are 8 dots. That shows 2 x 4 = 8. The rows are one factor and the dots in a row are the other factor.

Key idea: An array shows rows times columns.

The order does not matter

Multiplication has a special rule called commutative. It means you can switch the order and get the same product.

3 x 4 = 12 and 4 x 3 = 12. Both are 12! So if you know one, you know the other.

Key idea: You can flip the factors and the product stays the same.

The rules for 0 and 1

• Any number times 0 is 0. 5 x 0 = 0. Zero groups means nothing at all.
• Any number times 1 is itself. 5 x 1 = 5. One group of five is just five.

Key idea: Times 0 is always 0, and times 1 keeps the number the same.

Watch out for

• Groups must be equal to multiply. Do not multiply groups of different sizes.
• Times 0 is 0, not the number itself. 7 x 0 = 0.
• Remember 3 x 4 and 4 x 3 give the same answer.

Recap

• Multiplication is a fast way to add equal groups.
• The factors are what you multiply. The product is the answer.
• An array shows multiplication as rows and columns.
• You can flip the factors and get the same product.
• Times 0 is 0. Times 1 keeps the number the same.

Sources

1. Khan Academy, intro to multiplication lessons (Khan Academy).
2. NCTM Illuminations, equal groups and arrays activities (National Council of Teachers of Mathematics).
3. PBS LearningMedia, multiplication learning resources.
4. Bedtime Math, group-counting problems for kids.

The big picture

Today we practice multiplication facts for 2s, 5s, and 10s. Knowing these by heart makes math quick and easy. It feels great to just know the answer!

What a multiplication fact is

A multiplication fact is a small times problem you can remember, like 2 x 3 = 6. When you know a fact by heart, you do not have to count it out each time.

You already know some! 2 x 1 = 2 and 5 x 1 = 5 are facts. We will learn many more.

Key idea: A fact is a times problem you know right away.

The 2s: just double it

To multiply by 2 means to double a number. To double means to make two of the same and add them.

• 2 x 1 = 2
• 2 x 2 = 4
• 2 x 3 = 6
• 2 x 4 = 8
• 2 x 5 = 10

See the pattern? The answers are 2, 4, 6, 8, 10. Those are the counting-by-twos numbers!

Key idea: Times 2 just means double the number.

Even numbers

An even number is a number you can split into two equal groups with none left over. 2, 4, 6, 8, and 10 are even.

All the answers in the 2s are even. That is a handy clue!

Key idea: Every answer in the 2s times table is an even number.

The 5s: skip count by fives

To multiply by 5, just skip count by fives. Skip count means to jump by the same number each time.

• 5 x 1 = 5
• 5 x 2 = 10
• 5 x 3 = 15
• 5 x 4 = 20
• 5 x 5 = 25

Here is a fun pattern. The answers always end in 5 or 0: 5, 10, 15, 20, 25. Just like counting nickels!

Key idea: The 5s always end in a 5 or a 0.

The 10s: the easiest of all

To multiply by 10, just put a 0 at the end of the other number. That is it!

• 10 x 1 = 10
• 10 x 2 = 20
• 10 x 3 = 30
• 10 x 4 = 40
• 10 x 5 = 50

Key idea: Times 10 puts a zero at the end of the number.

Forgot a fact? Skip count to find it

If you forget a fact, you can always find it. Say you forget 5 x 4. Skip count by fives four times: 5, 10, 15, 20. The answer is 20!

Or use a pattern you know. A pattern is a rule that repeats. Since the 5s end in 5 or 0, you know 5 x 4 will end in 0.

Key idea: You can always skip count to figure out a fact you forget.

Watch out for

• Times 2 is double, not plus 2. 2 x 5 = 10, not 7.
• The 5s end in 5 or 0. If you get 5 x 3 = 16, that is wrong, because it must end in 5 or 0.
• Times 10 adds one zero, not two. 10 x 4 = 40, not 400.

Recap

• A fact is a times problem you know by heart.
• Times 2 means double the number, and the answers are even.
• The 5s end in 5 or 0, like counting nickels.
• Times 10 puts a zero at the end.
• If you forget, skip count to find the answer.

Sources

1. Khan Academy, multiplication facts and patterns lessons (Khan Academy).
2. NCTM Illuminations, skip counting and times table activities (National Council of Teachers of Mathematics).
3. PBS LearningMedia, multiplication fact practice resources.
4. Bedtime Math, playful times problems for kids.

The big picture

Today we learn about division. Division shares things into equal groups. It is useful when you split treats fairly with friends!

What division means

To divide means to share into equal groups. We use the sign for division to write it.

Say you have 6 cookies and 3 friends. You share them fairly. Each friend gets 2 cookies. We write 6 divided by 3 = 2.

Key idea: Division shares a group into equal parts.

The parts of a division

The answer to a division is called the quotient. It tells how many are in each equal share.

In 6 divided by 3 = 2, the quotient is 2. Each friend gets 2 cookies.

Key idea: The quotient is the answer to a division problem.

Making equal shares

Equal shares means every group gets the same amount, with none left over. That is what makes sharing fair.

Let us share 8 blocks between 2 kids. Give one to each, then one to each, until they are gone. Each kid ends up with 4. So 8 divided by 2 = 4.

Key idea: Fair sharing means each group gets an equal amount.

Division and multiplication are opposites

Division and multiplication are inverse operations. That means they undo each other, like opposites.

You know 3 x 2 = 6. So 6 divided by 3 = 2, and 6 divided by 2 = 3. They all use the same three numbers: 2, 3, and 6.

Key idea: Multiplying and dividing undo each other.

Fact families

A fact family is a group of facts that use the same numbers. The numbers 2, 3, and 6 make one fact family.

• 2 x 3 = 6
• 3 x 2 = 6
• 6 divided by 3 = 2
• 6 divided by 2 = 3

If you know one fact in the family, you can find the others!

Key idea: A fact family uses the same three numbers in four facts.

Turn division into a missing-factor question

Here is a great trick. To solve a division, ask a multiplication question with a missing factor. A missing factor is the number you are trying to find.

To find 12 divided by 4, ask: 4 times what equals 12? Count: 4, 8, 12. That is three fours. So the missing factor is 3, and 12 divided by 4 = 3.

Key idea: To divide, ask what number times the divisor makes the total.

Watch out for

• The shares must be equal. If groups have different amounts, the sharing is not fair.
• Order matters in division. 6 divided by 2 is not the same as 2 divided by 6.
• Use the fact family. If you know the times fact, the division is easy.

Recap

• Division shares a group into equal parts.
• The quotient is the answer.
• Division and multiplication are opposites.
• A fact family uses the same three numbers.
• To divide, ask a missing-factor multiplication question.

Sources

1. Khan Academy, intro to division and fact families lessons (Khan Academy).
2. NCTM Illuminations, sharing and equal groups activities (National Council of Teachers of Mathematics).
3. PBS LearningMedia, division learning resources.
4. Bedtime Math, fair-sharing problems for kids.

The big picture

Today we learn what a fraction is. A fraction is a part of a whole thing. It is fun because it helps us share pizza, pie, and candy fairly!

What a fraction means

A fraction is a way to name part of a whole. Think of one pizza cut into equal slices. One slice is a fraction of the whole pizza.

We write a fraction with two numbers, one on top and one on the bottom, like 1 over 2.

Key idea: A fraction names part of a whole.

The top and bottom numbers

The bottom number is the denominator. It tells how many equal parts the whole is cut into.

The top number is the numerator. It tells how many of those parts we have.

In the fraction 1 over 2, the denominator is 2 (the whole has 2 parts) and the numerator is 1 (we have 1 part).

Key idea: The bottom shows the total parts. The top shows how many we have.

One half

One half, written 1 over 2, means one of two equal parts. If you cut a sandwich into 2 equal pieces and take one, you have one half.

Picture a circle split down the middle into 2 equal parts. Color one part. That colored part is 1 over 2.

Key idea: One half is one of two equal parts.

The parts must be equal

Equal parts means all the pieces are the same size. Fractions only work when the parts are equal.

If you cut a cookie into one big piece and one tiny piece, those are not halves. Halves must be the same size.

Key idea: Fraction parts must all be the same size.

Reading some fractions

Let us read a few together.

• 1 over 3 means 1 part out of 3 equal parts. We say one third.
• 1 over 4 means 1 part out of 4 equal parts. We say one fourth, or one quarter.
• 3 over 4 means 3 parts out of 4 equal parts. We say three fourths.

The bigger the bottom number, the more pieces the whole is cut into, so each piece is smaller. One fourth is smaller than one half!

Key idea: A bigger bottom number means smaller pieces.

When a fraction makes one whole

When the top and bottom numbers are the same, the fraction equals one whole. A whole means the complete thing, all the parts together.

If a pie has 4 parts and you have all 4, that is 4 over 4, which is 1 whole pie. Yum!

Key idea: When the top equals the bottom, you have one whole.

Watch out for

• The parts must be equal. Different-sized pieces are not fractions.
• A bigger bottom number does not mean a bigger piece. It means smaller pieces.
• The top counts the parts you have. The bottom counts all the parts.

Recap

• A fraction names part of a whole.
• The denominator (bottom) shows the total equal parts.
• The numerator (top) shows how many parts you have.
• Parts must be equal in size.
• When the top equals the bottom, the fraction is one whole.

Sources

1. Khan Academy, intro to fractions lessons (Khan Academy).
2. NCTM Illuminations, fraction and equal parts activities (National Council of Teachers of Mathematics).
3. PBS LearningMedia, fraction learning resources.
4. Bedtime Math, sharing and fraction problems for kids.

The big picture

Today we add and subtract fractions that have the same bottom number. It is easy and fun. You just work with the top numbers!

Fractions with the same bottom

Like fractions are fractions that have the same denominator. That means the bottom numbers match.

1 over 4 and 2 over 4 are like fractions, because both have a 4 on the bottom. They both come from a whole cut into 4 equal parts.

Key idea: Like fractions have the same bottom number.

Adding: add the tops, keep the bottom

To add like fractions, we add the tops and keep the bottom the same. The bottom does not change because the pieces are still the same size.

Let us add 1 over 4 plus 2 over 4.

1. Add the top numbers: 1 + 2 = 3.
2. Keep the bottom the same: 4.
3. The answer is 3 over 4.

Picture a pizza cut into 4 slices. You have 1 slice, then get 2 more. Now you have 3 slices out of 4, which is 3 over 4.

Key idea: To add, add the tops and keep the bottom the same.

Why the bottom stays the same

The bottom tells the size of the pieces. When you add fourths to fourths, the pieces are still fourths. So the bottom stays 4.

Think of it like adding apples. 1 apple plus 2 apples is 3 apples. The word apple does not change. Here, fourths do not change either.

Key idea: The bottom stays the same because the pieces stay the same size.

Subtracting: take away the tops

To subtract like fractions, we take away the top numbers and keep the bottom the same.

Let us do 3 over 5 minus 1 over 5.

1. Subtract the tops: 3 - 1 = 2.
2. Keep the bottom the same: 5.
3. The answer is 2 over 5.

You had 3 pieces out of 5 and ate 1. Now you have 2 out of 5, which is 2 over 5.

Key idea: To subtract, take away the tops and keep the bottom.

When the answer is one whole

Remember, when the top and bottom match, you have one whole. So sometimes adding fractions makes a whole.

Let us add 1 over 3 plus 2 over 3. Add the tops: 1 + 2 = 3. Keep the bottom: 3. The answer is 3 over 3, which is 1 whole!

That makes sense. If a pie has 3 parts and you gather all 3, you have the whole pie.

Key idea: If the tops add up to the bottom, the answer is one whole.

One more example

Let us try 2 over 6 plus 3 over 6.

1. Add the tops: 2 + 3 = 5.
2. Keep the bottom: 6.
3. The answer is 5 over 6.

Key idea: Same steps work for any like fractions.

Watch out for

• Do not add the bottom numbers. Only add the tops. 1 over 4 plus 2 over 4 is 3 over 4, not 3 over 8.
• These easy steps only work when the bottoms are the same.
• When the top equals the bottom, remember that is one whole.

Recap

• Like fractions have the same bottom number.
• To add, add the tops and keep the bottom the same.
• To subtract, take away the tops and keep the bottom the same.
• The bottom stays the same because the pieces stay the same size.
• When the top equals the bottom, the answer is one whole.

Sources

1. Khan Academy, adding and subtracting fractions with like denominators (Khan Academy).
2. NCTM Illuminations, fraction operations activities (National Council of Teachers of Mathematics).
3. PBS LearningMedia, fraction learning resources.
4. Bedtime Math, fraction sharing problems for kids.

The big picture

Today we learn about decimals. A decimal is another way to show a part of a whole. It is handy for money and for measuring!

What a decimal is

A decimal is a number that shows parts smaller than one, using a little dot. That dot is called the decimal point.

The number 0.5 is a decimal. The dot separates the whole numbers on the left from the parts on the right.

Key idea: A decimal uses a dot to show parts of a whole.

The tenths place

The first place right after the dot is the tenths place. It means the whole is cut into 10 equal parts.

So 0.1 is one tenth, the same as the fraction 1 over 10. And 0.3 means 3 tenths, or 3 over 10.

Picture a chocolate bar with 10 equal squares. One square is 0.1 of the bar.

Key idea: The first place after the dot counts tenths.

Connecting decimals and fractions

Decimals and fractions are two ways to name the same part. Look how they match.

• 0.1 is the same as 1 over 10.
• 0.5 is the same as 5 over 10, which is one half.
• 0.7 is the same as 7 over 10.

Key idea: A decimal in tenths matches a fraction over 10.

The hundredths place

The second place after the dot is the hundredths place. It means the whole is cut into 100 equal parts.

So 0.01 is one hundredth, the same as 1 over 100. And 0.25 means 25 hundredths, or 25 over 100. That is like 25 cents out of a whole dollar!

Key idea: The second place after the dot counts hundredths.

Comparing decimals

To compare decimals, line up the decimal points and look at the tenths place first.

Which is bigger, 0.7 or 0.3? Both have a 0 for the whole. Look at the tenths: 7 is bigger than 3. So 0.7 is bigger.

Here is another. Compare 0.6 and 0.6. The tenths are the same, so they are equal.

Key idea: To compare, line up the dots and check the tenths first.

A worked example with money

Money uses decimals every day. Half of a dollar is 0.50 dollars, or 50 cents. A quarter is 0.25 dollars, or 25 cents.

Which is more, 0.50 or 0.25? Look at the tenths: 5 is bigger than 2. So 0.50 is more. Fifty cents beats twenty-five cents!

Key idea: Decimals help us read and compare money.

Watch out for

• Line up the decimal points before you compare. Do not just count digits.
• The first place after the dot is tenths, not hundredths.
• 0.5 and 0.50 are the same amount. Extra zeros at the end do not change the value.

Recap

• A decimal uses a dot to show parts of a whole.
• The first place after the dot is tenths.
• The second place after the dot is hundredths.
• Decimals in tenths match fractions over 10.
• To compare, line up the dots and check the tenths first.

Sources

1. Khan Academy, intro to decimals and place value lessons (Khan Academy).
2. NCTM Illuminations, decimals and fractions activities (National Council of Teachers of Mathematics).
3. PBS LearningMedia, decimal learning resources.
4. Bedtime Math, money and decimal problems for kids.

The big picture

Today we learn to count money. We will use pennies, nickels, dimes, and quarters. It is fun because you can buy your favorite treats!

Cents and dollars

A cent is the smallest unit of money we count. A dollar is worth 100 cents. So it takes 100 cents to make one dollar.

We use a cent sign or a dollar sign to write money. Ten cents can be written as 10 cents or as 0.10 dollars.

Key idea: One dollar is the same as 100 cents.

Meet the coins

Here are the four coins and how much each is worth.

• A penny is worth 1 cent.
• A nickel is worth 5 cents.
• A dime is worth 10 cents.
• A quarter is worth 25 cents.

A dime is small but it is worth more than a big nickel. Size does not tell you the value!

Key idea: Penny is 1, nickel is 5, dime is 10, quarter is 25.

Counting a group of coins

To count coins, start with the biggest ones and add as you go. A smart trick is to skip count.

Say you have 3 dimes. Skip count by tens: 10, 20, 30. So 3 dimes are 30 cents.

Say you have 4 nickels. Skip count by fives: 5, 10, 15, 20. So 4 nickels are 20 cents.

Key idea: Skip count coins that are the same to add them fast.

Counting mixed coins

What if you have different coins? Add them one group at a time. Let us count 1 quarter, 2 dimes, and 1 penny.

1. Start with the quarter: 25 cents.
2. Add 2 dimes. That is 20 more. 25 + 20 = 45 cents.
3. Add 1 penny. That is 1 more. 45 + 1 = 46 cents.
4. The total is 46 cents.

Key idea: Add the biggest coins first, then the smaller ones.

Making the same amount different ways

You can make the same amount with different coins. Here are three ways to make 25 cents.

• 1 quarter is 25 cents.
• 2 dimes and 1 nickel is 10 + 10 + 5 = 25 cents.
• 5 nickels is 5 + 5 + 5 + 5 + 5 = 25 cents.

Key idea: Different coins can add up to the same amount.

Writing money with a dollar sign

When we have a dollar or more, we use a dollar sign and a decimal point. The numbers after the dot are the cents.

One dollar and fifty cents is written as 1.50 dollars. The 1 is the dollar and the 50 is the cents. Two quarters make that 50 cents!

Key idea: After the dot in a money amount comes the cents.

Watch out for

• A dime is worth more than a nickel, even though it is smaller.
• Skip count by the right number: fives for nickels, tens for dimes.
• Remember 100 cents make a dollar, not 10 cents.

Recap

• One dollar is 100 cents.
• Penny is 1, nickel is 5, dime is 10, quarter is 25 cents.
• Skip count coins that are the same to add fast.
• Add the biggest coins first, then the smaller ones.
• Write money with a dollar sign and a dot before the cents.

Sources

1. Khan Academy, counting money and coins lessons (Khan Academy).
2. NCTM Illuminations, money and place value activities (National Council of Teachers of Mathematics).
3. PBS LearningMedia, counting money resources.
4. Bedtime Math, money problems for kids.

The big picture

Today we learn to measure length and to tell time. Measuring tells us how long or how far. Telling time tells us when. Both help us every day!

What measuring means

To measure means to find out how big something is using a tool. We measure length, which is how long something is from one end to the other.

We use a ruler or a tape to measure length. Think of measuring how long your pencil is.

Key idea: Measuring tells us how long or how far something is.

Inches and centimeters

We measure length with units. An inch is one unit. A centimeter is another, smaller unit. Both are marked on a ruler.

A centimeter is smaller than an inch. So a crayon might be about 3 inches long, or about 8 centimeters long. Same crayon, different units!

Key idea: Inches and centimeters are units we use to measure length.

Start at zero

Here is a super important rule. Always line up the end of the object with the 0 on the ruler, not the 1.

If your pencil starts at 0 and ends at 5, it is 5 units long. If you start at 1 by mistake, your answer will be wrong.

Key idea: Always start measuring from the 0 mark.

Feet are made of inches

When something is long, we can use a foot. One foot is made of 12 inches.

So if a ribbon is 12 inches long, that is the same as 1 foot. If it is 24 inches, that is 2 feet, because 12 + 12 = 24.

Key idea: There are 12 inches in one foot.

Telling time on a clock

A clock with hands has two pointers. The short hand points to the hour. The long hand points to the minutes.

When the long hand points straight up to the 12, it is an even hour. If the short hand points to 3 and the long hand points to 12, the time is 3 oclock.

Key idea: The short hand shows the hour and the long hand shows the minutes.

Half past the hour

An hour is made of 60 minutes. Half of 60 is 30. So halfway through an hour is 30 minutes.

When the long hand points straight down to the 6, it is half past the hour. That means 30 minutes have gone by.

If the short hand is between 4 and 5, and the long hand is on the 6, the time is half past 4, which we can say as 4:30.

Key idea: When the long hand points to 6, it is half past the hour.

Watch out for

• Start your ruler at 0, not at 1.
• The short hand is the hour, the long hand is the minutes. Do not mix them up.
• One foot is 12 inches, and one hour is 60 minutes.

Recap

• Measuring tells how long something is.
• We measure length with inches and centimeters.
• Always start measuring from the 0 mark.
• There are 12 inches in a foot and 60 minutes in an hour.
• The short hand shows the hour, and half past means the long hand points to 6.

Sources

1. Khan Academy, measuring length and telling time lessons (Khan Academy).
2. NCTM Illuminations, measurement and time activities (National Council of Teachers of Mathematics).
3. PBS LearningMedia, measurement and clock resources.
4. Bedtime Math, measuring and time problems for kids.

The big picture

Today we learn about shapes. We will name them, find the distance around them, and find how much space they cover. Shapes are all around you!

What a flat shape is

A 2D shape is a flat shape you can draw on paper. 2D means it has two directions, across and up and down, but no thickness.

A drawing of a square on paper is a 2D shape. A block you can hold is not flat, so it is different.

Key idea: A 2D shape is flat, like a shape drawn on paper.

Sides and corners

A side is one straight edge of a shape. A corner is where two sides meet. A corner is also called a vertex.

Here are some shapes and their sides and corners.

• A triangle has 3 sides and 3 corners.
• A square has 4 sides that are all the same length and 4 corners.
• A rectangle has 4 sides and 4 corners, but the sides are not all equal.

Key idea: A side is a straight edge, and a corner is where two sides meet.

Perimeter is the distance around

The perimeter is the distance all the way around a shape. To find it, add up the lengths of all the sides.

Picture a square garden where each side is 3 feet long. Add all 4 sides: 3 + 3 + 3 + 3 = 12. The perimeter is 12 feet.

Think of walking all the way around the edge of the garden. The distance you walk is the perimeter.

Key idea: Perimeter is the distance around, found by adding all the sides.

A perimeter example with a rectangle

Let us find the perimeter of a rectangle. It is 5 across the top and bottom, and 2 up each side.

1. Top is 5 and bottom is 5.
2. Left side is 2 and right side is 2.
3. Add them all: 5 + 5 + 2 + 2 = 14.
4. The perimeter is 14 units.

Key idea: Add every side, even the ones that are the same.

Area is the space inside

The area is how much space is inside a shape. We measure area in square units, which are little squares that fill the shape.

For a rectangle, we find area by multiplying the length by the width. That is faster than counting every little square.

Key idea: Area is the space inside a shape.

An area example

Let us find the area of a rectangle that is 4 long and 3 wide.

1. Multiply the length by the width: 4 x 3.
2. 4 x 3 = 12.
3. The area is 12 square units.

You can check by picturing 3 rows of 4 little squares. Count them and you get 12 squares. It matches!

Key idea: Area of a rectangle is length times width.

Perimeter and area are different

Perimeter is the distance around the edge. Area is the space inside. They are not the same!

Perimeter is like the fence around a yard. Area is like the grass that fills the yard.

Key idea: Perimeter goes around the edge, area fills the inside.

Watch out for

• For perimeter, add every single side. Do not skip any.
• For area of a rectangle, multiply. Do not add.
• Do not mix up perimeter (around) and area (inside).

Recap

• A 2D shape is flat, with sides and corners.
• A side is a straight edge, and a corner is where two sides meet.
• Perimeter is the distance around, found by adding all sides.
• Area is the space inside, measured in square units.
• Area of a rectangle is length times width.

Sources

1. Khan Academy, shapes, perimeter, and area lessons (Khan Academy).
2. NCTM Illuminations, geometry and measurement activities (National Council of Teachers of Mathematics).
3. PBS LearningMedia, shapes and area resources.
4. Bedtime Math, shape and measuring problems for kids.

The big picture

Today we solve word problems and read graphs. A word problem is a math story. Graphs are pictures of numbers. Both help us use math in real life!

What a word problem is

A word problem is a math question told as a little story. Your job is to find the numbers and figure out what to do with them.

Here is one. Sam has 3 apples. Mom gives him 2 more. How many apples does Sam have now?

Key idea: A word problem is a math story you solve step by step.

Choosing the right operation

An operation is a math action like adding, subtracting, multiplying, or dividing. To solve a word problem, you pick the right one.

In the apple story, Sam gets more apples, so we add. 3 + 2 = 5. Sam has 5 apples.

Key idea: First decide which operation the story needs.

Clue words help you choose

Clue words are words in the story that hint at which operation to use. They are like little helpers.

• Words like in all, more, and together often mean add.
• Words like left, fewer, and take away often mean subtract.
• Words like each and groups of often mean multiply.
• Words like share and split evenly often mean divide.

Key idea: Look for clue words to pick the operation.

A subtraction word problem

Let us try one. Mia has 8 stickers. She gives 3 to her friend. How many are left?

1. Find the numbers: 8 and 3.
2. See the clue word: left means subtract.
3. Subtract: 8 - 3 = 5.
4. Mia has 5 stickers left.

Key idea: Read carefully, find the numbers, then do the math.

A multiplication word problem

Here is one more. There are 3 baskets. Each basket has 4 eggs. How many eggs in all?

1. Find the numbers: 3 baskets and 4 eggs each.
2. The clue word each with equal groups means multiply.
3. Multiply: 3 x 4 = 12.
4. There are 12 eggs in all.

Key idea: Equal groups in a story often means multiply.

Reading a bar graph

A bar graph uses bars to show how many. A taller bar means more. A shorter bar means less.

Say a bar graph shows fruit picked. The apple bar reaches 5 and the pear bar reaches 3. That means 5 apples and 3 pears were picked. To compare, we see 5 is more than 3, so more apples were picked.

Key idea: In a bar graph, a taller bar means more.

Reading a picture graph

A picture graph uses little pictures to show how many. Each picture stands for one thing. You count the pictures to find the number.

If a picture graph shows 4 star pictures for Monday and 2 star pictures for Tuesday, then 4 stars were earned Monday and 2 on Tuesday. Monday had more, because 4 is more than 2.

Key idea: In a picture graph, count the pictures to find the number.

Watch out for

• Read the whole story before you choose an operation.
• Clue words are helpers, but always check that your answer makes sense.
• On a graph, look at the numbers on the side to read each bar correctly.

Recap

• A word problem is a math story you solve step by step.
• Choose the right operation: add, subtract, multiply, or divide.
• Clue words hint at which operation to use.
• A bar graph shows how many with bars, taller means more.
• A picture graph shows how many with pictures you count.

Sources

1. Khan Academy, word problems and reading graphs lessons (Khan Academy).
2. NCTM Illuminations, problem solving and data activities (National Council of Teachers of Mathematics).
3. PBS LearningMedia, word problem and graph resources.
4. Bedtime Math, story-based math problems for kids.