Mathematics: Number Sense and Operations – Grade 1

Counting is like following a path with numbers, and there are many different paths you can take! When you count by ones, you're taking small steps: 1, 2, 3, 4, 5... But did you know you can also take bigger steps by skip counting? Let's explore these exciting counting adventures! 🚶‍♀️🚶‍♂️

When you count forward, you're adding one more each time. Starting from any number, like 15, you can count: 16, 17, 18, 19, 20... It's like climbing stairs - each step takes you one higher! 📈

When you count backward, you're taking one away each time. From 20, you could count back: 19, 18, 17, 16, 15... This is like walking down stairs - each step takes you one lower! 📉

A 120 chart is a special tool that shows all the numbers from 1 to 120 in rows. When you count forward on this chart, you move to the right until you reach the end of a row, then you start on the left side of the next row down. When you count backward, you do the opposite - move left, and when you reach the beginning of a row, jump to the right side of the row above! This helps you see how numbers connect to each other.

Skip counting by 2s means you count every other number: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. It's like hopping on every other stepping stone across a stream! 🪨 This pattern helps you practice counting pairs of things, like your two hands, two feet, or two eyes.

When you skip count by 2s, notice the pattern: all the numbers end in 0, 2, 4, 6, or 8. These are called even numbers. The numbers you skip (1, 3, 5, 7, 9) are called odd numbers and they all end in 1, 3, 5, 7, or 9.

Skip counting by 5s means counting: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100. This is like counting by the fingers on one hand! ✋ Every number when you skip count by 5s ends in either 0 or 5.

Skip counting by 5s is super helpful when counting nickels (which are worth 5 cents each) or when counting groups of 5 objects. If you have 4 groups of 5 stickers each, you can skip count: 5, 10, 15, 20 stickers total! 🌟

Counting forward is actually the same as adding! When you count from 8 to 12 (8, 9, 10, 11, 12), you're really doing: $8 + 1 = 9$, $9 + 1 = 10$, $10 + 1 = 11$, $11 + 1 = 12$. This is called counting on, and it's a strategy for adding numbers.

Counting backward is the same as subtracting! When you count back from 15 to 11 (15, 14, 13, 12, 11), you're really doing: $15 - 1 = 14$, $14 - 1 = 13$, $13 - 1 = 12$, $12 - 1 = 11$. This is called counting back, and it's a strategy for subtracting numbers.

Skip counting helps you understand multiplication later! When you skip count by 2s five times (2, 4, 6, 8, 10), you're actually finding $5 \times 2 = 10$. Cool, right? 😎

The 120 chart is like a map for numbers. When you want to find what comes after 47, you can find 47 on the chart and look to the right to see 48. When you want to count by 10s from 23, you can start at 23 and move straight down to see 33, 43, 53, and so on.

Number lines are another great tool. They show numbers in order from left to right, like houses on a street. Smaller numbers are on the left, and bigger numbers are on the right. When you count forward, you move to the right. When you count backward, you move to the left.

The more you practice these counting patterns, the easier they become! Start with small numbers and work your way up. Try counting forward from 67 to 75, or backward from 89 to 81. Practice skip counting while you're walking - take two steps and say '2', two more steps and say '4', and keep going!

Remember, counting is the foundation for all the amazing math you'll learn. Every time you count, you're building your number sense and getting ready for more exciting math adventures! 🎉

Numbers can be written in different ways, just like you can wear different outfits for different occasions! 👗👔 Each way of writing a number shows something special about it. Let's explore how numbers can dress up in standard form, expanded form, and word form!

Every number from 0 to 100 can be written in three different ways:

• Standard form: The regular way we write numbers (like 47)
• Expanded form: Breaking the number into its place value parts (like $40 + 7$)
• Word form: Writing the number using words (like forty-seven)

Think of it like describing the same person in different ways - you might say "my friend," "the girl with brown hair," or "Sarah." They're all talking about the same person, just like these three forms are all talking about the same number!

Standard form is the way you normally see numbers written. When you see 25, 63, or 89, those are all in standard form. It's the quickest and most common way to write numbers.

For numbers 0 to 9, standard form is just one digit: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

For numbers 10 to 100, standard form uses two digits. The first digit tells you how many groups of ten you have, and the second digit tells you how many extra ones you have.

Place value is like a number's address - it tells you where each digit lives and what its job is! In a two-digit number, there are two places:

• The tens place (on the left)
• The ones place (on the right)

In the number 73:

• The 7 is in the tens place, so it represents 7 groups of ten (which equals 70)
• The 3 is in the ones place, so it represents 3 individual ones

This is super important! The digit 3 means something very different depending on where it lives. In the number 32, the 3 represents 30 (three tens). But in the number 23, the 3 represents just 3 (three ones). It's the same digit, but its location changes its value! 🏠

Expanded form breaks a number apart to show the value of each digit. It's like taking apart a LEGO creation to see all the individual pieces! 🧱

For the number 56:

• Standard form: 56
• Expanded form: $50 + 6$

The expanded form shows that 56 is really 50 (five tens) plus 6 (six ones). Here are more examples:

• 34 = $30 + 4$
• 81 = $80 + 1$
• 60 = $60 + 0$
• 7 = $0 + 7$ (though we usually just write 7 for single digits)

Base ten blocks help you see place value with your eyes! There are two types:

• Unit cubes (ones): Small cubes that represent 1
• Ten rods (tens): Long pieces made of 10 unit cubes stuck together

When you build the number 43 with base ten blocks, you use:

• 4 ten rods (representing 40)
• 3 unit cubes (representing 3)

This helps you see that 43 really is $40 + 3$! You can touch and count the blocks to understand expanded form better.

Word form writes numbers using words instead of digits. This is how you would say the number out loud:

• 15 → fifteen
• 62 → sixty-two
• 89 → eighty-nine
• 100 → one hundred

Some numbers have tricky spellings! Here are the most important ones to remember:

• 11 → eleven (not "ten-one")
• 12 → twelve (not "ten-two")
• 13 → thirteen, 14 → fourteen, 15 → fifteen, 16 → sixteen, 17 → seventeen, 18 → eighteen, 19 → nineteen
• 20 → twenty, 30 → thirty, 40 → forty, 50 → fifty, 60 → sixty, 70 → seventy, 80 → eighty, 90 → ninety

Sometimes students mix up the order of digits when writing numbers. If you hear "fifteen," you might think to write 51 because you hear "five" first. But fifteen is actually 15! 🤔

Here's a helpful trick: fifteen means "ten plus five," so it's 1 ten and 5 ones = 15. Fifty-one means "fifty plus one," so it's 5 tens and 1 one = 51.

Another common mistake is writing expanded form incorrectly. For 68, some students write $6 + 8$ instead of $60 + 8$. Remember, the 6 is in the tens place, so it represents 60 (six tens), not just 6!

All three forms are just different ways to show the same number:

• Standard form is the fastest to write and read
• Expanded form helps you understand place value
• Word form is how you say the number out loud

Practice switching between all three forms! If someone says "thirty-four," can you write it in standard form (34) and expanded form ($30 + 4$)? If you see 91, can you say it in word form (ninety-one) and write it in expanded form ($90 + 1$)? 🎯

Understanding different number forms helps in everyday life! When you:

• Write your age (standard form)
• Count money: $\$47$ is 4 tens ($\$40$) plus 7 ones ($\$7$) (expanded form)
• Tell someone your address number (word form)
• Fill out forms that ask for your house number

Mastering these three forms makes you a number expert who can read, write, and understand numbers in any situation! 🌟

Numbers are like puzzles that can be taken apart and put back together in many different ways! 🧩 When you compose a number, you're putting pieces together to make it. When you decompose a number, you're taking it apart to see what pieces it's made of. This is one of the most important skills in mathematics because it helps you understand how numbers really work!

Composing means putting parts together to make a whole number. For example, if you have 3 groups of ten and 7 individual ones, you can compose them to make 37.

Decomposing means breaking a whole number into smaller parts. For example, you can decompose 37 into 3 tens and 7 ones, or into 2 tens and 17 ones, or even into 37 ones!

Think of it like building with blocks. You can build the same tower using different combinations of blocks - maybe 2 big blocks and 3 small blocks, or 1 big block and 8 small blocks. The tower is the same height, but you used different pieces! 🏗️

Here's where it gets really exciting: the same number can be made in many different ways! Let's look at the number 45:

• $4 \text{ tens} + 5 \text{ ones} = 45$
• $3 \text{ tens} + 15 \text{ ones} = 45$
• $2 \text{ tens} + 25 \text{ ones} = 45$
• $1 \text{ ten} + 35 \text{ ones} = 45$
• $0 \text{ tens} + 45 \text{ ones} = 45$

All of these equal 45! This is because 10 ones can always be traded for 1 ten, and 1 ten can always be traded for 10 ones. It's like trading 10 pennies for 1 dime - you have the same amount of money! 💰

Base ten blocks make this concept easy to see and touch:

• Unit cubes represent ones
• Ten rods represent tens (they're made of 10 unit cubes stuck together)

To show 52 with base ten blocks:

• Standard way: 5 ten rods + 2 unit cubes
• Different way: 4 ten rods + 12 unit cubes (trade 1 ten rod for 10 unit cubes)
• Another way: 3 ten rods + 22 unit cubes
• Yet another way: 52 unit cubes

When you physically trade 1 ten rod for 10 unit cubes, you can see that you still have the same total amount! 👀

When you decompose numbers, you can write equations to show your work. An equation uses the equal sign ($=$) to show that two things have the same value.

For the number 73:

• $73 = 7 \text{ tens} + 3 \text{ ones}$
• $73 = 70 + 3$
• $73 = 6 \text{ tens} + 13 \text{ ones}$
• $73 = 60 + 13$

The equal sign means "same as" or "has the same value as." So when you write $70 + 3 = 73$, you're saying that 70 plus 3 has the same value as 73. It's like saying "This pile of blocks" = "That pile of blocks" because they both have the same total! ⚖️

Decomposing numbers helps you with:

Addition: If you need to add $47 + 8$, you can think: "I have 47, which is 4 tens and 7 ones. Adding 8 more ones gives me 4 tens and 15 ones. But 15 ones = 1 ten and 5 ones, so I have 5 tens and 5 ones = 55!"

Subtraction: If you need to subtract $52 - 7$, you can think: "I have 52, which is 5 tens and 2 ones. I can't take away 7 ones from just 2 ones, so I'll trade 1 ten for 10 ones. Now I have 4 tens and 12 ones. I can take away 7 ones, leaving me with 4 tens and 5 ones = 45!"

Understanding money: $\$67$ can be thought of as 6 ten-dollar bills and 7 one-dollar bills, or 5 ten-dollar bills and 17 one-dollar bills, or even 67 one-dollar bills! 💵

A place value chart has columns for tens and ones, helping you organize your thinking:

This shows 46. But you could also show 46 as:

This still represents 46, just decomposed differently! The chart helps you see that the same number can live in different arrangements.

This flexible thinking about numbers shows up everywhere:

At the store: If you buy something that costs $\$38$ and you pay with a $\$50$ bill, the cashier might think: "$\$50$ is 5 tens. $\$38$ is 3 tens and 8 ones. I need to give change of $\$12$, which is 1 ten and 2 ones."

Counting collections: If you have 83 stickers, you could organize them as 8 groups of 10 plus 3 extra, or 7 groups of 10 plus 13 extra, depending on how you want to count or share them! 🌟

Building things: If you're building a tower and need 29 blocks, you could grab 2 sets of 10 blocks plus 9 individual blocks, or 1 set of 10 blocks plus 19 individual blocks.

Try decomposing different numbers in multiple ways. Start with smaller numbers like 23 or 34, then work up to larger numbers like 67 or 89. Use base ten blocks, draw pictures, or just think about it in your head. The more ways you can think about the same number, the stronger your number sense becomes! 💪

Remember: there's no single "right" way to decompose a number. The best way depends on what you're trying to do with that number. Flexible thinking about numbers will help you become a mathematical problem-solver! 🚀

Numbers have their own special order, just like kids lining up by height or books arranged on a shelf! 📚 Learning to put numbers in order and compare them helps you understand which numbers are bigger, smaller, or the same. You'll also learn to place numbers on a number line, which is like giving each number its own special spot on a math street! 🏠

Numbers follow a special pattern where each number is exactly one more than the number before it: 0, 1, 2, 3, 4, 5... This continues all the way up to 100 and beyond! When we order numbers, we arrange them from smallest to largest or from largest to smallest.

Ascending order means arranging numbers from smallest to largest, like climbing stairs: 23, 45, 67, 89.

Descending order means arranging numbers from largest to smallest, like walking down stairs: 89, 67, 45, 23.

Think of numbers like people of different heights standing in line. The shortest person (smallest number) would be first in ascending order, and the tallest person (largest number) would be first in descending order! 👥

A number line is like a street where each number has its own address! Numbers are arranged from left to right, with smaller numbers on the left and larger numbers on the right. The distance between each number is exactly the same, like houses evenly spaced on a street.

When you plot a number on a number line, you find its exact spot and mark it. Here's how:

1. Find the number on the line
2. Make a mark (like a dot or small line) at that spot
3. Sometimes write the number above or below the mark

For example, on a number line from 0 to 20, the number 15 would be plotted three-quarters of the way from 0 to 20. The number 5 would be plotted one-quarter of the way from 0 to 20.

When we compare numbers, we look at them to see which one is bigger, smaller, or if they're the same. We use special words to describe these relationships:

• Greater than: One number is bigger than another (78 is greater than 45)
• Less than: One number is smaller than another (32 is less than 56)
• Equal to: Two numbers are exactly the same (25 is equal to 25)
• Between: A number falls in the middle of two other numbers (50 is between 40 and 60)

You can remember this by thinking about your age. If you're 7 years old, you're greater than someone who is 5, less than someone who is 9, and equal to another 7-year-old! 🎂

Comparison symbols are special signs that show the relationship between numbers:

• Greater than ($>$): The opening points to the larger number
• Less than ($<$): The opening points to the larger number
• Equal to ($=$): Both sides have the same value

Here's a helpful trick: the symbol always "eats" the bigger number! The big opening is like a mouth that wants to eat the larger number. 🍽️

Examples:

• $67 > 43$ (67 is greater than 43)
• $29 < 85$ (29 is less than 85)
• $50 = 50$ (50 equals 50)

When comparing two-digit numbers, start by looking at the tens place first:

• If the tens digits are different, the number with the larger tens digit is greater
• If the tens digits are the same, look at the ones place
• The number with the larger ones digit is greater

For example, comparing 47 and 52:

• Look at tens place: 4 tens vs. 5 tens
• Since 5 > 4, we know that 52 > 47

For another example, comparing 73 and 78:

• Look at tens place: 7 tens vs. 7 tens (they're the same!)
• Look at ones place: 3 ones vs. 8 ones
• Since 8 > 3, we know that 78 > 73

Base ten blocks help you see why one number is greater than another! When you build 46 with base ten blocks (4 ten rods + 6 unit cubes) and compare it to 39 (3 ten rods + 9 unit cubes), you can see that 46 has more ten rods, making it the larger number.

This visual comparison helps when numbers are close together. Even though 39 has more unit cubes (9 vs. 6), the extra ten rod in 46 makes it larger overall. One ten rod equals 10 unit cubes, so that extra ten rod is worth more than the 3 extra unit cubes! 🔍

On a number line, numbers that are further to the right are always greater than numbers to the left. This makes comparison easy:

• If you plot 34 and 57 on the same number line, 57 will be to the right of 34
• This shows you visually that 57 > 34
• The distance between them on the number line shows how much greater 57 is than 34

When you need to order three or more numbers, use the same comparing strategies:

1. Compare pairs of numbers
2. Arrange them from smallest to largest (or largest to smallest)
3. Double-check your work by making sure each number is smaller than the next one

For example, to order 45, 23, 67, and 51:

• First, identify the smallest: 23
• Next smallest: 45
• Next smallest: 51
• Largest: 67
• Final order: 23, 45, 51, 67

Comparing and ordering numbers happens everywhere in real life:

Sports scores: Which team won? The team with the higher score! If Team A scored 42 points and Team B scored 38 points, then 42 > 38, so Team A won! ⚽

Shopping: Which item costs more? If a toy costs $\$25$ and a book costs $\$18$, then $25 > 18$, so the toy costs more.

Measuring height: Who is taller? If you are 48 inches tall and your friend is 52 inches tall, then 52 > 48, so your friend is taller! 📏

Temperature: Which day was warmer? If Monday was 73°F and Tuesday was 68°F, then 73 > 68, so Monday was warmer.

The more you practice plotting, ordering, and comparing numbers, the stronger your number sense becomes. Number sense is like having a "feeling" for numbers - you start to automatically know that 67 is much closer to 70 than to 60, or that 85 is a pretty big number when you're only counting to 100.

This number sense will help you estimate, check if your answers make sense, and solve problems more efficiently. It's like developing a superpower for understanding numbers! 🦸‍♀️🦸‍♂️

Learning your addition and subtraction facts is like learning to tie your shoes or ride a bike - once you practice enough, you can do it without even thinking about it! 🚲 This special skill is called automaticity, and it means you can recall math facts quickly and accurately. Let's explore how these number facts work together and why they're so important!

Addition facts to 10 are all the ways to add two numbers that give you a sum of 10 or less. Here are all the addition facts with sums up to 10:

Sum of 0: $0 + 0 = 0$ Sum of 1: $0 + 1 = 1$, $1 + 0 = 1$ Sum of 2: $0 + 2 = 2$, $1 + 1 = 2$, $2 + 0 = 2$ Sum of 3: $0 + 3 = 3$, $1 + 2 = 3$, $2 + 1 = 3$, $3 + 0 = 3$ Sum of 4: $0 + 4 = 4$, $1 + 3 = 4$, $2 + 2 = 4$, $3 + 1 = 4$, $4 + 0 = 4$ Sum of 5: $0 + 5 = 5$, $1 + 4 = 5$, $2 + 3 = 5$, $3 + 2 = 5$, $4 + 1 = 5$, $5 + 0 = 5$

And this pattern continues all the way up to sums of 10! There are 55 total addition facts within 10. 🔢

Here's something amazing: every addition fact has a related subtraction fact! If you know that $3 + 4 = 7$, then you also know that $7 - 4 = 3$ and $7 - 3 = 4$. It's like knowing that if you can walk from your house to the park, you can also walk from the park back to your house! 🏠🌳

These related facts form fact families. A fact family shows all the addition and subtraction facts that use the same three numbers. For example, the fact family for 2, 5, and 7 includes:

• $2 + 5 = 7$
• $5 + 2 = 7$
• $7 - 5 = 2$
• $7 - 2 = 5$

All four facts belong to the same family because they use the same three numbers: 2, 5, and 7! 👨‍👩‍👧‍👦

Addition and subtraction are opposite operations, also called inverse operations. This means they undo each other! If you start with 6, add 3 to get 9, then subtract 3, you're back to 6. It's like taking 3 steps forward and then 3 steps back - you end up where you started! 👣

This relationship helps you check your work:

• If you think $4 + 5 = 9$, you can check by calculating $9 - 5$ to see if you get 4
• If you think $8 - 3 = 5$, you can check by calculating $5 + 3$ to see if you get 8

Automaticity doesn't mean you have to memorize facts without understanding them. Instead, you build automaticity by:

1. Understanding patterns: Notice that $5 + 3 = 8$ and $3 + 5 = 8$ (addition is commutative - order doesn't matter)
2. Seeing relationships: If you know $6 + 2 = 8$, you automatically know $8 - 2 = 6$
3. Using strategies: Start with strategies like counting on, then practice until you don't need to count anymore
4. Regular practice: The more you use these facts, the more automatic they become

Count On: Start with the larger number and count up. For $7 + 2$, start at 7 and count: "8, 9." So $7 + 2 = 9$. 🔢

Doubles: Some facts are easy because both numbers are the same: $3 + 3 = 6$, $4 + 4 = 8$, $5 + 5 = 10$. These are called doubles facts!

Near Doubles: If you know doubles, you can figure out near doubles. If $4 + 4 = 8$, then $4 + 5$ is one more than that, so $4 + 5 = 9$! 🎯

Make 10: This strategy helps with larger facts. For $7 + 5$, think: "7 + 3 = 10, and I still have 2 more to add, so 10 + 2 = 12."

Decomposing: Break apart one number to make the problem easier. For $6 + 7$, think: "6 + 6 + 1 = 12 + 1 = 13."

Two-color counters are perfect for exploring fact families! Take 8 counters. You might have 5 red and 3 blue, showing $5 + 3 = 8$. Flip 2 red counters to blue, and now you have 3 red and 5 blue, showing $3 + 5 = 8$. You can see that you still have 8 counters total! 🔴🔵

Ten frames help you visualize numbers up to 10. A ten frame is a 2×5 grid where you place dots or counters. When you see 7 dots in a ten frame, you can quickly see that it's 3 away from 10, which helps with many strategies.

These basic facts appear everywhere in daily life!

Money: If you have 6 pennies and find 4 more, you have $6 + 4 = 10$ pennies, which equals 1 dime! 💰

Time: If it's 2 o'clock and you wait 3 hours, it will be $2 + 3 = 5$ o'clock. ⏰

Games: If you score 4 points in the first round and 6 points in the second round, your total is $4 + 6 = 10$ points! 🎮

Food: If you eat 3 grapes 🍇 and then eat 2 more, you've eaten $3 + 2 = 5$ grapes total.

When you know your facts automatically, you can focus on solving more complex problems instead of spending time on basic calculations. It's like knowing your letters automatically so you can focus on reading whole words and stories! 📚

Automaticity also helps you:

• Solve word problems faster
• Check if your answers make sense
• Build confidence in mathematics
• Prepare for more advanced operations like multiplication

Some students think automaticity means speed, but it's really about accuracy and reliability. It's better to be correct every time than to be fast but make mistakes. Speed comes naturally with practice! 🏃‍♀️

Remember that understanding comes before memorization. If you understand why $6 + 4 = 10$, memorizing it becomes much easier. Don't just memorize - understand the relationships between numbers!

The best way to build automaticity is through distributed practice - a little bit of practice every day rather than trying to learn everything at once. Play math games, use flashcards, solve problems, and look for these facts in everyday situations. Before you know it, these facts will be as automatic as saying your own name! 🌟

Remember: every mathematician started by learning these basic facts. Mastering them is your first step toward becoming a mathematical thinker who can solve amazing problems! 🚀

Now that you've mastered facts to 10, it's time to stretch your mathematical muscles and work with bigger numbers! 💪 Adding and subtracting with sums up to 20 opens up a whole new world of problem-solving opportunities. You'll learn reliable strategies that help you solve problems confidently, even when the numbers get bigger. Let's explore these exciting mathematical adventures! 🗺️

Procedural reliability means you can use a method or strategy that works every time, even if you're not super fast yet. It's like learning to ride a bike - at first you might go slowly and wobble a bit, but you can still get where you're going! 🚲 The speed and smoothness come with practice.

For addition and subtraction within 20, you don't need to have automatic recall like you do for facts within 10. Instead, you need reliable strategies that you can count on to give you the correct answer.

Not all problems are the same! There are different problem types that help you understand when to add or subtract:

Adding To: You start with some objects and add more. "Maria has 8 stickers. Her teacher gives her 5 more stickers. How many stickers does Maria have now?" $8 + 5 = 13$ stickers. 🌟

Putting Together: You have two groups and want to find the total. "There are 7 boys and 6 girls in the class. How many students are there altogether?" $7 + 6 = 13$ students. 👧👦

Taking From: You start with some objects and take some away. "Jake has 15 toy cars. He gives 7 to his brother. How many toy cars does Jake have left?" $15 - 7 = 8$ toy cars. 🚗

Comparing: You want to find the difference between two amounts. "Team A scored 14 points. Team B scored 9 points. How many more points did Team A score?" $14 - 9 = 5$ more points. ⚽

Count On Strategy: Start with the larger number and count up by the smaller number. For $9 + 4$, start at 9 and count: "10, 11, 12, 13." So $9 + 4 = 13$. This works well when one number is small! 🔢

Make Ten Strategy: This is like finding a shortcut! For $8 + 5$, think: "8 + 2 = 10, and I still have 3 more to add (since 5 = 2 + 3), so 10 + 3 = 13." Breaking numbers apart to make 10 first often makes problems easier! 🎯

Doubles Plus Strategy: If you know your doubles facts, you can solve near-doubles easily. For $7 + 8$, think: "7 + 7 = 14, and 8 is one more than 7, so 7 + 8 = 14 + 1 = 15." 🎪

Decomposing Strategy: Break apart numbers to make the problem easier. For $6 + 9$, you might think: "6 + 10 - 1 = 16 - 1 = 15" or "6 + 4 + 5 = 10 + 5 = 15." 🧩

Count Back Strategy: Start at the larger number and count backward. For $13 - 4$, start at 13 and count back: "12, 11, 10, 9." So $13 - 4 = 9$. This works best when the number you're subtracting is small! ⬅️

Count Up Strategy: Start at the smaller number and count up to the larger number. For $15 - 8$, start at 8 and count up: "9, 10, 11, 12, 13, 14, 15." That's 7 jumps, so $15 - 8 = 7$. This is great when the numbers are close together! ➡️

Think Addition Strategy: Use the relationship between addition and subtraction. For $16 - 9$, think: "What plus 9 equals 16?" Since $7 + 9 = 16$, you know that $16 - 9 = 7$. This connects to your known addition facts! 🔄

Make Ten Strategy for Subtraction: For $13 - 5$, think: "13 - 3 = 10, and I still need to subtract 2 more (since 5 = 3 + 2), so 10 - 2 = 8." 🎲

Number lines are fantastic tools for addition and subtraction! They help you visualize your thinking and check your work.

For addition like $7 + 6$:

1. Start at 7 on the number line
2. Jump forward 6 spaces: 8, 9, 10, 11, 12, 13
3. You land on 13, so $7 + 6 = 13$

For subtraction like $15 - 8$:

1. Start at 15 on the number line
2. Jump backward 8 spaces: 14, 13, 12, 11, 10, 9, 8, 7
3. You land on 7, so $15 - 8 = 7$

You can also use number lines to count up for subtraction! 📏

Base ten blocks help you see place value while adding and subtracting. For $12 + 7$:

• Start with 1 ten rod and 2 unit cubes (representing 12)
• Add 7 more unit cubes
• You now have 1 ten rod and 9 unit cubes
• Since you have less than 10 ones, your answer is 19!

For $18 - 5$:

• Start with 1 ten rod and 8 unit cubes (representing 18)
• Take away 5 unit cubes
• You're left with 1 ten rod and 3 unit cubes
• So $18 - 5 = 13$!

Sometimes when adding, you get more than 10 ones and need to regroup! For $9 + 7$:

• $9 + 7 = 16$
• 16 has 1 ten and 6 ones
• You can see this with base ten blocks: 16 unit cubes can be regrouped as 1 ten rod and 6 unit cubes

For subtraction, sometimes you need to "borrow" or regroup. Don't worry - you'll explore this more deeply as you continue learning! 🔄

As you practice these strategies, you'll start doing some of the steps in your head rather than counting everything out loud. This is called mental math! For example:

• $8 + 6$: "8 + 2 = 10, plus 4 more = 14"
• $17 - 9$: "17 - 7 = 10, minus 2 more = 8"

Developing mental math skills makes you a more efficient problem solver! 🧠

These skills help you solve everyday problems:

At the store: You have $\$12$ and want to buy something that costs $\$7$. How much change will you get? $12 - 7 = 5$, so you'll get $\$5$ in change! 💵

Playing games: You scored 8 points in round one and 9 points in round two. What's your total score? $8 + 9 = 17$ points! 🎮

Time: It's 3:00 PM and you need to wait 45 minutes for your friend. What time will it be? Well, 45 minutes is close to an hour, so it will be close to 4:00 PM! ⏰

The best mathematicians know many different strategies and choose the one that works best for each problem! Consider:

• $9 + 3$: Count on (start at 9, count up 3)
• $8 + 8$: Use doubles ($8 + 8 = 16$)
• $7 + 9$: Make ten ($7 + 3 + 6 = 10 + 6 = 16$)
• $15 - 2$: Count back (15, 14, 13)
• $14 - 6$: Think addition ("What plus 6 equals 14? It's 8!")

Flexibility in strategy choice is a sign of mathematical thinking! 🎭

Remember, you don't need to be fast to be successful. Focus on:

1. Understanding the problem
2. Choosing a strategy that makes sense to you
3. Using the strategy carefully
4. Checking your answer

Speed comes naturally with practice. The most important thing is that you can solve the problem reliably and explain your thinking! 🌟

With these strategies in your mathematical toolkit, you're ready to tackle any addition or subtraction problem within 20. Keep practicing, stay curious, and enjoy discovering all the different ways numbers can work together! 🚀

Understanding how numbers relate to each other is like discovering secret pathways between number neighbors! 🏘️ When you know how to find one more, one less, ten more, and ten less than any number, you're building powerful place value understanding that will help you with many math problems. These relationships create amazing patterns that make mathematics both predictable and beautiful! ✨

One more than a number means adding 1 to it. One less than a number means subtracting 1 from it. These are the smallest steps you can take on the number path! 👣

For any two-digit number like 47:

• One more: $47 + 1 = 48$
• One less: $47 - 1 = 46$

Think of it like being on a number line and taking exactly one step forward (one more) or one step backward (one less). You're always moving to the very next number! 📏

Ten more than a number means adding 10 to it. Ten less than a number means subtracting 10 from it. These are bigger jumps that follow special patterns! 🦘

For the same number 47:

• Ten more: $47 + 10 = 57$
• Ten less: $47 - 10 = 37$

Notice something amazing: when you add or subtract 10, only the tens digit changes! The ones digit stays exactly the same. This is because 10 is made up of 1 ten and 0 ones, so you're only affecting the tens place! 🎯

Here's where it gets really exciting! When you understand place value, these relationships create beautiful patterns:

For one more/one less: Usually only the ones digit changes

• $34 + 1 = 35$ (ones digit goes from 4 to 5)
• $58 - 1 = 57$ (ones digit goes from 8 to 7)

Special case: When the ones digit is 9 or 0, both digits might change!

• $29 + 1 = 30$ (we need to regroup: 9 ones + 1 one = 10 ones = 1 ten + 0 ones)
• $40 - 1 = 39$ (we need to regroup: 4 tens + 0 ones = 3 tens + 10 ones, then subtract 1)

For ten more/ten less: Only the tens digit changes!

• $25 + 10 = 35$ (tens digit goes from 2 to 3, ones digit stays 5)
• $73 - 10 = 63$ (tens digit goes from 7 to 6, ones digit stays 3)

A hundreds chart is like a map that shows all the numbers from 1 to 100 organized in rows of 10. This tool makes the patterns crystal clear! 🗺️

 1  2  3  4  5  6  7  8  9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
...

On a hundreds chart:

• One more: Move one space to the right
• One less: Move one space to the left
• Ten more: Move straight down to the next row
• Ten less: Move straight up to the previous row

Try finding 45 on the chart:

• One more: 46 (move right)
• One less: 44 (move left)
• Ten more: 55 (move down)
• Ten less: 35 (move up)

Number lines show these relationships in a straight line format. For one more and one less, you make small jumps. For ten more and ten less, you make big jumps of 10! 🦘

On a number line from 20 to 80:

20 --- 30 --- 40 --- 50 --- 60 --- 70 --- 80

If you start at 40:

• One more: tiny hop to 41
• Ten more: big jump to 50
• One less: tiny hop to 39
• Ten less: big jump to 30

These relationships are really just special cases of addition and subtraction!

• One more = adding 1
• One less = subtracting 1
• Ten more = adding 10
• Ten less = subtracting 10

But the key difference is that you don't need to count or use complex strategies. Once you understand the patterns, you can find these numbers instantly! ⚡

Knowing these relationships makes you a mental math superhero! 🦸‍♀️🦸‍♂️

For estimation: If you know that $37 + 9$ is close to $37 + 10 = 47$, you can estimate that the answer is close to 47 (actually it's 46).

For checking answers: If you calculate $52 + 8 = 60$, you can check by thinking: "60 is ten more than 50 and eight more than 52, so that makes sense!"

For problem solving: If you have $\$43$ and earn $\$10$ more, you know immediately that you have $\$53$ because ten more than 43 is 53!

Most of the time, these patterns work simply, but there are a few special cases:

Crossing decades for one more:

• $29 + 1 = 30$ (tens digit changes from 2 to 3)
• $39 + 1 = 40$ (tens digit changes from 3 to 4)
• $99 + 1 = 100$ (becomes a three-digit number!)

Crossing decades for one less:

• $30 - 1 = 29$ (tens digit changes from 3 to 2)
• $40 - 1 = 39$ (tens digit changes from 4 to 3)

At the boundaries for ten more/less:

• $95 + 10 = 105$ (becomes a three-digit number!)
• $5 - 10$ = This takes us below 0, which you'll explore in later grades

These relationships show up everywhere in daily life:

Ages: If you're 7 years old now, next year you'll be one more year old (8), and in ten years you'll be ten more years old (17)! 🎂

Time: If it's 2:30 PM and you wait one more minute, it will be 2:31 PM. If you wait ten more minutes, it will be 2:40 PM! ⏰

Money: If you have $\$25$ and find $\$1$, you have one more dollar ($\$26$). If someone gives you $\$10$, you have ten more dollars ($\$35$)! 💰

Temperature: If it's 68°F and the temperature drops one degree, it becomes 67°F (one less). If it drops ten degrees, it becomes 58°F (ten less)! 🌡️

House numbers: If you live at house number 45, the next house might be 47 (two more), and the house on the next block up might be 145 (one hundred more)! 🏠

Here are fun ways to practice these relationships:

Number of the Day: Pick any two-digit number and find its one more, one less, ten more, and ten less neighbors. Create a cross pattern:

    65 (ten more)
64  55  56 (one less, original, one more)
    45 (ten less)

Hundreds Chart Exploration: Point to any number on a hundreds chart and quickly identify its four neighbors without counting!

Mental Math Challenges: Give yourself problems like "What's ten more than 37?" and see how fast you can answer using these patterns!

The more you work with these relationships, the stronger your number sense becomes. You start to feel how numbers are connected and related. You might notice that:

• Numbers ending in the same digit are always ten apart (25, 35, 45, 55...)
• Consecutive numbers differ by exactly one (56, 57, 58, 59...)
• Round numbers (like 30, 40, 50) are ten apart from each other

This deep understanding of number relationships will serve you well as you tackle more complex mathematics! 🌟

Remember: mathematics is all about patterns and relationships. When you master these simple but powerful connections between numbers, you're building the foundation for all your future mathematical adventures! 🚀

Adding a two-digit number and a one-digit number is like being a mathematical architect - you need to understand how the building blocks (tens and ones) work together to create larger structures! 🏗️ This is where your place value understanding really shines, and you'll discover the exciting concept of regrouping, which is like trading blocks to make your mathematical building more organized! ✨

When you add a two-digit number like 47 and a one-digit number like 6, you're combining place values. The two-digit number has both tens and ones, while the one-digit number only has ones. Your job is to figure out how these pieces fit together! 🧩

$47 + 6 = ?$

Think of 47 as:

• 4 tens (which equals 40)
• 7 ones

When you add 6 more ones, you have:

• 4 tens (still 40)
• 7 ones + 6 ones = 13 ones

But wait! 13 ones is more than 10, so you need to regroup!

Regrouping (also called "carrying" or "trading") happens when you have 10 or more ones. Since 10 ones equals 1 ten, you can trade them! It's like having 10 pennies and trading them for 1 dime - you have the same value, just organized differently! 💰

For our example $47 + 6$:

• 13 ones = 1 ten + 3 ones
• So you have: 4 tens + 1 ten + 3 ones = 5 tens + 3 ones = 53
• Therefore: $47 + 6 = 53$

You need to regroup when the ones digits add up to 10 or more:

Examples that need regrouping:

• $28 + 5$: $8 + 5 = 13$ (13 ≥ 10, so regroup!)
• $46 + 7$: $6 + 7 = 13$ (13 ≥ 10, so regroup!)
• $39 + 4$: $9 + 4 = 13$ (13 ≥ 10, so regroup!)

Examples that don't need regrouping:

• $32 + 5$: $2 + 5 = 7$ (7 < 10, no regrouping needed!)
• $61 + 3$: $1 + 3 = 4$ (4 < 10, no regrouping needed!)
• $45 + 2$: $5 + 2 = 7$ (7 < 10, no regrouping needed!)

Base ten blocks make regrouping visual and hands-on! 🧱

For $35 + 8$:

1. Start with 3 ten rods and 5 unit cubes (representing 35)
2. Add 8 more unit cubes
3. Now you have 3 ten rods and 13 unit cubes
4. Trade 10 unit cubes for 1 ten rod
5. You end up with 4 ten rods and 3 unit cubes = 43!

The physical act of trading helps you understand that $35 + 8 = 43$.

Count On Strategy: Start with the larger number and count up by the smaller number. For $57 + 4$, start at 57 and count: "58, 59, 60, 61." So $57 + 4 = 61$. This works well when the one-digit number is small! 📊

Break Apart Strategy: Break the one-digit number into parts that are easier to work with. For $48 + 7$:

• Think: "48 + 2 = 50, and I still have 5 more to add (since 7 = 2 + 5)"
• So: "50 + 5 = 55"
• Therefore: $48 + 7 = 55$

Place Value Strategy: Add the ones separately, then add to the tens. For $34 + 9$:

• Ones: $4 + 9 = 13$ (that's 1 ten and 3 ones)
• Tens: $30 + 10 = 40$ (the original 3 tens plus the new 1 ten)
• Total: $40 + 3 = 43$

Mental Math Strategy: For $29 + 6$, think: "29 is close to 30, so 30 + 6 = 36, but I added one too many, so 36 - 1 = 35." 🧠

Number lines help you visualize the addition process! For $43 + 8$:

1. Start at 43 on the number line
2. Make 8 jumps forward: 44, 45, 46, 47, 48, 49, 50, 51
3. You land on 51, so $43 + 8 = 51$

You can also make bigger jumps! Jump to the next ten first (43 → 50 is 7 jumps), then add the remaining 1 jump to get 51. 🦘

A place value chart organizes your thinking:

Tens | Ones
  4  |  7     (47)
  +  |  6     (+ 6)
  ---|---
  ?  |  ?     (answer)

Step by step:

1. Add the ones: $7 + 6 = 13$
2. 13 ones = 1 ten + 3 ones
3. Add the tens: $4 + 1 = 5$ tens
4. Final answer: 5 tens and 3 ones = 53

These skills help solve everyday problems:

Shopping: You have $\$27$ and find $\$8$ more in your pocket. How much do you have total? $27 + 8 = 35$, so you have $\$35$! 🛒

Games: You scored 34 points yesterday and 9 points today. What's your total score? $34 + 9 = 43$ points! 🎮

Collections: You have 56 baseball cards and your friend gives you 7 more. How many cards do you have now? $56 + 7 = 63$ cards! ⚾

Time: If you read for 45 minutes on Monday and 8 minutes on Tuesday, how many minutes did you read total? $45 + 8 = 53$ minutes! 📚

Forgetting to regroup: Some students might calculate $46 + 8$ as "46 + 8 = 414" by just putting the 14 at the end. Remember that 14 ones must be regrouped as 1 ten and 4 ones! ❌

Adding to the wrong place: Make sure you add the one-digit number to the ones place, not the tens place. $23 + 4 = 27$, not $43$! ✅

Regrouping when you don't need to: If $32 + 5 = 37$, you don't need to regroup because $2 + 5 = 7$, which is less than 10. 🤔

As you practice, you'll get faster and more efficient:

1. Start by using concrete materials like base ten blocks
2. Move to drawings of tens and ones
3. Progress to mental strategies like breaking apart numbers
4. Eventually develop automaticity for easier combinations

Don't rush this process! Each stage builds understanding for the next. 🌱

Mastering two-digit plus one-digit addition prepares you for:

• Adding two two-digit numbers
• Adding larger numbers with multiple digits
• Understanding the standard addition algorithm
• Solving more complex word problems

The place value understanding and regrouping concepts you're learning now are the foundation for all future addition work! 🏗️

The best mathematicians have many strategies and choose the one that fits the problem best:

• $73 + 2$: Count on (easy with small numbers)
• $48 + 7$: Break apart to make 50, then add the rest
• $29 + 6$: Add 1 to make 30, then add 5 more
• $56 + 4$: Simple addition without regrouping

Experiment with different strategies and find the ones that make the most sense to you! The goal is to solve problems accurately and efficiently while understanding what you're doing. 🎯

Remember: every expert mathematician once learned these same skills. Keep practicing, stay curious, and enjoy discovering the patterns and relationships that make addition both logical and beautiful! 🌟

Subtraction is like being a mathematical detective - sometimes you have to think creatively and use different strategies to solve the mystery! 🕵️‍♀️ When you subtract a one-digit number from a two-digit number, you're exploring the exciting world of taking apart numbers and discovering new ways to think about mathematical relationships. Let's unlock the secrets of subtraction together! 🔓

When you see a problem like $52 - 7$, you're being asked: "If you start with 52 and take away 7, how many are left?" Sometimes this is straightforward, but sometimes you need to be clever about how you approach it! 🤔

Let's think about this in terms of place value:

• 52 means 5 tens and 2 ones
• You want to subtract 7 ones
• But wait... you only have 2 ones to start with!
• This is where the magic of regrouping comes in! ✨

You need to regroup when the ones digit of the number you're subtracting is larger than the ones digit of the number you're starting with:

Examples that need regrouping:

• $43 - 8$: Can't take 8 from 3, so you need to regroup!
• $61 - 5$: Can't take 5 from 1, so you need to regroup!
• $30 - 4$: Can't take 4 from 0, so you need to regroup!

Examples that don't need regrouping:

• $47 - 3$: Can take 3 from 7, no regrouping needed!
• $89 - 6$: Can take 6 from 9, no regrouping needed!
• $55 - 2$: Can take 2 from 5, no regrouping needed!

Regrouping in subtraction means "borrowing" or trading 1 ten for 10 ones. It's like breaking open a $\$10$ bill to get 10 $\$1$ bills when you need to pay for something! 💵

For $52 - 7$:

1. Start with 5 tens and 2 ones
2. You can't take 7 from 2, so trade 1 ten for 10 ones
3. Now you have 4 tens and 12 ones (4 + 12 = 52, same amount!)
4. Take 7 from 12: $12 - 7 = 5$
5. You're left with 4 tens and 5 ones = 45
6. So $52 - 7 = 45$! ✅

Base ten blocks make this process visible and tactile! 🧱

For $41 - 6$:

1. Start with 4 ten rods and 1 unit cube
2. You need to take away 6 unit cubes, but you only have 1
3. Trade 1 ten rod for 10 unit cubes
4. Now you have 3 ten rods and 11 unit cubes
5. Take away 6 unit cubes
6. You're left with 3 ten rods and 5 unit cubes = 35

Physically moving the blocks helps you understand that the total value stays the same during regrouping! 👀

Count Back Strategy: Start at the larger number and count backward. For $48 - 5$, start at 48 and count back: "47, 46, 45, 44, 43." This works well when the number you're subtracting is small! ⬅️

Count Up Strategy (Think Addition): Start at the smaller number and count up to the larger number. For $53 - 8$, start at 8 and count up: "How much do I need to add to 8 to get 53?" Count: 9, 10, 11... all the way to 53. That's 45 jumps, so $53 - 8 = 45$! This is especially good when the numbers are close together. ➡️

Decompose the Subtrahend Strategy: Break apart the number you're subtracting. For $64 - 7$:

• Think: "64 - 4 = 60, and I still need to subtract 3 more (since 7 = 4 + 3)"
• So: "60 - 3 = 57"
• Therefore: $64 - 7 = 57$ 🎯

Add Up Strategy: Start from the number you're subtracting and add up to the minuend. For $72 - 9$:

• Start at 9: "9 + 1 = 10, 10 + 60 = 70, 70 + 2 = 72"
• Total added: $1 + 60 + 2 = 63$
• So: $72 - 9 = 63$ 🔢

Number lines are fantastic tools for visualizing subtraction! You can use them in two different ways:

Method 1 - Count Back: For $56 - 8$, start at 56 and jump back 8 spaces: 55, 54, 53, 52, 51, 50, 49, 48. You land on 48! 📏

Method 2 - Count Up: For $56 - 8$, start at 8 and count up to 56. How many jumps? From 8 to 10 is 2 jumps, from 10 to 50 is 40 jumps, from 50 to 56 is 6 jumps. Total: $2 + 40 + 6 = 48$ jumps! 🦘

Here's a powerful connection: every subtraction problem is really a missing addend problem! When you see $73 - 5$, you can think: "What number plus 5 equals 73?" 🤔

If you know that $68 + 5 = 73$, then you automatically know that $73 - 5 = 68$! This connects subtraction to your addition knowledge and can make some problems much easier.

For example:

• $45 - 9 = ?$ becomes "What plus 9 equals 45?"
• Since $36 + 9 = 45$, you know $45 - 9 = 36$!

A place value chart helps organize your thinking systematically:

Tens | Ones
  5  |  2     (52)
  -  |  7     (- 7)
  ---|---
  ?  |  ?     (answer)

Step by step for $52 - 7$:

1. Can you take 7 from 2? No!
2. Regroup: 5 tens becomes 4 tens + 1 ten = 4 tens + 10 ones
3. Now you have 4 tens and 12 ones (10 + 2 = 12)
4. Subtract: $12 - 7 = 5$ ones
5. Answer: 4 tens and 5 ones = 45

Subtraction with regrouping appears in many everyday situations:

Money: You have $\$42$ and spend $\$8$ on a toy. How much money do you have left? $42 - 8 = 34$, so you have $\$34$ left! 💰

Time: You need to wait 35 minutes, and 7 minutes have already passed. How much more time do you need to wait? $35 - 7 = 28$ more minutes! ⏰

Games: You had 61 points and lost 9 points for a wrong answer. How many points do you have now? $61 - 9 = 52$ points! 🎮

Collections: You had 84 stickers and gave 6 to your friend. How many stickers do you have left? $84 - 6 = 78$ stickers! 🌟

Forgetting to regroup: Some students try to subtract larger numbers from smaller ones in the ones place, getting impossible results like $52 - 7 = 55$. Always check if you need to regroup first! ❌

Making errors during regrouping: Double-check that your regrouping keeps the same total value. If you start with 52, after regrouping you should still have 52 (just organized as 4 tens and 12 ones). ✅

Subtracting in the wrong direction: $73 - 8$ is NOT $8 - 73$. The larger number comes first in subtraction notation! 🔄

As you practice, you'll start to see patterns and develop mental shortcuts:

• $50 - 3 = 47$ (no regrouping needed)
• $50 - 8 = 42$ (think: "50 - 10 + 2 = 40 + 2 = 42")
• $64 - 6 = 58$ (ones digits: 4 - 6 needs regrouping, so 14 - 6 = 8)

These mental strategies make you faster and more confident! 🧠

To master this skill:

1. Start with concrete materials: Use base ten blocks to physically see regrouping
2. Practice with drawings: Draw tens and ones, crossing out what you take away
3. Use place value charts: Organize your thinking systematically
4. Develop mental strategies: Work toward doing some steps in your head
5. Apply to real problems: Use these skills to solve everyday situations

Remember: understanding comes before speed! Focus on making sense of what you're doing, and speed will develop naturally with practice. 🌱

The regrouping and place value concepts you're learning now prepare you for:

• Subtracting larger multi-digit numbers
• Understanding the standard subtraction algorithm
• Working with decimals and fractions
• Solving complex word problems

You're building a strong mathematical foundation that will serve you well throughout your educational journey! 🏗️

Remember: every mathematician has worked through these same concepts. Keep practicing, stay curious, and celebrate your progress as you master these important subtraction skills! 🌟