Mathematics: Number Sense and Operations – Grade K

Counting is one of the most important math skills you'll ever learn! When you count, you're finding out exactly how many things are in a group. Let's explore how to count objects and write the numbers that match.

When you count, you're doing something very special called one-to-one correspondence. This fancy term means that each object gets exactly one number, and each number goes with exactly one object. Think of it like giving each of your stuffed animals 🧸 a unique name - each animal gets one name, and each name belongs to one animal!

Here's how to count like a math expert:

1. Point to each object as you say its number
2. Say the numbers in order: 1, 2, 3, 4, 5...
3. Count each object only once - no double counting!
4. The last number you say tells you how many objects there are in total

Cardinality is a big word that means "how many in total." When you count a group of 7 blocks 🧱🧱🧱🧱🧱🧱🧱, the number 7 doesn't just mean "the seventh block" - it means "there are 7 blocks altogether!"

Let's practice: If you count your fingers and say "1, 2, 3, 4, 5" on one hand, the 5 doesn't just mean the pinky finger. It means you have 5 fingers total on that hand! 🖐️

Here's something amazing: when you rearrange objects, the total number stays exactly the same! This is called conservation of number.

Imagine you have 6 crayons 🖍️ lined up in a row. You count them: 1, 2, 3, 4, 5, 6. Now, what happens if you arrange them in a circle? You still have 6 crayons! What if you scatter them on the table? Still 6 crayons! The arrangement changes, but the total amount never changes.

This is one of the most important discoveries in math. Once you count a group and find out there are 8 toys, you don't need to count again if someone just moves them around. The answer is still 8!

Numbers have special symbols that we write to show how many things we have. These symbols are called numerals. When you see the symbol "4," it represents the idea of "four things."

Let's practice writing numerals for different amounts:

• 🍎 = 1 apple = write 1
• 🍎🍎🍎 = 3 apples = write 3
• 🍪🍪🍪🍪🍪🍪🍪 = 7 cookies = write 7

Objects can be arranged in many ways, and you need to be ready to count them all:

Line arrangements: Objects in a straight row are easy to count because you can point to each one from left to right.

Circular arrangements: When objects are in a circle, pick a starting point and count around. Put your finger on your starting object so you remember where you began!

Rectangular arrays: Sometimes objects are arranged in rows and columns, like desks in a classroom. You can count across each row, then move to the next row.

Scattered arrangements: When objects are scattered randomly, count systematically. You might count from left to right, or group them as you count to keep track.

To be a good counter, you need strategies to make sure you don't skip any objects or count them twice:

1. Touch and count: Point to or touch each object as you count it
2. Move objects: As you count each one, move it to a "counted" pile
3. Cross out or mark: If you're counting pictures, make a small mark on each one as you count
4. Count in patterns: For scattered objects, count all objects of one color first, then another color

• Double counting: Make sure you don't count the same object twice
• Skipping objects: Go slowly and systematically so you don't miss any
• Skipping numbers: Practice the counting sequence so you don't skip "4" and go straight from "3" to "5"
• Forgetting the total: Remember that the last number you say is the total amount

Counting helps you in many everyday situations:

• Snack time: Count how many crackers you have 🍘
• Cleanup time: Count toys to make sure you found them all 🧸
• Art projects: Count how many stickers you want to use ⭐
• Game time: Count game pieces to make sure none are missing 🎲
• Getting dressed: Count buttons on your shirt 👕

Practicing counting every day will make you faster and more accurate. Soon you'll be able to look at a small group and know how many there are without even counting - this special skill is called subitizing!

Now that you know how to count groups of objects, let's learn the opposite skill: when someone tells you a number, you can count out exactly that many objects! This is like being a math magician 🎩 - someone says a number, and you can make a group with exactly that many things.

A target number is the exact amount you need to count out. If someone says "count out 7 blocks," then 7 is your target number. Your job is to count objects one by one until you reach exactly 7.

Think of target numbers like a goal in a game 🥅. You keep counting until you "score" by reaching your target!

Here's the step-by-step process for counting out objects:

1. Listen carefully to the target number
2. Start with zero objects in your counting area
3. Count slowly: "1" (take one object), "2" (take another), "3" (take another)...
4. Stop when you reach the target - don't keep going!
5. Double-check by counting your group again

This is super important! When you're counting out objects, you need to know exactly when to stop. If your target is 5:

• Count: "1, 2, 3, 4, 5" - STOP!
• Don't count "6" or any more
• You have exactly 5 objects

It's like a countdown in reverse! Instead of "5, 4, 3, 2, 1, blast off!" you're counting "1, 2, 3, 4, 5, mission complete!" 🚀

The Container Strategy: Start with a container that has more objects than your target number. If you need 8 crayons, make sure your container has at least 10 crayons 🖍️.

The Moving Strategy: As you count each object, move it from the "not counted" pile to the "counted" pile. This helps you keep track visually.

The Touching Strategy: Point to or touch each object as you count it, then leave it in place.

The Organizing Strategy: Arrange your counted objects in a line or pattern. This makes it easy to see how many you have and helps you double-check.

A ten frame is a special tool that looks like a rectangle divided into 10 spaces (2 rows of 5 spaces each). Ten frames help you organize your counting and see patterns in numbers.

When counting out objects using a ten frame:

• Fill the top row first (spaces 1-5)
• Then fill the bottom row (spaces 6-10)
• For numbers bigger than 10, use a second ten frame

For example, if you need to count out 7 objects:

• Put 5 objects in the top row ●●●●●
• Put 2 objects in the bottom row ●●○○○
• You can easily see you have 7 objects!

Skipping Numbers: Sometimes you might count "1, 2, 4, 5" and skip 3. Practice the counting sequence often: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20.

Going Over the Target: If you need 6 objects but you count out 8, you went too far! Always listen carefully to the target number and stop when you reach it.

Losing Track: If you lose track of where you are in counting, start over. It's better to restart than to guess!

To count out objects successfully, you need to recognize numbers when you see or hear them:

Hearing Numbers: When someone says "fourteen," you know that means 14 Seeing Numbers: When you see the symbol "9," you know that means nine Number Order: You know that 12 comes after 11 and before 13

Counting out exact amounts is useful in many situations:

Art Time: "Get 5 paintbrushes from the supply bin" 🖌️ Snack Time: "Take 8 goldfish crackers" 🐠 Game Time: "Each player needs 3 game pieces" 🎲 Helping at Home: "Set out 4 forks for dinner" 🍽️ School Supplies: "Get 10 crayons for your project" 🖍️

Start with smaller numbers (1-5) and gradually work up to larger numbers (6-20). The more you practice, the more confident you'll become! Remember:

• It's okay to count slowly and carefully
• Double-checking your work is smart, not slow
• Every expert started as a beginner
• Practice makes counting automatic

Counting out objects is something adults do every day:

• Giving exact change at a store 💰
• Setting places at the dinner table 🍽️
• Distributing supplies at work 📝
• Following recipes in cooking 🍪

When you master counting out exact amounts, you're learning a life skill that you'll use forever!

Position words are special words that tell us exactly where something is in a line or group! 📍 These words help us describe the order of things, just like describing who's first, second, or third in line for the slide at recess.

Ordinal numbers tell us about position or order. They're different from regular counting numbers (called cardinal numbers). Here's the difference:

• Cardinal numbers tell us "how many": 1, 2, 3, 4, 5 (There are 5 ducks)
• Ordinal numbers tell us "which position": first, second, third, fourth, fifth (The yellow duck is third)

Think of it this way: if you're counting your toys, you use cardinal numbers to find out how many you have. But if you're describing which toy you want, you use ordinal numbers to tell someone its position!

In kindergarten, you'll learn these five important position words:

1. First - the very beginning position
2. Second - the position right after first
3. Third - the position after second
4. Fourth - the position after third
5. Fifth - the position after fourth

This is super important: position depends on which direction you're looking and where you start! 🧭

From the left: 🐱🐶🐭🐹🐰

• The cat is first from the left
• The dog is second from the left
• The mouse is third from the left

From the right: 🐱🐶🐭🐹🐰

• The rabbit is first from the right
• The hamster is second from the right
• The mouse is third from the right

See how the mouse is third from both directions? That's because it's right in the middle!

Here's something really cool: when you rearrange objects, their positions change, but the total number stays the same!

Original line: 🍎🍌🍊🍇🥝 (5 fruits)

• Apple is first
• Banana is second
• Orange is third

Rearranged line: 🍇🥝🍎🍌🍊 (still 5 fruits!)

• Now grape is first
• Kiwi is second
• Apple is third

The apple moved from first position to third position, but we still have exactly 5 fruits total!

Line Up Time: "You're third in line for lunch!" 🍕 Story Time: "In the story, the first little pig built with straw" 📚 Race Time: "Sarah finished second in the running race!" 🏃‍♀️ Cleanup Time: "Put the blocks in the first bin and the crayons in the second bin" 📦 Getting Dressed: "Put on your socks first, then your shoes second" 👟

One great way to learn position words is through movement and games:

Stair Steps: Build stairs with blocks. The first step is number 1, the second step is number 2, and so on. You can walk up the stairs and call out each position! 🪜

Daily Schedule: Talk about your day using position words: "First we have circle time, second we have snack, third we have centers."

Parade Games: Line up stuffed animals or toys and have them "march" in a parade. Call out who's in each position!

Position words help you solve problems and follow directions:

Following Recipes: "First crack the eggs, second add the milk, third stir everything together" 🥚🥛

Building Projects: "First put the big block on the bottom, second add the medium block, third put the small block on top" 🧱

Art Projects: "First draw your picture, second color it in, third add glitter if you want" 🎨

It's important not to get confused between position words and counting words:

When you count objects: "1, 2, 3, 4, 5 - there are 5 crayons" When you describe position: "The red crayon is first, the blue crayon is second"

Counting tells you how many. Position words tell you which one.

The Position Challenge: Have someone arrange 5 different colored blocks in a line. Without looking, have them tell you "Point to the third block" or "Which color is in the fourth position?"

Musical Positions: Play music and have children march in a line. When the music stops, call out positions: "Who is second in line? Who is fourth?"

Story Positions: Read stories and talk about what happens first, second, third in the story. This helps with reading comprehension too! 📖

Once you're comfortable with first through fifth, you can explore beyond:

• Many students can learn sixth, seventh, eighth, and beyond
• Practice with longer lines of objects
• Try positions from different starting points (bottom to top, right to left)

Forgetting the Direction: Always ask "first from which side?" or "starting from where?" Mixing Up Position and Count: Remember that "third" means position, not "there are 3 things" Thinking Positions Never Change: Understand that when you rearrange objects, positions change but the total count doesn't

Position words are tools that help you be precise when describing where things are. They're like giving each object an address in the line - very useful for clear communication! 🏠

Comparing is like being a detective who figures out which group has more, which has less, or if they have the same amount! 🔍 This skill helps you make decisions every day, from sharing snacks fairly to knowing if you have enough supplies for a project.

When we compare two groups, we use three special terms:

Less than: One group has fewer objects than the other

• 3 apples is less than 5 apples 🍎🍎🍎 < 🍎🍎🍎🍎🍎

Equal to: Both groups have exactly the same number of objects

• 4 cookies equals 4 cookies 🍪🍪🍪🍪 = 🍪🍪🍪🍪

Greater than: One group has more objects than the other

• 7 blocks is greater than 4 blocks 🧱🧱🧱🧱🧱🧱🧱 > 🧱🧱🧱🧱

The best way to compare groups is called one-to-one correspondence or matching. Here's how it works:

1. Line up objects from both groups
2. Match each object from Group A with one object from Group B
3. See what's left over

Example: Compare 6 teddy bears 🧸 with 4 toy cars 🚗

• Match them up: 🧸-🚗, 🧸-🚗, 🧸-🚗, 🧸-🚗
• Two teddy bears have no cars to match with: 🧸🧸
• Conclusion: There are more teddy bears than cars

This is really important: bigger objects don't always mean more objects! 🎈

Imagine you have:

• Group A: 3 big balloons 🎈🎈🎈
• Group B: 5 small balloons 🎈🎈🎈🎈🎈

Even though Group A has bigger balloons, Group B has more balloons (5 is greater than 3). Size doesn't matter when we're counting quantity!

Another way to compare is to count each group and then compare the numbers:

1. Count Group A: "1, 2, 3, 4, 5, 6, 7" = 7 objects
2. Count Group B: "1, 2, 3, 4, 5" = 5 objects
3. Compare the numbers: 7 is greater than 5
4. Conclusion: Group A has more objects

The counting sequence helps us know which numbers are bigger: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Numbers that come later in the sequence are greater than numbers that come earlier:

• 8 is greater than 5 (because 8 comes after 5 in counting)
• 3 is less than 7 (because 3 comes before 7 in counting)
• 12 equals 12 (same number!)

Snack Time: "Do I have enough crackers to give one to each friend?" 🍘

• Count friends: 6
• Count crackers: 8
• Compare: 8 is greater than 6, so yes, you have enough!

Art Time: "Which color crayon do we have the most of?" 🖍️

• Count red crayons: 4
• Count blue crayons: 7
• Count yellow crayons: 3
• Compare: Blue has the most (7 is the largest number)

Cleanup Time: "Are there more blocks or more books to put away?" 📚🧱

• Count blocks: 9
• Count books: 6
• Compare: 9 is greater than 6, so there are more blocks

Sometimes we want to make groups have the same amount (equal). Here's how:

Adding to the smaller group:

• Group A has 3 stickers ⭐⭐⭐
• Group B has 5 stickers ⭐⭐⭐⭐⭐
• Add 2 stickers to Group A: ⭐⭐⭐⭐⭐
• Now both groups are equal!

Taking away from the larger group:

• Group A has 8 marbles 🔵🔵🔵🔵🔵🔵🔵🔵
• Group B has 5 marbles 🔵🔵🔵🔵🔵
• Remove 3 marbles from Group A: 🔵🔵🔵🔵🔵
• Now both groups are equal!

When you compare groups, it's important to explain how you know your answer:

Good explanations:

• "I know 8 is greater than 5 because when I count, 8 comes after 5"
• "I matched each apple with each orange, and there were 2 apples left over, so there are more apples"
• "I counted 6 pencils and 6 erasers, so they are equal"

Comparing helps you understand adding and taking away:

Addition Connection: "7 is greater than 5 because 5 + 2 = 7" Subtraction Connection: "9 is greater than 6 because 9 - 6 = 3 (there are 3 more)"

Ten Frames: Put objects in ten frames to compare visually

• Group A fills 1 full ten frame (10 objects)
• Group B fills half a ten frame (5 objects)
• Easy to see that Group A has more!

Number Lines: Place numbers on a number line

• Numbers to the right are greater
• Numbers to the left are less
• Same position means equal

"More, Less, or Equal" Game: Show two groups of objects and have children identify the relationship using comparison words.

"Make It Equal" Challenge: Give unequal groups and challenge children to make them equal by adding or removing objects.

"Estimation Challenge": Before counting, guess which group has more, then count to check!

Comparison skills help you make smart decisions, share fairly with friends, and understand the relationships between numbers. These skills will help you in math for years to come! 🌟

Learning to count to 100 is like climbing a mountain - it seems big at first, but when you take it step by step, you'll reach the top! 🏔️ And when you learn to count by tens, it's like taking an express elevator instead of walking up every single step!

When you count by ones, you're saying each number in order: 1, 2, 3, 4, 5... all the way to 100! Each number is exactly one more than the number before it, and exactly one less than the number after it.

Think of counting like a number train 🚂: each number is a train car connected to the ones before and after it. The train goes: 1-2-3-4-5-6-7-8-9-10-11-12... and each car is connected to the next!

The teen numbers (11-19) can be tricky because their names don't always match their pattern:

• Eleven and twelve have special names (not "ten-one" or "ten-two")
• Thirteen through nineteen follow a pattern: thir-teen, four-teen, fif-teen

Practice saying them in order: 11 (eleven), 12 (twelve), 13 (thirteen), 14 (fourteen), 15 (fifteen), 16 (sixteen), 17 (seventeen), 18 (eighteen), 19 (nineteen), 20 (twenty).

One of the trickiest parts of counting is crossing from one group of tens to the next:

• 19 → 20 (from nineteen to twenty)
• 29 → 30 (from twenty-nine to thirty)
• 39 → 40 (from thirty-nine to forty)

It's like reaching the end of one street and turning onto the next street! The pattern is: when you get to 9 in the ones place, the next number starts a new group of tens.

Counting by tens is like taking giant steps instead of baby steps! 👣

By ones: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (10 steps) By tens: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 (10 steps to get to 100!)

Here's the complete counting-by-tens sequence: 10 (ten), 20 (twenty), 30 (thirty), 40 (forty), 50 (fifty), 60 (sixty), 70 (seventy), 80 (eighty), 90 (ninety), 100 (one hundred)

Counting by tens helps you:

• Count large groups faster: Instead of counting 40 individual items, count 4 groups of 10
• Understand money: Dimes are worth 10 cents each! 💰
• See number patterns: Notice how 23 is 20 plus 3 more
• Prepare for bigger math: Addition and subtraction are easier when you think in tens

Once you know the counting sequence, you can start counting from any number!

Starting from 15: 15, 16, 17, 18, 19, 20, 21, 22... Starting from 47: 47, 48, 49, 50, 51, 52, 53... Starting from 83: 83, 84, 85, 86, 87, 88, 89, 90, 91...

This is like jumping into the middle of the number train and continuing the journey! 🚂

Counting backward is like counting forward, but in reverse! It's great practice for understanding that numbers can go in both directions.

From 10: 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 From 15: 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5... From 20: 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10...

Counting backward helps you understand that each number is one less than the number before it.

As you practice counting, you'll develop number sense - a feeling for how numbers work:

• Number neighbors: 7 comes between 6 and 8
• Number distance: 15 is 5 more than 10
• Number patterns: Every 10th number ends in 0 (10, 20, 30, 40...)
• Number landmarks: 50 is halfway to 100

Counting Collections: Gather small objects like beans, buttons, or pasta pieces. Count out groups of 10 and see how many groups you can make!

Ten Frames: Use ten frames (grids with 10 spaces) to practice counting by tens. Fill one frame = 10, fill two frames = 20, etc.

Number Charts: Use a 100-chart to practice finding number patterns. You can:

• Color all the numbers that end in 5
• Find all the teen numbers
• Practice counting by tens by coloring 10, 20, 30, etc.

Counting Days: Count the days on a calendar 📅 Counting Exercises: Count jumping jacks or claps Counting Collections: Count your toys, books, or stuffed animals Counting Money: Count pennies by ones, dimes by tens 🪙 Counting Food: Count grapes, crackers, or cereal pieces

Rhythm and Rhyme: Say numbers with a beat or melody Movement: March or clap while counting Visual Patterns: Notice that counting by tens always ends in 0 Chunking: Break 1-100 into smaller chunks (1-20, 21-40, 41-60, etc.)

Mixing up teen numbers: Practice 13, 14, 15, 16, 17, 18, 19 until they're automatic Skipping numbers: Count slowly and point to a number chart Crossing tens: Practice the tricky transitions like 29→30, 39→40 Backward counting: Start with small ranges (10→1) before trying larger ones

Counting skills prepare you for future math:

• Counting on: To add 5 + 3, start at 5 and count on: "6, 7, 8"
• Counting back: To subtract 8 - 2, start at 8 and count back: "7, 6"
• Skip counting: Counting by 2s, 5s, and 10s helps with multiplication later

Celebrate your counting achievements! 🎉

• Milestone 1: Counting to 20 without mistakes
• Milestone 2: Counting by tens to 100
• Milestone 3: Counting forward from any number 1-50
• Milestone 4: Counting backward from 20 to 0
• Milestone 5: Counting all the way to 100!

Remember, every mathematician started by learning to count. These counting skills are the foundation for all the amazing math you'll learn in the future! 🌟

Teen numbers are special! They're like number sandwiches 🥪 - they have ten as the "bread" and some ones as the "filling." Understanding how teen numbers are built will help you with math for years to come!

Teen numbers are the numbers from 10 to 20: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. What makes them special is that they all contain one group of ten plus some extra ones.

Think of it like this:

• 13 = 1 group of ten + 3 extra ones
• 17 = 1 group of ten + 7 extra ones
• 19 = 1 group of ten + 9 extra ones

Ten is a very important number because it's like a complete set. Think about:

• Your hands: 10 fingers 🤚🤚
• A ten frame: 10 spaces ▢▢▢▢▢ ▢▢▢▢▢
• A pack of gum: often 10 pieces

When we have 10 of something, we can think of it as one group of ten instead of counting each individual item.

Let's look at how each teen number breaks down:

10 = 1 ten + 0 ones ("ten and zero ones") 11 = 1 ten + 1 one ("ten and one ones") 12 = 1 ten + 2 ones ("ten and two ones") 13 = 1 ten + 3 ones ("ten and three ones") 14 = 1 ten + 4 ones ("ten and four ones") 15 = 1 ten + 5 ones ("ten and five ones") 16 = 1 ten + 6 ones ("ten and six ones") 17 = 1 ten + 7 ones ("ten and seven ones") 18 = 1 ten + 8 ones ("ten and eight ones") 19 = 1 ten + 9 ones ("ten and nine ones") 20 = 2 tens + 0 ones ("two tens and zero ones")

A ten frame is a rectangular grid with 10 spaces arranged in 2 rows of 5. It's one of the best tools for understanding teen numbers!

For the number 14:

●●●●●  (5 in top row)
●●●●○  (4 in bottom row, 1 empty)

You can see: 1 full ten frame (10) + 4 extra ones = 14

For the number 17:

●●●●●  (5 in top row)
●●●●●  (5 in bottom row - full ten frame)

●●○○○  (2 extra ones in second ten frame)
○○○○○

You can see: 1 full ten frame (10) + 7 extra ones = 17

The way we say teen numbers gives us clues about their structure:

Clear connections:

• Thirteen = "thir-teen" (sounds like "three-ten")
• Fourteen = "four-teen" (sounds like "four-ten")
• Fifteen = "fif-teen" (sounds like "five-ten")

Tricky ones:

• Eleven and twelve don't follow the pattern (they're special!)
• But they still mean "1 ten + 1 one" and "1 ten + 2 ones"

You can build teen numbers using any objects! Here's how:

Method 1 - Grouping:

• Get 15 blocks 🧱
• Make one group of 10 blocks
• Count the leftover blocks: 5
• Say: "15 is 1 ten and 5 ones"

Method 2 - Adding:

• Start with 10 crayons 🖍️
• Add 3 more crayons
• Count the total: 13
• Say: "10 and 3 more makes 13"

You can draw teen numbers too:

For 16:

• Draw a circle with 10 dots inside (this represents 1 ten)
• Draw 6 individual dots outside the circle (these represent 6 ones)
• Write: 16 = 1 ten and 6 ones

Using Tally Marks:

• Bundle tally marks in groups of 10
• For 14: |||| |||| |||| (one bundle of 10 + 4 individual marks)

This is important: there are different ways to show the same teen number!

For the number 12:

• 1 ten and 2 ones ✓
• 12 ones ✓
• 10 + 2 ✓
• 6 + 6 ✓
• 8 + 4 ✓

All of these equal 12, but thinking in tens and ones is most helpful for understanding place value!

Once you understand that teen numbers start with 10, you can count on:

To make 13: Start with 10, count on 3 more: "10... 11, 12, 13" To make 18: Start with 10, count on 8 more: "10... 11, 12, 13, 14, 15, 16, 17, 18"

This is much easier than counting from 1 every time!

Egg cartons: A dozen eggs = 10 + 2 🥚 Crayons: A box of 16 crayons = 10 + 6 🖍️ Stickers: A sheet of 15 stickers = 10 + 5 ⭐ Books: 18 books on a shelf = 10 + 8 📚 Flowers: A bouquet of 19 flowers = 10 + 9 🌸

"How Many Ways?" Game: Find different ways to make teen numbers

• 14 = 10 + 4
• 14 = 7 + 7
• 14 = 8 + 6
• 14 = 5 + 9

"Ten Plus" Game: Someone says "ten plus four" and you answer "fourteen!"

"Break Apart" Game: Someone says "seventeen" and you answer "one ten and seven ones!"

Teen numbers connect to money too! 💰

• 1 dime (10 cents) + 3 pennies = 13 cents
• 1 dime (10 cents) + 7 pennies = 17 cents

This helps you see tens and ones in a practical way!

Understanding teen numbers prepares you for bigger numbers:

• 23 = 2 tens and 3 ones
• 47 = 4 tens and 7 ones
• 89 = 8 tens and 9 ones

Once you understand how teen numbers work, all two-digit numbers follow the same pattern!

Thinking teens are just big ones: Remember that 15 is not just "a lot of ones" - it's specifically 1 ten and 5 ones Forgetting the ten: When you see 13, don't just think "3" - think "10 and 3" Mixing up the words: Practice saying "1 ten and 6 ones" clearly so you don't mix up the numbers

Understanding tens and ones in teen numbers is like learning the secret code of how numbers work! This knowledge will help you with addition, subtraction, and understanding bigger numbers as you continue your math journey. 🚀

A number line is like a number highway 🛣️ where numbers live in order from smallest to largest! Learning to use number lines helps you see how numbers relate to each other and makes comparing numbers super easy.

A number line is a straight line with numbers marked at equal distances. It shows the order of numbers and helps you see which numbers are bigger, smaller, or equal. Think of it as a ruler for numbers!

A simple number line for 0-10 looks like this:

0---1---2---3---4---5---6---7---8---9---10

Each dash (—) represents one step between numbers, and each number has its own position.

To read a number line correctly:

1. Find the starting number (often 0, but not always!)
2. Notice the direction: usually left to right, small to large
3. Count the spaces between hash marks to see what each mark represents
4. Look for patterns in how numbers are arranged

Number lines can look different depending on what you're learning:

Horizontal (left to right):

0---1---2---3---4---5---6---7---8---9---10

Vertical (bottom to top):

10
 |
 9
 |
 8
 |
 7
 |
 6

Different starting points:

5---6---7---8---9---10---11---12---13---14---15

To find a number on a number line:

1. Start at the beginning (usually the left end)
2. Count the marks until you reach your target number
3. Point to the exact position of your number

For example, to find 7 on a 0-10 number line:

• Start at 0
• Count: 1, 2, 3, 4, 5, 6, 7
• Point to the mark labeled 7

Number lines often have missing numbers that you need to fill in. This helps you practice the counting sequence!

Example: 12, 13, ___, 15, ___, 17

• The missing numbers are 14 and 16
• You know this because you count: 12, 13, 14, 15, 16, 17

Strategy: Count from the number before the blank to figure out what's missing.

Number lines make comparing easy because of one simple rule: Numbers to the RIGHT are GREATER than numbers to the LEFT 👉

0---1---2---3---4---5---6---7---8---9---10
    ↑           ↑
    2           6

• 6 is to the right of 2, so 6 is greater than 2
• 2 is to the left of 6, so 2 is less than 6

Number lines help you find neighboring numbers:

One More: Move one step to the RIGHT

• One more than 8 is 9
• One more than 15 is 16

One Less: Move one step to the LEFT

• One less than 12 is 11
• One less than 20 is 19

Number lines show you how far apart numbers are:

0---1---2---3---4---5---6---7---8---9---10
        ↑               ↑
        3               7

• From 3 to 7: count the steps: 4, 5, 6, 7 = 4 steps
• So 7 is 4 more than 3
• Or 3 is 4 less than 7

Thermometers: Show temperature on a vertical number line 🌡️ Rulers: Measure length using a horizontal number line 📏 Clocks: Hours are arranged like a circular number line 🕐 Age: Your age increases along an invisible number line each year 🎂 Addresses: House numbers often follow number line order on streets 🏠

"Number Line Hop": Place a number line on the floor and hop to different numbers when called out!

"What's Missing?": Cover some numbers on a number line and guess what's hidden.

"Greater or Less?": Point to two numbers and decide which is greater using the "right is greater" rule.

"How Far?": Count the steps between two numbers to find the distance.

You can make your own number lines:

Tape Number Line: Use masking tape on the floor with number cards Paper Number Line: Draw a line and mark numbers with a ruler Outdoor Number Line: Use chalk on sidewalk or playground Human Number Line: Have children stand in order holding number cards

Number lines help solve problems:

"Maria started at 5 and took 3 steps forward. Where did she end up?"

• Start at 5 on the number line
• Count 3 steps right: 6, 7, 8
• She ended up at 8!

"Tom was at 12 and went back 4 steps. Where is he now?"

• Start at 12 on the number line
• Count 4 steps left: 11, 10, 9, 8
• He's at 8!

Remember these important rules:

• Moving RIGHT = numbers get BIGGER (increasing)
• Moving LEFT = numbers get SMALLER (decreasing)
• Same position = EQUAL numbers

Counting spaces instead of marks: Make sure you're pointing to the number marks, not the spaces between them Starting from 1 instead of 0: Some number lines start at 0, some at 1 - always check! Mixing up left and right: Remember "Right is Greater" (bigger numbers) Forgetting direction: Always check which way the numbers increase

Number lines prepare you for:

• Addition: Moving right on the number line
• Subtraction: Moving left on the number line
• Negative numbers: Number lines can extend to the left of 0
• Fractions: Number lines can show parts between whole numbers
• Measurement: Understanding scale and distance

Number lines help you develop number sense by:

• Seeing the order and pattern of numbers
• Understanding that numbers have neighbors
• Recognizing that some numbers are "closer together" or "farther apart"
• Building mental pictures of where numbers "live"

Number lines are powerful tools that will help you throughout your math journey. They make abstract number relationships visible and concrete! 🎯

Addition and subtraction are two of the most important math operations you'll ever learn! They help you solve problems every single day, from counting your toys 🧸 to figuring out how many crackers you'll have left after sharing with friends 🍪.

Addition means putting together or combining two or more groups to find the total. When you add, you're finding out "how many altogether?"

Think of addition like:

• Putting two puzzle pieces together to make a bigger picture 🧩
• Adding more blocks to your tower to make it taller 🧱
• Getting more stickers to add to your collection ⭐

The plus sign (+) tells you to add, and the equals sign (=) tells you the total.

Example 1: You have 3 red apples 🍎🍎🍎 and someone gives you 2 green apples 🍏🍏. How many apples do you have altogether?

• Start with 3 red apples
• Add 2 green apples
• Count all together: 🍎🍎🍎🍏🍏 = 5 apples total
• We write: 3 + 2 = 5

Subtraction means taking away or separating from a group to find what's left. When you subtract, you're finding out "how many are left?" or "what's the difference?"

Think of subtraction like:

• Eating some of your cookies and seeing how many are left 🍪
• Giving away some toys and counting what remains 🧸
• Using up some crayons and seeing how many you still have 🖍️

The minus sign (-) tells you to subtract or take away.

Example 1: You start with 8 stickers ⭐⭐⭐⭐⭐⭐⭐⭐ and you give 3 stickers to your friend. How many stickers do you have left?

• Start with 8 stickers
• Take away 3 stickers: ⭐⭐⭐
• Count what's left: ⭐⭐⭐⭐⭐ = 5 stickers
• We write: 8 - 3 = 5

Addition happens in different ways:

Put Together/Take Apart:

• You have 4 toy cars 🚗🚗🚗🚗 and 2 toy trucks 🚚🚚
• Put them together: 4 + 2 = 6 vehicles total

Add To:

• You start with 5 marbles 🔵🔵🔵🔵🔵
• Your friend gives you 3 more marbles 🔵🔵🔵
• Now you have: 5 + 3 = 8 marbles

Both Addends Unknown:

• You have 7 crayons total 🖍️
• Some are red and some are blue
• Maybe 3 red + 4 blue = 7, or 2 red + 5 blue = 7!

Subtraction also happens in different ways:

Take From:

• You start with 9 grapes 🍇🍇🍇🍇🍇🍇🍇🍇🍇
• You eat 4 grapes 🍇🍇🍇🍇
• You have left: 9 - 4 = 5 grapes

Take Apart:

• You have 6 flowers 🌸🌸🌸🌸🌸🌸
• 2 are pink and the rest are yellow
• How many are yellow? 6 - 2 = 4 yellow flowers

Compare:

• Sarah has 7 books 📚
• Tom has 4 books 📚
• How many more books does Sarah have? 7 - 4 = 3 more books

Manipulatives (objects you can touch and move) help you see what's happening in addition and subtraction:

Counting Bears: Use teddy bear counters to physically put together or take apart groups Blocks: Build with blocks, adding more or removing some Beans or Buttons: Small objects are perfect for grouping and counting Fingers: Your hands are always available for counting! 🤚

You can draw pictures to show math problems:

Addition Drawing:

• Draw 3 circles: ○○○
• Draw 4 more circles: ○○○○
• Count all circles: ○○○○○○○ = 7 total
• Write: 3 + 4 = 7

Subtraction Drawing:

• Draw 6 stars: ⭐⭐⭐⭐⭐⭐
• Cross out 2 stars: ✗✗⭐⭐⭐⭐
• Count what's not crossed out: 4 stars left
• Write: 6 - 2 = 4

Addition on a Number Line (moving RIGHT): To solve 4 + 3:

• Start at 4 on the number line
• Jump 3 spaces to the right: 5, 6, 7
• You land on 7, so 4 + 3 = 7

Subtraction on a Number Line (moving LEFT): To solve 8 - 2:

• Start at 8 on the number line
• Jump 2 spaces to the left: 7, 6
• You land on 6, so 8 - 2 = 6

An equation is like a math sentence that shows what equals what:

• 5 + 2 = 7 ("Five plus two equals seven")
• 9 - 4 = 5 ("Nine minus four equals five")

Equations can be written different ways:

• 3 + 4 = 7 ✓
• 7 = 3 + 4 ✓ (same thing, just flipped!)
• 4 + 3 = 7 ✓ (order doesn't matter in addition!)

Addition and subtraction are opposite operations - they undo each other:

Fact Family Example: 3, 4, and 7

• 3 + 4 = 7 (putting together)
• 4 + 3 = 7 (same thing, different order)
• 7 - 3 = 4 (taking apart)
• 7 - 4 = 3 (taking apart the other way)

All four equations use the same three numbers!

Math operations help solve everyday problems:

Snack Time Problem: "You have 5 crackers. Your mom gives you 3 more. How many crackers do you have now?"

• Start with: 5 crackers
• Add: 3 more crackers
• Total: 5 + 3 = 8 crackers

Sharing Problem: "You have 10 stickers. You give 4 to your sister. How many do you have left?"

• Start with: 10 stickers
• Take away: 4 stickers
• Left: 10 - 4 = 6 stickers

As you work with addition and subtraction, you'll notice patterns:

• Adding 0 doesn't change a number: 5 + 0 = 5
• Subtracting 0 doesn't change a number: 7 - 0 = 7
• Adding 1 gives you the next number: 6 + 1 = 7
• Subtracting 1 gives you the previous number: 6 - 1 = 5

There's often more than one way to solve the same problem:

To solve 6 + 2:

• Count all: 1,2,3,4,5,6,7,8
• Count on: Start at 6, count "7, 8"
• Use fingers: Hold up 6 fingers, then 2 more
• Draw pictures: Draw 6 dots, then 2 more dots
• Use a number line: Start at 6, jump right 2 spaces

All methods give the same answer: 8!

Remember:

• It's okay to use objects, drawings, or fingers
• Taking your time to understand is more important than being fast
• Every method that gets the right answer is good
• Practice makes these operations feel more natural

Addition and subtraction are the building blocks for all future math. Understanding these concepts deeply now will help you succeed in mathematics for years to come! 🌟

Now that you understand what addition and subtraction mean, it's time to become a strategy expert! 🎯 Having reliable strategies means you can solve problems quickly and confidently, choosing the best method for each situation.

A reliable strategy is a method you can use consistently to get the correct answer. It's like having a toolbox 🧰 - you want tools that work every time and help you choose the right one for each job!

Reliable strategies are:

• Accurate: They give you the right answer
• Efficient: They don't take too long
• Flexible: You can use them in different situations
• Understandable: You know why they work

Count All Strategy: This is often the first strategy children learn. You count every object to find the total.

For 4 + 3:

• Count the first group: "1, 2, 3, 4"
• Count the second group: "1, 2, 3"
• Count all together: "1, 2, 3, 4, 5, 6, 7"
• Answer: 7

Count On Strategy (More Efficient!): Instead of counting everything, start with the larger number and count on.

For 4 + 3:

• Start with the larger number: 4
• Count on 3 more: "5, 6, 7"
• Answer: 7

This is faster because you don't recount the first group!

Count Back Strategy: For subtraction, start with the larger number and count backward.

For 8 - 3:

• Start at 8
• Count back 3: "7, 6, 5"
• Answer: 5

Here's a super important discovery: it's always easier to start with the bigger number!

Less Efficient: 2 + 7

• Start with 2, count on 7: "3, 4, 5, 6, 7, 8, 9" (7 counts)

More Efficient: 7 + 2

• Start with 7, count on 2: "8, 9" (2 counts)

Both give the same answer (9), but starting with 7 is much faster! 🏃‍♀️

The commutative property means you can add numbers in any order and get the same answer:

• 3 + 5 = 8
• 5 + 3 = 8

This is incredibly useful! If you see 2 + 8, you can think "8 + 2" and count on just 2 from 8.

Important Note: This only works for addition, not subtraction!

• 8 - 3 = 5 ✓
• 3 - 8 = ? (This doesn't work in kindergarten!)

Your fingers are amazing math tools! 🖐️🖐️ Here are smart ways to use them:

For 6 + 2:

• Hold up 6 fingers (use both hands)
• Add 2 more fingers
• Count the total: 8

Strategy Tip: To show 6 quickly, hold up 5 fingers on one hand and 1 on the other (5+1=6).

Doubles are addition facts where both numbers are the same:

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

Doubles are easy to remember and help with other facts:

• If 4 + 4 = 8, then 4 + 5 = 9 (one more than the double)
• If 3 + 3 = 6, then 3 + 2 = 5 (one less than the double)

Making 10 is one of the most powerful strategies! Since 10 is such an important number, facts that make 10 become automatic:

• 1 + 9 = 10
• 2 + 8 = 10
• 3 + 7 = 10
• 4 + 6 = 10
• 5 + 5 = 10

These "number bonds to 10" help with bigger numbers later!

A fact family is a group of related addition and subtraction facts that use the same three numbers.

Example: 2, 3, 5 fact family

• 2 + 3 = 5
• 3 + 2 = 5
• 5 - 2 = 3
• 5 - 3 = 2

When you know one fact, you automatically know the others! 🤝

Ten frames help you see number relationships:

For 7 + 2:

●●●●●●●○○○  (7 filled, 3 empty)
●●○○○○○○○○  (2 more filled)

You can see that 7 + 2 fills 9 spaces, so 7 + 2 = 9!

Count Back (for small numbers): For 6 - 2: Start at 6, count back 2: "5, 4"

Count Up (when numbers are close): For 8 - 6: "How many from 6 to 8?" Count up: "7, 8" = 2

Take Apart: For 9 - 4: Think "9 is 4 and what?" The answer is 5!

Smart mathematicians choose strategies based on the numbers:

For 8 + 1: Just think "one more than 8 is 9" (no counting needed!) For 5 + 5: Use the doubles fact For 3 + 6: Flip to 6 + 3 and count on 3 from 6 For 10 - 1: Think "one less than 10 is 9" For 8 - 6: Count up from 6 to 8

The goal is procedural reliability - being able to solve problems accurately using strategies you trust. Eventually, some facts become automatic (you just know them), but strategies help when you need them!

Practice Activities:

• Flash cards with strategy reminders
• Dice games for practicing combinations
• Dominoes for seeing number combinations visually
• Card games that practice fact families

Always counting from 1: Learn to count on from the larger number Not checking work: Use a different strategy to verify your answer Giving up too quickly: If one strategy is hard, try another! Rushing: Accuracy is more important than speed in kindergarten

At the Store: "I have 5 dollars and need 8 dollars for a toy. How much more do I need?" (8 - 5 = 3)

Playing Games: "I have 4 points and just scored 3 more. What's my total?" (4 + 3 = 7)

Cooking: "The recipe needs 6 cups and we've added 4. How many more do we need?" (6 - 4 = 2)

As you practice, your strategy toolbox grows:

• Level 1: Count all, use fingers
• Level 2: Count on, count back
• Level 3: Use doubles, make 10, fact families
• Level 4: Automatic recall for some facts

Every mathematician uses strategies! Even adults use mental math strategies for larger numbers. Building reliable strategies now gives you a strong foundation for all future math learning.

Remember: The best strategy is the one that makes sense to you and helps you get the right answer! 🎯