Mathematics: Algebraic Reasoning – Grade K

Numbers are like friends who love to play together! Some number friends have a very special relationship – when they come together, they always make exactly 10. Let's explore these magical number partnerships! 🤝

Ten is like a special treasure box that can hold different combinations of smaller numbers. When you have 10 fingers ✋✋, you can show this in many ways! You might hold up 6 fingers on one hand and 4 on the other. Or maybe 7 fingers and 3 fingers. No matter how you split them up, you still have 10 fingers total!

Let's discover all the number friends that make 10:

• 1 and 9 are friends that make 10 (1 + 9 = 10) 🍎🍎🍎🍎🍎🍎🍎🍎🍎➕🍎
• 2 and 8 are friends that make 10 (2 + 8 = 10) 🎈🎈➕🎈🎈🎈🎈🎈🎈🎈🎈
• 3 and 7 are friends that make 10 (3 + 7 = 10) 🌟🌟🌟➕🌟🌟🌟🌟🌟🌟🌟
• 4 and 6 are friends that make 10 (4 + 6 = 10) 🧸🧸🧸🧸➕🧸🧸🧸🧸🧸🧸
• 5 and 5 are friends that make 10 (5 + 5 = 10) 🍪🍪🍪🍪🍪➕🍪🍪🍪🍪🍪

Here's something really cool about number friends: it doesn't matter which friend comes first! If 3 + 7 = 10, then 7 + 3 = 10 too. It's like saying "Maria and Jake" or "Jake and Maria" – they're still the same two friends!

This is super helpful because once you learn that 4 + 6 = 10, you automatically know that 6 + 4 = 10. You've learned two facts at the same time! 🎉

A ten frame is like a special parking lot for numbers! It has exactly 10 spaces arranged in two rows of 5. When you put counters in a ten frame, you can easily see how many more you need to make 10.

Imagine you have 6 red counters in your ten frame. How many empty spaces are left? 4 spaces! So 6 + 4 = 10. If you fill those 4 spaces with blue counters, you can see that 6 red + 4 blue = 10 total counters.

Here's where it gets even more interesting! When you know that 3 + 7 = 10, you also know subtraction facts:

• If you have 10 and take away 3, you're left with 7 (10 - 3 = 7)
• If you have 10 and take away 7, you're left with 3 (10 - 7 = 3)

It's like having a bag of 10 marbles 🔴🔴🔴🔴🔴🔴🔴⚪⚪⚪. If 7 marbles are red and 3 are white, then taking away the 3 white marbles leaves you with 7 red marbles!

You can practice making 10 everywhere! Use your fingers, count toys, or even use snacks. If you have 8 goldfish crackers 🐟, how many more do you need to make 10? Try it with different starting numbers and see if you can find the missing partner quickly!

Remember, learning these number partnerships will help you become faster and more confident with math. When you know that 6 + 4 = 10, solving problems becomes as easy as recognizing old friends! 👫

Just like you can build the same tower with different colored blocks, you can make the same number using different combinations! Every number has its own special family of addition combinations. Let's explore how to become number combination detectives! 🔍

Think about the number 6. How many different ways can you make it using two smaller numbers? Let's find out!

• 1 + 5 = 6 (1 red block + 5 blue blocks) 🔴🔵🔵🔵🔵🔵
• 2 + 4 = 6 (2 red blocks + 4 blue blocks) 🔴🔴🔵🔵🔵🔵
• 3 + 3 = 6 (3 red blocks + 3 blue blocks) 🔴🔴🔴🔵🔵🔵
• 4 + 2 = 6 (4 red blocks + 2 blue blocks) 🔴🔴🔴🔴🔵🔵
• 5 + 1 = 6 (5 red blocks + 1 blue block) 🔴🔴🔴🔴🔴🔵

Wow! The number 6 can be made in 5 different ways using addition. Each way is like a different outfit for the same number!

Some numbers have a very special way to be made – using doubles! Doubles are when both numbers in the addition are exactly the same:

• 2 + 2 = 4 (like having 2 cookies in each hand) 🍪🍪 + 🍪🍪
• 3 + 3 = 6 (like having 3 stickers on each page) ⭐⭐⭐ + ⭐⭐⭐
• 4 + 4 = 8 (like having 4 crayons in each box) 🖍️🖍️🖍️🖍️ + 🖍️🖍️🖍️🖍️
• 5 + 5 = 10 (like having 5 fingers on each hand) ✋ + ✋

Doubles are super easy to remember because both parts are the same!

Let's look at some smaller numbers and find all their combinations:

Number 4:

• 1 + 3 = 4 🐸 + 🐸🐸🐸
• 2 + 2 = 4 🐸🐸 + 🐸🐸
• 3 + 1 = 4 🐸🐸🐸 + 🐸

Number 5:

• 1 + 4 = 5 🦋 + 🦋🦋🦋🦋
• 2 + 3 = 5 🦋🦋 + 🦋🦋🦋
• 3 + 2 = 5 🦋🦋🦋 + 🦋🦋
• 4 + 1 = 5 🦋🦋🦋🦋 + 🦋

Remember how we learned that 3 + 7 = 10 and 7 + 3 = 10? This works for all numbers! When you know that 2 + 5 = 7, you automatically know that 5 + 2 = 7 too. It's like knowing that "peanut butter and jelly" makes the same sandwich as "jelly and peanut butter"! 🥜🍇

You can discover number combinations using lots of different things:

With two-color counters: Flip some counters to show different color combinations. If you have 7 counters total, you might see 3 red and 4 yellow, or 1 red and 6 yellow!

With toy animals: If you have 8 toy animals, you could have 5 dogs and 3 cats, or 2 dogs and 6 cats, or any other combination that adds up to 8!

With drawings: Draw circles, stars, or smiley faces. For the number 9, you might draw 4 circles and 5 stars, or 7 circles and 2 stars!

Here's a fun game to practice: Have someone put some objects under a cup and some objects next to the cup. If there are 6 objects total and you can see 4 objects, how many are hidden under the cup? 2 objects! Because 4 + 2 = 6.

Learning different ways to make numbers helps your brain become flexible and strong! When you see the number 8, your brain might think "Oh, that could be 3 + 5, or 4 + 4, or 6 + 2!" This makes you faster at solving math problems and helps you understand that math is full of interesting patterns and relationships.

The more combinations you discover, the more you'll start to see patterns. Even numbers always have a double (like 4 = 2 + 2), while odd numbers don't. Some numbers have more combinations than others. These patterns will help you become a math detective! 🕵️‍♀️

Math isn't just numbers on paper – it's everywhere around you! Every day, you solve addition and subtraction problems without even realizing it. Let's learn how to become everyday math problem solvers! 🌍

In the Classroom: You have 3 pencils in your pencil box 📝📝📝. Your teacher gives you 2 more pencils 📝📝. How many pencils do you have now? You can solve this by:

• Using objects: Count out 3 real pencils, add 2 more, then count them all
• Drawing pictures: Draw 3 pencils, draw 2 more, then count your drawings
• Writing an equation: 3 + 2 = 5

At Snack Time: You have 8 goldfish crackers 🐟🐟🐟🐟🐟🐟🐟🐟. You eat 3 of them 🐟🐟🐟. How many crackers are left? You can solve this by:

• Using objects: Start with 8 crackers, take away 3, count what's left
• Drawing and crossing out: Draw 8 crackers, cross out 3, count the remaining ones
• Writing an equation: 8 - 3 = 5

When you hear a story problem, you become a math detective! 🔍 Here's what to listen for:

Action Words for Addition:

• "More" (You had 4 stickers, then got 3 more)
• "Altogether" (How many toys do you have altogether?)
• "In total" (How many books are there in total?)
• "Add" or "Put together" (Add the red and blue blocks)

Action Words for Subtraction:

• "Take away" or "Remove" (Take away 2 cookies)
• "Left" or "Remaining" (How many are left?)
• "Ate," "Lost," "Gave away" (She ate 3 grapes)
• "How many more" (How many more do you need?)

Way 1: Using Real Objects 🧸 When Maria has 2 toy cars and gets 4 more toy cars, you can:

• Get 2 toy cars and put them on your desk
• Add 4 more toy cars to the group
• Count all the cars: 1, 2, 3, 4, 5, 6. Maria has 6 toy cars!

Way 2: Drawing Pictures 🎨 For the same problem, you can:

• Draw 2 cars: 🚗🚗
• Draw 4 more cars: 🚗🚗🚗🚗
• Count all your drawn cars: 6 cars total!

Way 3: Writing Equations ✏️ You can write the math sentence: 2 + 4 = 6 This tells the whole story in just one line!

Example: Jake had 9 balloons 🎈🎈🎈🎈🎈🎈🎈🎈🎈. The wind blew away 4 balloons 💨. How many balloons does Jake have left?

Using objects: Start with 9 balloons, take away 4, count what's left: 5 balloons Drawing: Draw 9 balloons, cross out 4, count the uncrossed ones: 5 balloons Equation: 9 - 4 = 5

Here's something amazing: addition and subtraction problems are like opposite twins! 👯‍♀️

If you know that 5 + 3 = 8, then you also know:

• 8 - 3 = 5 (take away 3 from 8, get 5)
• 8 - 5 = 3 (take away 5 from 8, get 3)

This connection helps you check your answers! If you solve 7 - 2 = 5, you can check by seeing if 2 + 5 = 7. If it does, you got it right! ✅

Step 1: Listen or Read Carefully 👂 What is happening in the story? Who are the characters? What are they doing with the objects?

Step 2: Find the Numbers 🔢 What numbers do you hear? Which number tells you how many you start with? Which number tells you what changes?

Step 3: Decide: Adding or Subtracting? 🤔 Are things coming together (addition) or going away (subtraction)?

Step 4: Solve Using Your Favorite Way ⭐ Use objects, drawings, or equations – whatever helps you think best!

Step 5: Check Your Answer ✓ Does your answer make sense? Can you tell the story using your answer?

At the Park: You see 6 birds on a tree 🐦🐦🐦🐦🐦🐦. Then 2 more birds fly to the tree 🐦🐦. How many birds are on the tree now?

In Your Room: You have 10 toy blocks 🧱🧱🧱🧱🧱🧱🧱🧱🧱🧱. You use 7 blocks to build a tower. How many blocks are left?

At the Store: Mom buys 4 apples 🍎🍎🍎🍎 and 5 oranges 🍊🍊🍊🍊🍊. How many pieces of fruit did she buy altogether?

Remember, every time you solve a real-world math problem, you're using the same skills that mathematicians use! You're learning to see math patterns in the world around you and becoming a confident problem solver. The more you practice, the easier it becomes to spot math problems and solve them quickly! 🌟

The equal sign (=) is one of the most important symbols in all of mathematics! But what does it really mean? Let's discover the secret power of the equal sign and learn how to be equation detectives! 🔍⚖️

Imagine a balance scale ⚖️ – you know, the kind where you put things on each side to see if they weigh the same. The equal sign works exactly like that! When we write 3 + 4 = 7, we're saying that the left side (3 + 4) weighs exactly the same as the right side (7).

Let's test this with real objects:

• Left side: 3 blocks + 4 blocks = 7 blocks total
• Right side: 7 blocks
• Are they the same? Yes! So 3 + 4 = 7 is TRUE ✅

An equation is true when both sides have exactly the same value – like having the same number of objects on each side of our balance scale!

True Equation Example: 5 + 2 = 7

• Left side: 🍎🍎🍎🍎🍎 + 🍎🍎 = 7 apples
• Right side: 🍎🍎🍎🍎🍎🍎🍎 = 7 apples
• Both sides have 7 apples, so this equation is TRUE! ✅

False Equation Example: 4 + 3 = 9

• Left side: 🐻🐻🐻🐻 + 🐻🐻🐻 = 7 bears
• Right side: 🐻🐻🐻🐻🐻🐻🐻🐻🐻 = 9 bears
• The sides don't match (7 ≠ 9), so this equation is FALSE! ❌

Method 1: Counting and Comparing For the equation 6 - 2 = 4:

1. Start with 6 toy cars 🚗🚗🚗🚗🚗🚗
2. Take away 2 cars 🚗🚗
3. Count what's left: 🚗🚗🚗🚗 (4 cars)
4. Compare with the right side: 4
5. They match! The equation is TRUE ✅

Method 2: Building Both Sides For the equation 2 + 5 = 7:

1. Build the left side: 2 red blocks + 5 blue blocks = 7 blocks total
2. Build the right side: 7 green blocks
3. Count both sides: Left = 7, Right = 7
4. They're the same! The equation is TRUE ✅

You don't always need real objects – you can draw pictures too!

For the equation 8 = 5 + 3:

• Draw the left side: 🌟🌟🌟🌟🌟🌟🌟🌟 (8 stars)
• Draw the right side: 🌟🌟🌟🌟🌟 + 🌟🌟🌟 (5 + 3 stars)
• Count both sides: Left = 8, Right = 8
• They match! This equation is TRUE ✅

Sometimes equations are written in ways that might surprise you! The equal sign doesn't always come at the end.

Standard form: 4 + 2 = 6 Flipped form: 6 = 4 + 2 Both are TRUE! The equal sign just shows that both sides are the same.

Think of it like saying "Maria has the same number of stickers as Jake." It doesn't matter if we say "Maria = Jake" or "Jake = Maria" – they still have the same amount!

Here's something very important: The equal sign does NOT mean "find the answer" or "do the math." It means "is the same as" or "has the same value as."

❌ Wrong thinking: "3 + 4 = ...now I need to find the answer" ✅ Correct thinking: "3 + 4 has the same value as whatever comes after the equal sign"

Sometimes you'll see equations that look confusing, but you can still check if they're true!

Example: 10 = 7 + 3 This might look backwards, but let's check:

• Left side: 10
• Right side: 7 + 3 = 10
• Do they match? Yes! So it's TRUE ✅

Example: 5 + 1 = 4 + 3 Both sides have addition! Let's solve both sides:

• Left side: 5 + 1 = 6
• Right side: 4 + 3 = 7
• Do they match? No! (6 ≠ 7) So it's FALSE ❌

When someone gives you an equation to check, follow these detective steps:

1. Look at both sides of the equal sign
2. Count or calculate the value of the left side
3. Count or calculate the value of the right side
4. Compare the two values
5. Decide: If they're the same, it's TRUE ✅. If they're different, it's FALSE ❌
6. Prove it using objects, drawings, or explaining your thinking

Situation: Emma says there are 3 red balloons and 4 blue balloons, which equals 8 balloons total. Is Emma correct?

Equation: 3 + 4 = 8 Check: 3 + 4 = 7, not 8 Answer: Emma is incorrect! The true equation would be 3 + 4 = 7 ✅

Understanding what the equal sign really means will help you throughout your math journey! It will help you:

• Solve more complex problems as you grow
• Understand that math is about relationships, not just answers
• Think more flexibly about numbers and equations
• Check your own work to make sure it makes sense

Remember, every time you see an equal sign, think "balance scale" or "same as." This will help you become a stronger mathematical thinker! 🧠💪