Mathematics: Algebraic Reasoning – Grade 1

When you have three or more numbers to add together, there are special tricks that can make your job much easier! These tricks are like having superpowers for math. 🦸‍♀️

Did you know that numbers love to move around? When you're adding, you can put the numbers in any order you want, and you'll always get the same answer! This is like having three friends - Maria, Jake, and Sam - line up for lunch. No matter what order they stand in, you still have the same three friends.

For example:

• $3 + 5 + 2 = 10$
• $5 + 3 + 2 = 10$
• $2 + 5 + 3 = 10$

See? Same numbers, same answer, just in different orders! 🔄

One of the most powerful tricks is looking for ways to make 10. Our number system loves the number 10, so making 10 first often makes adding much easier!

Let's try $8 + 7 + 2$:

• First, look for numbers that make 10: $8 + 2 = 10$
• Then add the remaining number: $10 + 7 = 17$
• So $8 + 7 + 2 = 17$! 🎯

Another example with $6 + 3 + 4$:

• Look for the 10: $6 + 4 = 10$
• Add what's left: $10 + 3 = 13$
• Answer: $6 + 3 + 4 = 13$

Sometimes you'll see numbers that are the same or almost the same. These are called doubles! Doubles are easy to remember:

• $4 + 4 = 8$
• $5 + 5 = 10$
• $6 + 6 = 12$

For near doubles (numbers that are close), you can use doubles to help:

• $5 + 6$ is like $5 + 5 + 1 = 10 + 1 = 11$
• $7 + 8$ is like $7 + 7 + 1 = 14 + 1 = 15$

Let's practice with some examples that use the rich information from your curriculum:

Example 1: Maria has $2$ apples 🍎🍎, her brother gives her $6$ more apples 🍎🍎🍎🍎🍎🍎, and her mom gives her $4$ more apples 🍎🍎🍎🍎. How many apples does Maria have altogether?

• We need to add: $2 + 6 + 4$
• Look for ways to make 10: $2 + 4 = 6$ (not 10), but $6 + 4 = 10$
• So: $6 + 4 = 10$, then $10 + 2 = 12$
• Maria has $12$ apples! 🍎

Example 2: In a dice game, you roll three dice and get $3$, $5$, and $4$. What's your total score?

• Add: $3 + 5 + 4$
• Look for 10: $3 + 5 = 8$ (not 10), but we can try $3 + 4 = 7$ (not 10)
• Try $5 + 4 = 9$ (close to 10!), so $5 + 4 + 1 = 10$
• Wait, let's be systematic: $3 + 5 = 8$, then $8 + 4 = 12$
• Total score: $12$ points! 🎲

These strategies work because of special properties of addition:

1. Commutative Property: You can add numbers in any order
2. Associative Property: You can group numbers in different ways

These aren't just tricks - they're mathematical superpowers that will help you for your entire life! The more you practice, the faster and more confident you'll become at adding numbers. 💪

Math isn't just numbers on paper - it's everywhere around you! From counting toys to sharing snacks, you use math every day. Let's learn how to turn real-world situations into math problems and solve them like a pro! 🌟

Before you start solving, you need to be a math detective 🔍. Ask yourself these important questions:

• What is happening in this problem?
• What do I already know?
• What am I trying to find out?
• Do I need to put things together (add) or take things away (subtract)?

There are three main ways to solve real-world problems, and you can use any one or even combine them!

1. Using Objects 🧸 You can use real things like blocks, toys, or your fingers to act out the problem.

2. Drawing Pictures 🎨 You can draw simple pictures to show what's happening in the problem.

3. Writing Math Sentences ✏️ You can write addition or subtraction equations to represent the problem.

Let's look at some addition problems from your everyday life:

Example 1: The Toy Collection Trevor had $16$ toy cars 🚗. He went to the toy store with his father. His father bought him $4$ more toy cars. How many toy cars does Trevor have now?

• What's happening? Trevor is getting more cars
• What we know: Started with $16$ cars, got $4$ more
• What we're finding: Total number of cars
• Operation: Addition (putting groups together)
• Math sentence: $16 + 4 = 20$
• Answer: Trevor has $20$ toy cars! 🚗🚗

Example 2: The Animal Farm There are chickens 🐔, sheep 🐑, and pigs 🐷 in a barn. There are $5$ chickens, $7$ sheep, and $5$ pigs. How many animals are in the barn altogether?

• What's happening? Counting all animals together
• What we know: $5$ chickens, $7$ sheep, $5$ pigs
• What we're finding: Total number of animals
• Operation: Addition (three groups)
• Strategy: Look for doubles! $5 + 5 = 10$, then $10 + 7 = 17$
• Math sentence: $5 + 7 + 5 = 17$
• Answer: There are $17$ animals in the barn! 🐔🐑🐷

Subtraction problems often involve taking away, giving away, or finding how many are left:

Example 3: The Stuffed Animal Collection Elliana had $19$ stuffed animals 🧸. She gave some away to her little cousin. Now Elliana has $11$ stuffed animals. How many stuffed animals did she give away?

• What's happening? Elliana gave some animals away
• What we know: Started with $19$, now has $11$
• What we're finding: How many she gave away
• Operation: Subtraction (finding the difference)
• Math sentence: $19 - ? = 11$
• Think: What number plus $11$ equals $19$? $11 + 8 = 19$
• Answer: Elliana gave away $8$ stuffed animals! 🧸

There are several types of problems you might see:

1. Result Unknown (easiest type)

• "Maria has $3$ stickers. Her friend gives her $5$ more. How many does she have now?"
• Math: $3 + 5 = ?$

2. Change Unknown (trickier)

• "Jamal had $12$ marbles. After playing a game, he has $8$ marbles. How many did he lose?"
• Math: $12 - ? = 8$

3. Start Unknown (trickiest)

• "Sophia bought some pencils. Her teacher gave her $6$ more. Now she has $14$ pencils. How many did she buy?"
• Math: $? + 6 = 14$

Here's your step-by-step guide to solving any real-world problem:

Step 1: Read the problem carefully (or listen if someone reads it to you) Step 2: Figure out what's happening - are things being combined or separated? Step 3: Identify what you know and what you need to find Step 4: Choose your method: objects, pictures, or math sentences Step 5: Solve the problem Step 6: Check if your answer makes sense in the real world

After solving, always ask yourself: "Does this answer make sense?"

• If someone gives you more toys, you should have more than you started with
• If you give toys away, you should have fewer than you started with
• Your answer shouldn't be bigger than the largest number in the problem (unless you're adding)
• Think about whether your answer seems reasonable for the situation

Remember, being a good problem solver takes practice! The more real-world problems you solve, the better you'll get at spotting what kind of math to use. Soon, you'll be solving problems automatically! 🎯

Get ready to learn one of the coolest math tricks ever! Did you know that every subtraction problem is secretly an addition problem in disguise? It's like having a math superpower! 🦸‍♂️

Addition and subtraction are called inverse operations because they are opposites that undo each other. Think of them like putting on your shoes and taking off your shoes - they do opposite things!

• If you put on your shoes and then take them off, you're back where you started
• If you add a number and then subtract the same number, you're back where you started

Every subtraction problem can be turned into an addition problem! Here's how:

When you see: $12 - 7 = ?$ You can think: "What number plus $7$ equals $12$?" So it becomes: $7 + ? = 12$

Let's try this magic with real examples:

Example 1: Katina has $14$ grapes 🍇. She gives $8$ of them to her brother Kevin. How many grapes does she have left?

• Subtraction way: $14 - 8 = ?$
• Addition magic: "What number plus $8$ equals $14$?"
• New problem: $8 + ? = 14$
• Thinking: I know that $8 + 6 = 14$
• Answer: Katina has $6$ grapes left! 🍇

Fact families are groups of related addition and subtraction facts that use the same three numbers. They're like number families where everyone is related!

For the numbers $6$, $7$, and $13$, the fact family is:

• $6 + 7 = 13$
• $7 + 6 = 13$
• $13 - 6 = 7$
• $13 - 7 = 6$

See how they're all connected? If you know one fact, you can figure out all the others! 🔗

Imagine numbers as parts and wholes:

• The whole is the biggest number
• The parts are the smaller numbers that make up the whole

For $11 - 5 = ?$:

• The whole is $11$
• One part is $5$
• The missing part is what we're looking for
• Think: "$5$ plus what equals $11$?"
• Answer: $5 + 6 = 11$, so $11 - 5 = 6$

Example 2: There were $15$ cookies 🍪 on a plate. After the party, there are $9$ cookies left. How many cookies were eaten?

• Original problem: $15 - ? = 9$
• Addition magic: "What number plus $9$ equals $15$?"
• New problem: $9 + ? = 15$
• Solution: $9 + 6 = 15$
• Answer: $6$ cookies were eaten! 🍪

Example 3: A puzzle has $20$ pieces. If $12$ pieces are already connected, how many pieces are still loose?

• Subtraction: $20 - 12 = ?$
• Addition thinking: "$12$ plus what equals $20$?"
• Addition problem: $12 + ? = 20$
• Solution: $12 + 8 = 20$
• Answer: $8$ pieces are still loose! 🧩

Sometimes addition is easier than subtraction! When you have a hard subtraction problem, you can:

1. Turn it into addition - often easier to think about
2. Use what you already know - if you know $7 + 5 = 12$, then you know $12 - 7 = 5$
3. Check your work - use the opposite operation to verify your answer
4. Build confidence - more ways to solve problems means more success!

After solving any subtraction problem, you can check your answer with addition:

• If $15 - 9 = 6$, then check: $6 + 9 = 15$ ✓
• If $12 - 4 = 8$, then check: $8 + 4 = 12$ ✓

This checking strategy helps you catch mistakes and builds your confidence! 💪

Sometimes students get confused about which numbers to use. Remember:

• In $12 - 7 = ?$, think "$7$ plus what equals $12$?"
• NOT "$12$ plus what equals $7$?" (This would be wrong!)

The key is to keep the same numbers but change how you think about the problem. You're not changing the math - just changing how you approach it to make it easier! 🧠

Welcome to Math Detective Academy! 🕵️‍♀️ Your job is to investigate math equations and determine if they're telling the truth or not. Just like a real detective looks for clues, you'll look for mathematical clues to solve the case!

The equal sign ($=$) is one of the most important symbols in math. But what does it really mean?

Many people think the equal sign means "the answer is coming next," but that's not quite right! The equal sign actually means:

• "The same as"
• "Is equal to"
• "Has the same value as"

Think of the equal sign like a balance scale ⚖️. Whatever is on the left side must weigh exactly the same as whatever is on the right side for the scale to balance.

Let's examine some equations like a detective:

Case 1: $8 = 8$

• Left side: $8$
• Right side: $8$
• Verdict: TRUE! ✅ Both sides have the same value.

Case 2: $9 - 1 = 7$

• Left side: $9 - 1 = 8$
• Right side: $7$
• Verdict: FALSE! ❌ $8$ does not equal $7$.

Case 3: $5 + 2 = 2 + 5$

• Left side: $5 + 2 = 7$
• Right side: $2 + 5 = 7$
• Verdict: TRUE! ✅ Both sides equal $7$.

Case 4: $1 = 9 - 8$

• Left side: $1$
• Right side: $9 - 8 = 1$
• Verdict: TRUE! ✅ Both sides equal $1$.

Strategy 1: The Calculation Method Solve both sides of the equation separately, then compare:

For $4 + 3 = 8 - 1$:

• Left side: $4 + 3 = 7$
• Right side: $8 - 1 = 7$
• Since $7 = 7$, this equation is TRUE! ✅

Strategy 2: The Balance Scale Method Imagine putting the left side on one side of a scale and the right side on the other side:

For $6 + 2 = 10 - 1$:

• Left scale: $6 + 2 = 8$
• Right scale: $10 - 1 = 9$
• The scales don't balance ($8 ≠ 9$), so this equation is FALSE! ❌

Strategy 3: The Object Method Use real objects like blocks or toys to represent each side:

For $3 + 4 = 2 + 5$:

• Left side: Get $3$ blocks, add $4$ more = $7$ blocks total
• Right side: Get $2$ blocks, add $5$ more = $7$ blocks total
• Same number of blocks on both sides, so TRUE! ✅

Case A: The Building Blocks Mystery Lee had $14$ building blocks. He shared $6$ blocks with his friend Remi. Someone wrote: $14 - 6 = 9$. Is this statement true or false?

• Investigation: $14 - 6 = ?$
• Calculation: $14 - 6 = 8$
• Comparison: $8 ≠ 9$
• Verdict: FALSE! ❌ Lee has $8$ blocks left, not $9$.

Case B: The Cookie Equation Tiffany says that $9 = 8 + 1$ is a true statement. Paulie says it's false. Who's right?

• Investigation: Check if $9$ equals $8 + 1$
• Right side calculation: $8 + 1 = 9$
• Comparison: $9 = 9$
• Verdict: Tiffany is correct! ✅ This is a TRUE statement.

Equations can look different, but the detective work is the same:

Format 1: Answer on the right

• $5 + 3 = 8$ (Traditional format)

Format 2: Answer on the left

• $8 = 5 + 3$ (Same equation, different arrangement)

Format 3: Operations on both sides

• $4 + 2 = 7 - 1$ (Both sides have operations)

Format 4: Multiple terms

• $2 + 3 + 1 = 4 + 2$ (More than two numbers)

Remember: No matter how the equation looks, your job is to check if both sides have the same value! 🔍

Mistake 1: Thinking the equal sign means "the answer comes next"

• Wrong thinking: "$7 = 4 + 3$ is backwards"
• Correct thinking: "$7$ has the same value as $4 + 3$, so it's true!"

Mistake 2: Not calculating both sides

• Always solve both sides completely before comparing

Mistake 3: Rushing through the investigation

• Take your time and double-check your work

Sometimes you'll see equations with the same numbers in different positions:

Investigation: Are $6 + 4 = 10$ and $4 + 6 = 10$ both true?

• First equation: $6 + 4 = 10$ ✅ TRUE
• Second equation: $4 + 6 = 10$ ✅ TRUE
• Discovery: You can add numbers in any order! This is the commutative property.

Congratulations! You're now a certified Math Detective! 🏅 Remember these key skills:

1. Always check both sides of the equation
2. Use calculation, balance thinking, or objects to investigate
3. Take your time and be thorough
4. The equal sign means "the same as," not "the answer is"

Keep practicing your detective skills, and soon you'll be able to spot true and false equations instantly! 🌟

Welcome to the exciting world of math puzzles! 🧩 In these special equations, one number is missing, and your job is to figure out what it should be. It's like being a detective solving a mystery, but instead of looking for clues, you're looking for the perfect number that makes the equation true!

A missing number equation has a special symbol (like $?$, $\_\_$, or a box $\square$) where a number should be. Your mission is to find the number that makes the equation true!

Examples:

• $9 + ? = 12$
• $\_ - 4 = 8$
• $15 = \square + 6$

The missing number can hide in different places:

Position 1: At the end (easiest)

• $5 + 3 = ?$
• Just add: $5 + 3 = 8$

Position 2: In the middle

• $5 + ? = 8$
• Think: "What plus $5$ equals $8$?"

Position 3: At the beginning

• $? + 3 = 8$
• Think: "What plus $3$ equals $8$?"

Position 4: After the equal sign

• $5 + 3 = ?$
• Calculate: $5 + 3 = 8$

Strategy 1: The Related Facts Method

Use what you know about fact families!

For $9 + ? = 12$:

• Think: "What number plus $9$ equals $12$?"
• Remember: If $9 + 3 = 12$, then the missing number is $3$
• Check: $9 + 3 = 12$ ✅

Strategy 2: The Opposite Operation Method

Use the opposite operation to find the missing number!

For $15 = ? + 6$:

• The equation shows addition, so use subtraction
• Think: $15 - 6 = ?$
• Calculate: $15 - 6 = 9$
• Check: $9 + 6 = 15$ ✅

For $? - 4 = 8$:

• The equation shows subtraction, so use addition
• Think: $8 + 4 = ?$
• Calculate: $8 + 4 = 12$
• Check: $12 - 4 = 8$ ✅

Strategy 3: The Balance Scale Method

Imagine the equation as a balance scale that must stay balanced:

For $10 = 7 + ?$:

• Left side has $10$
• Right side has $7$ plus something
• To balance: $7 + 3 = 10$
• Missing number: $3$

Example 1: The Marble Mystery Emelio has some marbles. He gives $7$ marbles to his friend. Now he has $9$ marbles left. How many marbles did Emelio start with?

• Equation: $? - 7 = 9$
• Strategy: Use addition (opposite of subtraction)
• Calculation: $9 + 7 = 16$
• Answer: Emelio started with $16$ marbles
• Check: $16 - 7 = 9$ ✅

Example 2: The Building Block Challenge Annette is solving $18 - ? = 14$. She says the missing number is $8$. Jessica says it's $4$. Who is correct?

• Annette's answer: If $? = 8$, then $18 - 8 = 10$ ❌ (Not $14$)
• Jessica's answer: If $? = 4$, then $18 - 4 = 14$ ✅ (Correct!)
• Jessica is right! The missing number is $4$.

Type 1: Missing Addend $6 + ? = 13$

• Think: "$6$ plus what equals $13$?"
• Use subtraction: $13 - 6 = 7$
• Answer: $7$

Type 2: Missing Minuend (first number in subtraction) $? - 5 = 7$

• Think: "What number minus $5$ equals $7$?"
• Use addition: $7 + 5 = 12$
• Answer: $12$

Type 3: Missing Subtrahend (second number in subtraction) $14 - ? = 9$

• Think: "$14$ minus what equals $9$?"
• Use subtraction: $14 - 9 = 5$
• Answer: $5$

Which equations are true when the unknown equals $8$?

a. $19 - ? = 9$ → $19 - 8 = 11$ ❌ (Not $9$) b. $18 - ? = 10$ → $18 - 8 = 10$ ✅ (Correct!) c. $? = 20 - 8$ → $8 = 20 - 8 = 12$ ❌ ($8 ≠ 12$) d. $? = 2 + 6$ → $8 = 2 + 6 = 8$ ✅ (Correct!) e. $4 + 5 = ?$ → $4 + 5 = 9$ ❌ ($8 ≠ 9$)

Answer: Equations b and d are true when $? = 8$.

Sometimes you need to think about addition problems to solve subtraction problems:

Challenge: What addition equation could help determine the unknown in $13 = ? - 4$?

This means: "What number minus $4$ equals $13$?"

Helpful addition equation: $13 + 4 = ?$ Solution: $13 + 4 = 17$ Answer: The missing number is $17$ Check: $17 - 4 = 13$ ✅

When you see a missing number equation:

1. Identify where the missing number is located
2. Choose your strategy (related facts, opposite operation, or balance scale)
3. Calculate to find the missing number
4. Check your answer by substituting it back into the original equation
5. Celebrate when the equation balances! 🎉

Mistake 1: Using the wrong operation

• For $? - 3 = 7$, don't do $7 - 3$; do $7 + 3$

Mistake 2: Not checking your answer

• Always substitute your answer back into the original equation

Mistake 3: Getting confused by the position

• Remember: $5 + ? = 8$ and $? + 5 = 8$ have the same answer!

With practice, you'll become a master at solving missing number puzzles! These skills will help you with more advanced math as you grow. Keep practicing, and soon you'll solve these problems as quickly as a superhero! 🦸‍♀️