Mathematics: Algebraic Reasoning – Grade 2

Let's learn how to solve word problems that involve adding and subtracting! Word problems are like puzzles where we need to figure out what's happening and what math operation will help us find the answer.

When you read a word problem, you need to be a detective and ask yourself important questions:

• What is this problem about? (What's happening in the story?)
• What do I know? (What numbers and information are given?)
• What am I trying to find? (What is the question asking for?)
• Do I need one step or two steps to solve this?

Let's practice with this problem: "Maria had 15 stickers 🌟. Her friend gave her 8 more stickers. Then she gave 6 stickers to her brother. How many stickers does Maria have now?"

One-step problems need only one operation to solve:

• "There are 23 birds in a tree. 15 more birds fly to the tree. How many birds are there now?"
• This needs just one addition: $23 + 15 = 38$

Two-step problems need two operations to solve:

• "John collected 23 leaves 🍃 on Monday and 35 leaves on Tuesday. At the end of Wednesday, he had 97 leaves total. How many leaves did he collect on Wednesday?"
• Step 1: Add Monday and Tuesday: $23 + 35 = 58$
• Step 2: Subtract from total: $97 - 58 = 39$

Different words in problems give us clues about which operation to use:

Addition clues: altogether, total, sum, more than, plus, combined, increased by Subtraction clues: difference, less than, minus, fewer, take away, decreased by, how many more

You can use pictures, drawings, or objects to help solve problems:

• Number lines: Jump forward for addition, backward for subtraction
• Drawings: Draw circles, dots, or pictures to represent the objects in the problem
• Ten frames: Use grids to organize your counting
• Base-10 blocks: Use blocks to represent tens and ones

Sometimes you might be given an equation like $76 = 11 + 65$ and asked to create a word problem. Here's one example: "At the school library, there are 11 fiction books 📚 and 65 non-fiction books on the new books shelf. How many books are there altogether?" This matches our equation because $11 + 65 = 76$.

After solving a problem, always check if your answer makes sense:

• Does your answer seem reasonable for the situation?
• Can you solve the problem a different way to check?
• If you added, can you subtract to get back to where you started?

When solving word problems:

1. Read the problem twice - once to understand the story, once to find the numbers
2. Underline or circle the important information
3. Cross out information you don't need
4. Draw a picture if it helps you visualize the problem
5. Write your equation
6. Solve step by step
7. Check that your answer makes sense in the original problem

Remember, becoming good at word problems takes practice! The more you work with them, the easier it becomes to see which operation to use and how to set up your solution. 🎯

Let's explore what makes an equation true or false! An equation is like a balance scale ⚖️ – both sides must be equal for it to be balanced and true.

The equal sign (=) means "the same as" or "has the same value as." It does NOT mean "the answer is." This is a very important difference!

Think of the equal sign as a balance scale. When we write $5 + 3 = 8$, we're saying that 5 + 3 has the same value as 8. Both sides of the equation "weigh" the same amount.

Let's look at some examples:

• $7 + 2 = 9$ is true because 7 + 2 equals 9
• $6 + 4 = 5 + 5$ is true because both sides equal 10
• $8 + 3 = 12$ is false because 8 + 3 equals 11, not 12

Sometimes we can use clever thinking to check if equations are true without doing all the math:

For $27 + 13 = 26 + 14$:

• On the left side: 27 + 13
• On the right side: 26 + 14
• Notice that 26 is one less than 27, but 14 is one more than 13
• Since we took away 1 and added 1, both sides are still equal! ✅

Equations can be written in different ways:

• $15 + 18 = 33$ (sum on the right)
• $33 = 15 + 18$ (sum on the left)
• $25 + 8 = 15 + 18$ (operations on both sides)

All of these forms are valid as long as both sides have the same value!

You can use objects to check if equations are true:

• Counting bears or blocks: Build each side of the equation and compare
• Two-color counters: Make groups to represent each side
• Number balance: Place objects on each side to see if it balances
• Drawings: Draw dots or pictures to represent each side

Let's check if $8 + 6 = 10 + 5$ is true:

• Left side: $8 + 6 = 14$
• Right side: $10 + 5 = 15$
• Since $14 ≠ 15$, this equation is false

Or we can think about it this way:

• We moved 2 from the 8 to make 10 (8 - 2 = 6, 6 + 2 = 8, so 8 becomes 6 and 6 becomes 8)
• Wait, that's not right! Let me think more carefully...
• We have $8 + 6$ on the left and $10 + 5$ on the right
• $8 + 6 = 14$ and $10 + 5 = 15$
• Since 14 and 15 are different, the equation is false

Mistake: Thinking the equal sign means "do something" or "the answer is" Better thinking: The equal sign means "is the same as"

Mistake: Only looking at one side of the equation Better thinking: Always check that both sides have the same value

Let's practice with $15 + 18 = 25 + 8$:

• Left side: $15 + 18 = 33$
• Right side: $25 + 8 = 33$
• Both sides equal 33, so this equation is true! ✅

We can also think about it this way:

• We moved 10 from 15 to 8 (15 - 10 = 5, but wait, that would make it 5 + 18 = 25 + 8)
• Actually, let me be more careful: 15 + 18 versus 25 + 8
• 25 is 10 more than 15, and 8 is 10 less than 18
• Since we added 10 to one number and subtracted 10 from another, the total stays the same! 🎯

Finding missing numbers in equations is like being a math detective! 🔍 You need to figure out what number makes both sides of the equation equal.

An unknown number is a missing value in an equation, often represented by symbols like:

• A blank: 5 + ___ = 12
• A box: $8 + ☐ = 15$
• A triangle: $△ + 6 = 14$
• A question mark: $? + 7 = 13$

The unknown number can be anywhere in the equation:

• Unknown as a second addend: 5 + ___ = 12
• Unknown as a first addend: ___ + 7 = 15
• Unknown as a sum: 6 + 8 = ___
• Unknown in subtraction: 15 - ___ = 8
• Unknown on either side: 4 + 9 = ___ + 6

Remember that addition and subtraction are related:

• If $5 + 7 = 12$, then $12 - 5 = 7$ and $12 - 7 = 5$
• For 8 + ___ = 15, think: "What plus 8 equals 15?"
• Since $15 - 8 = 7$, the unknown is 7

Compensation means adjusting numbers to keep equations balanced: For 30 + 43 = 26 + ___:

• Notice that 26 is 4 less than 30
• So we need to add 4 more to 43 to keep it balanced
• $43 + 4 = 47$
• Check: $30 + 43 = 73$ and $26 + 47 = 73$ ✅

For 12 + ___ = 19:

• Start at 12 on a number line
• Count how many jumps it takes to get to 19
• $12 → 13 → 14 → 15 → 16 → 17 → 18 → 19$
• That's 7 jumps, so the unknown is 7!

For ___ + 6 = 11:

• Draw 11 circles: ○○○○○○○○○○○
• Circle 6 of them to represent the known addend
• Count the remaining circles: 5
• So the unknown is 5!

For equations like 16 + 37 = ___ + 38:

Method 1 - Calculate both known sides:

• Left side: $16 + 37 = 53$
• Right side needs: ___ + 38 = 53
• So: $53 - 38 = 15$
• The unknown is 15!

Method 2 - Use compensation:

• Compare 37 and 38: 38 is 1 more than 37
• So we need 1 less than 16 to balance
• $16 - 1 = 15$
• The unknown is 15! ✅

Always check your answer by substituting it back into the original equation:

• For 8 + ___ = 15 with answer 7:
• Check: $8 + 7 = 15$ ✅ Correct!

Think of equations like a balance scale:

• For 4 + 5 = ___ + 7
• Left side weighs: $4 + 5 = 9$
• Right side needs to weigh 9 too
• If we already have 7, we need $9 - 7 = 2$ more
• So the unknown is 2!

For 54 - ___ = 32 - 15:

• First, solve the right side: $32 - 15 = 17$
• Now we have: 54 - ___ = 17
• Think: "54 minus what equals 17?"
• $54 - 17 = 37$
• So the unknown is 37!
• Check: $54 - 37 = 17$ and $32 - 15 = 17$ ✅

Practice with related problems to see patterns:

• $10 + 15 = 25$
• $9 + 16 = 25$ (1 less + 1 more = same sum)
• $8 + 17 = 25$ (2 less + 2 more = same sum)
• This helps you see how compensation works! 🧠

Let's explore the fascinating world of even and odd numbers! These special number categories have unique patterns that we can see and understand through equal groups and pairs. 👫

An even number can be split into two equal groups with no objects left over. It's like having a perfect partner for every object!

Let's look at the number 8:

• We can make two groups of 4: 🟦🟦🟦🟦 and 🟦🟦🟦🟦
• We can write this as: $4 + 4 = 8$
• Every object has a partner, so 8 is even!

Other examples of even numbers:

• 6: $3 + 3 = 6$ (🟢🟢🟢 and 🟢🟢🟢)
• 10: $5 + 5 = 10$ (five pairs of objects)
• 12: $6 + 6 = 12$ (six pairs of objects)

An odd number can be split into two equal groups with one left over. There's always one object that doesn't have a partner!

Let's look at the number 9:

• We can make two groups of 4 with 1 left over: 🟦🟦🟦🟦 and 🟦🟦🟦🟦 and 🟦
• We can write this as: $4 + 4 + 1 = 9$
• One object doesn't have a partner, so 9 is odd!

Other examples of odd numbers:

• 7: $3 + 3 + 1 = 7$ (three pairs plus one left over)
• 11: $5 + 5 + 1 = 11$ (five pairs plus one left over)
• 15: $7 + 7 + 1 = 15$ (seven pairs plus one left over)

Two-color counters method:

1. Get the number of counters you want to test
2. Try to pair them up by putting two counters together
3. If every counter has a partner → even
4. If one counter is left without a partner → odd

Circle pairing method:

1. Draw the number of circles you want to test
2. Draw lines connecting pairs of circles
3. Count if any circles are left unpaired

Example with 10: ○-○ ○-○ ○-○ ○-○ ○-○ (all paired = even) Example with 13: ○-○ ○-○ ○-○ ○-○ ○-○ ○-○ ○ (one unpaired = odd)

Here's something amazing about doubles (adding a number to itself): Doubles always make even numbers! 🎯

• $1 + 1 = 2$ (even)
• $2 + 2 = 4$ (even)
• $3 + 3 = 6$ (even)
• $4 + 4 = 8$ (even)
• $5 + 5 = 10$ (even)

This happens because when you double a number, you're making perfect pairs!

We can also use arrays (objects arranged in rows and columns) to see even and odd patterns:

Even number arrays:

• 6 objects: ●●● or ●● ●●● ●● ●●
• Both arrangements show 6 can be split evenly!

Odd number arrays:

• 7 objects: ●●● ● or ●● ●●● ●● ●● ●
• There's always one object that can't make a complete pair!

Skip counting by 2s helps us see the even number pattern: $2, 4, 6, 8, 10, 12, 14, 16, 18, 20...$

All these numbers are even because they're made by counting pairs!

Here's a helpful pattern to remember:

• Even numbers end in 0, 2, 4, 6, or 8
• Odd numbers end in 1, 3, 5, 7, or 9

But it's important to understand why this pattern works, not just memorize it. The pattern works because of how our number system is built with groups of 10!

Even number situations:

• Shoes come in pairs 👟👟
• Hands on people (2 hands per person)
• Eyes on animals 👀
• Wheels on bicycles 🚲

Odd number situations:

• Tricycle wheels 🚲 (3 wheels)
• Fingers on one hand ✋ (5 fingers)
• Days in a week (7 days)
• Players on a baseball team (9 players)

1. Partner Check: Give yourself a number between 1-20. Use objects to see if everyone can have a partner.

2. Number Sort: Make two boxes labeled "Even" and "Odd." Sort number cards into the correct boxes.

3. Draw and Check: Draw objects for numbers and circle them in pairs to check even or odd.

4. Real-World Hunt: Look around your classroom or home for groups of objects and determine if the totals are even or odd! 🔍

Repeated addition is like counting by the same number over and over again! 🔢 It's a powerful tool that helps us count large groups of objects quickly and efficiently. Arrays make this even easier by organizing objects in neat rows and columns.

Repeated addition means adding the same number multiple times. Instead of counting one by one, we can count by groups!

For example:

• Instead of counting: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
• We can count by 3s: 3, 6, 9, 12 (much faster!)
• This is the same as: $3 + 3 + 3 + 3 = 12$

An array is a way of arranging objects in equal rows and equal columns. Think of it like seats in a movie theater or ice cubes in an ice cube tray! 🧊

Parts of an array:

• Rows: Objects going across (horizontally →)
• Columns: Objects going down (vertically ↓)
• Total: All the objects in the array

Let's say we have 12 stickers 🌟 and want to arrange them in an array:

Array 1: 3 rows of 4

🌟 🌟 🌟 🌟
🌟 🌟 🌟 🌟  
🌟 🌟 🌟 🌟

• 3 rows
• 4 stickers in each row
• Repeated addition: $4 + 4 + 4 = 12$

Array 2: 4 rows of 3

🌟 🌟 🌟
🌟 🌟 🌟
🌟 🌟 🌟
🌟 🌟 🌟

• 4 rows
• 3 stickers in each row
• Repeated addition: $3 + 3 + 3 + 3 = 12$

Both arrays have the same total (12), but they're organized differently!

When we see an array or equal groups, we can write equations to show our thinking:

For 5 groups of 4 apples 🍎:

• Groups: 🍎🍎🍎🍎 | 🍎🍎🍎🍎 | 🍎🍎🍎🍎 | 🍎🍎🍎🍎 | 🍎🍎🍎🍎
• Equation: $4 + 4 + 4 + 4 + 4 = 20$
• We can also write: "5 groups of 4 equals 20"

Here's something fantastic about arrays: you can rearrange the rows and columns, but the total stays the same!

Let's explore with 24 objects:

• 6 rows of 4: $4 + 4 + 4 + 4 + 4 + 4 = 24$
• 4 rows of 6: $6 + 6 + 6 + 6 = 24$
• 3 rows of 8: $8 + 8 + 8 = 24$
• 2 rows of 12: $12 + 12 = 24$

All different arrangements, same total! This is because we're not changing how many objects we have, just how we organize them.

Arrays are everywhere around us:

• Egg cartons: 2 rows of 6 eggs = $6 + 6 = 12$ eggs 🥚
• Muffin tins: 3 rows of 4 = $4 + 4 + 4 = 12$ muffins 🧁
• Ice cube trays: 2 rows of 7 = $7 + 7 = 14$ ice cubes 🧊
• Classroom desks: 4 rows of 5 = $5 + 5 + 5 + 5 = 20$ desks

Number lines help us visualize repeated addition by making equal jumps:

For $3 + 3 + 3 + 3 = 12$:

0 --3--> 3 --3--> 6 --3--> 9 --3--> 12

We make 4 jumps of 3 to reach 12!

Using counters or blocks:

1. Make equal groups with the same number in each group
2. Count how many groups you have
3. Count how many objects are in each group
4. Write the repeated addition equation
5. Find the total

Example with 15 objects in groups of 5:

• Group 1: ●●●●● (5 objects)
• Group 2: ●●●●● (5 objects)
• Group 3: ●●●●● (5 objects)
• Equation: $5 + 5 + 5 = 15$
• Total: 15 objects

Sometimes objects are scattered and need to be organized:

Scattered: ● ● ● ● ● ● ● ● ● ● ● ● Organized into 3 groups of 4:

• Group 1: ●●●●
• Group 2: ●●●●
• Group 3: ●●●● Equation: $4 + 4 + 4 = 12$

The total doesn't change – we just made it easier to count!

Word Problem: "A teacher is setting up chairs for a class party. She puts 5 chairs at each of 4 tables. How many chairs are there altogether?"

Solution steps:

1. Identify the equal groups: 4 tables with 5 chairs each
2. Draw or visualize: Table 1 (🪑🪑🪑🪑🪑), Table 2 (🪑🪑🪑🪑🪑), etc.
3. Write the equation: $5 + 5 + 5 + 5 = 20$
4. Create an array if helpful:

🪑 🪑 🪑 🪑 🪑
🪑 🪑 🪑 🪑 🪑
🪑 🪑 🪑 🪑 🪑
🪑 🪑 🪑 🪑 🪑

5. Answer: 20 chairs total

1. Look for equal groups in problems and real-life situations
2. Draw arrays when objects can be arranged in rows and columns
3. Use objects to build understanding before moving to abstract equations
4. Check your work by counting the total a different way
5. Practice with different numbers to see patterns and relationships 🎯