Mathematics: Algebraic Reasoning – Grade 3

The distributive property is like having a mathematical superpower that helps you solve multiplication problems by breaking them into smaller, easier parts! 🦸‍♂️ Instead of trying to multiply big numbers all at once, you can split them up and work with numbers you already know.

The distributive property allows you to multiply a number by a sum by multiplying that number by each part of the sum, then adding the results. For example, if you want to find $4 \times 72$, you can rewrite it as $4 \times (70 + 2)$, which becomes $(4 \times 70) + (4 \times 2)$.

This works because when you have $4$ groups of $72$ items, it's the same as having $4$ groups of $70$ items plus $4$ groups of $2$ items. The total is still the same!

When you see a two-digit number like $24$, you can think of it as $20 + 4$. This is called expanded form. Here's how the distributive property works with $3 \times 24$:

$3 \times 24 = 3 \times (20 + 4) = (3 \times 20) + (3 \times 4) = 60 + 12 = 72$

You can visualize this with arrays or drawings. Imagine $3$ rows of $24$ dots. You could split this into $3$ rows of $20$ dots plus $3$ rows of $4$ dots!

Along with the distributive property, you'll use other helpful properties:

Commutative Property: You can multiply numbers in any order. $6 \times 9 = 9 \times 6$

Associative Property: When multiplying three numbers, you can group them differently. $(2 \times 3) \times 4 = 2 \times (3 \times 4)$

These properties let you rearrange multiplication problems to make them easier to solve. If you know that $6 \times 9$ is tricky, you might decompose $6$ into $4 + 2$, giving you $(4 \times 9) + (2 \times 9) = 36 + 18 = 54$.

Parentheses are like mathematical containers that show which operations to do first. When you write $(4 \times 70) + (4 \times 2)$, the parentheses tell you to multiply inside each pair first, then add the results.

The distributive property helps in everyday situations. If you're buying $4$ packages of stickers and each package has $23$ stickers, you can think:

• $4 \times 23 = 4 \times (20 + 3) = (4 \times 20) + (4 \times 3) = 80 + 12 = 92$ stickers total!

To better understand these concepts, you can use:

• Base-ten blocks: Show tens and ones separately
• Arrays: Draw rectangles split into parts
• Area models: Draw rectangles divided by place value

These tools help you see why the distributive property works and make abstract math more concrete and understandable.

Real-world problems often require more than one step to solve, just like following a recipe or building something step-by-step! 👩‍🍳 Learning to break down complex problems into smaller parts is a valuable skill that helps in math and everyday life.

Before jumping into calculations, it's important to understand what's happening in the problem. Ask yourself these key questions:

• What is the story telling me?
• What information do I already know?
• What am I trying to find out?
• What do the numbers represent in this situation?

For example: "Keisha and Diego are selling pies for a fundraiser. Each pie costs $\$5$. If Keisha sells $15$ pies and Diego sells $5$ pies, how much money did they earn for the fundraiser?"

Here, you know the price per pie ($\$5$), the number of pies each person sold ($15$ and $5$), and you need to find the total money earned.

Some problems require two operations to solve completely. In the pie example:

1. First step: Find the total number of pies sold: $15 + 5 = 20$ pies
2. Second step: Calculate total money: $20 \times \$5 = \$100$

You could also solve it differently:

1. Calculate Keisha's earnings: $15 \times \$5 = \$75$
2. Calculate Diego's earnings: $5 \times \$5 = \$25$
3. Add them together: $\$75 + \$25 = \$100$

Different types of models can help you organize your thinking:

Drawings and Pictures: Sketch the situation to visualize what's happening. For the pie problem, you might draw circles for pies and group them by seller.

Equations: Write mathematical expressions that represent each step. Use the equation $(15 + 5) \times \$5$ to show the complete solution.

Manipulatives: Use counters, blocks, or other objects to act out the problem. This is especially helpful when the numbers are smaller.

Before solving, estimate what a reasonable answer might be. In the pie example, if each pie costs about $\$5$ and they sold about $20$ pies, the answer should be around $\$100$. This helps you check if your final answer makes sense.

You can create high and low estimates:

• Low estimate: $20 \times \$4 = \$80$
• High estimate: $20 \times \$6 = \$120$
• Your answer should fall between these values!

Three Reads Strategy:

1. First read: What is this problem about?
2. Second read: What are the important numbers and information?
3. Third read: What is the question asking me to find?

Work Backwards: Sometimes it helps to think about what the final answer should look like, then work backwards to figure out the steps.

Remaining Amount Problems: "There are $30$ paintbrushes. Antwan puts $6$ brushes on each of $4$ tables. How many brushes are left for the counter?"

• Step 1: $6 \times 4 = 24$ brushes used
• Step 2: $30 - 24 = 6$ brushes remaining

Total Collection Problems: "Three students bring canned goods. Uriel brings $4$ cases, Paola brings $6$ cases, and Mika brings $5$ cases. Each case has $8$ cans. How many total cans?"

• Step 1: $4 + 6 + 5 = 15$ total cases
• Step 2: $15 \times 8 = 120$ total cans

After solving, always check:

• Does my answer make sense in the context?
• Did I answer the question that was asked?
• Does my answer fall within my estimated range?
• Can I solve the problem a different way to verify?

Did you know that every division problem is actually a multiplication problem in disguise? 🕵️‍♀️ Learning to see this connection will make you much stronger at both multiplication and division!

Division and multiplication are inverse operations, which means they "undo" each other. When you see a division problem like $56 ÷ 7 = ?$, you can think of it as asking: "What number times $7$ equals $56$?"

So $56 ÷ 7 = ?$ becomes $7 \times ? = 56$

This is much easier to solve because you can use your multiplication facts! You know that $7 \times 8 = 56$, so $56 ÷ 7 = 8$.

A fact family is a group of related multiplication and division facts that use the same three numbers. For example, the numbers $3$, $7$, and $21$ make this fact family:

• $3 \times 7 = 21$
• $7 \times 3 = 21$
• $21 ÷ 7 = 3$
• $21 ÷ 3 = 7$

Notice how all four facts use the same three numbers, just in different positions! This shows how multiplication and division are connected.

Arrays help you visualize why division and multiplication are related. If you draw an array with $3$ rows and $7$ columns, you have $21$ total dots.

• Looking at it as $3 \times 7$: $3$ rows of $7$ dots each
• Looking at it as $21 ÷ 7$: $21$ dots arranged in groups of $7$
• Looking at it as $21 ÷ 3$: $21$ dots arranged in $3$ equal rows

The same array shows all these relationships!

When you encounter a division problem, follow these steps:

1. Rewrite the division as a missing factor problem
2. Think about which multiplication fact will help
3. Solve using your known multiplication facts
4. Check by multiplying your answer

For example, to solve $42 ÷ 6$:

1. Rewrite as: $6 \times ? = 42$
2. Think: "What times $6$ equals $42$?"
3. Answer: $7$ (because $6 \times 7 = 42$)
4. Check: $42 ÷ 6 = 7$ ✓

To become fluent with division, practice writing complete fact families. Start with a multiplication fact you know well, like $4 \times 8 = 32$, then write all four related facts:

• $4 \times 8 = 32$
• $8 \times 4 = 32$
• $32 ÷ 8 = 4$
• $32 ÷ 4 = 8$

This strategy helps in everyday situations. If you have $48$ stickers to share equally among $6$ friends, you can think: "$6$ times what number equals $48$?" Since $6 \times 8 = 48$, each friend gets $8$ stickers!

Counters, base-ten blocks, and other manipulatives help you see these relationships. You can physically group objects to show division, then rearrange them to show the related multiplication facts.

Remember, when division seems difficult, turn it into a multiplication problem you already know how to solve! This strategy will make you more confident with both operations.

Becoming a mathematical truth detective means learning to evaluate equations carefully and explain your reasoning! 🔍 The equal sign is much more powerful than just meaning "the answer is" – it shows that two sides are perfectly balanced.

The equal sign (=) doesn't mean "the answer is." Instead, it means "the same as" or "is equivalent to." Think of it like a balance scale – both sides must have the same value to be balanced (true).

For example:

• $3 \times 4 = 12$ is true because $3 \times 4$ equals $12$
• $3 \times 4 = 2 \times 6$ is also true because both sides equal $12$
• $3 \times 4 = 2 \times 5$ is false because $12$ does not equal $10$

To determine if an equation is true or false:

1. Calculate the value of the left side
2. Calculate the value of the right side
3. Compare the results
4. Explain your reasoning

Let's check: $27 ÷ 3 = 3 \times 3$

• Left side: $27 ÷ 3 = 9$
• Right side: $3 \times 3 = 9$
• Since $9 = 9$, the equation is true

True Equations:

• $2 \times 12 = 4 \times 6$ (both sides equal $24$)
• $16 ÷ 2 = 36 ÷ 9$ (both sides equal $4$)
• $18 = 3 \times 6$ (both sides equal $18$)

False Equations:

• $2 \times 3 = 4 \times 6$ ($6$ does not equal $24$)
• $20 ÷ 4 = 12 ÷ 3$ ($5$ does not equal $4$)
• $15 = 2 \times 6$ ($15$ does not equal $12$)

Unlike simple addition problems you may have seen before, equations can have the answer on the left side too:

• $24 = 6 \times 4$ (answer on the left)
• $6 \times 4 = 24$ (answer on the right)
• $8 = 32 ÷ 4$ (quotient on the left)
• $32 ÷ 4 = 8$ (quotient on the right)

All of these equations are true! The equal sign works in both directions.

You can use visual tools to check equations:

T-Charts: Draw a chart with two columns. Put counters or drawings representing the left side in one column and the right side in the other. If both columns have the same amount, the equation is true.

Arrays: Draw arrays to represent both sides of the equation, then count the total dots in each array.

Base-Ten Blocks: Use blocks to build both sides of the equation and compare the totals.

When you determine if an equation is true or false, always explain why:

Good explanations include:

• "This equation is true because both sides equal $12$."
• "The left side equals $8$ and the right side equals $6$, so this equation is false."
• "I calculated $4 \times 5 = 20$ and $2 \times 10 = 20$, so the equation is true."

Don't assume the equal sign always means "the answer comes next." In $18 = 3 \times 6$, the $18$ isn't the "answer" – it's just one side of a balanced equation.

Always calculate both sides independently before comparing. Don't just look at the numbers and guess!

When practicing, try writing the same equation in different ways:

• $3 \times 7 = 21$
• $21 = 3 \times 7$
• $7 \times 3 = 21$
• $21 = 7 \times 3$

This helps reinforce that the equal sign means "the same as" regardless of which side has which expression.

Sometimes equations have missing pieces, just like puzzles with blank spaces! 🧩 Learning to find these unknown values helps you become a mathematical problem-solver who can figure out what number belongs in any position.

An unknown value is a missing number in an equation, often represented by a symbol like $?$, a letter like $n$, or a box like $□$. Your job is to figure out what number makes the equation true.

For example:

• $6 \times ? = 42$
• $n ÷ 5 = 8$
• $3 = □ ÷ 12$

The unknown can be in any position – as a factor, product, dividend, divisor, or quotient!

The best strategy for finding unknown values is to use fact families and inverse operations. Remember that multiplication and division are opposites, so you can use one to solve the other.

For $7 \times n = 56$:

1. Think: "What times $7$ equals $56$?"
2. Use the related division fact: $56 ÷ 7 = 8$
3. Therefore, $n = 8$
4. Check: $7 \times 8 = 56$ ✓

Let's practice finding unknowns in various positions:

Unknown Factor: $4 \times ? = 32$

• Think: $32 ÷ 4 = 8$
• Answer: $? = 8$

Unknown Product: $5 \times 9 = ?$

• Think: $5 \times 9 = 45$
• Answer: $? = 45$

Unknown Dividend: $? ÷ 6 = 7$

• Think: $6 \times 7 = 42$
• Answer: $? = 42$

Unknown Divisor: $35 ÷ ? = 5$

• Think: $35 ÷ 5 = 7$
• Answer: $? = 7$

Unknown Quotient: $48 ÷ 8 = ?$

• Think: $48 ÷ 8 = 6$
• Answer: $? = 6$

Don't be confused when the unknown appears on the left side of the equal sign! The equal sign works both ways.

For $3 = n ÷ 12$:

1. This means $n ÷ 12 = 3$
2. Think: "What number divided by $12$ equals $3$?"
3. Use multiplication: $3 \times 12 = 36$
4. Therefore, $n = 36$
5. Check: $36 ÷ 12 = 3$ ✓

Arrays help you visualize unknown values. For $4 \times ? = 20$:

• Draw $4$ rows
• You need $20$ total dots
• How many dots in each row? $20 ÷ 4 = 5$
• So $? = 5$

Manipulatives like counters can help too. Use them to act out the problem and find the missing piece.

Often, you can solve the same problem using different fact family relationships. For $72 ÷ ? = 9$:

Method 1: Think division

• $72 ÷ 9 = 8$, so $? = 8$

Method 2: Think multiplication

• $9 \times ? = 72$
• $9 \times 8 = 72$, so $? = 8$

Both methods give the same answer!

1. Identify what type of unknown you have (factor, product, dividend, etc.)
2. Choose the inverse operation or related fact
3. Calculate the unknown value
4. Check by substituting your answer back into the original equation
5. Verify that the equation is now true

Unknown value problems appear in real life:

• "I bought $6$ packs of pencils and got $42$ pencils total. How many pencils were in each pack?" ($6 \times ? = 42$)
• "There are $35$ students who need to form equal teams of $5$. How many teams will there be?" ($35 ÷ 5 = ?$)

Unknowns can be represented many ways:

• Letters: $x$, $n$, $a$, $b$
• Symbols: $?$, $□$, $○$
• Words: "What number?"

All of these mean the same thing – find the missing value that makes the equation true!

Numbers have personalities! Some like to share equally (even numbers), while others always have a little extra (odd numbers). Understanding these number personalities helps you recognize patterns and make predictions about larger numbers! 🔢

Even numbers can be divided into two equal groups with nothing left over. When you have an even number of objects, you can pair them up perfectly with no singles remaining.

Odd numbers always have one object left over when you try to divide them into two equal groups. No matter how you arrange them, there's always one that doesn't have a partner!

Here's an amazing discovery: you can tell if any number is even or odd just by looking at the ones digit (the last digit)!

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

This works for any number, no matter how big! The number $387$ is odd because it ends in $7$. The number $1,248$ is even because it ends in $8$.

This pattern works because of place value. Every number can be broken down into tens and ones:

• $243 = 240 + 3$
• $186 = 180 + 6$

Since any multiple of $10$ (like $240$ or $180$) is always even, the ones digit determines whether the whole number is even or odd!

You can use different tools to see even and odd patterns:

Dot Patterns: Draw dots and try to pair them up

• $6$ dots: ●● ●● ●● (pairs perfectly = even)
• $7$ dots: ●● ●● ●● ● (one left over = odd)

Tally Marks: Group tally marks by twos

• $8$: |||| |||| (groups of 2 = even)
• $9$: |||| |||| | (one extra = odd)

Arrays: Try to make rectangles with 2 rows

• $12$ objects make a $2 \times 6$ rectangle (even)
• $13$ objects can't make a complete $2$-row rectangle (odd)

Even numbers are multiples of 2, which means they're in the $2$ times table:

• $2 \times 1 = 2$ (even)
• $2 \times 2 = 4$ (even)
• $2 \times 3 = 6$ (even)
• $2 \times 4 = 8$ (even)

If a number is not a multiple of $2$, then it's odd.

Even and odd numbers appear everywhere:

• Pairing up: If there are $15$ students and they need to work in pairs, one student won't have a partner ($15$ is odd)
• Sharing equally: $20$ cookies can be shared equally between $2$ children ($20$ is even)
• House numbers: Many streets have even numbers on one side and odd numbers on the other

The ones-digit rule works for numbers up to $1,000$ and beyond:

• $461$ ends in $1$ → odd
• $789$ ends in $9$ → odd
• $642$ ends in $2$ → even
• $1,000$ ends in $0$ → even

Don't look at other digits! In the number $883$, you might notice the $8$s in the hundreds and tens places, but since it ends in $3$, the number is odd.

Only the ones digit matters for determining if a number is even or odd.

When asked if a number is even or odd:

1. Look at the ones digit
2. Check if it's $0, 2, 4, 6, 8$ (even) or $1, 3, 5, 7, 9$ (odd)
3. Explain your reasoning using the pattern
4. Verify by thinking about whether it divides evenly by $2$

Understanding even and odd numbers prepares you for learning about:

• Divisibility rules in Grade 4
• Prime and composite numbers in later grades
• Number theory concepts in advanced mathematics

These foundational patterns help you see the beauty and logic in mathematics!

Multiples are like mathematical families where numbers are related through multiplication! 👨‍👩‍👧‍👦 Learning about multiples helps you see patterns, understand relationships between numbers, and prepares you for more advanced math concepts.

A multiple of a number is what you get when you multiply that number by any whole number. Think of multiples as the "multiplication children" of a number.

For example, the multiples of $5$ are:

• $5 \times 1 = 5$
• $5 \times 2 = 10$
• $5 \times 3 = 15$
• $5 \times 4 = 20$
• And so on...

So $5, 10, 15, 20, 25, 30...$ are all multiples of $5$.

Multiples are exactly what you get when you skip-count! When you skip-count by $3$s ($3, 6, 9, 12, 15...$), you're listing the multiples of $3$.

This makes multiples easy to find – just keep adding the number to itself:

• Multiples of $4$: $4, 8, 12, 16, 20, 24...$ (keep adding $4$)
• Multiples of $7$: $7, 14, 21, 28, 35, 42...$ (keep adding $7$)

Method 1: Multiplication To check if $45$ is a multiple of $5$, ask: "What times $5$ equals $45$?"

• $5 \times 9 = 45$
• Since $9$ is a whole number, $45$ is a multiple of $5$!

Method 2: Division To check if $45$ is a multiple of $5$, divide: $45 ÷ 5 = 9$

• Since $9$ is a whole number with no remainder, $45$ is a multiple of $5$!

You can create multiples using counters or other objects:

To find multiples of $8$:

1. Make one group of $8$ counters → $1 \times 8 = 8$
2. Add another group of $8$ → $2 \times 8 = 16$
3. Add another group of $8$ → $3 \times 8 = 24$
4. Continue until you reach $12 \times 8 = 96$

Your list: $8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96$

The Smallest Multiple: The smallest multiple of any number is the number itself.

• Smallest multiple of $6$ is $6$ (because $6 \times 1 = 6$)
• Smallest multiple of $12$ is $12$ (because $12 \times 1 = 12$)

Infinite Multiples: Every number has infinitely many multiples because you can keep multiplying by larger and larger numbers.

Zero is Special: $0$ is a multiple of every number because any number times $0$ equals $0$.

Multiples help solve everyday problems:

Equal Groups: "Can $27$ students form equal teams of $3$?"

• Check if $27$ is a multiple of $3$: $27 ÷ 3 = 9$ ✓
• Yes! They can form $9$ teams of $3$ students each.

Packaging: "Do $56$ cookies fit exactly into boxes of $8$?"

• Check if $56$ is a multiple of $8$: $56 ÷ 8 = 7$ ✓
• Yes! You need exactly $7$ boxes.

Different numbers create different patterns:

Multiples of 2: $2, 4, 6, 8, 10, 12...$ (all even numbers!) Multiples of 5: $5, 10, 15, 20, 25, 30...$ (end in $0$ or $5$) Multiples of 10: $10, 20, 30, 40, 50...$ (end in $0$)

A hundreds chart helps you see multiple patterns. Color all multiples of $3$ one color, all multiples of $4$ another color. You'll see amazing patterns emerge!

When determining if a number is a multiple:

1. Try division: Does the number divide evenly?
2. Try multiplication: Can you multiply by a whole number to get it?
3. Use skip-counting: Is the number in the skip-counting sequence?
4. Check your work: Use a different method to verify

Understanding multiples prepares you for:

• Factors (numbers that multiply to give a product)
• Prime and composite numbers in Grade 4
• Least common multiples in later grades
• Divisibility rules for quick multiple identification

Number patterns are like mathematical music – they follow rules and create beautiful sequences that help us predict what comes next! 🎵 Learning to identify, create, and extend patterns develops your algebraic thinking and problem-solving skills.

A number pattern (or sequence) is a list of numbers that follows a specific rule. Each number in the pattern is called a term, and we use ordinal numbers (1st, 2nd, 3rd...) to describe their positions.

For example, in the pattern $6, 12, 18, 24, 30...$:

• $6$ is the 1st term
• $12$ is the 2nd term
• $18$ is the 3rd term
• And so on...

Patterns can use any of the four operations:

Addition Patterns: Add the same number each time

• Rule: "Add $4$" → $3, 7, 11, 15, 19...$

Subtraction Patterns: Subtract the same number each time

• Rule: "Subtract $5$" → $50, 45, 40, 35, 30...$

Multiplication Patterns: Multiply by the same number each time

• Rule: "Multiply by $2$" → $1, 2, 4, 8, 16...$

Division Patterns: Divide by the same number each time

• Rule: "Divide by $3$" → $81, 27, 9, 3, 1...$

To identify a pattern:

1. Look for what changes between consecutive terms
2. Check if the same change happens throughout
3. Describe the rule in words
4. Test your rule on the next few terms

Example: $20, 17, 14, 11, 8...$

• From $20$ to $17$: subtract $3$
• From $17$ to $14$: subtract $3$
• From $14$ to $11$: subtract $3$
• Rule: "Subtract $3$"
• Next terms: $5, 2, -1...$

To create a pattern, you need:

1. A starting number
2. A rule for getting the next term

Example: Start with $500$ and subtract $35$ each time

• 1st term: $500$
• 2nd term: $500 - 35 = 465$
• 3rd term: $465 - 35 = 430$
• 4th term: $430 - 35 = 395$
• 5th term: $395 - 35 = 360$

Once you know the rule, you can find any term in the sequence:

Pattern: $2, 6, 18, 54...$ (Rule: "Multiply by $3$")

• 5th term: $54 \times 3 = 162$
• 6th term: $162 \times 3 = 486$
• 7th term: $486 \times 3 = 1,458$

The same pattern can be described in different ways, and that's perfectly fine! For the pattern $6, 12, 18, 24...$:

• "Add $6$"
• "Multiply by the position number, then multiply by $6$"
• "List the multiples of $6$"
• "Skip-count by $6$s"

All these descriptions are correct! Classroom discussions can compare different ways of seeing the same pattern.

A hundreds chart is a powerful tool for seeing patterns:

• Color multiples of $3$: $3, 6, 9, 12...$ (every 3rd number)
• Color the pattern $5, 15, 25, 35...$ (add $10$, starting from $5$)
• Look for diagonal, horizontal, or vertical patterns

Saving Money: Bailey saves $\$6$ every day

• Day 1: $\$6$
• Day 2: $\$12$
• Day 3: $\$18$
• Day 6: $6 \times \$6 = \$36$

Growing Collections: Trading cards, stickers, or other collectibles often follow patterns based on how many you get each week or month.

Using exact mathematical language prevents confusion:

• Term: A number in the sequence
• Position: Where the term appears (1st, 2nd, 3rd...)
• Value: The actual number at that position
• Rule: How to get from one term to the next

For $6, 12, 18...$ → The value of the 3rd term is $18$.

Finding Missing Terms: $34, 30, 26, ?, ?$

• Rule: "Subtract $4$"
• 4th term: $26 - 4 = 22$
• 5th term: $22 - 4 = 18$

Finding Future Terms: What's the 10th term in $3, 6, 9, 12...$?

• Rule: "Add $3$" or "Multiples of $3$"
• 10th term: $3 \times 10 = 30$

Pattern work builds algebraic thinking:

• Recognizing relationships between numbers
• Using rules to make predictions
• Understanding variables and functions
• Developing logical reasoning skills

These skills prepare you for more advanced mathematics in middle and high school!